SSC CGL formula sheet

Every formula and shortcut trick from the notes, on one page. Print it (Ctrl/Cmd + P) or save as PDF — it's formatted for paper.

Quantitative Aptitude

Number System

Number types, place value & counting

Multiples of k from 1 to n

\left\lfloor \frac{n}{k} \right\rfloor

Multiples of k from a to b

\left\lfloor \frac{b}{k} \right\rfloor - \left\lfloor \frac{a-1}{k} \right\rfloor

Inclusion–exclusion

n(A \cup B) = n(A) + n(B) - n(A \cap B)

A ∩ B = multiples of LCM

Sum of first n natural numbers

\frac{n(n+1)}{2}

Sum of squares

1^2+2^2+\cdots+n^2 = \frac{n(n+1)(2n+1)}{6}

Sum of cubes

1^3+2^3+\cdots+n^3 = \left[\frac{n(n+1)}{2}\right]^2

First n odd / even numbers

1+3+\cdots+(2n-1) = n^2,\quad 2+4+\cdots+2n = n(n+1)

AP: terms and sum

n = \frac{l-a}{d}+1,\quad S = \frac{n}{2}(a+l)

⚡ Floor-division counting. Count multiples of k up to b, subtract the multiples up to a-1. No listing needed.

⚡ Place value − face value. Place value = digit × its position value. Difference = digit × (position value − 1).

⚡ AP sum = terms × middle term. For any AP, sum = number of terms × average of first and last term. Saves writing the formula.

Divisibility rules

Divisibility by 11

\left(\sum \text{odd-place digits}\right) - \left(\sum \text{even-place digits}\right) \in \{0, \pm 11, \pm 22, \dots\}

Composite divisor

pq \mid N \iff p \mid N \text{ and } q \mid N, \quad \gcd(p,q)=1

Difference of powers

(a-b) \mid (a^n - b^n) \text{ for all } n

Difference of even powers

(a+b) \mid (a^n - b^n) \text{ when } n \text{ is even}

Sum of odd powers

(a+b) \mid (a^n + b^n) \text{ when } n \text{ is odd}

abcabc form

\overline{abcabc} = \overline{abc} \times 1001 = \overline{abc} \times 7 \times 11 \times 13

⚡ Split into co-prime factors. Never split 12 as 2 × 6 or 72 as 4 × 18 — the factors must be co-prime. Test each factor's rule separately.

⚡ The 1001 family. Any six-digit number of the form abcabc equals abc \times 1001, so it is always divisible by 7, 11, 13 (and 77, 91, 143, 1001).

⚡ Sum of odd powers. a^n + b^n with odd n is divisible by a+b — look at the options for a+b first.

Remainders & remainder theorem

Division algorithm

N = d \times q + r,\quad 0 \le r < d

Product rule

\text{Rem}\left(\frac{a \times b}{d}\right) = \text{Rem}\left(\frac{R_a \times R_b}{d}\right)

Fermat's little theorem

a^{p-1} \equiv 1 \pmod{p},\quad p \text{ prime},\ \gcd(a,p)=1

Divisor-multiple rule

N \equiv r \pmod{D},\ d \mid D \ \Rightarrow\ N \equiv r \pmod{d}

Base one more than divisor

(ad+1)^n \equiv 1 \pmod{d}

Base one less than divisor

(ad-1)^n \equiv (-1)^n \pmod{d}

remainder 1 if n even, d − 1 if n odd

⚡ Make the base ±1. Write the base as (multiple of divisor) ± 1. Then the power collapses to \pm 1.

⚡ Negative remainders for products. When the numbers are just below the divisor, use small negative remainders.

⚡ Divisor is a multiple — just reduce. If the second divisor is a factor of the first, divide the old remainder by the new divisor. If it is NOT a factor, the answer cannot be found this way.

Unit digit & cyclicity

Cyclicity rule

\text{u.d.}(a^n) = \text{u.d.}(a^{r}),\ r = n \bmod 4,\ (r = 0 \Rightarrow r = 4)

Cycles of 2, 3, 7, 8

2:\,2,4,8,6 \quad 3:\,3,9,7,1 \quad 7:\,7,9,3,1 \quad 8:\,8,4,2,6

Factorials

n! \equiv 0 \pmod{10} \text{ for } n \ge 5

⚡ Last two digits of the exponent. For mod 4, only the last two digits of the exponent matter (100 is a multiple of 4).

⚡ Even × 5 = 0. If the product contains an even factor and a factor ending in 5, the unit digit is 0 — no cycle work needed.

⚡ Factorial sums: only the first four terms. From 5! onwards every factorial ends in 0, so the unit digit of 1! + 2! + … + n! (n ≥ 4) is always 3.

Factors, prime factorisation & trailing zeros

Number of factors

d(N) = (a+1)(b+1)(c+1)

Sum of factors

\sigma(N) = \frac{p^{a+1}-1}{p-1} \cdot \frac{q^{b+1}-1}{q-1} \cdot \frac{r^{c+1}-1}{r-1}

Even factors

a \cdot (b+1)(c+1)

when p = 2 with exponent a

Product of all factors

N^{d(N)/2}

Trailing zeros in n!

\left\lfloor \frac{n}{5} \right\rfloor + \left\lfloor \frac{n}{25} \right\rfloor + \left\lfloor \frac{n}{125} \right\rfloor + \cdots

Highest power of prime p in n!

\sum_{k \ge 1} \left\lfloor \frac{n}{p^k} \right\rfloor

⚡ Even factors: force one 2. If N = 2^a \times (\text{odd part}), even factors = a \times (factors of odd part).

⚡ Successive division by 5. Keep dividing by 5 (dropping remainders) and add the quotients.

⚡ Sum of factors as bracket product. Write one bracket per prime: (1 + p + p² + …). Multiply.

Fractions, decimals & recurring decimals

Pure recurring

0.\overline{ab} = \frac{ab}{99},\quad 0.\overline{abc} = \frac{abc}{999}

Mixed recurring

0.a\overline{bc} = \frac{abc - a}{990}

Terminating test

\frac{p}{q} \text{ (lowest terms) terminates} \iff q = 2^m \, 5^n

Cross-multiplication

\frac{a}{b} > \frac{c}{d} \iff ad > bc \quad (b, d > 0)

⚡ 9s-and-0s rule. Numerator: whole block after the point minus the non-repeating part. Denominator: one 9 per repeating digit, then one 0 per non-repeating digit.

⚡ Complement comparison. When fractions are close to 1, compare 1 - \text{fraction}: the smallest gap gives the largest fraction.

⚡ Terminating check in one glance. Cancel first, then look only at the denominator's primes.

Simplification

BODMAS, 'of', brackets & vinculum

'of' before division

a \div b \text{ of } c = \frac{a}{b \times c}

÷ and × left to right

a \div b \times c = \frac{a}{b} \times c

Removing a bracket after minus

a - (b - c) = a - b + c

Continued fraction (bottom-up)

a + \cfrac{1}{b + \cfrac{1}{c}} = a + \frac{c}{bc + 1}

⚡ 'of' binds tighter than ÷. Convert every 'of' into a bracketed product before touching ÷.

⚡ Continued fractions bottom-up. Start with the lowest fraction, invert and add step by step.

⚡ Digit-sum (casting out 9s) option check. Digit sum of a product = digit sum of (digit sum × digit sum). Kills wrong options in long multiplication without computing.

Algebraic identities in numerical simplification

Square of sum / difference

(a \pm b)^2 = a^2 \pm 2ab + b^2

Difference of squares

a^2 - b^2 = (a+b)(a-b)

Sum of cubes

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Difference of cubes

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Cube of sum

(a+b)^3 = a^3 + b^3 + 3ab(a+b)

Three cubes

a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)

Zero-sum case

a + b + c = 0 \Rightarrow a^3 + b^3 + c^3 = 3abc

⚡ Pattern-match the fraction. If the numerator has cubes and the denominator has three terms with a middle product, it is a sum/difference-of-cubes identity.

⚡ a + b + c = 0 ⇒ cubes = 3abc. Check if the three bases add to zero before cubing anything.

⚡ Products around a round number. (n - d)(n + d) = n^2 - d^2.

Surds & indices

Product / quotient

a^m \cdot a^n = a^{m+n},\quad \frac{a^m}{a^n} = a^{m-n}

Power of power

(a^m)^n = a^{mn},\quad (ab)^n = a^n b^n

Zero & negative index

a^0 = 1,\quad a^{-n} = \frac{1}{a^n}

Fractional index

a^{p/q} = \sqrt[q]{a^p}

Rationalisation

\frac{1}{\sqrt{a} \pm \sqrt{b}} = \frac{\sqrt{a} \mp \sqrt{b}}{a - b}

Root of a surd

\sqrt{a \pm 2\sqrt{b}} = \sqrt{x} \pm \sqrt{y},\ x + y = a,\ xy = b,\ x > y

Reciprocal of a unit surd

x = a + \sqrt{b},\ a^2 - b = 1 \Rightarrow \frac{1}{x} = a - \sqrt{b}

⚡ Common base, then equate powers. Write both sides as powers of the same prime.

⚡ Split a + 2√b. Find two numbers with sum a and product b; the root is √(larger) ± √(smaller).

⚡ LCM of root orders to compare. Raise every surd to the LCM of the orders so all become integers.

Square roots & cube roots

Product rule

\sqrt{ab} = \sqrt{a}\,\sqrt{b},\quad \sqrt[3]{ab} = \sqrt[3]{a}\,\sqrt[3]{b}

Infinite radical (plus)

\sqrt{n + \sqrt{n + \sqrt{n + \cdots}}} = \frac{1 + \sqrt{1 + 4n}}{2}

Infinite radical (minus)

\sqrt{n - \sqrt{n - \sqrt{n - \cdots}}} = \frac{-1 + \sqrt{1 + 4n}}{2}

Infinite nested product

\sqrt{x\sqrt{x\sqrt{x \cdots}}} = x

Finite nested product

\sqrt{x\sqrt{x\sqrt{x}}} = x^{7/8}

n roots: exponent (2ⁿ − 1)/2ⁿ

⚡ Cube root by unit digit + leading group. Last digit of the root from the cube-digit map; first digit = largest cube not exceeding the leading group (digits left of the last three).

⚡ Square root of a perfect square. Unit digit gives two candidates; compare with the square of the middle value (…5) to pick one.

⚡ n = k(k + 1) radicals. Factor n as two consecutive integers. Plus-signs give the larger one, minus-signs the smaller.

Approximation

Percentage swap

x\% \text{ of } y = y\% \text{ of } x

Near-square root

\sqrt{n^2 + k} \approx n + \frac{k}{2n}

Percent-fraction anchors

12.5\% = \tfrac{1}{8},\ 16.67\% = \tfrac{1}{6},\ 33.33\% = \tfrac{1}{3},\ 37.5\% = \tfrac{3}{8}

⚡ Round and compute. Round each term to the nearest convenient number and compute mentally.

⚡ Swap the percentage. If x% of y is awkward, compute y% of x instead.

HCF & LCM

HCF & LCM: definitions and core relations

Product relation (two numbers)

\text{HCF} \times \text{LCM} = a \times b

Other number

b = \frac{\text{HCF} \times \text{LCM}}{a}

Co-prime case

\gcd(a,b) = 1 \Rightarrow \text{LCM}(a,b) = ab

HCF divides every difference

\gcd(a,b) \mid (a-b)

LCM multiple of HCF

\gcd(a,b) \mid \text{LCM}(a,b)

⚡ Ratio split. Numbers in ratio a : b (a, b co-prime) with HCF h are ha and hb; LCM = hab; sum = h(a+b); difference = h(b-a).

⚡ Product ÷ HCF = LCM in one step. Any question giving two of {product, HCF, LCM, one number} resolves by the product relation.

⚡ Divisor pairs of (product ÷ HCF). For 'how many pairs of numbers' questions, split product/HCF into co-prime factor pairs.

Finding HCF & LCM (incl. fractions and decimals)

HCF by factors

\text{HCF} = p_1^{\min} \cdot p_2^{\min} \cdots \text{(common primes, lowest powers)}

LCM by factors

\text{LCM} = p_1^{\max} \cdot p_2^{\max} \cdots \text{(all primes, highest powers)}

HCF of fractions

\text{HCF}\!\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\gcd(a, c)}{\text{LCM}(b, d)}

LCM of fractions

\text{LCM}\!\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{LCM}(a, c)}{\gcd(b, d)}

Two-number division method

\text{LCM} = \frac{a \times b}{\gcd(a,b)}

⚡ LCM row of prime powers. Write each number as a row of prime powers; LCM = read column maxima, HCF = read common minima. Three numbers take under 30 seconds.

⚡ Fractions: numerators ↔ denominators cross rule. HCF of fractions = HCF of tops over LCM of bottoms; LCM is the mirror image.

⚡ Decimals: shift the point. Multiply everything by the power of 10 needed to clear decimals, solve as integers, shift the point back.

Standard word problems (tiles, bells, groups, divisible numbers)

Same remainder r

N = \text{LCM}(d_1, d_2, \dots) \times k + r

Different remainders, N ≡ −c

N = \text{LCM} \times k - c \text{ when } r_i = d_i - c

Largest tile count

\text{tiles} = \frac{L \times W}{h^2},\ h = \gcd(L, W)

Greatest n-digit multiple

\text{answer} = \underbrace{99\ldots9}_{n} - \left(\underbrace{99\ldots9}_{n} \bmod \text{LCM}\right)

⚡ Subtract remainders, then HCF. 'Same remainder' questions: HCF of (each given number minus its remainder). Remainder unknown: HCF of the pairwise differences.

⚡ LCM + r. 'Least number leaving remainder r with each divisor': add r to the LCM.

⚡ Bells: add the LCM to the start time. Convert all intervals to the same unit, take the LCM, add to the given time.

Two-step LCM/HCF cases (extra condition, N-digit bounds)

Extra divisibility condition

N = Lk + r,\ N \equiv 0 \pmod p \ \Rightarrow\ Lk \equiv -r \pmod p

Reconstruction

ab = \frac{\text{LCM}}{h},\quad \gcd(a, b) = 1

Pair count

\#\{(a,b): ab = M,\ \gcd(a,b)=1,\ a \le b\}

⚡ Solve the little congruence for k. Once N = LCM·k + r, only k's remainder mod p matters — test k = 1, 2, 3, … until it clicks.

⚡ N-digit multiple scan. For least n-digit: divide the smallest n-digit number by the LCM, go to the next multiple. For greatest: reduce the largest n-digit number by its remainder.

⚡ Sum or difference picks the pair. With LCM/HCF and a sum (or difference) given, list co-prime pairs of LCM/HCF and match the sum.

Percentage

Percentage meaning & conversions

Per cent of a number

x\% \text{ of } N = \frac{xN}{100}

Fraction to per cent

\frac{p}{q} \times 100\%

Per cent to fraction

x\% = \frac{x}{100}

Commutativity

x\% \text{ of } y = y\% \text{ of } x

Recovering the base

N = \frac{\text{part} \times 100}{\text{percentage}}

⚡ The fraction table IS the shortcut. Convert percentages to fractions before multiplying: 12.5% of 640 = 1/8 × 640 = 80. One line, no calculator.

⚡ Swap the percentages. Use x\% of y = y\% of x when the swapped pair is friendlier.

⚡ Part-to-whole in one division. 'p% of a number is k' ⇒ number = 100k/p. Look at options: only one will make the product land on k.

Percentage increase / decrease & successive change

Percentage change

\%\ \text{change} = \frac{\text{final} - \text{initial}}{\text{initial}} \times 100

Successive change

a + b + \frac{ab}{100}

use signs: + for increase, − for decrease

Same % up and down

+x\% \text{ then } -x\% \Rightarrow \text{net} = -\frac{x^2}{100}\%

Multiplier form

\text{final} = \text{initial} \times \frac{100 + a}{100} \times \frac{100 - b}{100}

Original value back

\text{initial} = \text{final} \times \frac{100}{100 + a}

⚡ Multiplier chaining. Replace every % by its fraction multiplier and multiply across all steps.

⚡ a + b + ab/100 in one line. Add, then adjust by the product term. Fastest when one change is small.

⚡ Reverse to the original. To undo +25%, divide by 5/4 (not multiply by 75/100).

'x% more than' ↔ 'x% less than' & chains

More → less

A = B\left(1 + \frac{x}{100}\right) \Rightarrow B = A\left(\frac{100}{100+x}\right),\ \text{less by } \frac{100x}{100+x}\%

Less → more

A = B\left(1 - \frac{x}{100}\right) \Rightarrow \text{B is } \frac{100x}{100-x}\% \text{ more than A}

Chain of percentages

A = \frac{a}{100}B,\ B = \frac{b}{100}C \Rightarrow A = \frac{ab}{100}\% \text{ of } C

⚡ Flip with 100x/(100±x). More→less uses 100+x in the denominator; less→more uses 100−x. The answer must be bigger than x for the less→more direction.

⚡ Assume the base = 100. Set the unmentioned quantity to 100 and read off both values.

⚡ Chain percentages by multiplying fractions. Convert each link to a fraction of the next and multiply.

Population growth, depreciation & elections

Growth / decay

P_n = P\left(1 \pm \frac{r}{100}\right)^n

Two different rates

P_2 = P\left(1 + \frac{r_1}{100}\right)\left(1 + \frac{r_2}{100}\right)

Depreciated value back

P = \text{present} \times \left(\frac{100}{100-r}\right)^n

Election margin

\text{margin} = (\text{winner share} - \text{loser share}) \times \text{valid votes}

⚡ Multiplier chips for n years. Chips: +10% → 11/10, −5% → 19/20. Multiply chips across years; the final fraction tells everything.

⚡ Work backwards with division. For 'value n years ago', divide by the chips instead of guessing.

⚡ Elections: everything on valid votes. Convert both candidates' data to fractions of valid votes; the total polled number then follows from the margin.

Marks, income–expenditure–savings & price–consumption

Pass marks equation

\frac{pM}{100} + f = \frac{qM}{100} - e

Savings identity

\text{savings} = \text{income} - \text{expenditure}

Price up r%, expenditure fixed

\text{reduce consumption by } \frac{100r}{100+r}\%

Price cut & extra quantity

\text{reduced price} = \frac{r\% \text{ of outlay}}{\text{extra quantity}},\quad \text{old price} = \frac{\text{reduced price}}{1 - r/100}

⚡ Marks gap = percentage gap. In two-candidate mark problems the difference of the two percentages times M equals the total mark gap (fail-by + exceed-by).

⚡ Income = 100 bookkeeping. Set income to 100, express savings and expenditure, then apply every change and read the answer off.

⚡ Price cut: r% of money buys the extra. A r% price cut saves r% of the outlay, and that saved money buys the extra quantity at the REDUCED price.

Ratio, Proportion, Partnership & Ages

Ratio basics & dividing amounts

Share of the total

\text{share} = \frac{\text{ratio term}}{\text{sum of terms}} \times N

Combining ratios

A:B = m:n,\ B:C = p:q \Rightarrow A:B:C = mp : np : nq

Cross-multiplication test

a:b > c:d \iff ad > bc

Multiplier method

\text{shares } ax, bx,\ (b-a)x = \text{given difference}

Duplicate / sub-duplicate

a:b \Rightarrow a^2:b^2 \text{ (duplicate)},\ \sqrt{a}:\sqrt{b} \text{ (sub-duplicate)}

⚡ Multiplier x solves everything. Translate 'numbers are in ratio a : b' into ax and bx. Every extra fact (sum, difference, product) becomes a small linear equation in x.

⚡ LCM bridging for three-term ratios. Scale A:B so its B-term equals the B-term of B:C, then read off A:B:C.

⚡ One share or difference known. Share = fraction × total. If one share is given, first recover the total, then any other share.

Proportion & proportional division of terms

Basic proportion

\frac{a}{b} = \frac{c}{d} \iff ad = bc

Fourth proportional

d = \frac{bc}{a}

Third proportional

c = \frac{b^2}{a}

Mean proportional

\text{mean} = \sqrt{ab}

Componendo & dividendo

\frac{a}{b} = \frac{c}{d} \Rightarrow \frac{a+b}{a-b} = \frac{c+d}{c-d}

Invertendo / alternando

\frac{b}{a} = \frac{d}{c},\quad \frac{a}{c} = \frac{b}{d}

⚡ Extremes × means. Set up the proportion, cross-multiply, done. For 'fourth proportional' keep the order exactly as written.

⚡ Mean proportional = geometric mean. Multiply the two numbers and take the square root. Perfect squares make it instant.

⚡ Componendo-dividendo for x : y. When (x + y) and (x − y) both appear, write the ratio and jump straight to x/y.

Partnership

Simple partnership

P_1 : P_2 = C_1 : C_2 \quad (\text{same time})

Compound partnership

P_1 : P_2 : P_3 = C_1T_1 : C_2T_2 : C_3T_3

Working partner

\text{profit} = \text{manager's cut} + \text{residual split by } C_iT_i

Capital change mid-year

\text{effective capital} = C_a t_1 + C_b t_2 + \cdots

⚡ Capital-months table. One row per partner: capital × months. The profit ratio is the row ratio — no other step.

⚡ Deduct the working partner's cut first. Salary/commission % applies to the whole profit; only the residue is shared by capital-time.

⚡ Split the year for capital changes. Add capital × months across the stretches of the year.

Problems on ages

Present = a x, b x

\frac{ax + n}{bx + n} = \frac{p}{q} \quad (\text{ratio after } n \text{ years})

Ratio n years ago

\frac{ax - n}{bx - n} = \frac{p}{q}

Constant difference

A - B \text{ is the same at every time}

Multiple of age

F = kS \Rightarrow \frac{F - n}{S - n} = \frac{p}{q}

⚡ Multiplier + shift equation. Ages = ax and bx now; add/subtract the shift; equate the new ratio; solve for x.

⚡ Use the constant difference. The gap between two ages never changes — compute it once and reuse it at any time point.

⚡ Two ratios bracket the answer. With 'n years ago' and 'n years hence' ratios, form both equations and eliminate x by subtraction.

Money ratios: income–expenditure, coins & mixed amounts

Savings equations

ax - py = s_1,\quad bx - qy = s_2

Coin value

\text{total value} = k \sum_i (n_i \times d_i)

Difference of shares

\text{given excess} = (b - a)x

⚡ Solve for x, y in two lines. Write both saving equations, subtract them to kill one variable.

⚡ Value per set for coins. Take one full set of coins in the given ratio, price it, and scale.

⚡ Sum-difference shortcuts. Sum of shares = (a + b)x, difference = (b − a)x — jump straight to x from whichever is given.

Average

Average & the sum bridge

Definition

\bar{x} = \frac{\sum x_i}{n} \iff \sum x_i = n\bar{x}

Shift property

\overline{x_i + k} = \bar{x} + k,\quad \overline{k x_i} = k\bar{x}

First n naturals / odd / even

\frac{n+1}{2},\quad n,\quad n+1

Squares / cubes

\frac{(n+1)(2n+1)}{6},\quad \frac{(n+1)^2}{4}

Consecutive / AP terms

\text{average} = \frac{\text{first} + \text{last}}{2} = \text{middle term}

⚡ Sum = average × n first. Never manipulate averages directly — flip to sums, adjust, divide back.

⚡ Uniform shift. Adding k to every number just adds k to the average.

⚡ Middle of consecutive terms. For evenly spaced numbers, the average is the central value — no addition needed.

Members joining or leaving

Joining member

\text{value} = A' + n(A' - A)

Leaving member

\text{value} = A' + n(A - A')

Replacement

\text{new} = \text{old} + n(A' - A)

Count change both ways

\text{total of newcomers} = \text{new total} - \text{old total}

⚡ Joining: own share + extras. New value = new average + (number of old members) × (average jump).

⚡ Replacement: n × jump added. Only the swapped item changes the sum, by exactly n × (average change).

⚡ Exclusion: subtract the sums. Old total − (new count × new average) = removed value.

Weighted average & two-group problems

Weighted mean

\bar{x} = \frac{\sum n_i \bar{x}_i}{\sum n_i}

Missing group average

\bar{x}_2 = \frac{N\bar{x} - n_1\bar{x}_1}{n_2}

Alligation cross

\frac{n_1}{n_2} = \frac{\bar{x}_2 - \bar{x}}{\bar{x} - \bar{x}_1}

⚡ Totals, not averages. Convert each group to a total, add, divide by the combined count.

⚡ Balance point intuition. The combined average divides the gap between the two group averages in the ratio n₂ : n₁ (inverse of sizes).

⚡ Newcomer pushing a known average. Combined-average questions with a single newcomer reduce to value = A' + n(A' − A).

Batsman problems, overlapping sums & multi-step sets

Batsman score

\text{score} = A' + (n-1)d

Batsman new average

A' = \frac{(n-1)A + \text{score}}{n}

Overlap subtraction

\text{(Thu)} - \text{(Mon)} = 3(b - a)

Split sums

n\bar{x} = \sum_{\text{chunks}} (\text{chunk total})

⚡ Batsman: work in totals. Old total + new score = new count × new average.

⚡ Overlap: subtract the sums. The shared days cancel, leaving the difference of the end days.

⚡ One variable for the unknown chunk. Name the smallest unknown x, express the rest from the given relations, and let the leftover sum close the equation.

Interest (SI & CI)

Simple Interest

Simple interest

SI = \frac{PRT}{100}

Amount

A = P + SI = P\left(1 + \frac{RT}{100}\right)

Recovering inputs

P = \frac{100\,SI}{RT},\quad R = \frac{100\,SI}{PT},\quad T = \frac{100\,SI}{PR}

n-times in T years

R = \frac{100(n-1)}{T}

Equal yearly interest

SI_{\text{per year}} = \frac{SI_{\text{total}}}{T}

⚡ Flip the formula, don't solve equations. Write SI = PRT/100 and cover the unknown — that IS the equation.

⚡ n-times ⇄ rate at SI. 'Becomes n times in T years' means interest earned = (n−1)P.

⚡ SI per year is constant. Divide the total interest by the number of years; everything else follows.

Compound Interest

Compound amount

A = P\left(1 + \frac{R}{100}\right)^T

Compound interest

CI = P\left[\left(1 + \frac{R}{100}\right)^T - 1\right]

Year-wise multipliers

A = P \times \frac{100 + r_1}{100} \times \frac{100 + r_2}{100} \times \cdots

Half-yearly compounding

A = P\left(1 + \frac{R}{200}\right)^{2T}

rate halves, periods double

Quarterly compounding

A = P\left(1 + \frac{R}{400}\right)^{4T}

⚡ Multiplier chips. One chip per year; multiply. For fraction rates pick chips that cancel.

⚡ Recover P by dividing the chip. If the amount after T years is known, divide by the chip chain to get the principal.

⚡ Half-yearly: halve the rate, double the count. Convert the problem, then run the chips.

CI vs SI: differences & doubling

2-year difference

CI - SI = P\left(\frac{R}{100}\right)^2

3-year difference

CI - SI = P\left(\frac{R}{100}\right)^2\left(3 + \frac{R}{100}\right)

CI doubling chain

2\times \text{ in } T \Rightarrow 2^k\times \text{ in } kT

SI multiple pace

n\times \text{ in } T \Rightarrow n'\times \text{ in } T',\ (n'-1) = (n-1)\frac{T'}{T}

SI and CI both given

P\left(\frac{R}{100}\right)^2 = CI_2 - SI_2

⚡ The P r²/10000 gap. For 2 years, the SI–CI difference alone fixes the product structure: two equations, two unknowns.

⚡ Doubling chain at CI. Every T years the money multiplies by the same factor — count the doublings.

⚡ Difference as first-year interest re-lent. For 2 years, think: CI - SI = (1st year interest) × (R/100).

Equal annual instalments

Present value (CI)

P = \sum_{k=1}^{n} \frac{x}{\left(1 + \frac{r}{100}\right)^k}

Simple interest instalment

P = nx - \frac{x\,r}{100} \cdot \frac{n(n-1)}{2}

Tabular step

\text{debt}_{k+1} = \text{debt}_k\left(1 + \frac{r}{100}\right) - x

⚡ Tabular year-by-year. Grow, subtract, repeat. Two or three rows finish the question.

⚡ Present-value fraction add. Discount each instalment by \left(1+\frac{r}{100}\right)^{-k} and add.

⚡ SI instalment formula. Use P = nx - \frac{xr}{100} \cdot \frac{n(n-1)}{2} directly for simple-interest loans.

Finding P, R, T from amount data

Principal from amount

P = \frac{A}{\left(1 + \frac{r}{100}\right)^T}

Rate from consecutive amounts

1 + \frac{r}{100} = \frac{A_{t+1}}{A_t} \quad (\text{CI})

Yearly SI from amounts

SI_{\text{year}} = A_{t+1} - A_t \quad (\text{SI})

Two-amount system

\frac{A_2}{A_1} = 1 + \frac{r}{100} \Rightarrow P = \frac{A_1}{1 + r/100}

⚡ Consecutive amounts → rate → principal. The ratio of consecutive CI amounts exposes the rate chip; divide once more for P.

⚡ SI: equal yearly steps. Check the amounts differ by a constant — that constant is the yearly interest.

⚡ Two amounts, two unknowns. Divide the equations: the ratio kills P and gives the chip.

Profit, Loss & Discount

Profit, Loss and CP–SP Basics

Profit / loss per cent

\text{P\%} = \frac{SP - CP}{CP} \times 100

Selling price

SP = CP\left(1 \pm \frac{x}{100}\right)

Cost price from SP

CP = \frac{SP}{1 \pm \frac{x}{100}} = \frac{100 \cdot SP}{100 \pm x}

No profit, no loss

SP = CP

⚡ Fraction ⇄ percentage swap. Convert the profit % to a fraction; SP/CP becomes a clean ratio.

⚡ Divide by the chip to get CP. SP given with a profit/loss %: undo the multiplier.

⚡ Two prices, same % — scale linearly. If the same article at another CP/SP keeps the same profit %, everything scales by the same factor.

Successive Changes & Equivalent Single Change

Two successive changes

\text{net} = a + b + \frac{ab}{100}

Same change twice (increase)

2x + \frac{x^2}{100}

Same change twice (decrease)

2x - \frac{x^2}{100}

Equivalent single discount

D = d_1 + d_2 - \frac{d_1 d_2}{100}

⚡ Multiply the chips. Chips work for any mix of increases and decreases, in any order.

⚡ The d₁ + d₂ − d₁d₂/100 formula. Two discounts collapse in one line.

⚡ Work backwards through chips. For 'find the second change given the net', divide the net chip by the first chip.

Marked Price, Discount and the CP–MP–SP Chain

Discount per cent

\text{D\%} = \frac{MP - SP}{MP} \times 100

Marked price for target gain

MP = CP \cdot \frac{100 + g}{100 - d}

SP through the chain

SP = CP\left(1 + \frac{m}{100}\right)\left(1 - \frac{d}{100}\right)

Multi-level markups

\text{Final} = \text{Base} \times \prod \left(1 + \frac{x_k}{100}\right)

⚡ Build the MP/CP ratio. Gain g% and discount d% together fix MP as a multiple of CP.

⚡ Markup chip then discount chip. One chip for the markup, one for the discount.

⚡ Trade-chain multiplication. Each middleman's markup is one chip.

Dishonest Dealer, False Weights & Claims

False weight gain

\text{gain\%} = \frac{\text{true weight} - \text{used weight}}{\text{used weight}} \times 100

Claimed loss + short weight

\text{multiplier} = \frac{100 - L}{100 - c}

Cheat at both ends

\text{multiplier} = \frac{100 + b}{100 - s}

True gain from multiplier

\text{gain\%} = (\text{multiplier} - 1) \times 100

⚡ The W−w over w rule. Loss to the customer is measured against what the dealer actually gave.

⚡ Claimed loss can still be a gain. Compare the money per TRUE gram, not per claimed gram.

⚡ Both-ends multiplier. Multiply the buy-side gain chip by the sell-side chip.

Same Selling Price: One Profit, One Loss

Net loss for equal ±x%

\text{net loss\%} = \frac{x^2}{100}

Reconstructed costs

CP_1 = \frac{S}{1 + x/100}, \quad CP_2 = \frac{S}{1 - x/100}

Overall position

\text{Loss} = CP_1 + CP_2 - 2S

⚡ The x²/100 reflex. Same SP, opposite equal % → straight to the loss formula.

⚡ Reconstruct both CPs, then total. Divide the common SP by each chip.

Mixtures & Alligation

Mixture Concentration & Amounts

Amount from ratio

\text{part} = \text{total} \times \frac{\text{part's share}}{\text{sum of shares}}

Concentration

C = \frac{\text{ingredient}}{\text{mixture}} \times 100\%

Adding water

C' = \frac{A}{M + w} \quad (A \text{ unchanged})

Removing mixture

\text{each ingredient} \to \text{same fraction removed}

⚡ Track the constant ingredient. Water added ⇒ milk unchanged. Everything hinges on the unchanged part.

⚡ Rebuild the per cent after dilution. Milk fixed, total grows — divide again.

⚡ Ratio change ⇒ one equation. Set the unchanged quantity against the wanted ratio.

Rule of Alligation

Alligation ratio

\frac{n_1}{n_2} = \frac{v_2 - \bar{v}}{\bar{v} - v_1}

Weighted average

\bar{v} = \frac{n_1 v_1 + n_2 v_2}{n_1 + n_2}

Cheaper : dearer

(d - m) : (m - c)

Water as free ingredient

\text{milk} : \text{water} = (m - 0) : (c - m), \ c > m

⚡ Cross over the distances. Dearer-minus-mean and mean-minus-cheaper swap sides — that's the whole rule.

⚡ Water costs zero. Dilution questions are alligation with one strength = 0.

⚡ Alligation on averages. Any average — salary, age, marks — alligates the same way.

Replacement & Repeated Operations

Repeated equal replacement

C\left(1 - \frac{r}{C}\right)^n

Milk : water after n ops

\left(1-\frac{r}{C}\right)^n : \left[1-\left(1-\frac{r}{C}\right)^n\right]

Proportional removal

\text{lost} = \text{removed volume} \times \frac{\text{ingredient share}}{\text{total}}

Chain of replacements

C \prod_k \left(1 - \frac{r_k}{C}\right)

⚡ The (1 − r/C)ⁿ shrink. One fraction per operation; multiply.

⚡ Two operations, wine : water. Square the shrink factor; the complement is the water.

⚡ Draw-and-refill with a ratio shift. Removing uniform mixture deletes each ingredient proportionally; only the drawn fraction matters.

Milk–Water Ratio & Profit by Adulteration

Adulteration profit

\text{profit\%} = \frac{\text{water}}{\text{milk}} \times 100

Ratio target

\frac{m}{w + x} = \text{wanted ratio} \Rightarrow x

Alloy rebuild

\text{metal} = \text{alloy weight} \times \frac{\text{share}}{\text{sum of shares}}

SP of mixture

\text{SP} = (\text{milk}) \times \text{milk price} + (\text{water}) \times 0

⚡ Water-over-milk is the profit. Adulteration profit = free litres per honest litre.

⚡ Hit the target ratio by addition. Keep the bigger quantity constant; solve for the addition.

⚡ Rebuild the alloy, then change one metal. Convert ratio to grams, adjust, re-ratio.

Mixing Two Mixtures

Blend of two mixtures

f = \frac{V_1 f_1 + V_2 f_2}{V_1 + V_2}

Volumes for target fraction

\frac{V_1}{V_2} = \frac{f_2 - f}{f - f_1}

Fraction from ratio

f_{\text{milk}} = \frac{m}{m + w}

⚡ Fractions first, then average. Ratio → milk fraction per vessel → weighted average.

⚡ Alligate the fractions. Target fraction sits between the two vessel fractions; distances give volumes.

⚡ Unequal volumes — weight the fractions. Multiply each vessel's milk fraction by its volume before adding.

Time & Work

Work Rates & the LCM Method

One-day work

\frac{1}{T}

Combined time (two workers)

T = \frac{ab}{a + b}

Combined time (three workers)

T = \frac{abc}{ab + bc + ca}

Work done

W = \text{rate} \times \text{time}

⚡ LCM units, not fractions. Set the job to the LCM of the individual times; rates become small integers.

⚡ The ab/(a+b) reflex. For exactly two workers, one formula — no addition of fractions.

⚡ Three workers through the LCM. Same method, three rates.

Efficiency, 'Twice as Good' & Ratio Cases

Efficiency ⇄ time

\frac{E_A}{E_B} = \frac{T_B}{T_A}

k-times worker

T_B = (k+1)T, \quad T_A = \frac{(k+1)T}{k}

Efficiency ratio from times

E_A : E_B = \frac{1}{T_A} : \frac{1}{T_B}

⚡ Rate units from the efficiency ratio. Let B = 1 unit/day, A = k units/day; total = (k+1)/day.

⚡ Invert, never scale. 'A is 3 times as good' ⇒ A's days = B's days ÷ 3.

⚡ Days-difference cases. 'B takes 24 days more than A' plus an efficiency ratio pins both times.

Pipes & Cisterns

Net filling rate

\frac{1}{T} = \sum \frac{1}{T_i^{\,+}} - \sum \frac{1}{T_j^{\,-}}

Fill + drain pair

T = \frac{ab}{b - a}

Work done in stage 1

W_1 = \text{rate}_1 \times t_1

Tank as LCM units

\text{tank} = \text{LCM of times; rates in units/hour}

⚡ Sign discipline. Empty pipes subtract. Write + and − before adding anything.

⚡ Stage the problem. Finish stage 1 first, then treat the remainder as a fresh tank.

⚡ Fill-and-drain pair formula. One inlet, one outlet: net time = ab/(b − a).

Men–Days–Hours Chain & Provisions

MDH chain

M_1 D_1 H_1 E_1 = M_2 D_2 H_2 E_2 \quad (W_1 = W_2)

Men × days constant

M_1 D_1 = M_2 D_2

Provisions remaining

\text{days} = \frac{M_1 (D_{\text{total}} - t)}{M_2}

Piece of work

\text{days for } \tfrac{k}{n} \text{ of work} = \frac{k}{n} T

⚡ Multiply resources, equate products. Everything inverse-proportional lands in the numerator; direct-proportional in the denominator.

⚡ Provisions after reinforcement. Stock left = original men × days left; then divide by the new headcount.

⚡ Work left after a share is done. First find what fraction remains, then apply men-days to that remainder.

Joining / Leaving Mid-work, Alternate Days & Wages

Work done by A in t days

\frac{t}{T_A}

Remaining work

1 - \frac{t}{T_A}

Two-day cycle

\frac{1}{T_A} + \frac{1}{T_B} \text{ per 2 days}

Wage split

w_A = \text{total} \times \frac{1/T_A}{1/T_A + 1/T_B}

⚡ Subtract the finished part. Whoever continues inherits only the remainder.

⚡ Count full cycles, then the tail. Alternate days: measure progress per 2-day cycle.

⚡ Wages follow work, not days alone. Rate ratio × equal days = work ratio.

Time, Speed & Distance

Speed, Distance, Time & Unit Conversion

Basic relation

S = \frac{D}{T}, \quad D = S \times T

km/h to m/s

\text{km/h} \times \frac{5}{18} = \text{m/s}

Equal-distance average

\bar{S} = \frac{2ab}{a + b}

General average speed

\bar{S} = \frac{D_1 + D_2}{T_1 + T_2}

⚡ Convert first, always. Train lengths are in metres, times in seconds — get to m/s before anything else.

⚡ Harmonic mean for round trips. Same distance out and back ⇒ 2ab/(a+b) in one line.

⚡ Early/late differences are time equations. Both speeds cover the SAME distance; equate the expressions.

Relative Speed

Relative speed

S_{\text{rel}} = S_1 \pm S_2

Meeting time

t = \frac{\text{initial gap}}{S_1 + S_2}

Overtake time

t = \frac{\text{gap or combined length}}{S_1 - S_2}

Time after meeting (opposite starts)

\frac{\text{speed of other}}{\text{own speed}} = \frac{\text{time to B from meet}}{\text{time to A from meet}}

⚡ Subtract for the same direction. Overtaking uses the difference; the length is both trains together.

⚡ Add for opposite directions. Closing speed is the sum even if one object is a walking man.

⚡ Distance flown till meeting. Find the meeting time first; any third object's distance = its speed × that time.

Trains Crossing Poles, Platforms & Trains

Pole / man

L_{\text{train}} = S \times t

Platform / bridge

L_{\text{train}} + L_{\text{platform}} = S \times t

Train vs train

L_1 + L_2 = S_{\text{rel}} \times t

Two-equation extraction

\frac{t_{\text{platform}}}{t_{\text{pole}}} = \frac{L + P}{L}

⚡ Pole first: it's the train's own length. The pole crossing IS the definition of the train's length in motion.

⚡ Subtract the pole equation. Platform time minus pole time covers exactly the platform.

⚡ Man in the other train. A man is a point — only the crossing train's length matters.

Boats & Streams

Effective speeds

u = b + s, \quad v = b - s

Boat & stream from trips

b = \frac{u + v}{2}, \quad s = \frac{u - v}{2}

Time for two legs

t = \frac{d_1}{b + s} + \frac{d_2}{b - s}

Swimmer/boat carried by stream

\text{drift} = s \times \text{time}

⚡ Halve the sum, halve the difference. Down and up speeds give boat and stream in one line each.

⚡ Extract u and v from trip times. Each trip is one equation; the pair is linear in 1/u and 1/v.

⚡ Two double-trip equations. Two journeys pin down both speeds exactly.

Races & Handicaps

Beating margin in metres

\frac{S_A}{S_B} = \frac{L}{L - x}

Beating margin in time

\text{B's remaining time} = t \Rightarrow S_B = \frac{x}{t}

Speeds from 'by x m or t s'

S_B = \frac{x}{t}, \quad S_A = S_B \cdot \frac{L}{L - x}

Start handicap

\text{B runs } L - \text{start}

⚡ Translate 'beats by x m' into a speed ratio. Same finishing time ⇒ distances are in the speed ratio.

⚡ 'By x metres or t seconds' reveals both speeds. B's last x metres took t seconds — that's B's speed for free.

⚡ Handle the handicap. A start shortens one runner's distance; recompute the ratio.

Algebra

Basic algebraic identities

Square of sum / difference

(a\pm b)^2=a^2\pm 2ab+b^2

Difference of squares

a^2-b^2=(a+b)(a-b)

Sum & difference of the two squares

(a+b)^2+(a-b)^2=2(a^2+b^2),\quad (a+b)^2-(a-b)^2=4ab

Cube of sum

(a+b)^3=a^3+b^3+3ab(a+b)

Cube of difference

(a-b)^3=a^3-b^3-3ab(a-b)

Sum of cubes

a^3+b^3=(a+b)(a^2-ab+b^2)=(a+b)^3-3ab(a+b)

Difference of cubes

a^3-b^3=(a-b)(a^2+ab+b^2)=(a-b)^3+3ab(a-b)

Square of trinomial

(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)

Fourth-power factorisation

a^4+a^2b^2+b^4=(a^2+ab+b^2)(a^2-ab+b^2)

Fourth powers from sum and product

a^4+b^4=(a^2+b^2)^2-2a^2b^2

⚡ Build higher powers from $a+b$ and $ab$. Memorise the chain: a^2+b^2=(a+b)^2-2ab, then a^3+b^3=(a+b)^3-3ab(a+b). For differences use (a-b)^2=(a+b)^2-4ab. No need to find a and b.

⚡ Recognise the identity inside decimals. \frac{p^3+q^3}{p^2-pq+q^2}=p+q and \frac{p^3-q^3}{p^2+pq+q^2}=p-q. If the denominator's middle sign is opposite to the numerator's sign, the identity fits.

⚡ Value-putting for expression options. When options are algebraic expressions, substitute small numbers (e.g. a=1, b=1 or a=2, b=1) in the question and in each option. Test a second pair if two options tie. Avoid values that make denominators zero.

⚡ Complete the squares when an equation equals zero. Group x-terms and y-terms, complete each square, and check that the leftover constants cancel. Then each square must be zero.

$x+\frac{1}{x}$ type expressions

Square

x^2+\frac{1}{x^2}=\left(x+\frac1x\right)^2-2=\left(x-\frac1x\right)^2+2

Cube (sum)

x^3+\frac{1}{x^3}=\left(x+\frac1x\right)^3-3\left(x+\frac1x\right)

if $x+\frac1x=k$ then it is $k^3-3k$

Cube (difference)

x^3-\frac{1}{x^3}=\left(x-\frac1x\right)^3+3\left(x-\frac1x\right)

Fourth power

x^4+\frac{1}{x^4}=\left(x^2+\frac{1}{x^2}\right)^2-2

Fifth power

x^5+\frac{1}{x^5}=\left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)-\left(x+\frac1x\right)

Link between sum and difference

\left(x+\frac1x\right)^2-\left(x-\frac1x\right)^2=4

Quadratic to reciprocal form

ax^2-bx+a=0\ \Rightarrow\ x+\frac1x=\frac ba

Mixed form

\left(px+\frac{1}{qx}\right)^2=p^2x^2+\frac{1}{q^2x^2}+\frac{2p}{q}

⚡ The ladder. From k=x+\frac1x: x^2+\frac{1}{x^2}=k^2-2, x^4+\frac{1}{x^4}=(k^2-2)^2-2, x^8+\frac{1}{x^8}=\ldots — keep squaring and subtracting 2. For the cube use k^3-3k.

⚡ Special values give cyclic powers. | If x+\frac1x= | Then | |---|---| | 2 | x=1 | | -2 | x=-1 | | 1 | x^3=-1 (so x^6=1) | | -1 | x^3=1 | | \sqrt3 | x^6=-1 (so x^{12}=1) | | \sqrt2 | x^4=-1 (so x^8=1) | Reduce every exponen…

⚡ Divide the quadratic by $x$. ax^2-bx+a=0 (equal first and last coefficients) means x+\frac1x=\frac ba. Spot equal end coefficients instantly.

⚡ Mixed form $px+\frac{1}{qx}$. Squaring gives a middle term 2\cdot px\cdot\frac{1}{qx}=\frac{2p}{q}, not 2.

$a^3+b^3+c^3-3abc$ and conditional identities

Main identity

a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)

Half form

a^3+b^3+c^3-3abc=\tfrac12(a+b+c)\left[(a-b)^2+(b-c)^2+(c-a)^2\right]

Zero-sum case

a+b+c=0\ \Rightarrow\ a^3+b^3+c^3=3abc

Equal-variables case

a^2+b^2+c^2=ab+bc+ca\ \Rightarrow\ a=b=c

Pairwise products

ab+bc+ca=\frac{(a+b+c)^2-(a^2+b^2+c^2)}{2}

Cyclic differences

(a-b)^3+(b-c)^3+(c-a)^3=3(a-b)(b-c)(c-a)

⚡ Hunt for a hidden zero sum. (x-y)+(y-z)+(z-x)=0 always, so the sum of their cubes is 3(x-y)(y-z)(z-x). Similarly (a^2-b^2)+(b^2-c^2)+(c^2-a^2)=0.

⚡ Close numbers: use the half form. When a,b,c differ by small amounts, (a-b)^2+(b-c)^2+(c-a)^2 is tiny and easy to compute.

⚡ Value-putting under a condition. If a condition like a+b+c=0 is given, pick numbers that satisfy it (e.g. 1,1,-2) and evaluate.

Surds: rationalisation and square roots of surds

Rationalising

\frac{1}{\sqrt a+\sqrt b}=\frac{\sqrt a-\sqrt b}{a-b}

Conjugate product

(\sqrt a+\sqrt b)(\sqrt a-\sqrt b)=a-b

Square root of a surd

\sqrt{a+2\sqrt b}=\sqrt m+\sqrt n,\ \text{where } m+n=a,\ mn=b

Square root (minus)

\sqrt{a-2\sqrt b}=\sqrt m-\sqrt n\ \ (m>n)

Product-1 pair

x=p+\sqrt q,\ p^2-q=1\ \Rightarrow\ \frac1x=p-\sqrt q,\ x+\frac1x=2p

Telescoping sum

\sum_{n=1}^{N-1}\frac{1}{\sqrt n+\sqrt{n+1}}=\sqrt N-1

⚡ Spot the product-1 conjugate. If x=p+\sqrt q and p^2-q=1, write \frac1x=p-\sqrt q at once. Then use x+\frac1x=2p or x-\frac1x=2\sqrt q and the reciprocal ladder.

⚡ Denest $\sqrt{a\pm2\sqrt b}$ by sum-product. Make the coefficient of the inner root 2 (e.g. 4\sqrt3=2\sqrt{12}), then find two numbers with sum a and product b.

⚡ Compare differences of roots. \sqrt a-\sqrt b=\frac{a-b}{\sqrt a+\sqrt b}. With the same a-b, the pair with the larger roots gives the smaller difference.

⚡ Telescoping rationalisation. Each \frac{1}{\sqrt n+\sqrt{n+1}}=\sqrt{n+1}-\sqrt n; the middle terms cancel, leaving last minus first.

Linear equations, graphs and polynomials

Unique solution

\frac{a_1}{a_2}\neq\frac{b_1}{b_2}

lines intersect

No solution

\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}

parallel lines

Infinitely many solutions

\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}

coincident lines

Intercept form

\frac xp+\frac yq=1

Area with the axes

\text{Area}=\frac12\cdot\left|\frac ca\right|\cdot\left|\frac cb\right|=\frac{c^2}{2|ab|}

Slope of ax+by+c=0

m=-\frac ab

Remainder theorem

f(x)\div(x-a)\Rightarrow R=f(a);\quad f(x)\div(px-q)\Rightarrow R=f\!\left(\tfrac qp\right)

Roots of a quadratic

\alpha+\beta=-\frac ba,\quad \alpha\beta=\frac ca

⚡ Area with axes from intercepts. Put y=0 for the x-intercept and x=0 for the y-intercept, then area =\frac12\times|p|\times|q|.

⚡ Remainder = put the zero of the divisor. No long division. For divisor x-2, put x=2; for 2x+1, put x=-\frac12.

⚡ Intersection point: test the options. For 'the lines meet at' questions, substitute each option into both equations — usually faster than solving.

Maxima and minima (AM ≥ GM, quadratics)

AM–GM

\frac{a+b}{2}\ge\sqrt{ab}\quad(a,b>0)

Min of ax + b/x

ax+\frac bx\ge 2\sqrt{ab},\ \text{at } x=\sqrt{\tfrac ba}

Vertex of a quadratic

x=-\frac{b}{2a},\quad \text{extreme value}=\frac{4ac-b^2}{4a}

Fixed sum

x+y=S\ \Rightarrow\ xy\le\frac{S^2}{4}

Fixed product

xy=P\ \Rightarrow\ x+y\ge 2\sqrt P

⚡ Equal-terms rule for $ax+\frac bx$. Minimum =2\sqrt{ab}. No calculus needed.

⚡ Complete the square. Write the quadratic as (x-h)^2+m; the minimum is m. For a negative leading coefficient write M-(x-h)^2; the maximum is M.

⚡ Fixed weighted sum → equalise the parts. If px+qy=S with x,y>0, the product (px)(qy) is greatest when px=qy=\frac S2.

Geometry

Lines and angles

Angles on a line / around a point

\text{on a line}=180^\circ,\quad \text{round a point}=360^\circ

Co-interior angles

(a+b)^\circ=180^\circ\ \text{for allied angles between parallels}

Complement / supplement

x+(90^\circ-x)=180^\circ\cdot\tfrac12,\quad \text{supplement}-\text{complement}=180^\circ-2\cdot\text{angle}

Parallel line pairs

\text{corresponding}=\text{alternate}=\text{vertically opposite (each pair equal)}

⚡ Co-interior gives the equation. If two allied angles are given as expressions in x, set their sum to 180^\circ, solve for x, then substitute back into the angle asked.

⚡ Supplement − complement one-liner. For an angle x: supplement =180-x, complement =90-x, so the difference is always 90^\circ, and the angle itself is \frac{180-d}{2} if the difference d is given.

Triangles and their centres

Angle sum & exterior angle

A+B+C=180^\circ,\quad \text{ext}=B+C

Centroid division

AG:GD=2:1\ \text{on median }AD

Incentre angle

\angle BIC=90^\circ+\frac{A}{2}

Circumcentre angle

\angle BOC=2A\ \text{(minor arc BC)}

Orthocentre angle

\angle BHC=180^\circ-A

Apollonius (median length)

m_a^2=\frac{2b^2+2c^2-a^2}{4}

Isosceles median

m_{\text{base}}=\sqrt{a^2-\left(\frac{\text{base}}{2}\right)^2}

Heron's area

K=\sqrt{s(s-a)(s-b)(s-c)},\ s=\frac{a+b+c}{2}

⚡ Centre angles from one input. The asked angle involves \angle A only. Read off which centre is drawn: bisectors → 90+\frac A2, perpendicular bisectors → 2A, altitudes → 180-A.

⚡ Centroid ratios without coordinates. The centroid cuts each median in 2:1 from the vertex. If the median is 3m long, the pieces are 2m and m.

⚡ Median by Apollonius or symmetry. For an isosceles triangle skip the formula: the median to the base is \sqrt{a^2-(b/2)^2}. Otherwise use m_a^2=\frac{2b^2+2c^2-a^2}{4}.

⚡ Heron with a standard semi-perimeter. For 13-14-15, s=21, area =\sqrt{21\cdot8\cdot7\cdot6}=84. Memorise the common Heron families: (13,14,15)→84, (3,4,5)→6, (5,12,13)→30, (6,8,10)→24, (7,24,25)→84, (9,12,15)→54, (10,24,26)→120.

Congruence, similarity and BPT

Similarity ratios

\frac{a_1}{a_2}=k,\quad \frac{P_1}{P_2}=k,\quad \frac{K_1}{K_2}=k^2

Areas from perimeters

\frac{K_1}{K_2}=\left(\frac{P_1}{P_2}\right)^2

BPT

DE\parallel BC\Rightarrow\frac{AD}{DB}=\frac{AE}{EC}

Midpoint theorem

D,E\ \text{midpoints}\Rightarrow DE=\frac{BC}{2}

Angle bisector theorem

\frac{BD}{DC}=\frac{AB}{AC}\ \text{(internal bisector of }A\text{)}

⚡ Square the perimeter ratio for areas. Perimeters in ratio p:q → areas in ratio p^2:q^2 (and back: side ratio =\sqrt{\text{area ratio}}).

⚡ Midpoint theorem shortcut. Spot the words "midpoints of two sides" → the joining segment is half the third side, parallel to it. Conversely, through the midpoint of one side, parallel to another → it bisects the third side.

⚡ BPT with a parallel drawn inside. When a line parallel to a side cuts the other two sides, write the ratio directly. If the segments on one side are AD:DB=2:3, the same ratio holds on the other side.

Pythagoras theorem and triplets

Pythagoras

h^2=p^2+b^2

Median to hypotenuse

m_{\text{hyp}}=\frac{h}{2}

Rectangle diagonal

d=\sqrt{l^2+b^2}

Triangle type test

a\ \text{longest}:\ a^2\gtrless b^2+c^2\Rightarrow \text{obtuse}/\text{acute}

⚡ Triplet recognition. See two numbers, recall the third: 12 and 13 → 5; 24 and 25 → 7; 15 and 17 → 8. Multiples scale: 6-8-10, 9-12-15, 10-24-26 are all 3-4-5 families.

⚡ Median to the hypotenuse is half of it. In any right triangle the median from the right angle equals half the hypotenuse (three equal pieces: the two half-hypotenuses and the median).

⚡ Converse: check $a^2$ vs $b^2+c^2$. To identify the right/obtuse/acute triangle, square only the longest side and compare.

Quadrilaterals and polygons

Quadrilateral angle sum

A+B+C+D=360^\circ

Cyclic quadrilateral

A+C=180^\circ,\quad B+D=180^\circ

Rhombus

K=\frac12 d_1d_2,\quad a=\sqrt{\left(\frac{d_1}{2}\right)^2+\left(\frac{d_2}{2}\right)^2}

Trapezium

K=\frac12(a+b)h

Parallelogram

K=bh

Regular polygon

\text{int}=\frac{(n-2)180^\circ}{n},\quad \text{ext}=\frac{360^\circ}{n},\quad \text{diagonals}=\frac{n(n-3)}{2}

Brahmagupta

K=\sqrt{(s-a)(s-b)(s-c)(s-d)}

⚡ Exterior angle → number of sides. For regular polygons the exterior angle is the fastest route: n=\frac{360^\circ}{\text{ext}}, and \text{ext}=180^\circ-\text{int}.

⚡ Rhombus side from diagonals. Half-diagonals form a right triangle with the side: side =\sqrt{(d_1/2)^2+(d_2/2)^2}. If the halves are 8 and 6, the side is a 6-8-10 triplet.

⚡ Diagonal count one-liner. n vertices, each joins n-3 others (not itself, not two neighbours), each diagonal counted twice: \frac{n(n-3)}{2}.

Circles: chords, tangents, secants and cyclic angles

Chord from distance

\ell=2\sqrt{r^2-d^2}

Equal chords

\ell_1=\ell_2\Rightarrow d_1=d_2

Tangent–secant

PT^2=PA\cdot PB

Intersecting chords

PA\cdot PB=PC\cdot PD

Cyclic quadrilateral

A+C=180^\circ,\ \text{ext}=\text{interior opposite}

Alternate segment

\angle\text{tangent-chord}=\angle\text{in alternate segment}

Common tangents (transverse)

L_T=\sqrt{d^2-(r_1+r_2)^2}

Common tangents (direct)

L_D=\sqrt{d^2-(r_1-r_2)^2}

⚡ Triplet inside a circle. r-d-\frac{\ell}{2} is a right triangle: radius 13 and distance 5 → half-chord 12 → chord 24. Circles reuse the same triplets as Pythagoras questions.

⚡ Tangent–secant: multiply the pieces. Tangent squared = external secant × whole secant. Read the "whole" as external + internal part.

⚡ Count common tangents from $d$ vs $r_1, r_2$. Compare d with r_1+r_2 and |r_1-r_2|: outside → 4, touching outside → 3, overlapping → 2, touching inside → 1, contained → 0.

⚡ Concentric chord trick. A chord of the outer circle tangent to the inner circle has length 2\sqrt{R^2-r^2} — the inner radius is the distance from the centre to the chord.

Mensuration (2D)

Areas of triangles

General triangle

K=\frac12 b h

Equilateral triangle

K=\frac{\sqrt3}{4}a^2,\quad h=\frac{\sqrt3}{2}a

Heron's formula

s=\frac{a+b+c}{2},\quad K=\sqrt{s(s-a)(s-b)(s-c)}

Radii of an equilateral triangle

r_{\text{in}}=\frac{a}{2\sqrt3},\quad R=\frac{a}{\sqrt3}

⚡ Equilateral area from the side table. \frac{\sqrt3}{4}=0.433. Memorise: side 4 → 4\sqrt3, side 6 → 9\sqrt3, side 8 → 16\sqrt3, side 12 → 36\sqrt3, side 10 → 25\sqrt3. The coefficient is just \left(\frac{\text{side}}{4}\right)^2.

⚡ Heron with a triplet shortcut. If the sides form a Pythagorean triple the triangle is right-angled: use \frac12\times legs directly. 13-14-15 is the classic non-right Heron case, area 84.

Areas of quadrilaterals

Rectangle

K=lb,\quad P=2(l+b),\quad d=\sqrt{l^2+b^2}

Square

K=a^2=\frac{d^2}{2},\quad d=a\sqrt2

Rhombus

K=\frac{d_1d_2}{2}

Trapezium

K=\frac{(a+b)}{2}h

Cyclic quadrilateral (Brahmagupta)

K=\sqrt{(s-a)(s-b)(s-c)(s-d)}

⚡ Square from its diagonal. Area =\frac{d^2}{2}, side =\frac{d}{\sqrt2}. Halve any square-area question that gives a diagonal.

⚡ Rhombus: the half-diagonals are a right triangle. Perimeter → side → half the other diagonal by Pythagoras → area by \frac12d_1d_2. Triplets (6,8,10) and (5,12,13) appear constantly.

Circles, sectors and rings

Circle

K=\pi r^2,\quad C=2\pi r=\pi d

Sector

K=\frac{\theta}{360}\pi r^2,\quad \ell=\frac{\theta}{360}\,2\pi r

Segment

K=\frac{\theta}{360}\pi r^2-\frac12 r^2\sin\theta

Ring / path

K=\pi(R^2-r^2)=\pi(R+r)(R-r)

Rolling wheel

\text{distance}=n\pi d

⚡ Sector as a fraction of the circle. \theta=60^\circ\to\frac16, 90^\circ\to\frac14, 45^\circ\to\frac18, 120^\circ\to\frac13 of the circle. No separate formula needed.

⚡ Ring: difference of squares. R^2-r^2=(R+r)(R-r): with a path of width w, R-r=w and R+r=2r+w — often a one-line answer.

⚡ Wheel revolutions = distance / circumference. Convert the diameter to metres first: d=70 cm =0.7 m gives a 2.2 m circumference with \pi=\frac{22}{7}.

Regular polygons and inscribed figures

Regular hexagon

K=\frac{3\sqrt3}{2}a^2=6\times\frac{\sqrt3}{4}a^2

Regular octagon

K=2(1+\sqrt2)a^2

Square in a circle

\text{diagonal}=\text{diameter}=2R

Circle in a square

d_{\text{circle}}=a

Circle in an equilateral triangle

r=\frac{a}{2\sqrt3}

⚡ Hexagon = six equilateral triangles. Multiply the equilateral area by 6: \frac{3\sqrt3}{2}a^2. Given the perimeter, divide by 6 first.

⚡ Inscribed square → the diameter is the diagonal. All four corners touch the circle, so the square's diagonal passes through the centre and equals the diameter.

Percentage change, similarity and re-bent shapes

Scaling

\ell\to k\ell\ \Rightarrow\ K\to k^2K,\quad P\to kP

Successive change in area

\Delta\%=a+b+\frac{ab}{100}

Equal perimeter

P_1=P_2\ \text{(re-bent wire)}

Side change from area change

\Delta a\%=\left(\sqrt{1+\frac{\Delta K}{100}}-1\right)\times100

⚡ Two-dimensional percentage change. Apply a+b+\frac{ab}{100} with signs. +20\% and -10\%: 20-10-\frac{200}{100}=+8\%.

⚡ Area change → side change via square root. The side multiplies by \sqrt{\text{area factor}}. An increase of 21% → factor 1.21 → side ×1.1 → +10%.

⚡ Re-bent wire: conserve the perimeter. Circumference = perimeter of the new shape. Compare the two areas computed from the common length L.

Mensuration (3D)

Cube and cuboid

Volume

V=lbh,\qquad V_{\text{cube}}=a^3

Lateral surface

LSA=2(l+b)h,\qquad LSA_{\text{cube}}=4a^2

Total surface

TSA=2(lb+bh+lh),\qquad TSA_{\text{cube}}=6a^2

Space diagonal

d=\sqrt{l^2+b^2+h^2},\qquad d_{\text{cube}}=a\sqrt3

⚡ Longest rod = space diagonal. A rod can poke corner-to-corner through the room: \sqrt{l^2+b^2+h^2}. Triplets like (3,4,12,13) exist: 3^2+4^2+12^2=169=13^2.

⚡ Painting/flooring picks the right area. Four walls → 2(l+b)h (ignore floor and roof). Full painting including roof and floor → TSA. Cost = area × rate.

Cylinder

Volume

V=\pi r^2h

Curved surface

CSA=2\pi rh

Total surface

TSA=2\pi r(r+h)

Hollow cylinder

V=\pi(R^2-r^2)h

Capacity

1\ \text{m}^3=1000\ \ell

⚡ 1 m³ = 1000 litres. Tank answers in litres: compute \pi r^2h in metres, then ×1000. Keep radii in the units the answer needs.

⚡ Hollow cylinder: difference of squared radii. Material volume =\pi(R^2-r^2)h — the cross-section is a ring.

Cone

Slant height

\ell=\sqrt{r^2+h^2}

Volume

V=\frac13\pi r^2h

Curved surface

CSA=\pi r\ell

Total surface

TSA=\pi r(\ell+r)

⚡ Triplet for the slant height. Check for (3,4,5) or (7,24,25) among r,h,\ell before computing a square root.

⚡ One-third of the cylinder. Equal base and height: V_{\text{cone}}=\frac13V_{\text{cylinder}} — useful directly in ratio questions and for conical heaps of sand/grain.

Sphere and hemisphere

Sphere

V=\frac43\pi r^3,\quad S=4\pi r^2

Hemisphere

V=\frac23\pi r^3,\quad S_{\text{curved}}=2\pi r^2,\quad S_{\text{total}}=3\pi r^2

Melting rule

V_{\text{before}}=V_{\text{after}}

Count of small spheres

\text{count}=\left(\frac{R}{r}\right)^3

⚡ Count small balls by the cube of the radius ratio. Number of small spheres =\left(\frac{R}{r}\right)^3. Radii in ratio 8:1 → 512 pieces, no π needed.

⚡ Hemisphere TSA is three circles. 3\pi r^2: two for the curved half (recall the sphere is 4\pi r^2) plus one for the base.

Prisms, pyramids and painted cubes

Prism volume

V=A_{\text{base}}\times h

Pyramid volume

V=\frac13A_{\text{base}}h

Painted-cube counts

3\text{f}=8,\ 2\text{f}=12(n-2),\ 1\text{f}=6(n-2)^2,\ 0\text{f}=(n-2)^3

⚡ Painted-cube formula table. Identify n = number of small cubes along one edge. Corners are fixed at 8; the rest depend on n-2.

⚡ Pyramid by one-third. Same base and height as a prism: the pyramid holds exactly one-third.

Trigonometry

Ratios and standard values

Reciprocal pairs

\text{cosec}\theta=\frac{1}{\sin\theta},\quad \sec\theta=\frac{1}{\cos\theta},\quad \cot\theta=\frac{1}{\tan\theta}

sin column

\sin0^\circ=0,\ \sin30^\circ=\frac12,\ \sin45^\circ=\frac{1}{\sqrt2},\ \sin60^\circ=\frac{\sqrt3}{2},\ \sin90^\circ=1

cos column

\cos\theta=\sin(90^\circ-\theta)

tan column

\tan0^\circ=0,\ \tan30^\circ=\frac{1}{\sqrt3},\ \tan45^\circ=1,\ \tan60^\circ=\sqrt3

⚡ The 1-2-3 memory trick for sin. Write \frac{\sqrt0}{2},\frac{\sqrt1}{2},\frac{\sqrt2}{2},\frac{\sqrt3}{2},\frac{\sqrt4}{2} for 0^\circ,30^\circ,45^\circ,60^\circ,90^\circ — the sin row falls out; the cos row is the same read backwards. Divide the t…

⚡ Reciprocal first, table second. When an expression has cosec/sec/cot, flip each to sin/cos/tan before touching the table.

Fundamental identities

Pythagorean identities

\sin^2\theta+\cos^2\theta=1,\quad 1+\tan^2\theta=\sec^2\theta,\quad 1+\cot^2\theta=\text{cosec}^2\theta

Fourth powers

\sin^4\theta+\cos^4\theta=1-2\sin^2\theta\cos^2\theta

Sixth powers

\sin^6\theta+\cos^6\theta=1-3\sin^2\theta\cos^2\theta

Product-to-sum bridge

\tan\theta=\frac{\sin\theta}{\cos\theta},\quad \cot\theta=\frac{\cos\theta}{\sin\theta}

⚡ Substitute $x=\sin^2\theta$. Everything above degree 4 collapses with x and 1-x. Memorise the two reduced forms for 4th and 6th powers.

⚡ Sec/tan pairs move as one. See \sec^2 and \tan^2 together → replace by 1 (or by each other via \sec^2=1+\tan^2). Same for cosec/cot.

Complementary angles

Complementary pairs

\sin(90-\theta)=\cos\theta,\ \tan(90-\theta)=\cot\theta,\ \sec(90-\theta)=\text{cosec}\theta

Paired products

\tan\theta\,\tan(90^\circ-\theta)=1,\quad \sin\theta\,\text{cosec}\theta=1

Angle-equation rule

\sin(A+\theta)=\cos(B+\theta)\Rightarrow A+B+2\theta=90^\circ

⚡ Pair the angles that add to 90°. In a long product, look for \theta and 90-\theta pairs first — each pair collapses to 1. The classic \tan1^\circ\tan2^\circ\cdots\tan89^\circ collapses completely.

⚡ Convert everything to one angle. Rewrite each term at its complementary partner until only one angle remains; a \sin^2+\cos^2 pair may then collapse.

⚡ Equation → sum to 90°. When \sin(\ldots)=\cos(\ldots), set the two bracketed angles to sum to 90^\circ and solve.

Value-putting and given-ratio questions

From sin to the rest

\sin\theta=\frac{p}{h}\Rightarrow \cos\theta=\frac{\sqrt{h^2-p^2}}{h},\ \tan\theta=\frac{p}{\sqrt{h^2-p^2}}

Standard triplets

(3,4,5),\ (5,12,13),\ (7,24,25),\ (8,15,17),\ (9,40,41)

Sum-product of a ratio and reciprocal

x+\frac1x=k\Rightarrow x^2+\frac{1}{x^2}=k^2-2

⚡ Draw the triplet triangle. Ratio given → sides in hand in 5 seconds. Check the triplet table before any algebra.

⚡ Reduce the asked expression. Rewrite the target in terms of the given relation. For x+\frac1x chains, square and subtract 2.

Maximum and minimum values

Ranges

0\le\sin\theta\le1,\quad \sec\theta\ge1,\quad \sin\theta+\cos\theta\in[1,\sqrt2]

Weighted square form

a\sin^2\theta+b\cos^2\theta\in[\min(a,b),\max(a,b)]

AM-GM forms

a\tan^2\theta+b\cot^2\theta\ge2\sqrt{ab},\quad a\sin^2\theta+b\,\text{cosec}^2\theta\ge2\sqrt{ab}

Combined sec-cosec

\sec^2\theta+\text{cosec}^2\theta=2+\tan^2\theta+\cot^2\theta\ge4

Linear combination

a\sin\theta+b\cos\theta\text{ has maximum }\sqrt{a^2+b^2}

⚡ Match the question to a range. Bounded pairs (\sin+\cos etc.) → memorised range. Weighted square sums → min of coefficients or 2\sqrt{ab}.

⚡ Weighted sin-cos squares are bounded by coefficients. 2\sin^2\theta+3\cos^2\theta=2+\cos^2\theta, so it ranges from 2 to 3 — min 2, max 3, no calculus.

Heights and Distances

Angles of elevation and depression

Master relation

\tan\theta=\frac{\text{height above eye level}}{\text{horizontal distance}}

Slant line

\sin\theta=\frac{h}{L},\quad \cos\theta=\frac{d}{L}\quad (L=\text{line-of-sight or ladder length})

Depression = elevation

\text{angle of depression at observer}=\text{angle of elevation at object}

⚡ Draw the horizontal through the eye. One sketch kills most errors: horizontal dashed line from the eye, the line of sight, and the right angle between them. Drop heights *below the eye* when the observer is elevated.

⚡ Depression at the top equals elevation at the bottom. Alternate interior angles: the depression from the observer equals the elevation measured at the object from ground level directly below — use whichever is stated.

Standard angles: 30°, 45°, 60°

45 degrees

\theta=45^\circ\Rightarrow d=h

30 degrees

\theta=30^\circ\Rightarrow d=\sqrt3\,h\ \Leftrightarrow\ h=\frac{d}{\sqrt3}

60 degrees

\theta=60^\circ\Rightarrow d=\frac{h}{\sqrt3}\ \Leftrightarrow\ h=\sqrt3\,d

Ladder (hypotenuse given)

h=L\sin\theta,\quad d=L\cos\theta

⚡ Match the angle to the multiplier. 45° → same; 30° → distance is \sqrt3 times the height; 60° → height is \sqrt3 times the distance. Write the multiplier before touching numbers.

⚡ Ladder questions are sine/cosine, not tangent. The ladder itself is the hypotenuse: height =L\sin\theta, foot distance =L\cos\theta. A 45^\circ ladder of length L\sqrt2 climbs exactly L.

Two observation points (two angles)

Same side

h=\frac{d}{\cot\alpha-\cot\beta}

Opposite sides

h=\frac{d}{\cot\alpha+\cot\beta}

30-60 pair

\cot30^\circ-\cot60^\circ=\frac{2}{\sqrt3}\Rightarrow h=\frac{d\sqrt3}{2}

Gap from a height

\text{gap}=h(\cot\alpha-\cot\beta)

⚡ Subtract the cotangents, don't solve two triangles. Write d_1=h\cot\alpha, d_2=h\cot\beta from each position; the difference of the distances is what is given. One linear equation in h.

⚡ Opposite sides means add. When the two observers are on opposite sides, the base distances *add up* to d: denominators add, not subtract.

Moving observers: speed and time

Distance walked

\text{distance}=\text{speed}\times\text{time}

Chain to height

v\,t=h(\cot\theta_1-\cot\theta_2)

⚡ Convert speed and time first. Reduce to a plain distance (e.g. 6\times6=36 m) and the problem becomes the standard two-angle pattern.

⚡ Read it backwards when height is given. If h is known and the speed is asked, compute the gap h(\cot\theta_1-\cot\theta_2) first, then divide by time.

Compound figures: buildings, pedestals, broken objects

Statue on pedestal

\text{statue}=d(\tan\beta-\tan\alpha)

Tower on building

\text{tower}=d\tan\theta_{\text{top}}-\text{building}

Broken tree

\text{tree}=\frac{b}{\cos\theta}+b\tan\theta

Building + tower from one height

\text{tower}=\text{building}+d(\tan\beta-\tan\alpha)

⚡ Angles differ by 15°? Expect a surd answer. 45° paired with 60° or 30° gives clean radicals; keep (\sqrt3-1) or (1+\sqrt3) factored — that exact form is what the options show.

⚡ Broken tree = broken + standing. Work the right triangle of the fallen part: horizontal b, angle \theta at the tip. Standing part =b\tan\theta, fallen part =\frac{b}{\cos\theta}; add them.

Statistics

Mean, weighted mean and combined mean

Mean

\bar{x}=\frac{x_1+x_2+\cdots+x_n}{n}=\frac{\sum x_i}{n}

Combined mean

\bar{x}=\frac{n_1\bar{x}_1+n_2\bar{x}_2}{n_1+n_2}

Weighted mean

\bar{x}=\frac{\sum w_i x_i}{\sum w_i}

Member replacement

\text{value brought in}=\text{old mean}+(\text{avg shift})\times n_{\text{new}}

⚡ Think in totals, not averages. Every average question is one subtraction away if you track \text{sum}=\text{mean}\times n. Convert both sides to sums before combining.

⚡ Average of equal-sized groups is the plain middle. For two groups of equal size the combined mean is just \frac{\bar x_1+\bar x_2}{2}; weighting only matters when sizes differ.

Median and mode of raw data

Median (odd)

\text{median}=\left(\frac{n+1}{2}\right)\text{th value}

Median (even)

\text{median}=\frac{(\frac{n}{2})\text{th}+(\frac{n}{2}+1)\text{th}}{2}

Mode

\text{mode}=\text{most frequent value}

Empirical relation

\text{mode}=3\,\text{median}-2\,\text{mean}

⚡ Sort, then count to the middle. Never find a median of unsorted data. For small sets, sort and walk in from both ends — the meeting point is the answer.

⚡ The 3M relation runs both ways. Given any two of mean/median/mode, the third is one substitution away — keep the form \text{mode}=3m-2\bar x in mind.

Median and mode of grouped data

Grouped median

\text{median}=L+\frac{\frac{n}{2}-cf}{f}\,h

Grouped mode

\text{mode}=L+\frac{f_1-f_0}{2f_1-f_0-f_2}\,h

Cumulative frequency

cf_i=f_1+f_2+\cdots+f_i

Grouped mean (midpoints)

\bar{x}=\frac{\sum f_i m_i}{\sum f_i},\quad m_i=\frac{L_i+U_i}{2}

⚡ Find the median class by cf, not by eye. Build the cf column, stop at the first cf \ge n/2. That row's lower limit is L. Everything else is substitution.

⚡ Modal class first, formula second. The modal class is just the tallest bar. If it sits at an edge, the missing neighbour counts 0 in the formula.

Range, variance and standard deviation

Range

\text{range}=x_{\max}-x_{\min}

Variance and SD

\sigma^2=\frac{1}{n}\sum(x_i-\bar{x})^2,\quad \sigma=\sqrt{\sigma^2}

Shift-scale rule

\text{SD}(a x+b)=|a|\,\text{SD}(x)

SD of 1 to n

\sigma=\sqrt{\frac{n^2-1}{12}}

⚡ Shifts never change spread. A question saying "5 is added to every number, what happens to SD?" answers itself: nothing. Only the multiplier scales SD.

⚡ Small set? Deviations from the mean. For 5-10 numbers, subtract the mean, square, average, take the root — keep squares exact with fractions.

Averages of special series

First n naturals

\sum_{1}^{n}k=\frac{n(n+1)}{2},\quad \text{mean}=\frac{n+1}{2}

Squares

\sum_{1}^{n}k^2=\frac{n(n+1)(2n+1)}{6}

Cubes

\sum_{1}^{n}k^3=\left[\frac{n(n+1)}{2}\right]^2

AP mean

\text{mean of AP}=\frac{\text{first}+\text{last}}{2}

⚡ Consecutive numbers: average = middle. For any run of consecutive terms (odd, even, naturals), the average equals the median — the middle term. No summation needed.

⚡ Squares and cubes via the sums. Divide the closed-form sum by n; memorise the n=10 anchor (mean of squares =38.5) as a sanity check.

Data Interpretation

Reading tables: totals, differences, ratios

Total of a row/column

\text{total}=x_1+x_2+\cdots+x_k

Difference

\text{difference}=|A-B|

Ratio of cells

\text{ratio}=\frac{x_{ij}}{x_{kl}}\ \text{(simplify the fraction)}

⚡ Simplify the fraction before choosing. Ratios like 450:550 reduce to 9:11 — reduce before matching options, the unreduced pair is a common distractor.

⚡ Sums of friendly numbers. Group digits that make round numbers (e.g. 60+70+80). DI tables are built so subtotals land on round values.

Percentage change and comparisons

Percentage change

\Delta\%=\frac{\text{new}-\text{old}}{\text{old}}\times100

Share of total

\text{share}=\frac{\text{part}}{\text{whole}}\times100\%

Percentage points

\text{pp difference}=\Delta\%_1-\Delta\%_2

⚡ Divide by the old value, always. Half of all DI percentage errors come from dividing by the new value. Anchor: "increase OVER X" uses X as the denominator.

⚡ Compare percentages as fractions of friendly numbers. Convert \frac{\text{change}}{\text{base}} to a fraction over a round base: \frac{60}{300}=\frac15=20\% beats decimal division.

Pie charts: degrees, shares and totals

Angle to value

\text{value}=\frac{\theta}{360^\circ}\times\text{total}

Percent to angle

\theta=\text{share}\times3.6^\circ\ \text{per percent}

Share from angle

\text{share}=\frac{\theta}{3.6^\circ}\%

⚡ Memorise 3.6° per percent. Percent ↔ degree is one multiplication: 22% spans 79.2^\circ; 36^\circ is 10% of the total. Most pie questions are one such step plus a multiply.

⚡ Two-sector combos first. When asked about two sectors together, add angles (or percentages) before converting to values — one multiplication instead of two.

Averages from data

Average of cells

\bar{x}=\frac{\text{sum of cells}}{\text{number of cells}}

Weighted average

\bar{x}=\frac{n_1\bar{x}_1+n_2\bar{x}_2}{n_1+n_2}

Per-period value

\text{per period}=\frac{\text{total}}{\text{number of periods}}

⚡ Scan for the shortcut row. DI tables often hide symmetry: values around a central number (320, 280, 360, 300, 340 all deviate by ±20/40 around 320). Spotting the centre makes the average mental arithmetic.

⚡ Weight by the group sizes, not the marks. In combined-average tables the *count* is the weight. Multiply across each row, add, divide by total count.

Growth rates and successive changes

Growth rate

g=\frac{\text{new}-\text{old}}{\text{old}}\times100

Successive change

\text{factor}=\left(1+\frac{a}{100}\right)\left(1+\frac{b}{100}\right)

Reverse growth

\text{initial}=\frac{\text{final}}{\text{factor}}

Profit percent from table

\text{profit}\%=\frac{\text{income}-\text{expenditure}}{\text{expenditure}}\times100

⚡ Multiply factors, never add percentages. Convert each change to a factor (+20\%\to1.2), multiply, convert back. +20\%+(-10\%)=+10\% is the classic wrong shortcut — the true net is +8\%.

⚡ Reverse-growth: divide by the factor. When the final value is given and the initial is asked, write the factor chain and divide once. Keep factors as fractions (1.65=\frac{33}{20}) for clean division.

General Intelligence & Reasoning

Analogy

The idea of analogy

⚡ Speak the relation as a full sentence before touching the options. Convert 'A : B' into a sentence like 'B is the young one of A' or 'each letter of B is 3 ahead of A'. Then test option pairs against the *same sentence*. Options written to look related but failing one word of the senten…

Letter / letter-cluster analogy

Shift rule

Q(B) = Q(A) + k \pmod{26}

Q = alphabet position; k is constant for a constant-shift analogy.

Opposite letter

Q(B) = 27 - Q(A)

Atbash: A↔Z, M↔N. Two-letter quick check: sum of positions = 27.

Rising shift

Q(B_i) = Q(A_i) + i

Position i gets shift i — common in 4-letter clusters (BDFH from ABCD).

⚡ Subtract positions, then copy the shift pattern. Write positions of A under A, positions of B under B, subtract slot by slot. The shift tuple you get (e.g. +3, +3, +3) is the whole question. Apply the same tuple to C and read the answer with EJOTY.

⚡ Spot Atbash by the 27-sum. If corresponding letters of A and B always add to 27 (C=3, X=24 → 27), the rule is opposite letters. Then answer = opposite of each letter of C — no shifting at all.

⚡ Reversal check: read B backwards. If B read backwards is a constant shift of A (or of A reversed), the rule is reversal ± shift. Apply: reverse C, then shift by the same k.

Word analogy

⚡ Two-question test on every option. Ask (1) is the option pair the same *type* of relation? (2) is it the same *specific* relation in the same *direction*? An option surviving question 1 usually dies at question 2 — that is where the exam separates candida…

⚡ Gender/age precision check. For animal pairs, decide whether the model pair is (a) adult:young, (b) male:female or (c) male-young:female-young. Options often mix a young-one with a female — only one matches the model exactly.

Number analogy

Linear rule

b = k a + r

Solve k and r mentally from a and b; verify on the pair before applying to c.

Square family

b \in \{a^2,\ a^2 \pm r,\ (a{+}1)^2,\ a(a{+}1)\}

Test squares first when b is close to a².

Digit sum rule

b = k \cdot S(a) + r

S(a) = digit sum; common when b is much smaller than a.

⚡ Size test picks the family instantly. Compare b with a: if b is a few times a → linear; if b ≈ a² → square family; if b is 10-40 → digit rule when a is 2-digit; if b is small with big a → digit sum.

⚡ Verify the rule on the model pair before the probe. Whatever rule you guess, check it reproduces b from a exactly. Guessing ×4+1 from 12 : 49 fails instantly (12×4+1 = 49 passes!) — then apply to c. The 2-second check prevents half the errors.

Number sets (triads) analogy

Chain rule

b = ka + r,\; c = kb + r

Same step applied twice.

Third from first two

c = ab,\; a^2 + b^2,\; k(a+b)

Test on every model set.

Power set

(a,\; a^2 \pm r,\; a^3 \pm r)

Near-squares and near-cubes.

⚡ Check the last number first. Build the rule from the model sets, then for each option compute only what the last number should be. Traps nearly always change the last number by 1 to 3, so this finds the answer in one pass.

Mixed clusters and same-relation selection

⚡ Solve tracks, then assemble. In a mixed cluster make one column for letters (positions/shifts) and one for numbers (their own series). Fill the answer character by character — each column is independent, so an error stays inside one column.

⚡ One sentence for all four pairs. In same-relation selection, form the relation sentence once from the model pair and tick/cross every option against the *same* sentence, in the same order. The first option that passes both halves is the answer.

Classification (Odd One Out)

The idea of odd one out

⚡ Write the majority rule, not the exception. Instead of hunting for the odd item, write the rule the *other three* follow ('three are primes'). Then confirm each of the three obeys it and the remaining one breaks it. This prevents choosing an option that is merely …

Word odd one out

⚡ Two-level drill: category, then property. First find the broad category word shared by three. If the fourth also fits that category, switch to a binary property (salt/fresh, input/output, base/derived, male/female) — one of the four will be the only one on the o…

Number odd one out

Divisibility by 11

|S_{odd} - S_{even}| \equiv 0 \pmod{11}

S = sums of alternate digits; e.g. 2728: (2+2)-(7+8) = -11 → divisible.

Perfect square endings

n^2 \in \{0,1,4,5,6,9\}

A square never ends in 2, 3, 7 or 8 — instant elimination.

Digit sum 9 rule

9 \mid n \iff S(n) \equiv 0 \pmod 9

Numbers with digit sum exactly 9: 45, 54, 63, 72, 81.

⚡ Last-digit scan for squares. Squares end only in 0, 1, 4, 5, 6, 9. If three numbers are squares, any option ending in 2/3/7/8 is instantly odd — no squaring needed.

⚡ Prime check by small divisors. To test primality divide by 2, 3, 5, 7 only (for numbers under 121). 33 = 3 × 11, 51 = 3 × 17, 57 = 3 × 19, 91 = 7 × 13 — CGL's favourite fake primes.

Letter-cluster odd one out

Step of a cluster

s_i = Q(L_{i+1}) - Q(L_i) \pmod{26}

Compute steps for every option; three must match as tuples.

Opposite-pair sum

Q(X) + Q(Y) = 27

Letter pairs whose positions add to 27 (AZ, BY, ..., MN).

⚡ Step tuples in one line. Under each cluster write its steps: ACE → (2,2), BDF → (2,2), GIK → (2,2), MOP → (2,1). The mismatched tuple is the answer — no naming of vowels needed.

⚡ Vowel count as tiebreaker. If step tuples coincide or the clusters differ in length, count vowels (A, E, I, O, U). Three clusters with exactly one vowel and one with none (or two) is a complete CGL pattern.

Pair odd one out

⚡ Relation sentence, then four ticks. Form the relation sentence from three pairs that clearly agree, then tick/cross all four options. In number pairs, compute k and r for each pair — three will match.

Number pairs and sets odd one out

Pair rule

b = f(a),\; f(a) \in \{a^2,\ a^3,\ a^2 \pm k,\ ka \pm r\}

Find f from one pair, check the other three.

Chain triad

(a,\ f(a),\ f(f(a)))

Same operation applied twice, e.g. ×3: (3, 9, 27).

Outer-to-middle triad

(a,\ g(a, c),\ c),\; g = ac \text{ or } a^2 + c^2

Middle number made from the two outer numbers.

First-two-to-third triad

(a,\ b,\ h(a, b)),\; h = k(a+b) \text{ or } ab \pm r

Third number made from the first two.

⚡ Size tells the operation. Compare sizes: if b is about a², think square; if c ≈ 3 × b and b ≈ 3 × a, think a ×3 chain; if the middle number is large and the outer ones small, think product of the outer numbers.

⚡ Test the rule on two options before trusting it. A rule found from one option may be a coincidence. Confirm it on a second option, then check the remaining two. The one that fails is the answer.

Series (Number & Letter)

How every series is cracked

⚡ Difference ladder in 10 seconds. Write the terms with gaps and subtract under them, twice if needed. A constant row anywhere fixes the whole rule; extend the row one step and climb back up to get the answer.

Number series

AP / ladder

a_n = a_1 + (n-1)d

d = common difference from the ladder.

Recurrence xk plus r

a_{n+1} = k \cdot a_n \pm r

Test k = 2, 3 first with r = ±1, ±2.

n-squared family

a_n = n^2 \pm k \text{ or } n^2 + n

Second difference 2 ⇒ n²; second difference 2k ⇒ k·n².

Geometric

a_n = a_1 \cdot r^{\,n-1}

Constant ratio r; check ×2, ×3, ×1.5.

⚡ Ratio test before exotic rules. Divide each term by the previous one. Constant ratio → GP. Ratio drifting by +1 each time → rising multipliers. Neither → back to the ladder.

⚡ Interleaved split for jumpy series. If the series jumps up-down-up-down, write odd-position terms in one row and even-position terms below. Each row is a friendly AP/GP; extend both, then read the term that falls on '?'

⚡ Cubes and powers hiding in the ladder. Memorise the signature ladders: 1, 8, 27, 64 (cubes), 1, 3, 5, 7 (odds), 2, 4, 8, 16 (powers of 2), 1, 4, 9, 16 (squares). When the first difference row matches a signature, the series is solved.

Letter series

Slot steps

s_i = Q(L_{i+1}) - Q(L_i) \pmod{26}

Steps per slot decide the rule; wrap Y→B counts as +3.

Two-track cluster

Q(\text{first}_i) = a + d_1 i,\quad Q(\text{second}_i) = b + d_2 i

Tracks are independent; solve both.

⚡ Step row under the letters. Write positions over the letters and steps between them. One clean step row (+2, +2, +2...) or a rising row (2, 3, 4, 5) ends the question.

⚡ Wrap-around fluency. After Z the alphabet restarts: Y + 4 = C. Practise from EJOTY anchors — every jump is at most 13, take the shorter direction.

Alphanumeric series

⚡ Two columns, two answers, one bolt. Make a letter column and a number column. Solve the letter column (step row), solve the number column, then write them side by side in the original order.

Wrong number in the series

⚡ Ladder with one broken rung. Compute all differences; all but one gap will follow the pattern (constant, rising, doubling). The term touching the odd gap(s) is the wrong one.

Repeating letter series — fill the blanks

⚡ Write the blocks one below another. Guess the block size from the total number of places (12 → try 3, 4 or 6). Write the blocks in rows; each column must hold one letter, so every blank copies the visible letter in its column.

⚡ Double letter = mirror turn. When you see 'rr' or 'cc' inside the line, the block probably reads forward and then backward (pqr rqp). Fill the second half as the first half reversed.

Coding - Decoding

The coding-decoding family

⚡ Difference tuple first. Write the sample word and its code with positions, subtract slot by slot. A flat tuple (+1, +1, +1) is a shift; a mixed tuple is a per-index rule; a reversed tuple pattern hints at reversal. Then apply the same tuple to …

Letter shift coding

Shift tuple

Q(code_i) = Q(word_i) + d_i \pmod{26}

Compute d_i from the sample; apply the identical tuple to the probe.

Decode = negative shift

Q(word_i) = Q(code_i) - d_i \pmod{26}

For decoding questions subtract the tuple instead of adding.

⚡ Anchor hops, not counting. For shift +k, jump from the nearest EJOTY anchor: to shift W by 4, note W = Y − 2, so W + 4 = Y + 2 = A. Two small hops beat counting W→X→Y→Z→A.

⚡ Decode questions: run the tuple backwards. When a coded word is given and the original is asked, subtract the tuple instead of adding. Reverse every hop.

Reversal and reverse-plus-shift coding

⚡ Backwards reading test. Read the code from right to left. If it equals the sample word, code new words by writing them backwards. If it is *almost* the sample with every letter off by the same k, code new words by reverse-then-shift.

Number coding

Position sum

\text{code} = \sum Q(w_i)

Q = A1..Z26 positions.

Affine code

\text{code} = k \cdot \sum Q(w_i) + r

Fit k and r from the sample pair.

Weighted sum

\text{code} = \sum i \cdot Q(w_i)

Letter i multiplies its position.

Reverse positions

Q^{rev}(c) = 27 - Q(c)

Z=1 ... A=26; small codes for long words.

⚡ One sample fixes the affine code. Compute the position sum S of the sample. If code = S → plain sum. If code − S is constant-looking (5, 10...) the rule is S + r. If code/S is a clean 2 or 3, the rule is k·S. If neither fits, try the weighted sum.

⚡ Digit-sum shortcut for 9-heavy checks. Position sums reduce nicely: letters at EJOTY anchors contribute 5/10/15/20/25. Sums are easy to verify by grouping anchors instead of adding letter by letter.

Symbol and digit coding

⚡ Common letters ↔ common symbols. When two coded words share letters, their codes share exactly the same number of symbols. Circle the shared letters and the shared symbols — they belong to each other. What is left over in each word belongs to the left-o…

⚡ Conditions before the table. In condition-type questions, first check the conditions on the letter group (first letter vowel? last letter consonant?). Only if no condition applies do you code straight from the table.

Substitution and sentence coding

⚡ Elimination table. Write each sentence and its code as two aligned rows. Common words are common code-words — tick them. The word asked about then either appears directly or is the lone unmatched item.

Language (sentence) coding with common words

⚡ Intersect, then subtract. For a word, take the code words of every sentence that contains it (intersect), then remove code words already fixed for other words. One code left = answer; two left = cannot be determined (unless the options force it).

Mathematical Operations

Operator puzzles and BODMAS

BODMAS order

B \to O \to D/M \to A/S

D and M share a rank and resolve left to right; so do A and S.

Substitution map

\{+,-,\times,\div\} \to \{+,-,\times,\div\}

A bijection: rewrite every operator before touching any number.

⚡ Rewrite the whole line, then compute. Under the given expression, write the substituted operators in the same positions. Only then evaluate with BODMAS. One rewritten line prevents every 'multiplied out of habit' error.

Sign substitution

⚡ Compute the trap values too. After the correct value, evaluate the original expression once. If that number sits among the options, you have confirmed the examiner's design and your own substitution in one stroke.

Interchanging signs and numbers

⚡ Swap table in the margin. Write 'a ↔ b', 'op1 ↔ op2' at the left, rewrite the expression once with all swaps applied, evaluate.

⚡ Fixed-order sweep for fix-the-equation. Sweep pairs left to right: (n1,n2), (n1,n3), (n1,n4), (n2,n3), (n2,n4), (n3,n4). One written value per row; the row matching the RHS is the answer. If no row matches, you mis-evaluated something — redo the row, not the m…

Balancing and operator-filling

⚡ Evaluate all, then eliminate. In pick-the-correct-equation, evaluate every option in a written value column. Exactly one row matches its RHS; the column doubles as your final check.

Hidden-rule and defined operations

Sum of squares

a \# b = a^2 + b^2

Very common hidden rule: 5 # 3 = 34.

Difference of squares

a @ b = a^2 - b^2 = (a+b)(a-b)

8 @ 2 = 60.

Product plus sum

a \$ b = ab + a + b

4 $ 3 = 19.

⚡ Rule ladder for hidden operations. Test in this order: a + b, a × b, a^2 + b^2, a^2 - b^2, (a+b)^2, (a-b)^2, ab + a + b. Stop at the first rule that fits all the examples.

Missing Number

How missing-number puzzles work

Rule confirmation

f(a_1,b_1)=c_1 \ \text{and} \ f(a_2,b_2)=c_2 \Rightarrow \text{rule } f

Two agreeing instances pin the rule; one instance pins nothing.

Common families

a+b,\ a-b,\ a \times b,\ ab \pm k,\ (a \pm b)k,\ a^2 \pm b,\ \text{digitsum}

Test sums/products first — they cover ~80% of CGL puzzles.

⚡ Two-row confirmation. Never answer from one fitted row. Write the candidate rule symbolically ('c = a×b + 1'), check it on the second complete row, then apply. Ten seconds of checking prevents the most common wrong answer in this topic — a ru…

Grids with row (or column) rules

⚡ Anchor on the largest cell. Write the largest cell of a row first and rebuild it: 22 = (4 + 7) × 2, 49 = 8² − 15, 13 = 26 ÷ 2. Building the largest from the rest exposes the rule faster than scanning left to right.

⚡ Transcribe the figure as rows. Copy the grid into your rough column as three horizontal rows, question mark included. Reading values off the figure mid-calculation is where transposition errors come from.

Digit-sum and digit-reversal rules

⚡ The mismatch trigger. If 30 seconds of standard rules fails on both rows, immediately test: (i) digit sums, (ii) reversed sum, (iii) product of digit sums. One of these fits in the large majority of 'stuck' puzzles.

⚡ Reversal sanity check. A reversed-digit answer must read backwards cleanly: reverse(57) = 75. If the sum ends in 0 the reversal is not defined for CGL purposes — such puzzles avoid those sums, so if your sum ends in 0 you have the wrong rule.

Blood Relations

How to solve blood-relation questions

Symmetry pairs

\text{father} \leftrightarrow \text{son},\ \text{mother} \leftrightarrow \text{daughter},\ \text{uncle} \leftrightarrow \text{nephew}

Every relation has a mirror: if A is B's uncle, B is A's nephew (or niece).

In-law map

\text{spouse of child} \to \text{son/daughter-in-law};\ \text{parent of spouse} \to \text{father/mother-in-law}

Sibling of spouse, or spouse of sibling → brother/sister-in-law.

Cousin

\text{cousin} = \text{child of a parent's sibling}

Aunt/uncle's children are cousins — never 'nephew'.

⚡ Two-column decoding. Write each statement as an arrow in the margin: 'Suresh's daughter is Pooja' becomes Suresh → Pooja (−). Arrows point parent → child. When every statement is an arrow, the tree draws itself and direction mistakes disap…

Multi-statement tree puzzles

⚡ Top-down reconstruction. Find the person who is nobody's child (the oldest named) and start there. Every other statement then attaches downward or sideways by marriage. A tree built top-down never needs revising.

⚡ Arrow-count for generations. Relation from arrow count: 1 down = son/daughter, 2 down = grandson/granddaughter, 1 up = father/mother, 2 up = grandfather/grandmother. Sideways through a marriage adds '-in-law'.

Pointing and introducing

⚡ Peel the phrase inside-out. Underline the innermost 'my _' and resolve outward, one layer per line. Then place speaker and target in a two-person sketch.

⚡ The self-reference alarm. Whenever a phrase like 'my father's only son' or 'my mother's only daughter' appears, test whether it resolves to the speaker. If the speaker's gender fits, it does — and the question is really about the speaker's own ch…

Coded (symbol) relations

⚡ Decode pairwise, tree under it. For each operator, write its one-line sentence beneath the string, then draw the tree. Two operators ≈ four people; three ≈ five.

⚡ Gender audit before answering. Before marking, check every needed gender is *pinned by a symbol*. If the asked relation needs a gender nobody fixed, the answer is the gender-neutral form (cousin, sibling-in-law phrasing) — or you have mis-decoded.

Direction & Distance

Direction conventions and displacement

Shortest distance

d = \sqrt{x^2 + y^2}

x = net east-west movement, y = net north-south movement.

Triple shortcut

3\text{-}4\text{-}5 \times k

Legs (3k, 4k) give distance 5k: check 6-8-10, 9-12-15, 12-16-20.

Turn directions

\text{left/right} = \text{facing}\pm90^\circ

Left/right are relative to current facing; about-turn is 180°.

⚡ Two-axis bookkeeping. Keep two running totals: EAST-WEST and NORTH-SOUTH. East and West cancel each other; North and South cancel each other. The endpoint is (EW, NS) and distance is the square root of the sum of squares.

⚡ Triple recognition. When the two net components come out as 3k and 4k (any order), the answer is 5k with no calculation: (6, 8) → 10, (9, 12) → 15, (12, 5) → 13.

Direction faced and net direction

Clockwise cycle

N \to E \to S \to W \to N

Every right turn moves one step along this cycle.

Anticlockwise cycle

N \to W \to S \to E \to N

Every left turn moves one step along this cycle.

Sign reading

(x>0,y>0) \Rightarrow NE

Signs of the net components give the quadrant.

⚡ Write the heading down each time. After every turn, overwrite a single letter (N/E/S/W). Three written letters beat one mental chain — and if a wrong option appears, each written heading is checkable.

Shadow questions

⚡ One-line conversion. Rewrite the shadow sentence as a facing sentence before doing anything else: morning + behind = facing East; evening + in front = facing East; morning + shadow on the left = facing South (shadow West is on the left of a …

Position of one point from another

⚡ Place the final reference first. Start the sketch with the person the question asks FROM ('direction of C from B' → draw B at origin), then attach the chain. The answer is simply where C lands on the page.

Order & Ranking

How ranking questions work

Mirror position

p' = n + 1 - p

Position from the other end.

Total from two ranks

n = a + b - 1

Same person, ranked from both ends.

Between count

\text{between} = |a - b| - 1

Two positions measured from the SAME end.

⚡ Draw the end-zones. Sketch a short row of boxes. Fill in what you know at the ends: 'a-th from left' means a−1 people before her. Counting the unknowns at each end replaces every formula with a picture.

Ranks and positions

⚡ Sanity-span check. The two ranks of one person must satisfy a + b = n + 1. Before marking a total, add the given ranks: if the sum is not total-plus-one, one of the numbers was misread.

People between two positions

⚡ Convert, then count. Never subtract ranks from different ends. Convert one with n + 1 − p, then |a − b| − 1.

Ordering and comparison chains

⚡ Longest-chain first. Find the person who appears in the most statements and build outwards from them. The chain assembles in one pass instead of three.

Seating Arrangement

How seating puzzles are set

Circular neighbour rule

L(i) = (i+1) \bmod n,\ R(i) = (i-1) \bmod n

Seats numbered clockwise, everyone facing centre. Both swap if facing outward.

Opposite seats

i \leftrightarrow (i + n/2) \bmod n

Only when n is even; odd tables have no opposite seat.

Viewer vs person

\text{person faces south} \Rightarrow \text{their left} = \text{your right}

Applies only to the person's own left/right, never to the row ends.

⚡ Pin the absolutes first. Extreme ends, exact middles, 'opposite', 'fourth from left' fix specific seats before any chaining. Fill those, and the remaining people usually fall into one or two slots.

Single-row arrangements

⚡ Chain links, don't guess gaps. 'Second to the right of A' pins B exactly two seats from A — mark it with an arrow as soon as you read it. A puzzle of six clues usually has four such arrows; the picture completes itself.

Round-table arrangements

⚡ Walk the circle in one direction. Take the longest adjacency chain and lay it out clockwise without thinking: 'D is to the immediate left of B, C to the immediate left of D, E to the immediate left of C' is just the clockwise string B → D → C → E. Close …

Exam strategy for puzzles

⚡ Two-configuration bail-out. When exactly two seatings survive all clues, check what the QUESTION asks: if both configurations give the same answer, mark it and move on — you never needed the extra clue you thought was missing.

Syllogism

What a syllogism is and how to test it

Transitivity of All

A \subset B \wedge B \subset C \Rightarrow A \subset C

'All A are B + All B are C → All A are C' is the only freely chained rule.

Conversion of All

\text{All } A \subset B \Rightarrow \text{Some } B \subset A

Valid (classes are assumed non-empty): 'Some B are A' follows.

Conversion of No / Some

\text{No } A \cap B \Leftrightarrow \text{No } B \cap A;\quad \text{Some } A \cap B \Leftrightarrow \text{Some } B \cap A

Both convert symmetrically.

⚡ Hunt for one counter-diagram. To kill a conclusion, you need just ONE diagram satisfying all statements that breaks it. Ask: 'can the two circles stay apart / can this element sit outside?' If yes, the conclusion fails.

Possibility conclusions

⚡ One supporting diagram wins for possibility. For possibility conclusions, flip your thinking: you are now the defence lawyer. Build one diagram where the possibility is true alongside all statements — done, it follows.

Either-or (complementary pairs)

⚡ Spot the negation pair. Scan the two conclusions for the word pairs (All … , Some … not) or (Some … , No …) about the SAME two terms. If found and neither follows individually, mark either-or — no diagram needed.

Only-a-few and definite-case rulings

⚡ Split 'only a few' on sight. Rewrite every 'Only a few A are B' as two arrows — ✓Some A are B, ✓Some A are not B — and feed both into your diagram. Everything else is ordinary syllogism.

Venn Diagrams

What a Venn diagram encodes

Two-set inclusion-exclusion

|A \cup B| = |A| + |B| - |A \cap B|

The intersection is counted twice on the right, so subtract once.

Only-regions

|A \setminus B| = |A| - |A \cap B|

'Only A' is A minus the overlap.

⚡ Label regions, don’t imagine them. Write the numbers INTO the regions: |A∩B| in the lens, only-A and only-B in the moons, neither outside. Sums become single looks instead of formula recalls.

Two-circle counting

Two-set inclusion-exclusion

|A \cup B| = |A| + |B| - |A \cap B|

The intersection is counted twice on the right, so subtract once.

Only-regions

|A \setminus B| = |A| - |A \cap B|

'Only A' is A minus the overlap.

Neither

\text{neither} = \text{total} - |A \cup B|

Everyone outside both circles.

⚡ Label regions, don’t imagine them. Write the numbers INTO the regions: the lens, the two moons, and outside. Sums become single looks instead of formula recalls.

Three-circle counting

Three-set union

|A \cup B \cup C| = \sum|A| - \sum|A \cap B| + |A \cap B \cap C|

Add singles, subtract pairs, re-add the triple.

Exactly two sets

\sum (|A \cap B|) - 3|A \cap B \cap C|

Sum of the three pairwise figures minus three times the triple.

Exactly one set

|A \cup B \cup C| - \text{exactly two} - \text{all three}

Strip the multi-set members from the union.

⚡ Triple-strip subtraction. Whenever a question says 'exactly two', your reflex is: pairwise sums overcount the core by 3. Compute pairwise sums, subtract 3×(all three), done.

Choosing the correct diagram

⚡ Test one pair at a time. Never judge the whole picture at once. Run necessity/impossibility on each pair; the correct option is the only one agreeing on every pair.

Dictionary Order & Alphabet

Dictionary order rules

⚡ Find the first split point. Write the words one under another and slide a finger left to right until the letters first differ. That single column decides the entire comparison — ignore all later letters.

Arranging words and picking slots

⚡ Slot-count from the left. Once sorted, number the words 1, 2, 3… and simply read the asked position. 'Comes last' = highest position; 'middle of five' = position 3.

Letter positions and shifts

Position from the right

p_{\text{right}} = 27 - p_{\text{left}}

The two positions of the same letter add to 27.

Shifted position

p' = p \pm k

Move right (+) or left (−) by k letters, clamped to 1–26.

⚡ EJOTY + tiny shift. Anchor at the nearest of E, J, O, T, Y and step. Position 18: T(20) − 2 = R. Two seconds, no counting.

Dictionary rank and word surgery

Rank of a word

\text{rank} = 1 + \sum_i c_i \times (r_i)!

At each letter: c_i = unused letters smaller than it, r_i = letters remaining after it.

⚡ Count smaller unused letters. Freeze the sorted letter list, and for each letter of the word (left to right) ask: how many still-unused letters are SMALLER? Multiply by the factorial of what remains, sum, +1.

Statement & Conclusion

Statement and conclusions

⚡ Extreme-word filter. Circle words like only, all, always, never, surely, best, entire, must. If the conclusion has one and the statement doesn't, it almost always fails.

⚡ Necessary vs sufficient. 'Only X can do Y' means X is required, not that X guarantees Y. Check which direction the conclusion runs.

Statement and assumptions

⚡ Negation test. Put 'not' in the assumption. Does the speaker still have a reason to say what they said? If not, the assumption is implicit.

Courses of action

⚡ Proportion check. Ask: would a sensible administrator actually do this tomorrow? Investigating, repairing, warning, supplying, treating usually pass; banning, shutting forever, blaming everyone fail.

Counting Figures

Counting triangles

Apex lines

T = \frac{(k+1)(k+2)}{2}

k lines from the apex to the base.

Apex lines + horizontal cuts

T = \frac{(k+1)(k+2)}{2}\times(h+1)

h lines parallel to the base crossing every apex line.

⚡ Square with both diagonals = 8. A square (or any rectangle) with both diagonals always has 4 small triangles meeting at the centre and 4 half-squares — 8 in total. Add midlines and it becomes 16: 8 smallest, 4 made of two, 4 half-squares.

⚡ Choose-two-lines view. In a triangle with lines from the apex, each triangle is fixed by choosing its left and right side among the lines through the apex: C(k + 2, 2).

Counting squares and rectangles

Rectangles in an m × n grid

R = \binom{m+1}{2}\binom{n+1}{2}

Pick two vertical and two horizontal lines.

Squares in an m × n grid

S = \sum_{k=1}^{\min(m,n)} (m-k+1)(n-k+1)

n × n grid: 1² + 2² + … + n².

⚡ Line-pair trick. A rectangle is decided by its two vertical sides and two horizontal sides — so count line pairs, never shapes.

Counting straight lines

⚡ Direction sweep. Sweep once for horizontals, once for verticals, once per slant direction; the running total is the answer.

Mirror & Water Images

Mirror images (vertical mirror)

⚡ Check the end characters first. In a vertical-mirror image the last character of the original stands first, flipped. Eliminate options by looking only at the first and last characters.

⚡ Symmetric-letter shortcut. A word made only of A H I M O T U V W X Y that reads the same backwards (a palindrome, e.g. TOOT, WOW) looks identical in a vertical mirror.

Water images (horizontal mirror)

⚡ Order stays, letters go upside down. Scan the options for the one that keeps the original order and has only asymmetric letters turned upside down.

Mirror image of a clock

Mirror time

T_{\text{image}} = 11{:}60 - T_{\text{actual}}

Works in both directions; for 12:xx use 23:60 − T.

⚡ 11:60 rule. Subtract the time from 11:60 digit-group-wise: hours from 11, minutes from 60.

Paper Folding & Cutting

Folding and punching: unfolding the pattern

⚡ Reflect, do not rotate. Each unfold is a mirror reflection across the fold line — a hole near the top-right, unfolded across a vertical midline, reappears near the top-left at the same height.

⚡ Layer count shortcut. Holes in the final sheet = punches × layers at the punched spot. Each full half-fold doubles the layers: 1 fold → 2, 2 folds → 4, 3 folds → 8.

Counting layers and holes

Holes after full half-folds

H = p \times 2^{n}

p punches through n full half-folds.

⚡ Doubling chain. Write 1 → 2 → 4 → 8 as you read each full fold; multiply by the number of punches at the end.

Embedded Figures

Finding the hidden part in a figure

⚡ Anchor on the slant. A slanting line has only two possible directions (/ or \). Check whether the main figure even has the option's slant in the right place; this rejects mirror-image distractors in seconds.

Which figure contains the given part

⚡ Eliminate by the missing line. Count the part's line directions (horizontal, vertical, /, \). Any option missing one of these directions is out immediately.

Cube & Dice

Dice: finding opposite faces from positions

Standard die

1+6 = 2+5 = 3+4 = 7

Opposite faces of a standard die.

⚡ Eliminate the neighbours. List every face seen with the asked face; strike them out. If only one face is left, it is the answer — no rotation needed.

⚡ Two common faces. Two views sharing two faces: the leftover faces are opposite.

Cube nets: folding a sheet into a cube

⚡ Skip one. Along a line of squares, skip one: 1st ↔ 3rd, 2nd ↔ 4th.

Painted cube cut into small cubes

Two faces painted

12(n-2)

Cubes on the edges, excluding corners.

One face painted

6(n-2)^2

Cubes in the middle of each face.

No face painted

(n-2)^3

Hidden inner cube.

⚡ Check the total. 8 + 12(n−2) + 6(n−2)² + (n−2)³ = n³. Use it to verify your counts.

Figure Series

Movement and rotation

Position after k steps

p_{k} = (p_{0} + k\,s) \bmod 8

s = step (positive = clockwise) on the 8 border places.

⚡ One element, one line. Write a tiny table: element → positions in frames 1-4 → step → next. Options that fail any row are out.

⚡ Angles as clock hours. Treat 45° as 1.5 clock-hours; an arrow turning 90° clockwise each step goes 12 → 3 → 6 → 9 → 12.

Adding, removing and changing elements

⚡ Count first. Count the elements in each frame. If the counts go 1, 2, 3, 4, only options with 5 survive; then check positions.

English Comprehension

Reading Comprehension

How to attempt an RC passage in SSC exams

⚡ Two-minute structure read. On the first read, do not hunt for facts. After each paragraph say its job in three words. Facts are cheap to re-find; structure is what saves you.

⚡ Vocab questions first, main idea last. Answer the replace-the-word question and line-specific details before the main-idea question — by then you will have re-touched half the passage and the theme is clear.

⚡ Line anchoring. Every detail question has a home line. Match a distinctive word in the question (a name, a number, an odd noun) to the passage, read two lines around it, and choose the option that paraphrases them.

Main idea, central theme and best-title questions

⚡ First and last sentence sketch. Read only the opening and closing sentences of the passage, plus the first sentence of each middle paragraph. That skeleton usually states or implies the theme — build your ten-word summary from it before looking at opti…

⚡ Count the mentions. The concept the author repeats in different words across paragraphs is the theme. If an option's core idea appears in only one paragraph, it is a detail, not the theme.

⚡ Title vs theme wording. Titles are noun phrases ('The quiet comeback of the public library'); theme options are full claims. Do not reject a title for lacking a verb — reject it for wrong scope.

Factual detail questions ('According to the passage…')

⚡ Paraphrase before you look. After locating the line, cover the options, answer aloud in your own words, then uncover. This one habit defeats most engineered wrong options.

⚡ Reversal scan. Before marking a detail option, check its direction words: only/always/all, may/often/some, can/cannot. Wrong options flip these while keeping the same topic words.

⚡ Option pairing. When two options say the same thing in different words, both are wrong — a fact has one meaning. Kill the pair, then judge the remaining two.

Inference and extrapolation questions

⚡ One-step rule. Correct inference = stated idea + one logical step. If you need two steps, extra facts, or a change of scale (one city → all countries), reject it.

⚡ Verbatim is a verdict. In an inference question, an option that repeats passage wording without adding the step is wrong by definition. Cross it out first.

⚡ Extremity filter. Inferences with *only, never, must, inevitably* are almost always one step too far. Moderate options (*may, tends to, is likely*) fit the must-be-true test better.

Vocabulary in context (replace-the-word / meaning questions)

⚡ Own-words first. Cover the options, replace the word with your own simple word, then find the option matching YOUR word. This kills the dictionary-meaning bait.

⚡ Tone match. The substitute must keep the sentence's attitude. In a critical sentence, choose the critical option; a neutral synonym that drains the criticism is wrong.

⚡ Two-line rule for phrases. For phrase-meaning questions, read one line before and after: figurative phrases get their meaning from the surrounding argument, not from the words themselves.

Tone, attitude and author's-opinion questions

⚡ Adjective audit. Sweep the passage for adjectives and adverbs with feeling (quietly, badly, remarkable, merely, confident). Three of them in the same direction fix the tone.

⚡ However counts double. The clause after *but/however/yet* carries the author's real position; the clause before it is the position being set aside. Tone and opinion questions key off the second half.

⚡ Degree matching. Match intensity, not just direction: 'sceptical' ≠ 'hostile', 'cautiously optimistic' ≠ 'confident'. Pick the option at the same temperature as the passage.

Cloze Test

What a cloze test is and how to attack it

⚡ Tense anchor first. Before touching any blank, underline the time expressions: 'Every morning...', 'Last month...', 'by evening'. Verb blanks and even prepositions obey these anchors.

⚡ Predict, then match. Cover the options with your hand, read the sentence, and whisper your own word. Options are written to include your word's exact meaning — anything far from it is usually wrong.

⚡ The read-aloud finish. After answering all five, read the passage silently as if it were written by someone else. Grammar errors and logic breaks become audible — re-check any blank where you stumble.

Grammar inside the passage: tense, agreement, articles

⚡ Strip-to-subject. Delete every phrase between the subject and the verb blank, then apply agreement to the bare skeleton.

⚡ Only-one-past trick. When a narrative blank offers just one past-tense option among present/gerund forms, the anchor already decided it — mark it and move on without re-reading the paragraph.

⚡ Say the article aloud. For a/an blanks, pronounce the *next word's first sound*: 'an hour' but 'a university'. Silent letters and 'yu' sounds decide it, not the first letter.

Vocabulary blanks: meaning, register and collocation

⚡ Name the missing idea. Before looking down, say what the sentence lacks in plain words: 'it needs the idea of CONSISTENCY here'. Then choose the option that says it — this defeats look-alike distractors.

⚡ Collocation pairs to bank. Memorise high-frequency pairs: gain momentum, pay attention, take measures, reach a decision, bear fruit, meet a deadline, break the news, strike a balance. Cloze nouns and verbs revolve around them.

⚡ Polarity check. Mark the sentence + or − before choosing. A praising sentence rejects negative words even when they are grammatically perfect.

Connector blanks: contrast, cause, result, addition

⚡ Opposite-direction test. Give each clause a + or − sign. Same signs → addition/result connector; opposite signs → contrast connector.

⚡ Partner hunt. Scan the sentence for the other half of a correlative pair. Seeing 'not only' fixes the blank to 'but also' — zero thinking needed.

⚡ Comma tell. 'clause, _ clause' with a single-word slot wants *yet/but/so/for/and*. Options like 'however/therefore' fit 'clause. _, clause' patterns instead.

Preposition and phrasal-verb blanks

⚡ Governor chant. Underline the word governing the blank and recite its pair: depend—on, afraid—of, good—at, solution—to. Ten pairs cover most paper prepositions.

⚡ Particle direction sense. Ask what the sentence does to the object: cancels it (*call off*), rejects it (*turn down*), erects it (*set up*). The particle encodes the action's direction.

⚡ Substitute the simple verb. Replace the phrasal option with a one-word verb in your head: look after = tend. The sentence must keep its meaning with the substitution; if not, that phrasal is wrong.

Error Spotting

The master checklist: SVA → tense → article → preposition

⚡ The 7-step sweep. Verb first: find the subject, check number. Then tense anchors. Then a/an/the by sound. Then preposition partners. Only then pronouns, pairs, and word choice.

⚡ Strip the middle. Delete phrases between subject and verb ('of my friends', 'of the mangoes', 'as well as the players') and read the bare skeleton — agreement errors become loud.

⚡ No-error discipline. When the paper offers 'No error', choose it only after the full sweep; do not pick it because the sentence 'sounds fine' at speed. Roughly one in five such items hides a sneaky SVA or preposition error.

Subject–verb agreement errors

⚡ Subject hunt backwards. From the verb, walk backwards past 'of...' phrases until you hit a noun or pronoun without a preposition in front — that is the subject that decides the verb.

⚡ Proximity rule flash. See *neither/nor* or *either/or* with two subjects? The verb matches the second (nearer) subject. Check that, and the item usually collapses instantly.

⚡ Quantity is singular. Sums of money, distances, time spans and weights take a singular verb even in plural form: 'Ten kilometres is a long walk', 'Five hundred rupees was the fine'.

Tense and sequence-of-tenses errors

⚡ Underline every anchor. Circle yesterday/ago/since/for/by the time/when/next week before judging any verb. Most tense questions answer themselves once the anchors are visible.

⚡ Will-check. Scan for *will* inside a *when/if/until* clause with future meaning — SSC plants this trap repeatedly.

⚡ Since/for sorting. *Since* + starting point (since Monday, since 2019, since childhood); *for* + duration (for ten years, for a week). Wrong partner = wrong part.

Article and preposition errors

⚡ Say it aloud. Articles are sound decisions. Pronounce the phrase — 'an university' fails the ear immediately ('yu' sound), 'a honest man' likewise.

⚡ Governor chant. Underline the word the preposition depends on and recite its partner: insist—on, angry—with (person), prefer—to, married—to. Wrong partner = the answer.

⚡ The-with-superlative scan. Superlative adjectives and ordinals demand *the*: 'the most intelligent girl', 'the first attempt'. A bare superlative is a hidden article error.

Pronoun and correlative-conjunction errors

⚡ Preposition + I? Kill it. Any 'between you and I', 'for he and I' construction is wrong in standard English — object form after a preposition, every time.

⚡ Did + base form. After *did/does/do*, the verb loses its tense: *No sooner did the bell ring*. A past form after *did* is a guaranteed error.

⚡ Partner visibility test. Cover the sentence except the pair words. If one half of a correlative appears without its partner, or with the wrong partner (*hardly... than*), that part is the error.

Comparison, redundancy and confusable-word errors

⚡ -er/-more clash scan. Look for *more* glued to a comparative (*more bigger*) or *most* glued to a superlative — instant error, no further thought needed.

⚡ Back-and-forth redundancy hunt. Two-word pairs where the second word repeats the first are planted answers: return BACK, repeat AGAIN, revert BACK. Train the eye to flinch at them.

⚡ that-of insertion test. In comparisons of two nouns' qualities, the second noun needs *that of/those of*: 'The roads of Delhi are wider than those of Patna'. Missing *those of* = comparison error.

Verb forms, question tags, conditionals and structure

⚡ Question-tag flip. Hear the sentence end: a statement + tag must flip sign — positive statement → negative tag, negative word (never, rarely, hardly) → positive tag.

⚡ Unreal-past conditional. After 'if' about the unreal past, English wants had + participle — 'would have' in the if-clause is the planted error.

⚡ Parallel lists. A list is parallel: whatever shape the first item takes (-ing, to + verb, bare noun), the rest must copy it.

Sentence Improvement

The elimination ladder: how to choose in 20 seconds

⚡ Anchor before options. Find the time word or helper verb near the highlight ('yesterday', 'did', 'each of the'), decide what the verb must look like, then pick the option matching YOUR form.

⚡ Death-note elimination. In your head, tag every option with its fatal flaw before choosing: 'plural verb', 'breaks inversion', 'wrong partner'. If no option has a flaw, the answer is 'No substitution required'.

⚡ Meaning guard. Between two grammatical survivors, choose the one that preserves the original meaning word for word — the exam rewards accuracy, not elegance.

Tense and sequence fixes

⚡ Yesterday test. See a finished-time anchor? Kill every perfect and continuous option instantly — one survivor usually remains.

⚡ Backshift + straight order. In reported questions after *asked/told*, two things must change: tense backshifts and the question flips to statement order ('why he had left'). Each option usually breaks exactly one of the two — check both.

⚡ Since/for health check. 'Since 2015 / for ten years' demands a perfect form with *been* if the action continues. Options without *have/has been* die at once.

Agreement and verb-form fixes

⚡ Cover the phrase, read the skeleton. Physically cover 'of the candidates'-type phrases with a finger and read what remains: 'The list _ displayed'. The agreeing verb is now obvious.

⚡ Helper-chain audit. did → base, have → participle, be → -ing/participle. When the highlight contains a helper, check the partner form first — that is where the planted error lives.

⚡ Two-subject sentence check. If two subjects are joined by *and*, the verb is plural — unless the pair names one thing ('bread and butter is'). SSC plants singular verbs before such pairs.

Comparison and structural fixes

⚡ that-of insertion. When two possessive phrases are compared and only the second is highlighted, 'that of / those of' is almost always the answer. Check number: singular → that of, plural → those of.

⚡ Negative-opener flip. See *Hardly/No sooner/Not only/Little* at sentence start? Expect auxiliary-before-subject order in the highlighted part: 'had I reached', 'did he arrive'. Un-inverted options die on rung 3.

⚡ -ing opener ownership. Ask WHO is doing the -ing action. The sentence's main subject must be that doer; if not, pick the option that makes the doer the subject.

Precision fixes: articles, prepositions, pronouns, word choice

⚡ Small-word sweep. Read ONLY the small words in the highlight (a/an/the, in/on/at, I/me/myself). A wrong one usually stands alone in its wrongness — fix it and finish.

⚡ myself firewall. Reflexives never replace plain object pronouns. Any option offering *myself* where *me* belongs is a decoy — kill it on sight.

⚡ Redundancy flinch. Train on the fatal pairs: return back, repeat again, revert back, final conclusion. When one appears highlighted, the shorter option is the key.

Structural fixes: inversion, dangling modifiers, parallelism

⚡ Opener triggers flip. Negative word first? Then expect 'auxiliary + subject': had I reached, did he arrive, did she sing. Choose the option in flipped order with the right partner (when/than).

⚡ Who is doing the -ing?. Name the doer of the opening action; if the main subject is not that doer, pick the option that promotes the doer to subject.

⚡ Shape-match across pairs. Cover the pair words and compare the shapes on both sides: 'not only [verbed] ... but also [verb]' — any mismatch in form is the fault being tested.

Fill in the Blanks

Collocation blanks: word partnerships that decide the answer

⚡ Governor-first reading. Read the sentence up to the blank, stop, and ask: 'What does this word usually take next?' Answer from memory, then look at the options — your answer is usually sitting there verbatim.

⚡ Topic veto. Collocation options in SSC are often same-suffix decoys (-tions, -ments). Kill them by topic first: a border dispute invites *talks/deliberations*, never *medications*.

⚡ Verb-object completeness. For verb blanks, complete the object in your head: 'attention is _' → paid. If your completing verb is an option, mark it and move on.

Phrasal-verb blanks: verb + particle as one unit

⚡ Name the action, then match the unit. Say the sentence's action in one plain word (cancelled, tolerated, abolished), then pick the phrasal unit that translates it. The wrong particles translate other actions.

⚡ Particle direction check. Ask what the sentence does to its object — kills it (off/out), raises it (up), lowers it (down), accepts it (in). Match the particle's direction to the verb's fate.

⚡ Simple-verb substitution. Replace each option with a one-word verb (call off = cancel). The option whose substitution keeps the sentence's meaning is the answer.

Verb-form and tense blanks

⚡ Preposition-to test. Check what owns the 'to'. Owned by a preposition-verb or noun → -ing; owned by an infinitive verb → base form. This one test answers the largest tense-form family.

⚡ By-the-time ladder. Two events with 'by the time' → earlier verb in past perfect, later in simple past. Fit the blank to its rung of the ladder.

⚡ Verb-category lists. Memorise the three mini-lists: -ing verbs (avoid, enjoy, mind, suggest, finish), to-verbs (decide, hope, refuse, plan), base-after (had better, would rather, let, make).

Connective and relative-pronoun blanks

⚡ Partner scan before choosing. Search the sentence for the other half of the pair. Seeing 'Hardly had' fixes the blank to *when*; seeing 'No sooner' fixes it to *than*. Zero deliberation needed.

⚡ He/him substitution for who/whom. Rephrase the clause with he/him: 'everyone trusts (him)' → whom; '(he) inspires everyone' → who. No ambiguity survives the substitution.

⚡ Despite/Although sorting. Full clause after the blank → Although/Though; noun or -ing → Despite/In spite of. The trap options swap these two families.

Confusable and homonym blanks

⚡ Slot first, meaning second. Decide noun/verb/adjective from the sentence frame; kill every option of the wrong class. Then choose by meaning among survivors — usually only two ever reach step two.

⚡ Context picture. Form a mental image of the sentence: a pile of paper in a godown → stationery; a parked van → stationary. The image picks the spelling instantly.

⚡ Triple-family cards. Make flash cards only for families of three or more: assure/ensure/insure, site/sight/cite, affect/effect/affective. Singles rarely appear in options.

Double blanks: two gaps, one sentence

⚡ Structure-pair recognition. too + adjective + to + base verb; so + adjective + that + clause; such + noun + that + clause; hardly...when; no sooner...than. Recognise the skeleton and both blanks fill at once.

⚡ Verb-agreement tiebreak. Plural verb with two subjects → Both...and; singular verb → Either...or / Neither...nor (matching the nearer noun). The verb alone often fixes the pair.

⚡ Fixed binomial bank. Revise binomials in pairs: safe and sound, part and parcel, null and void, first and foremost, ways and means. Options split genuine binomials with a wrong partner — the genuine one is the answer.

Synonyms & Antonyms

The root-word method: decode words you have never seen

⚡ Split-then-eliminate. Chop the unknown word at its joins (bene-vol-ent), translate the parts you know, and eliminate options contradicting them. One confident root kills two options.

⚡ Family clustering. Learn words in root families, not alphabetical order: dict family (dictate, verdict, indomitable? no — edict, indict), loqu family (eloquent, loquacious). One sitting per family.

⚡ Positive/negative split. Roots carry emotional charge: bene-, eu-, am- are warm; mal-, dys-, phob-, mis- are cold. When stuck, mark the word's charge and choose an option of matching charge.

Choosing the nearest meaning: elimination by class, degree, charge

⚡ Overlap test. The right synonym shares the core idea, not the neighbourhood. Ask 'can one replace the other in most sentences?' — if the sentence changes meaning, it is an associate, not a synonym.

⚡ Charge matching. Label the given word + or − (notorious −, celebrated +) and pick the option of the same sign. Half of synonym options fall to charge alone.

⚡ Degree ruler. Place the word and both surviving options on an intensity line (mild → extreme). The option at the same mark is the answer.

Antonyms: prefix flips and real opposites

⚡ Mini-sentence contradiction. Put the word and each candidate in the same short sentence ('The manager praised / criticised the report'). The option producing a genuine contradiction is the antonym.

⚡ Prefix flip check. Before scanning options, guess the flip yourself (conformity → nonconformity). If your guess is among the options, mark it — the options are built around the same flip.

⚡ Absence is not opposition. Kill options that merely lack the quality ('careful' for 'brave'). An antonym fights the word; it does not stand beside it.

Exam strategy for vocabulary questions

⚡ Two-pass vocabulary. Answer every certain vocabulary item in the first pass (15 s each); in the second pass, return to unknowns and root-split them with the remaining time.

⚡ Charge guess with options. Options reveal the axis: if options are (kind, cruel, wealthy, talkative), the axis is temperament — now guess the word's charge from its sound (mal-, dys- negative; bene-, eu- positive) and pick accordingly.

⚡ The 2-4 year echo. Words repeat across years. Solving the last 5 years' synonym/antonym questions is the single highest-yield vocabulary exercise — the bank above already encodes that history.

One Word Substitution

Suffix families: decode the word by its ending

⚡ Family filter. Classify the stem first: is it a killing word, a fear, a study, a ruler-system, a person-type, a place? Reject every option from a different family before reading further.

⚡ Root + suffix stack. Split rare answers: nycto-phobia, entomo-logy, carto-grapher. Two known parts beat one memorised word.

⚡ Counting-prefix drill. One evening per prefix set: mono-, bi-, tri-, poly-, pan-, omni-. Write each with three familiar words and the family stays for life.

One word for people: traits, trades and types

⚡ Object-anchor the trade. Never learn 'cobbler' alone — learn 'cobbler → shoes'. Stems name the object ('one who repairs shoes'), and your anchor matches it instantly.

⚡ Contrast pairs together. Store opposites and neighbours in one card: garrulous ↔ taciturn; miser ↔ spendthrift; novice ↔ veteran. The stem often points to one side of a pair.

⚡ Omni-triplet. One prefix (omni- = all), three endings: -potent (power), -scient (knowing), -present (place). One card covers all three SSC favourites.

One word for places and groups

⚡ Stored-object anchor. Places are defined by what is inside: bees→apiary, birds→aviary, grain→granary, coins→mint, corpses→mortuary. Match the object, not the sound.

⚡ Animal-first matching. For group stems, name the animal first, then recall its partner noun. Options scramble the five common pairs; the animal decides.

⚡ Near-pair cards. Make cards for one-letter pairs: apiary/aviary, granary/granite, monastery/convent (men/women). These are where marks leak.

Government systems, studies and word-craft

⚡ Ruler-first matching. Underline WHO rules in the stem (people, rich, officials, one, God) and match to democracy/plutocracy/bureaucracy/autocracy/theocracy. The stem hands you the answer's head word.

⚡ Verb-link the negatives. Chain each 'cannot-be' word to its verb: read→illegible, believe→incredible, see→invisible. Stems always quote the verb.

⚡ Dead-word precision. About the dead: praise → eulogy, mourning-poem → elegy, tomb-writing → epitaph. Three cards, three certain marks.

Exam strategy for one-word substitution

⚡ Specificity rule. When two options fit the family, the stem's detail decides: 'brother' → fratricide, not homicide; 'twice a year' → biannual, not annual.

⚡ Stem-object underline. Underline the noun the word must carry (bees, grain, king, tomb). One underline removes half the options.

⚡ Mock-word ledger. Keep one running sheet titled 'OWS from mocks'. Most repeats come from your own past mistakes, not from unknown words.

Idioms & Phrases

Why literal meaning is always wrong — and how to find the picture

⚡ Literal-first elimination. Scan the options and kill every one you could photograph. The remaining two or three are the true contest.

⚡ Picture-to-abstract translation. Name the picture's key action in the abstract: escape→reveal, cold water→trouble, midnight oil→late work, blue (sky)→unexpected.

⚡ Plug-back sentence. Fit each surviving option into your own sentence using the idiom. The one that sounds like ordinary English is the key.

Animal and body-part idioms

⚡ Animal-character match. Recall each animal's stock character (rat = cheat, wolf = hunger, bull = force). The idiom's meaning is the animal's character plus the verb.

⚡ Body-function match. Ear = listen, eye = watch, foot = stand firm, hand = help/work. Map the named part to its function before reading options.

⚡ Opaque-idiom cards. Ten idioms have lost their logic (bite the dust, cold shoulder, red tape). Card them separately — they are the only real memory load.

Colour and food idioms

⚡ Colour association. Blue = surprise, red = anger or bureaucracy, green = envy, white = useless grandness, black = disgrace. Half the colour idioms decode from this alone.

⚡ Food-judgement fit. Food idioms are judgements in disguise: cake→easy, nut→stubborn, pickle→difficulty, potato→controversy. Match the judgement, not the ingredient.

⚡ Pair-learning. Learn confusable idioms in pairs on one card: beans/milk, elephant/lion's share, red tape/red-letter day. The option list usually contains both members.

Idioms for situations and types of people

⚡ Family surfacing. Classify the stem's demand: sudden? trouble? effort? defeat? person-type? Recall the family's four members; the right one is usually among the options, and the wrong options are from *other* families.

⚡ Allusion anchors. A few idioms hide a story: Achilles' heel (the hero's one weak spot), bell the cat (the mouse that dared), pass the buck (shifting the deal-marker). One-line stories fix meanings permanently.

⚡ Outcome test. For action idioms, ask what changes at the end: throw in the towel → the fight stops (give up); turn the tables → positions reverse; face the music → consequences arrive.

Phrasal verbs (verb + particle phrases)

⚡ Particle meanings. *off* often means stop or cancel (call off, put off); *out* means finish or remove (run out, put out); *up* means complete or give up; *after* means follow or care for (look after, take after).

⚡ Same verb, four particles. When all four options share one verb (break down / out / up / in), say the sentence with each and keep the one whose meaning fits the whole sentence.

⚡ One-word swap. Replace the phrasal verb with a single formal word to check it: put out = extinguish, call off = cancel, put up with = tolerate, come across = meet by chance.

Exam strategy: banking 300 idioms in 30 days

⚡ Sentence production. For every idiom, speak a sentence about your own life ('I burned the midnight oil before my SSC mock'). Personal sentences are recalled under pressure; list-readings are not.

⚡ Family-first revision. Revise by bank (animals → body → colours → food → situations → people). Each bank is one mental room; in the exam you 'enter the room' and the members appear.

⚡ Bleeder list. Fail an idiom twice in self-tests? It goes to a 15-item list revised weekly. Most permanent marks come from this list, not from new words.

Spelling

i before e — and the exceptions that fill the paper

⚡ The c-test. Is there a c right before the pair? c → 'ei' (receive). No c → usually 'ie' (believe). Ten exceptions override: weird, seize, height, foreign, sovereign, leisure, either, neither, their, protein.

⚡ Rhyme-anchor pairs. height–weight rhyme and must match; believe–achieve match; receive–deceive match. Fix one of each pair forever and its partner comes free.

⚡ Say it in syllables. Pronounce the word slowly in your head — be-lieve, re-ceive. The mouth half-confirms what the eye doubts.

Doubling consonants and one-letter betrayals

⚡ Stress decides doubling. Say the word. If the last syllable is stressed, double before a vowel-suffix (begin→beginning, occur→occurred). If not, don't (benefit→benefited).

⚡ Double-double checklist. Five words need TWO doubled letters: accommodate (c,m), committee (m,t), embarrass (r,s), occurrence (c,r), millennium (l,n). Count them.

⚡ Letter census. In the options, count the repeated letters. accommodate must show 2 c's and 2 m's; embarrass 2 r's and 2 s's. Eliminate on census failure.

-able / -ible, -ance / -ence, -ant / -ent

⚡ Root trace. Strip the suffix and look at the root: exist→exist-ence, maintain→maintenance (ai shortens), resist→resist-ance, persist→persist-ence. The root's own spelling decides.

⚡ -fer always -ence. Words from 'fer' nouns take -ence: reference, preference, difference, inference. Verbs keep one f or double by stress: refer→referring, but reference.

⚡ Scient family. One family handles four words: sufficient, efficient, ancient, science (and conscience = con + science). All -cient/-ience.

The trap-word bank: silent letters and inherited mistakes

⚡ The n-test for -ment words. government = govern+ment, environment = environ+ment: the root's final n must survive. Ask 'what is the root?' — the letter reappears.

⚡ Three -ceed, one -sede. proceed, succeed, exceed = the only -ceed words; supersede = the only -sede. All other 'seed-sound' words are -cede.

⚡ No- doubling plurals. criteria, phenomena, stimuli, data already are plurals — any option adding -s to them is instantly wrong.

Exam strategy: how to judge a spelling you have never seen

⚡ Difference-first reading. Don't read four full words — read the one letter-position where they differ. Options are built to differ in a single spot; that spot is the entire question.

⚡ Syllable count. Count syllables: mischievous (3), restaurant (2-3 as res-tau-rant), rhythm (2). A wrong spelling often forces a syllable that cannot be pronounced.

⚡ Write-from-memory drill. Recognition is not recall. Cover the bank, write the words, mark misses, repeat tomorrow — misses halve each cycle.

Active & Passive Voice

Voice basics: the three moves

⚡ Tense handover. The helping verb must reproduce the ORIGINAL tense: wrote→was, writes→is, will write→will be, has written→has been. Any tense change in the options is an error planted on purpose.

⚡ V3-only law. After be/been/being, the main verb is always V3 (written, given, built). 'Is wrote', 'was wrote', 'is build' are impossible — eliminate on sight.

⚡ Pronoun mirror. Subject pronoun moves to by-phrase in object case (he→by him); object pronoun moves to subject slot in subject case (me→I).

Tense-by-tense conversions in action

⚡ Row-recall. Identify the active tense, then recite its passive row: present cont. → is/are being + V3; past perf. → had been + V3; future → will be + V3. Options break exactly one cell of the row.

⚡ Re-agreement sweep. After converting, make is/are/was/were/has/have agree with the NEW subject — check this before anything else.

⚡ being vs been slot. Continuous in the active → being. Perfect in the active → been. Modal → neither (plain be).

Modals, imperatives, infinitives and questions

⚡ Let-anchor. An imperative stem (no subject, base verb) signals the Let + object + be + V3 answer. Any option without 'Let' or a modal cannot be right.

⚡ By whom inversion. Who→questions convert to 'By whom + was/were/is + new subject + V3'. Check the word order first — options scramble it.

⚡ Request formula. Please → 'You are requested to …'; order → 'Let … be …'; advice → 'You are advised to …' or '… should be …'.

Reverse conversions: passive to active

⚡ First-auxiliary tense. Read only the first auxiliary to fix the tense: was→past, is→present, has/had→perfect, will→future, is being→present continuous, was being→past continuous.

⚡ Agent hunt. Find the by-phrase; that noun/pronoun becomes the subject. No by-phrase → they/people/someone.

⚡ Object-case exit check. The final pronoun must be in object case: 'They praised him', not 'he'. Many options fail exactly here.

Exam strategy: the 20-second conversion protocol

⚡ Fault-class elimination. Every wrong option breaks exactly one rule: V-form, being/been, tense, agreement, pronoun. Name the fault — the option dies by its name.

⚡ Five-row priority. Master simple present, simple past, present perfect, present continuous, modals first — they cover 80% of paper items.

⚡ Write, don't speak. Voice errors live on paper. Convert in writing daily; the hand learns the skeleton faster than the ear.

Direct & Indirect Speech

Reporting verbs and the mechanics of the shift

⚡ Verb-tone match. Read the quoted words first for TONE: order, advice, request, question, joy, sorrow. Then pick the reporting verb that carries that tone — the verb choice alone often eliminates two options.

⚡ said-to vs told. said + to + person; told + person. Any 'said me' or 'told to me' option is dead on arrival.

⚡ Connective test. Statement → that. Yes/no question → if/whether (never that). Wh-question → the wh-word itself (never if).

Backshift: the tense conversion table

⚡ One-step-back ladder. Recite the ladder: present→past, present cont.→past cont., perfect→past perfect, past→past perfect, will→would, can→could, may→might, must→had to.

⚡ Frozen-five. could, would, should, might, ought to never change. Spot one in the quote → that option's tense is already correct; any shift is an error.

⚡ Truth test. Is the quoted statement a fact of nature, a habit or a proverb? Then no backshift — said that the earth REVOLVES.

Pronouns, and the time/place word table

⚡ SONA sweep. Mark the three pronoun anchors before touching anything else: 1st person → speaker, 2nd person → listener, 3rd person → unchanged. Most wrong options botch exactly one of these.

⚡ Yesterday/tomorrow pair. yesterday → the previous day; tomorrow → the next (following) day. The pair travels together — if one shifted and the option leaves the other, it's wrong.

⚡ Present-reporter exemption. Reporting verb 'says' (present) → nothing shifts: tense AND time words freeze. Check the reporting verb before applying any table.

Questions, commands and exclamations

⚡ Skeleton-first. Classify the quote (question/command/request/exclamation) and write its skeleton (asked if… / told … to … / exclaimed with joy that …). Then pour tense, pronoun and time-word shifts into the skeleton.

⚡ Do-support removal. Do/did in direct questions is a dummy — it disappears in indirect speech: 'Do you know' → if I knew, never 'if I did know'.

⚡ Exclamation-verb map. Hurrah → exclaimed with joy; Alas → exclaimed with sorrow; Bravo → applauded; Good morning → wished; Thank you → thanked. One-to-one map, no thinking needed.

Exam strategy: the four-check protocol

⚡ One-fault elimination. Name the fault of each wrong option — verb, tense, pronoun or time-word. If you can't name a fault, look again; if two options share none, you've misjudged the tone verb.

⚡ Reverse-mode anchors. Indirect→direct: the phrase 'the previous day' signals quotes needed with 'yesterday'; 'asked if' signals a yes/no question with inverted order and dummy 'do' if present tense returns.

⚡ Written drill. Narration lives in punctuation and word order — practise on paper, never orally; the hand catches what the ear forgives.

Sentence Rearrangement

The PQRS method: label, anchor, chain, eliminate

⚡ Pairs before strings. Do not read the four option strings first. Build 2–3 mandatory pairs from P/Q/R/S; then check which strings respect them. Strings die by broken links, not by vibes.

⚡ S1–S6 frame test. After chaining, read S1 + your chain + S6 as one paragraph. If any 'it/this/they' lacks its noun, the chain is wrong.

⚡ Adjacent-swap distractors. Wrong strings usually swap two adjacent parts. Find the ONE swap each wrong string commits and name its broken pair — that is your verification.

Opening and closing sentence clues

⚡ Pronoun-orphan test. Any part starting with he/she/it/they/this/these needs its noun earlier. If no earlier part in a string supplies the noun, that string dies.

⚡ Connector ban. But/However/Therefore/Thus/So cannot open. Two of four options often open with such parts — free elimination.

⚡ S6 pull-back. Read S6 and ask 'what sentence must come just before it?' — usually the one with Thus/That is why or the summarising judgement. Pin it, then build backwards.

Mandatory pairs: the links that never break

⚡ Noun-first scan. List the nouns/pronouns across P, Q, R, S. Circle pronouns and 'the'-nouns; draw arrows to the part that introduces each noun. The arrows ARE the answer skeleton.

⚡ This/That noun hunt. For every 'this + abstract noun', find which part contains that abstract noun. The order is forced.

⚡ a→the cascade. The part containing 'a + noun' must come before every part containing 'the + same noun'. Instant orderer for food-and-trick passages.

Time-sequence chains

⚡ Timeline first. Write the 4–6 events as a numbered list, then map numbers to letters. The string writes itself.

⚡ Tense radar. Past perfect = earlier; past continuous = background; simple past = the event line. Use tenses to order events when time words are missing.

⚡ By-time capping. 'By midnight/By dawn' closes the phase it describes — it follows, never opens, the event chain.

Exam strategy: elimination, budget and Tier 2 jumbles

⚡ Layered elimination. Layer 1: opener ban. Layer 2: mandatory pair. Layer 3: S6 check. Each layer is one question; two layers usually finish the item.

⚡ Local-difference test. Surviving strings differ in one part's position. Isolate that part and test only its two neighbours.

⚡ Odd-sentence radar. For odd-one-out items, mark each sentence's KEY noun. Four share it; the impostor only orbits it.

Grammar Essentials (Handbook)

Parts of speech: nouns, pronouns, adjectives, adverbs

⚡ Pronoun case test: cover the other name. When a pronoun is paired with a noun (*Ravi and I / Ravi and me*), cover the noun and read the sentence with the pronoun alone. *The teacher praised ~~Ravi and~~ me* ✓ because *praised me* sounds right; *praised I* does …

⚡ Noun trap scan. In any error-spotting sentence, circle every noun and ask two questions: *Is it countable?* and *Is it one of the special groups (uncountable / always plural / cattle-type)?* About one in five SSC noun errors is *informa…

⚡ Few vs a few: the half-glass rule. *few / little* = the glass is empty (negative feeling); *a few / a little* = there is something (positive feeling). If the sentence continues with a negative idea (*so he could not...*), pick *few/little*; if pos…

Tenses and sequence of tenses

⚡ Signal-word scan. Before reading options, underline the time signal: *since/for* → perfect; *yesterday/ago/last/in 2019* → simple past; *by + future time* → future perfect; *when/if/as soon as + future meaning* → simple present; *before/a…

⚡ The 'no will after when/if' reflex. If a clause begins with *when, if, unless, until, as soon as, before, after, once* and talks about the future, strike out *will/shall* in that clause. The *will* belongs only to the main clause.

Subject-verb agreement (all rules and exceptions)

⚡ Strike-through method. Draw a line through every prepositional phrase (*of the boys, in the box, with his friends*) and every add-on phrase (*as well as..., along with..., together with...*). What remains before the verb is the true su…

⚡ Nearest-noun rule for or/nor pairs. For *either...or, neither...nor, not only...but also, or, nor*: look only at the subject closest to the verb. Tip: put the plural subject second so the plural verb sounds natural.

⚡ Number vs A number. *The number* = one figure (singular). *A number* = many (plural). Memory hook: *A = Abundant*.

Articles (a, an, the, zero article) and determiners

⚡ Say it aloud. For a/an, whisper the next word. If your mouth starts with a vowel sound (*on-est, em-el-ay, ex-ray*), use an; if it starts with *y* or *w* sound (*you-niversity, won-day*), use a.

⚡ The geography rule of thumb. Plural or water or chain → THE (the Alps, the Ganga, the Indian Ocean, the Andamans). Single peak, single lake, city, most countries → no article (Mount Abu, Lake Dal, Delhi, Nepal).

Prepositions and conjunctions

⚡ Partner check for correlatives. The moment you see *hardly, scarcely, no sooner, not only, neither, either, both, whether, lest, though*, scan ahead for its fixed partner. A wrong or missing partner is the error ~80% of the time.

⚡ Verb-preposition memory by meaning. Group collocations by the idea: stopping (prevent/refrain/abstain/desist from), dependence (depend/rely/count on), skill (good/expert/adept at; proficient in), Latin comparatives (senior/j…

Modals, conditionals and the subjunctive

⚡ If-clause match table. Read the main clause first, then fix the *if*-clause: will → present; would + V1 → past/were; would have + V3 → had + V3. The *if*-clause is always one step 'further back' in time than the main verb.

⚡ Deduction ladder. Certain positive → *must have*; certain negative → *can't/couldn't have*; unsure → *may/might have*; duty missed → *should have*. Evidence words (*wet road, empty plate, lights off*) point to must have.

Degrees of comparison and question tags

⚡ 'Other' check in comparative sentences. Whenever you see *than any / than all*, ask: *Is the subject part of the group after 'than'?* If yes, *other* (or *else*) is mandatory. *Kolkata is bigger than any city in Nepal* ✓ (different group); *...than any city in…

⚡ Tag in two steps. Step 1: find the auxiliary (or supply *do/does/did*). Step 2: flip the polarity, remembering that *never, hardly, seldom, few, little, nobody, nothing* make the statement negative, so the tag is positive.

Non-finites (infinitive, gerund, participle), inversion and parallelism

⚡ Is 'to' a preposition? Test with a noun. Replace the verb after *to* with a noun. If a noun fits (*I look forward to the holiday*), then *to* is a preposition and the verb must be V-ing. If only a verb fits (*I want to ~~the holiday~~*), use to + V1…

⚡ Negative opener = question order. If a sentence begins with a negative or restrictive word (*never, seldom, hardly, no sooner, not only, only then, little, nowhere*), immediately check that the auxiliary comes before the subject. Missing inversion is…

General Awareness

Ancient Indian History

Indus Valley (Harappan) Civilisation

⚡ Four extremes — "Many Donkeys Are Slow". Manda (North, J&K) · Daimabad (South, Maharashtra) · Alamgirpur (East, UP) · Sutkagendor (West, Baluchistan).

⚡ Site-to-find pairing. Lothal = Launch (dockyard) · Kali-bangan = Kheti (ploughed field) · Mohenjo = Majestic Bath (Great Bath) · Chanhu = Chain of beads (bead factory) · Dholavira = Drop of water (reservoirs).

Vedic Age & Vedic Literature

⚡ Punjab rivers west→east: "Very Angry Parrots Visit Sutlej". Vitasta = Jhelum → Asikni = Chenab → Parushni = Ravi → Vipas = Beas → Sutudri = Sutlej. Same order as the modern J-C-R-B-S.

⚡ Veda → priest: "Rig-Hotri, Sama-Udgatri, Yajur-Adhvaryu, Atharva-Brahma". Think HUAB: Hotri recites, Udgatri sings, Adhvaryu performs, Brahma supervises.

⚡ Rigveda mandala anchors: 3-7-9-10. 3 = Gayatri, 7 = Ten Kings war, 9 = Soma, 10 = Purusha Sukta (varnas).

Mahajanapadas & Rise of Magadha

⚡ Capital pairs by first letter. Kosala–shravasti? No: use "Vatsa-Kaushambi, Vajji-Vaishali" (both V) and "Avanti-Ujjain, Anga-Champa" (both start with A). Kuru-Indraprastha, Gandhara-Taxila (G-T like 'GT road' ending in the north-we…

⚡ Magadha order: "Hari Shishu Nanda". Haryanka → Shishunaga → Nanda → Maurya. Councils: Ajatashatru (1st) is Haryanka, Kalashoka (2nd) is Shishunaga.

Buddhism & Jainism

⚡ Councils — places "Raja Vaishali Pata Kashmir", kings alternate "A-K-A-K". Places: Rajagriha → Vaishali → Pataliputra → Kashmir. Kings: Ajatashatru, Kalashoka, Ashoka, Kanishka. Presidents: "Maha-Saba-Moggali-Vasu".

⚡ Buddha's life symbols: "Lotus → Horse → Tree → Wheel → Stupa". Birth, renunciation, enlightenment, first sermon, death — in that order.

⚡ Tirthankara symbols: "Bull begins, Lion ends". 1st Rishabha = bull, 22nd = conch, 23rd Parshva = snake (Serpent), 24th Mahavira = lion.

Mauryan Empire

⚡ Edict numbers: "2-Hospital, 5-Officer, 12-Tolerance, 13-Kalinga". Think of a hospital ward 2, officer rank 5, the 12 faiths tolerated, and the unlucky 13 (war).

⚡ Mauryan dates chain: 322 → 185 → 73 BCE. Maurya starts 322, Shunga starts 185 (Pushyamitra), Kanva starts 73 (Vasudeva Kanva).

Post-Mauryan Age: Shungas to Kushanas

⚡ "57 minus, 78 plus". Vikram Samvat = 57 BCE (subtract), Shaka = 78 CE (add). To convert: CE = VS − 57; CE = Shaka + 78.

⚡ Satavahana = "S-P-G-H". Simuka founder · Pratishthana capital · Gautamiputra greatest · Hala poet.

Gupta Age & Harshavardhana

⚡ Gupta sequence: "Sri Ghat, Chandra-Samudra-Chandra, Kumar-Skand". Sri Gupta → Ghatotkacha → Chandragupta I → Samudragupta → Chandragupta II → Kumaragupta I → Skandagupta.

⚡ Prashasti authors: "H for Samudra, R for Pulakeshin". Harishena → Allahabad (Prayag) Prashasti of Samudragupta. Ravikirti → Aihole inscription of Pulakeshin II. Banabhatta → Harshacharita (biography, not an inscription).

⚡ Pilgrims in order: "Fa → Hiuen → I" (F-H-I). Fa-Hien (Chandragupta II) → Hiuen Tsang (Harsha) → I-Tsing (after Harsha). Alphabetical and chronological together.

Sangam Age & South Indian Dynasties

⚡ Sangam emblems: "Bow-Tiger-Fish = Chera-Chola-Pandya" (C-C-P). Alphabetical Bow, Tiger... simpler: Chera Bow (cheer with a bow), Chola Tiger (roar), Pandya Fish (Madurai's Meenakshi = 'fish-eyed').

⚡ Temple ↔ dynasty quick map. Kailasa Ellora = Rashtrakuta (Krishna I) · Kailasanatha Kanchi = Pallava (Rajasimha) · Brihadeeswara = Chola (Rajaraja I) · Virupaksha Pattadakal = Chalukya. Note 'Kailasa' vs 'Kailasanatha'.

⚡ Imperial Cholas: "VAPRR". Vijayalaya → Aditya I → Parantaka I → Rajaraja I → Rajendra I.

Ancient Literature & Authors

⚡ Drama trio: "Shudraka's Cart, Bhasa's Dream, Vishakha's Seal". *Mrichchhakatika* = little clay cart (Shudraka) · *Swapnavasavadatta* = dream of Vasavadatta (Bhasa) · *Mudrarakshasa* = Rakshasa's seal/ring (Vishakhadatta).

⚡ Medicine pair: "Charaka Cures, Sushruta Stitches". Charaka Samhita = general medicine (Kanishka's physician); Sushruta Samhita = surgery (plastic surgery, cataract).

Medieval Indian History

Early Medieval India (c. 750–1206)

⚡ Tripartite = "PPR fight for K". Pala, Pratihara, Rashtrakuta → Kannauj. East, West and South all wanted the centre.

⚡ Tarain: 1 to Prithviraj, 2 to Ghori. 1191 → 1st battle → Prithviraj wins. 1192 → 2nd battle → Ghori wins. The second number always goes to the invader.

Delhi Sultanate (1206–1526)

⚡ Dynasty order — "Sabka Khana Tum Sab Lo". Slave → Khalji → Tughlaq → Sayyid → Lodi. Starting years 1206 – 1290 – 1320 – 1414 – 1451 end at 1526 (Panipat).

⚡ Who did what — first letter hooks. Iltutmish = Iqta + Investiture; Balban = Bow down (sijda/paibos); Alauddin = All prices fixed; Muhammad bin Tughlaq = Move capital + Money tokens; Firoz = Fields & canals.

Vijayanagara & Bahmani Kingdoms

⚡ Vijayanagara dynasties — "Some Silly Tigers Attack". Sangama → Saluva → Tuluva → Aravidu. Krishnadevaraya sits in the third (Tuluva — "Top king").

⚡ Deccan five — "BiG AB B". Bijapur (Adil), Golconda (Qutb), Ahmadnagar (Nizam), Bidar (Barid), Berar (Imad). The first letters of the dynasty names: A-Q-N-B-I.

Mughal Empire & Sher Shah Suri

⚡ Mughal order — "Bahut Hi Accha Jalebi Shahi Aur". Babur → Humayun → Akbar → Jahangir → Shah Jahan → Aurangzeb. Sher Shah sits between Humayun's two reigns (1540–55).

⚡ Three Panipats — 26, 56, 61. 1526 Babur beats Ibrahim Lodi · 1556 Akbar (Bairam Khan) beats Hemu · 1761 Ahmad Shah Abdali beats the Marathas. Gap pattern: +30 years, then +205 years.

Marathas & Sikh Gurus

⚡ Guru order — "Nana Aur Amar Ram Arjun, Har-Har-Har Tegh Gobind". Nanak, Angad, Amar Das, Ram Das, Arjan → the three Hars (Hargobind, Har Rai, Har Krishan) → Tegh Bahadur → Gobind Singh.

⚡ Chauth vs Sardeshmukhi. Chauth = Chaar ana in the rupee (1/4); Sardeshmukhi = 10% — the Sardeshmukh claimed a 'tenth' as hereditary head.

Bhakti & Sufi Movements

⚡ Philosophy trio — "RaVi MaD". Ramanuja → Vishishtadvaita · Madhva → Dvaita · (Shankara → Advaita).

⚡ Language of the epic. Tulsidas = Awadhi (A for Ayodhya's Ram) · Surdas = Braj (Bal-Krishna).

Medieval Books & Foreign Travellers

⚡ Mughal chroniclers — "Abul = Ain, Badauni = Bitter". Abul Fazl wrote the official, praising Akbarnama/Ain; Badauni wrote the bitter, critical *Muntakhab*.

⚡ Travellers by century. 11th Al-Biruni → 13th Marco Polo → 14th Ibn Battuta → 15th Conti & Abdur Razzaq → 16th Paes & Nuniz → 17th Hawkins, Roe, Bernier, Tavernier.

Modern Indian History

Europeans in India & British Expansion

⚡ Plassey then Buxar — "57 buys, 64 seals". 1757 Plassey gave the British a foothold (Clive vs Siraj); 1764 Buxar sealed control (three Indian rulers beaten) and brought the Diwani (1765).

⚡ Governor-General reforms — "Cornwallis Settles, Wellesley Subsidises, Bentinck bans Sati, Ripon Rules locally". Permanent Settlement 1793 → Subsidiary Alliance 1798 → Sati abolition 1829 → Local self-government 1882.

Revolt of 1857 and Peasant & Tribal Uprisings

⚡ 1857 pairs — "Kanpur Nana, Lucknow Lady, Jhansi Rani, Bihar Kunwar". Kanpur–Nana Saheb · Lucknow–Begum Hazrat Mahal (the Lady) · Jhansi–Rani Lakshmibai · Bihar (Jagdishpur)–Kunwar Singh, the 80-year-old.

⚡ Tribal movements by letter. Santhal = Sidhu (1855) · Munda = Birsa (Millennium's end, 1899–1900).

Socio-Religious Reform Movements

⚡ 1875 — the year of three. Arya Samaj (Dayanand), Theosophical Society (Blavatsky–Olcott) and the Aligarh MAO school (Syed Ahmad Khan) all date to 1875.

⚡ Founder hooks. Ram Mohan = Reform of sati (Brahmo, 1828) · Dayanand = Discover the Vedas (Arya) · Phule = People of lower castes (Satyashodhak) · Vivekananda = Vedanta service (RK Mission, 1897).

Congress, Moderates & Extremists (1885–1916)

⚡ Five-six-seven-nine: "Split, League, Split, Separate". 1905 Bengal split → 1906 Muslim League → 1907 Congress split (Surat) → 1909 separate electorates (Morley–Minto).

⚡ Home Rule twins. Tilak first (April, Belgaum) → Besant next (Sept, Madras). "TAB then BSM".

Gandhian Era & Road to Independence (1915–1947)

⚡ Gandhi's first three — "C-A-K, 17-18-18". Champaran 1917 (indigo, Bihar) → Ahmedabad 1918 (mill workers, hunger strike) → Kheda 1918 (peasants, revenue).

⚡ Session presidents that repeat in exams. Lahore 1929 = Jawaharlal (Purna Swaraj) · Karachi 1931 = Patel (Fundamental Rights) · Tripuri 1939 = Bose. "J-P-B" from 29 to 39.

⚡ Dandi numbers. 12 March start · 6 April salt law broken · 78 marchers · about 385 km (240 miles) · from Sabarmati Ashram.

Newspapers, Journals & Books of the Freedom Era

⚡ Gandhi's papers — "YNH". Young India · Navajivan · Harijan (plus *Indian Opinion* from South Africa). Anything else in the options is someone else's paper.

⚡ Jail-written classics. Tilak → *Gita Rahasya* in Mandalay; Nehru → *Discovery of India* in Ahmednagar Fort.

Art & Culture

Classical Dances & Their Exponents

⚡ Kerala has two, the rest one each. Kerala = Kathakali + Mohiniyattam. Every other classical dance has one home state: TN-Bharatanatyam, AP-Kuchipudi, Odisha-Odissi, Manipur-Manipuri, Assam-Sattriya, UP-Kathak.

⚡ Exponent hooks. Birju = Bol of Kathak · Kelucharan = Odissi (Konark poses) · Rukmini Devi = Revived Bharatanatyam (Kalakshetra) · Vempati = Village Kuchipudi · Vallathol = Kerala Kalamandalam (Kathaka…

Folk Dances by State

⚡ North-East quick five — "Cheraw Mizo, Hoja Tripura, Nong Khasi, Wang Garo, Bagu Bodo". Cheraw → Mizoram · Hojagiri → Tripura · Nongkrem → Meghalaya (Khasi) · Wangala → Meghalaya (Garo) · Bagurumba → Assam (Bodo).

⚡ Men vs women in Punjab. Bhangra = Bhaiya (men) · Giddha = Girls (women).

Music, Instruments & Maestros

⚡ Carnatic Trinity — "Tea, Milk, Sugar". Tyagaraja · Muthuswami Dikshitar · Syama Sastri. Purandara Dasa is the *father*, not part of the trinity.

⚡ Instrument families — "Tata Sings, Avan Drums, Ghana Clangs". Tata = strings, Sushira = air (wind), Avanaddha = skin drums, Ghana = solid (bells, cymbals, ghatam).

Paintings, Textiles & Handicrafts

⚡ "Kalam in Andhra, Patta in Odisha, Phad in Rajasthan". Kalamkari → Andhra (the pen = *kalam*) · Pattachitra → Odisha (cloth = *patta*) · Phad → Rajasthan (Pabuji scroll). Madhubani → Bihar's Mithila.

⚡ Bidri = Bidar. The craft is named after its town — Bidriware from Bidar (Karnataka). Same trick: Patola from Patan, Chanderi from Chanderi.

Temple & Rock-cut Architecture

⚡ "Shikhara North, Gopuram South, Vesara in Between". See a gopuram → Dravida. See a curved shikhara → Nagara. A star-shaped Hoysala plan → Vesara (Deccan).

⚡ Odisha pagodas. Black Pagoda = Konark (dark stone, Sun temple) · White Pagoda = Puri (whitewashed Jagannath).

Festivals & Fairs

⚡ Nagaland December — "Hornbill closes the year". Hornbill Festival runs 1–10 December (Nagaland Statehood Day is 1 December).

⚡ Two cattle fairs. Pushkar = Pink city state (Rajasthan) camels; Sonepur = Saran/Bihar cattle & elephants.

UNESCO Heritage & Cultural Institutions

⚡ WHS count ladder. Remember the last few by year: 2021 → 40 (Dholavira), 2023 → 42 (Hoysala), 2024 → 43 (Moidams), 2025 → 44 (Maratha forts), 2026 → 45 (Sarnath).

⚡ Three Akademis — "Sangeet first, then Sahitya & Lalit together". SNA 1952, then Sahitya and Lalit Kala both in 1954.

Indian Polity & Constitution

Making of the Constitution & borrowed features

⚡ Committee-chairman pairing. Nehru took the *Union/States/Constitution* committees (the 'national' subjects), Patel took *Provinces and Rights* (the 'practical' subjects), Prasad took *procedure and flag*, Ambedkar took the *Drafting…

⚡ Borrowed features speed-pair. UK-Parliament • USA-Rights • Ireland-Duties-of-state (DPSP) • Canada-Centre-strong • Australia-List-3 • USSR-Duties-of-citizens • France-Republic • Germany-Emergency-rights • Japan-law-procedure.

Preamble, Parts and the 12 Schedules

⚡ Schedule story-line. 1-Names, 2-Salaries, 3-Oaths, 4-RS seats, 5-Areas(SC/ST), 6-Tribal NE, 7-Three Lists, 8-Languages, 9-Land laws, 10-Defection, 11-Panchayat(29), 12-Municipality(18).

⚡ Preamble keyword count. Sovereign-Socialist-Secular-Democratic-Republic = 5 descriptors; J-L-E-F = Justice-Liberty-Equality-Fraternity, with only Justice split three ways (social-economic-political).

Fundamental Rights, DPSP and Fundamental Duties

⚡ FR article blocks. 14-18 Equality • 19-22 Freedom • 23-24 Exploitation • 25-28 Religion • 29-30 Culture-Education • 32 Remedies. Count blocks as 5+4+2+4+2+1.

⚡ Duty amendment pair. 42nd gave 10 duties; 86th added the 11th (education of 6-14 year-olds). Same 86th also inserted Art. 21A — one amendment, twin gifts.

Union and State Executive

⚡ Electoral college contrast. President = elected MPs + elected MLAs (no nominated). Vice-President = ALL MPs of both Houses, including the 12 nominated. 'Vice' reaches where 'President' does not.

⚡ Pardon ladder. President's Art. 72 pardons court-martial and death sentences; the Governor's Art. 161 covers neither. 'Governor = softer ladder'.

Parliament and the Judiciary

⚡ Committee trio numbers. PAC 22 (opposition chair), CPU 22 (15+7), Estimates 30 (all LS). 'Estimates = 30, exclusively LS.'

⚡ Writ first letters. H-M-P-C-Q: Habeas(body), Mandamus(command), Prohibition(stop-before), Certiorari(quash-after), Quo warranto(by what authority). Prohibition stops a case early; certiorari quashes after.

Important Constitutional Amendments

⚡ Amendment decade clusters. 1950s structural (1st, 7th) • 1970s power (24, 42, 44) • 1980s politics (52 anti-defection, 61 voting age) • 1990s grass-roots (73, 74) + language (71, 92) • 2000s rights (86, 91, 97) • 2010s-20s economy-equality (101 GS…

⚡ 42 vs 44 tug-of-war. Whatever the 42nd tightened, the 44th loosened: term 6→5 years, 'internal disturbance'→'armed rebellion', property FR→300A legal right.

Panchayati Raj, Municipalities and Emergency

⚡ Committee tier-count rhyme. Balwant-3 (1957), Ashok-2 (1977), Rao-district (1985), Singhvi-Constitution (1986). Decade rhyme: 57-77-85-86.

⚡ Emergency digits. 352-National (3 proclamations: 62, 71, 75), 356-President's Rule (Kerala 1959 first), 360-Financial (zero uses). Approvals: 1 month + every 6 months.

Constitutional & Statutory Bodies (with current heads)

⚡ Article blocks for bodies. 76-AG • 280-FC • 148-CAG • 315-UPSC • 324-EC. Descending ladder: 324 > 315 > 280 > 148 > 76 reads 'EC-UPSC-FC-CAG-AG'.

⚡ Statutory vs constitutional filter. If the source is a dated Act (1993 NHRC, 2003 CVC, 2005 CIC, 2013 Lokpal) it is statutory; if the source is an Article number it is constitutional; NITI Aayog (Cabinet resolution, 2015) is neither.

Indian Geography

Location, extent, neighbours and islands

⚡ Tropic of Cancer states — 'GRM CJ WTM'. West to east: Gujarat, Rajasthan, Madhya Pradesh, Chhattisgarh, Jharkhand, West Bengal, Tripura, Mizoram. Read it as 'Great Royal Mango, Cool Juice, With Tasty Mangoes'. Odisha and Bih…

⚡ Border order: 'Big Cats Prowl Near My Big Area'. Longest to shortest border — Bangladesh > China > Pakistan > Nepal > Myanmar > Bhutan > Afghanistan.

⚡ Channel numbers go down as you go south-west. Ten Degree (Andaman | Nicobar) → Nine Degree (Minicoy | Lakshadweep) → Eight Degree (Minicoy | Maldives). The Andaman group has the bigger number.

Himalayas, peaks, passes, plateau, coasts and lakes

⚡ Pass → state by first letter group. Z-K-K (Zoji, Khardung, Karakoram) = Ladakh; R-S-B (Rohtang, Shipki, Baralacha) = Himachal; L-M (Lipulekh, Mana) = Uttarakhand; N-J (Nathu, Jelep) = Sikkim; Bomdi = Arunachal; Pal…

⚡ Highest-peak ladder. India claimed: K2 → India administered: Kangchenjunga → South India: Anamudi → Nilgiris: Doddabetta → Aravallis: Guru Shikhar → Satpura: Dhupgarh.

⚡ Lake superlatives: 'Fresh Wular, Brackish Chilika, Salty Sambhar, Floating Loktak'. Say it as one chain. Examiners swap the adjectives between the four lakes.

Rivers, dams and waterfalls

⚡ Panch Prayag order — 'Very Nice Kids Read Daily'. Vishnuprayag → Nandprayag → Karnaprayag → Rudraprayag → Devprayag (upstream to downstream). At Devprayag the river becomes the Ganga.

⚡ West-flowers: 'Nobody Takes Mahi Seriously'. Narmada, Tapi, Mahi, Sabarmati flow into the Arabian Sea and form estuaries. Narmada and Tapi flow through rift valleys.

⚡ Waterfall ↔ river pairs. Jog–Sharavati, Chitrakote–Indravati, Dudhsagar–Mandovi, Athirappilly–Chalakudy, Shivanasamudra/Hogenakkal–Kaveri, Dhuandhar–Narmada.

Monsoon, local winds and soils

⚡ Black soil rich/poor list. Rich in 'LIMP' — Lime, Iron, Magnesium, Potash. Poor in 'NPH' — Nitrogen, Phosphorus, Humus.

⚡ Shower names by crop. Mango showers → mango (Kerala); Blossom showers → coffee (Karnataka); Kalbaisakhi → tea/jute/rice (Bengal, Assam). Western disturbances → wheat (rabi) in the north-west.

Agriculture, revolutions, minerals and ports

⚡ Revolution colours. Colour of the product: White = milk, Blue = water/fish, Yellow = oilseeds (mustard flower), Silver = eggs (shell shine), Round = potato, Green = grains/fields.

⚡ Kharif = 'Monsoon crops need water'. Water-loving rice, jute, cotton → kharif. Cool-weather wheat, gram, mustard → rabi. Summer melons → zaid.

National parks, tiger reserves and wildlife

⚡ Animal-first recall. Rhino → Kaziranga; Lion → Gir; Cheetah → Kuno; Snow leopard → Hemis; Hangul → Dachigam; Sangai → Keibul Lamjao; Nilgiri tahr → Eravikulam; Lion-tailed macaque → Silent Valley; Saltwater cr…

⚡ Year chain 36-73-86-92. 1936 first NP (Hailey/Corbett) → 1973 Project Tiger → 1986 first biosphere reserve (Nilgiri) → 1992 Project Elephant.

World Geography

Solar system, Earth's motions, latitudes and time

Time difference from longitude

\Delta t = \Delta\lambda \times 4\ \text{minutes}

1° of longitude = 4 min; 15° = 1 hour

Local time

T_{\text{local}} = T_{\text{GMT}} \pm \frac{\lambda}{15}\ \text{h}

+ for east longitudes, − for west

⚡ Planet superlatives in one line. Mercury smallest–nearest, Venus hottest–brightest–backward spin, Mars red, Jupiter largest (Ganymede biggest moon), Saturn rings–lightest, Uranus on its side, Neptune farthest–windiest.

⚡ IST from the meridian. India's meridian is 82½°E → 82.5 × 4 = 330 minutes = 5 h 30 min ahead of GMT.

Earth's interior, earthquakes, rocks and volcanoes

⚡ Discontinuity order: 'Come Make Good Lunch'. Conrad (inside crust) → Moho (crust|mantle) → Gutenberg (mantle|core) → Lehmann (outer|inner core).

⚡ Metamorphic pairs: 'Lime-Marble, Sand-Quartz, Shale-Slate, Granite-Gneiss'. The first letters of the new rock are easy: Marble from Limestone is the most asked (Taj Mahal marble).

Atmosphere, pressure belts, planetary and local winds

⚡ Layer order: 'The Strong Man Throws Eggs'. Troposphere → Stratosphere (ozone, jets) → Mesosphere (coldest, meteors) → Thermosphere (ionosphere, radio) → Exosphere.

⚡ Local wind by country. Chinook–USA/Canada, Foehn–Alps, Sirocco–Sahara→Italy, Harmattan–West Africa, Mistral–France, Khamsin–Egypt, Santa Ana–California, Loo–India.

Oceans, currents, straits and canals

⚡ Cold current → desert pairs. Humboldt–Atacama, Benguela–Namib, California–Sonoran. West coasts + cold current = dry air = desert.

Continents, superlatives, grasslands and tribes

⚡ Grasslands: 'Pretty Pam Visits Steppe Down'. Prairies (N America), Pampas (Argentina), Veld (S Africa), Steppes (Eurasia), Downs (Australia).

⚡ Lake superlatives. Caspian largest (salt), Superior largest freshwater, Baikal deepest, Titicaca highest navigable, Dead Sea lowest.

Nicknames and boundary lines

⚡ Sun pair. Rising Sun = Japan (east of Asia, first sunrise); Midnight Sun = Norway (Arctic summer).

⚡ Line → neighbours: 'Radcliffe-Rift, McMahon-Mountains, Durand-Afghan'. Radcliffe partitioned India–Pakistan, McMahon runs along the Himalayas with China, Durand divides Pakistan–Afghanistan.

Indian Economy

National income, growth and base years

NNP

NNP = GNP - \text{Depreciation}

GNP already includes net factor income from abroad

GDP deflator

\text{Deflator} = \frac{\text{Nominal GDP}}{\text{Real GDP}} \times 100

divide nominal by (deflator/100) to get real

Market price bridge

MP = FC + \text{Indirect taxes} - \text{Subsidies}

moving between factor cost and market price

⚡ G-D-N ladder. GDP (territory) + NFIA = GNP; GNP − depreciation = NNP. 'Territory → Nation → Net'.

⚡ New base-year pairs. Production takes financial years: GDP and IIP = 2022-23; prices of consumers take a calendar year: CPI = 2024; wholesale moved to 2022-23 with PPI twins in June 2026.

Five Year Plans and NITI Aayog

⚡ Plan-era story line. 1-farm (1951) → 2-machines (Mahalanobis) → 3-wars → holiday (66-69) → 4-stability + bank nationalisation → 5-Garibi Hatao → rolling (Janata) → 6-7 growth → 8-reforms (1992) → 9-12 inclusive → NITI (2015).

⚡ Plan-break tags. Two gaps: Plan Holiday 1966-69 (wars/drought) and Annual Plans 1990-92 (transition after the 8th was delayed).

Money, banking and the RBI

⚡ Corridor arithmetic. Repo is the middle: SDF = repo − 0.25 (floor), MSF = Bank Rate = repo + 0.25 (ceiling). With repo 5.25 → 5.00 / 5.25 / 5.50.

⚡ Nationalisation anchors. 1935 born, 1949 nationalised (RBI), 1955 SBI, 1969 fourteen, 1975 RRBs, 1980 six.

⚡ MPC fixed points. 6 members; 4 meetings a year minimum; target CPI 4% ± 2%; Governor's casting vote.

Inflation and price indices

⚡ Index-compiler pairs. CPI-IIP-GDP = MoSPI/NSO; WPI-PPI = DPIIT's Office of Economic Adviser. 'M for MoSPI, W for DPIIT(OEA)'.

⚡ Core = strip the volatile. Core inflation = headline − food − fuel. Stagflation = stagnation + inflation together.

Budget and fiscal policy

Fiscal deficit

FD = \text{Total Expenditure} - \text{Total Receipts (excl. borrowings)}

equals government borrowing requirement

Revenue deficit

RD = \text{Revenue Expenditure} - \text{Revenue Receipts}

revenue items only

Primary deficit

PD = FD - \text{Interest Payments}

strips out past borrowing costs

⚡ Budget 2026-27 number sheet. Spend 53.5 • Receipts 36.5 • Borrow 16.9 (₹ lakh crore); FD 4.3% • RD 1.5% • PD 0.7%; Capex 12.2; Debt 55.6%.

⚡ Article trio for money matters. 112-Budget • 266-Consolidated Fund • 267-Contingency Fund; 265-no tax without law.

Taxation and GST

⚡ GST article trio. 246A power • 269A inter-state • 279A Council. 101st Amendment, 1 July 2017.

⚡ GST 2.0 slab story. Two working slabs — 5 (merit) and 18 (standard) — with 40 kept aside for sin/luxury; 12% and 28% abolished on 22 Sep 2025.

⚡ Tax-year switch. Income-tax Act, 2025 → effective 1 April 2026; one 'tax year' replaces previous/assessment year pair; 536 sections, 23 chapters.

Government schemes and missions

⚡ Launch-year clusters. 2014: Jan Dhan, Make in India • 2015: BBBP, APY, MUDRA, PMAY • 2016: Ujjwala • 2018: Ayushman • 2019: KISAN • 2020: SVANidhi, PLI • 2023: Vishwakarma, Drone Didi • 2024: Surya Ghar • 2026: VB-G RAM G live (125 days).

⚡ MGNREGA → RAM G swap. 100 → 125 days; 15-day → weekly wages; Act of 2025, in force 1 July 2026.

Regulators, exchanges and international organisations

⚡ Regulator HQ pairs. RBI-SEBI-NABARD = Mumbai; IRDAI = Hyderabad; SIDBI = Lucknow; PFRDA = New Delhi.

⚡ Bretton-Woods twins. IMF + World Bank both born at Bretton Woods (1944), both in Washington DC; WTO (1995, Geneva) replaced GATT (1947).

⚡ Asian banks map. ADB 1966 → Manila; AIIB 2016 → Beijing; NDB (BRICS) 2015 → Shanghai. India founding member in all three.

Physics

SI units, conversions and measuring instruments

Kilowatt-hour

1\ \text{kWh} = 3.6 \times 10^{6}\ \text{J}

1 'unit' of electricity

Horsepower

1\ \text{hp} \approx 746\ \text{W}

Light year

1\ \text{ly} \approx 9.46 \times 10^{15}\ \text{m}

a unit of distance

⚡ Base-unit roll call: 'Mighty Kings Seldom Answer Kind Monks Correctly'. Metre, Kilogram, Second, Ampere, Kelvin, Mole, Candela — the 7 SI base units. Anything else (newton, joule, volt, watt…) is derived.

⚡ 'Meter' by what it measures. Hygro = humidity (hygiene/water), Hydro = liquid density, Lacto = milk, Anemo = wind (Greek *anemos*), Sphygmo = pulse/BP, Pyro = fire/high temperature, Seismo = earthquake, Alti = altitud…

Motion, gravitation, work-energy, fluids and levers

First equation of motion

v = u + at

Second equation of motion

s = ut + \tfrac{1}{2}at^{2}

Third equation of motion

v^{2} = u^{2} + 2as

Newton's second law

F = ma = \frac{\Delta p}{\Delta t}

p = mv

Law of gravitation

F = \frac{G m_1 m_2}{r^{2}}

G = 6.67 × 10⁻¹¹ N m² kg⁻²

Kinetic and potential energy

KE = \tfrac{1}{2}mv^{2},\quad PE = mgh

Work and power

W = Fs\cos\theta,\quad P = \frac{W}{t}

1 W = 1 J/s

Pressure

P = \frac{F}{A},\quad P_{\text{liquid}} = h\rho g

unit pascal

⚡ Lever class by the middle item: 'FLE = 1-2-3'. Whatever sits in the middle decides the class: Fulcrum → 1, Load → 2, Effort → 3. Nutcracker has the load (nut) in the middle → class 2; tongs are squeezed in the middle → class 3.

⚡ Squares in energy. KE ∝ v²: double the speed → 4× KE; triple → 9×. Momentum ∝ v: double speed → 2× momentum.

Heat, temperature and sound

Temperature scales

\frac{C}{100} = \frac{F-32}{180} = \frac{K-273}{100}

−40 °C = −40 °F

Heat absorbed

Q = mc\,\Delta T

c = specific heat

Latent heat

Q = mL

no temperature change during phase change

Wave speed

v = f\lambda

speed = frequency × wavelength

Echo distance

d = \frac{v\,t}{2}

sound travels to the wall and back

⚡ Sound speed order: 'Steel Shouts, Water Whispers, Air Awaits'. Solids fastest, liquids next, gases slowest; vacuum — no sound at all (astronauts use radio).

⚡ Pitch–frequency, Loudness–amplitude. 'Pitch = Frequency' (PF like 'Provident Fund'), 'Loudness = Amplitude' (LA like 'Los Angeles').

Light — mirrors, lenses, eye, dispersion and scattering

Power of a lens

P = \frac{1}{f\,(\text{m})} = \frac{100}{f\,(\text{cm})}

unit dioptre

Lens formula

\frac{1}{f} = \frac{1}{v} - \frac{1}{u}

Cartesian sign convention

Mirror formula

\frac{1}{f} = \frac{1}{v} + \frac{1}{u}

f = R/2

Refractive index

n = \frac{c}{v}

c = 3 × 10⁸ m/s

⚡ Eye defect lens: 'My-Con, Hyper-Vex'. Myopia → Concave; Hypermetropia → convex. Short-sighted people cannot see far — a diverging lens pushes the image back onto the retina.

⚡ Mirror by job. Need a wide view → convex (vehicle rear view). Need an enlarged or focused image/beam → concave (shaving, dentist, headlight).

Electricity, magnetism and the EM spectrum

Ohm's law

V = IR

Series resistance

R_s = R_1 + R_2 + \cdots

Parallel resistance

\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots

two resistors: R₁R₂/(R₁+R₂)

Electric power

P = VI = I^{2}R = \frac{V^{2}}{R}

Joule heating

H = I^{2}Rt

Resistance of a wire

R = \rho\frac{L}{A}

ρ = resistivity

⚡ Fleming hands: 'Left for Motor, Right for Generator' (L-M, R-G). Remember as the alphabet — L comes before M; G is 'generated' by the Right hand.

⚡ EM spectrum order: 'Good X-rays Use Visible Infra-Micro Radios'. Gamma → X-ray → UV → Visible → IR → Microwave → Radio: wavelength increases, frequency and energy decrease.

Nuclear physics, inventions and scientists

Mass–energy equivalence

E = mc^{2}

Einstein

⚡ Fission vs fusion. Fission = split (heavy → lighter, reactors, atom bomb). Fusion = fuse (light → heavier, Sun, hydrogen bomb).

⚡ Particle discoverers: 'Tom Ran Chasing'. Electron — Thomson (1897); nucleus/proton — Rutherford; neutron — Chadwick (1932).

Chemistry

Matter, separation methods and atomic structure

Moles from mass

n = \frac{m}{M}

M = molar mass in g/mol

Number of particles

N = n \times N_A,\quad N_A = 6.022 \times 10^{23}

Avogadro number

Boyle's law

P_1V_1 = P_2V_2

temperature constant

Charles's law

\frac{V_1}{T_1} = \frac{V_2}{T_2}

pressure constant, T in kelvin

Ideal gas equation

PV = nRT

R = 8.314 J mol⁻¹ K⁻¹

Maximum electrons in a shell

2n^{2}

K = 2, L = 8, M = 18

⚡ Iso-words by the letter. Isotopes — same atomic number (same protons, 'P for isoto-P-es'). Isobars — same A (mass number). Isotones — same neutrons.

⚡ Sublimation set: 'Camphor, Naphthalene, Iodine, Ammonium chloride, Dry ice' — 'CNIAD'. If an option is one of these, it is the sublimating substance.

Periodic table, elements and record-holders

⚡ Crust order: 'OSAIC' — Oxygen, Silicon, Aluminium, Iron, Calcium. So the most abundant element is O and the most abundant metal is Al.

⚡ Two liquids at room temperature. Mercury — metal; Bromine — non-metal. Gallium and caesium melt just above room temperature.

Acids, bases, pH, common compounds and reactions

\text{pH} = -\log_{10}[\text{H}^{+}]

pH + pOH = 14 at 25 °C

Neutralisation

\text{Acid} + \text{Base} \rightarrow \text{Salt} + \text{H}_2\text{O}

Setting of Plaster of Paris

\text{CaSO}_4\cdot\tfrac{1}{2}\text{H}_2\text{O} + 1\tfrac{1}{2}\,\text{H}_2\text{O} \rightarrow \text{CaSO}_4\cdot 2\text{H}_2\text{O}

PoP → gypsum

⚡ Kitchen acid map: 'Vinegar-Acetic, Lemon-Citric, Tamarind-Tartaric, Tomato-Oxalic, Curd-Lactic, Apple-Malic, Ant-Formic'. Pairs that start alike help: Tamarind–Tartaric, Apple–mAlic (think 'mAlus', the apple genus), Formica (Latin for ant)–Formic.

⚡ Phenolphthalein goes 'Pink in Base'. 'Phenol → Pink' only in a base; in acid it stays colourless.

⚡ Vitriol colours by metal. Blue — copper (CuSO₄ crystals are blue), Green — iron (ferrous), White — zinc; Oil of vitriol — H₂SO₄.

Metals, ores, metallurgy and alloys

Gold purity

\text{Purity}\,(\%) = \frac{\text{carat}}{24} \times 100

22 ct ≈ 91.67%

⚡ Reactivity series: 'Please Stop Calling Me A Zebra, I Like Her Calling Me Smart Goat'. Potassium, Sodium, Calcium, Magnesium, Aluminium, Zinc, Iron, Lead, Hydrogen, Copper, Mercury, Silver, Gold.

⚡ Brass vs Bronze. Br-a-ss has Zinc (think 'brass band with Zing'); Bronze has Tin (bronze medal — 'third'/tin). Both are copper-based.

Carbon compounds, fuels, polymers, industry and pollution

Alkane / alkene / alkyne

C_nH_{2n+2},\quad C_nH_{2n},\quad C_nH_{2n-2}

Haber process

N_2 + 3H_2 \rightleftharpoons 2NH_3

iron catalyst

Complete combustion of methane

CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O

⚡ Process → product: 'Haber-Ammonia, Contact-Sulphuric, Ostwald-Nitric, Solvay-Soda'. Pair the catalysts too: Haber — Fe, Contact — V₂O₅, Ostwald — Pt.

⚡ Japanese diseases: 'Mina-Mercury, Itai-Cadmium'. Minamata — Mercury; Itai-itai ('ouch-ouch', bone pain) — Cadmium.

Biology

Cell, organelles and genetics

⚡ Organelle nicknames: 'Power-Mito, Suicide-Lyso, Protein-Ribo, Packing-Golgi'. These four nicknames answer most organelle questions.

⚡ Cell discoverers in order. Hooke saw it (1665, dead cork) → Leeuwenhoek saw it alive → Brown saw the nucleus → Schleiden–Schwann made the theory → Virchow said cells come from cells.

Human body — organs, blood, digestion, glands

Body mass index

\text{BMI} = \frac{\text{mass (kg)}}{\text{height (m)}^{2}}

18.5–24.9 is the normal range (WHO)

⚡ Blood group giving: 'O gives, AB takes'. O has no A/B antigens → can give to all. AB has no anti-A/anti-B antibodies → can take from all.

⚡ Brain parts: 'Cerebrum thinks, Cerebellum balances, Medulla keeps you alive'. Thinking/memory — cerebrum; balance/posture — cerebellum; heartbeat/breathing — medulla oblongata.

Vitamins, minerals and nutrition

⚡ Fat-soluble vitamins: 'KADE' (or 'ADEK'). A, D, E, K dissolve in fat and are stored in the body; B and C dissolve in water.

⚡ Deficiency pairs: 'A-Andha (night blind), B1-Beriberi, C-sCurvy, D-Deformed bones (rickets), K-Klotting'. Add B3 — Pellagra ('3 D's) and B12 — Pernicious anaemia (cobalt).

Diseases, pathogens, vectors and vaccines

⚡ Bacterial diseases: 'TB, Cholera, Typhoid, Diphtheria, Tetanus, Leprosy, Plague, Pertussis'. If the disease is in this list it is bacterial; the rest of the common ones (polio, measles, AIDS, dengue, rabies, chickenpox, hepatitis) are viral. Malaria and kala-azar are protozoan.

⚡ Mosquito map: 'Anopheles-Malaria, Aedes-Dengue, Culex-Filaria'. Aedes also spreads chikungunya, yellow fever and Zika (day-biting). Only the female mosquito bites for blood.

Classification, plants, hormones and animal facts

Photosynthesis

6CO_2 + 6H_2O \xrightarrow[\text{chlorophyll}]{\text{sunlight}} C_6H_{12}O_6 + 6O_2

O₂ comes from water

Aerobic respiration

C_6H_{12}O_6 + 6O_2 \rightarrow 6CO_2 + 6H_2O + \text{energy (ATP)}

⚡ Underground but not a root. Potato, ginger, turmeric, onion grow underground but are stems (they have nodes/buds or 'eyes'). Carrot, radish, sweet potato, beetroot are true roots.

⚡ Five kingdoms: 'My Pet Fish Plays Alone'. Monera, Protista, Fungi, Plantae, Animalia — Whittaker, 1969.

Branches of biology and ecology

10% law of energy transfer

E_{n} = E_{1} \times (0.1)^{\,n-1}

E₁ = energy at the producer level, n = trophic level

⚡ Culture words by root. Api = bee (apiary), Seri = silk, Pisci = fish (Pisces), Viti = vine/grapes, Pomo = fruit (pomegranate), Olera = vegetables.

⚡ Study-of roots. Ornitho = bird, Ento = insect, Ichthyo = fish, Herpeto = creeping animals, Myco = fungus, Phyco = seaweed/algae, Onco = tumour, Nephro = kidney.

Static GK

Books & Authors

⚡ Booker ladder — "Rushdie 81, Roy 97, Desai 06, Adiga 08". Four India-connected Booker winners in order: Midnight's Children → God of Small Things → Inheritance of Loss → White Tiger. Geetanjali Shree (2022) won the International Booker (for translated fiction).

⚡ Sports book hooks. Sunny Days = Sunil Gavaskar · Race of My Life = Milkha (the 'Flying Sikh' raced) · Unbreakable = Mary Kom (boxer) · Ace = Sania (tennis ace) · Test of My Life = Yuvraj (cancer battle).

Sports: Trophies, Terms, Players & Venues

⚡ Badminton cups — "Thomas the man, Uber the woman". Thomas Cup = men's team, Uber Cup = women's team, Sudirman = mixed. Tennis: Davis = Dudes (men), Billie Jean King = women.

⚡ Team size — "Basket 5, Volley 6, Kabaddi 7". Count up: basketball 5, volleyball 6, kabaddi 7; then cricket/football/hockey 11.

Awards & Honours

⚡ First Bharat Ratnas — "RRR 1954". Rajagopalachari, Radhakrishnan, Raman — three Rs in 1954.

⚡ Award decade ladder. 50s Bharat Ratna (54) → 60s Arjuna (61), Jnanpith (65), Phalke (69) → 80s Dronacharya (85) → 90s Khel Ratna (91–92).

Important Days

⚡ Birthday days — pair the person. 12 Jan Vivekananda → youth · 29 Aug Dhyan Chand → sports · 5 Sep Radhakrishnan → teachers · 15 Sep Visvesvaraya → engineers · 14 Nov Nehru → children · 22 Dec Ramanujan → maths.

⚡ Two 'Feb–Nov' science/constitution anchors. 28 Feb = Raman effect (Science Day). 26 Nov = Constitution adopted (26 Jan = enforced → Republic Day).

National Symbols & Firsts

⚡ Women firsts — "Sarojini Governs, Sucheta Chiefs". Governor → Sarojini Naidu (1947) · Chief Minister → Sucheta Kriplani (1963). Both from UP.

⚡ Flag numbers — "3:2 and 24". Length : width = 3 : 2; chakra spokes = 24; designer = Pingali Venkayya.

Organisations, Capitals, Currencies & Parliaments

⚡ Geneva vs Rome vs Nairobi. Geneva = health, trade, labour (WHO, WTO, ILO). Rome = food (FAO, WFP, IFAD). Nairobi = environment (UNEP) — the only major UN HQ in Africa/Global South.

⚡ Neighbour currencies — "Taka, Ngul, Kyat". East side of India: Bangladesh Taka, Bhutan Ngultrum, Myanmar Kyat. Nepal, Sri Lanka and Pakistan use the rupee.

Current Affairs — How to Prepare

How current-affairs questions are framed

⚡ Pairing drill. Read any news item and write its two halves — e.g. 'X award → field', 'Y summit → host'. Exams ask the halves, not the story.

Award categories to track — structure, not winners

⚡ Ladder memory. Civilian: Ratna > Vibhushan > Bhushan > Shri. Sports: Khel Ratna > Arjuna (players) > Dronacharya (coaches) — teachers below students is the joke that fixes the order.

Sports events and how tournament news is asked

⚡ Cycle split. Four-yearly: Olympics, Winter, CWG, Asian Games, FIFA, ODI WC. Two-yearly: T20 WC, Thomas/Uber. Annual: IPL, Ranji, Khelo India.

Summits, groupings and organisations to track

⚡ HQ trio. SCO — Beijing; SAARC — Kathmandu; BIMSTEC — Dhaka — the three most-swapped headquarters. ASEAN sits at Jakarta as the fourth.

Important days — national and UN framework

⚡ Person-day strings. Chain them by date: 12 Jan Vivekananda → 1 Jul B.C. Roy → 29 Aug Dhyan Chand → 5 Sep Radhakrishnan → 31 Oct Patel → 14 Nov Nehru → 22 Dec Ramanujan.

Indexes and reports — index ↔ publisher

⚡ UN vs NGO split. UN bodies: UNDP (HDI), SDSN (Happiness), WIPO (Innovation), IMF (WEO). NGOs: Hunger = Concern + Welthungerhilfe, Peace = IEP, Press = RSF, ASER = Pratham, Corruption = TI.

Computer Knowledge

Computer Fundamentals

Five generations of computers

⚡ Generation ladder mnemonic. Vacuum tube -> Transistor -> IC -> Microprocessor -> AI = "Very Tiny ICs Made AI". Five words = five generations, in order.

⚡ Firsts trio. ENIAC 1946 = first electronic general-purpose computer; UNIVAC-I 1951 = first *commercial* one; Intel 4004 1971 = first *microprocessor*. Remember 1946 -> 1951 -> 1971 as "built, sold, shrunk".

Types of computers

⚡ Size ladder. "Some Ministers Must Promise Speed" - Supercomputer > Mainframe > Minicomputer > PC (Micro) > Smartphone/embedded, largest to smallest.

⚡ PARAM pin. PARAM = India's supercomputer series, built by C-DAC, Pune; PARAM 8000 came in 1991. Pair it with 'Vijay Bhatkar = father of Indian supercomputing'.

Basic organisation: CPU, memory, I/O

⚡ FDES chant. "Fat Dogs Eat Snacks" = Fetch, Decode, Execute, Store - the machine cycle order. Every instruction, every time.

⚡ ALU vs CU. ALU = Actual Labour Unit (does the sums); CU = traffic Constable, Untouched by maths (only directs). If a question says 'performs calculations' -> ALU; 'coordinates/controls' -> CU.

Pioneers and programming-language levels

⚡ Compiler vs Interpreter. Compiler = Complete (whole program first, fast execution, shows all errors together); Interpreter = Instant (line by line, stops at first error). C/C++ -> compiler; Python/JS -> interpreter.

⚡ Who-does-what chain. Babbage Built (the idea), Ada Added (first program), Turing Thought (theory), von Neumann Noted (stored program), Berners-Lee Browsed (WWW).

Core abbreviations and full forms

⚡ ROM family ladder. P -> EP -> EEPROM: each generation gets *easier* to erase - PROM (once, at manufacture/programming), EPROM (erase with UV light), EEPROM (erase electrically, in circuit). "UV for EPROM, Electricity for EEPROM."

⚡ OCR-OMR-MICR trio. OCR reads Characters (printed text -> editable text), OMR reads Marks (exam bubbles), MICR reads Ink on cheques (banks). Characters, Marks, Ink = CRM.

Hardware, Memory & Number Systems

Memory hierarchy, RAM/ROM and cache

⚡ Volatile = Vanishes. Volatile memories Vanish when power goes: registers, cache, RAM. Everything you *keep files on* (ROM, SSD, HDD, pen drive) is non-volatile.

⚡ ROM family eraser. PROM (once) -> EPROM (Erase with Photons = UV) -> EEPROM (Electrically Erasable). The more E's, the more Electrical.

⚡ Cache position. Cache is the CPU's personal notepad between CPU and RAM: L1 inside the core (fastest), then L2, then L3 shared. Any question 'memory between CPU and main memory' = cache.

Memory units and conversions

Step multiplier

1\ \text{unit}_{next} = 2^{10} = 1024\ \text{units}_{prev}

KB->MB->GB->TB->PB all multiply by 1024

Byte identity

1\ \text{Byte} = 8\ \text{bits},\quad 1\ \text{Nibble} = 4\ \text{bits}

half a byte = nibble

Common ladders

1\ \text{GB} = 1024\ \text{MB} = 2^{20}\ \text{KB} = 2^{30}\ \text{B}

4 GB = 4096 MB = 2^22 KB

bits vs bytes

1\ \text{MB} = 8\ \text{Mb}

capital B = byte, small b = bit (speeds are usually bits)

⚡ x1024 ladder. Every step up multiplies by 1024 (2^10), never 1000 in SSC convention. 4 GB = 4 x 1024 = 4096 MB. Half a KB = 512 B.

⚡ Bit-vs-byte speed check. Line speeds are in bits (Mbps), file sizes in bytes (MB). To convert a download speed to MB/s, divide by 8: 8 Mbps = 1 MB/s.

Number systems: binary, octal, decimal, hexadecimal

Positional value

N = \sum d_i \cdot b^{i}

digit d_i at position i (from 0, rightmost), base b

Decimal to base b

N = (\ldots r_2 r_1 r_0)_b\ \text{from repeated division by } b

read remainders bottom to top

Grouping shortcuts

1\ \text{octal digit} \leftrightarrow 3\ \text{bits},\quad 1\ \text{hex digit} \leftrightarrow 4\ \text{bits}

group binary from the RIGHT; pad with leading zeros

⚡ 3-4 grouping. Octal = 3-bit groups, Hex = 4-bit groups, always made from the right (pad the left with zeros). Binary -> octal/hex needs no division at all.

⚡ Hex letter wheel. A=10, B=11, C=12, D=13, E=14, F=15. So 2F = 2x16+15 = 47; FF = 15x16+15 = 255 = 11111111. 'F fills: F=15, FF=255'.

⚡ Power-of-2 positions. Memorise 1,2,4,8,16,32,64,128 - then any binary->decimal is just picking positions: 11010111 = 128+64+16+8+4+2+1 = 215. No working needed beyond addition.

Input and output devices

⚡ Mark-Character-Ink. OMR = Marks (bubbles), OCR = Characters (text), MICR = Ink on cheques (banks). Marks, Characters, Ink - match the noun in the question.

⚡ Impact = hit. If the printer physically *hits* paper (dot matrix, daisy wheel) it is impact - noisy but makes carbon copies. Inkjet/Laser/Thermal never touch with force: non-impact.

Ports and connectors

⚡ Serial = Single lane. Serial sends a Single stream (1 bit at a time, slow, long distance); Parallel sends a Packet (whole byte across 8 wires, fast, short distance - old printers).

⚡ VGA vs HDMI. VGA = Video only (15 pins, analog); HDMI = Home-theatre cable: audio + video, digital. If the question says 'sound and picture together' -> HDMI.

Storage and backup media

⚡ Optical capacity ladder. CD 700 MB -> DVD 4.7 GB -> Blu-ray 25 GB (single layer). Roughly: CD 0.7, DVD 4.7, BR 25 - '700, 4.7, 25'.

⚡ SSD vs HDD one-liner. SSD = no moving parts (flash), silent, shock-proof, faster, costlier per GB. HDD = spinning platters, cheap bulk storage. 'No moving parts' in a question = SSD.

Software & Operating Systems

Types of software: system, application, utility

⚡ SUApp test. Ask: does it Serve the machine (system), Unclog/maintain it (utility), or do the user's App-job (application)? Windows = system, antivirus = utility, Excel = application.

⚡ Licence 4-word key. Freeware = Free forever (closed source); Shareware = Sample first (trial); Open source = Open code; Proprietary = Pay.

Operating system functions, types and booting

⚡ PMF-SUIE services. OS gives six services: Process, Memory, File, Security, UI, Error handling. 'Manage + platform' answers are OS; anything computing payroll is application.

⚡ Boot words. Cold boot = Current was off (switched on from cold); Warm boot = Without power cut (restart). POST always comes first: 'Power On, Self-Test, then load'.

Windows OS features

⚡ PnP = Plug 'n' Pray-free. Plug and Play means the OS auto-detects and auto-configures new hardware - no driver hunt. The phrase 'automatically detected' in a question = PnP.

⚡ Task Manager 3-2-Esc. Fastest route to Task Manager: Ctrl+Shift+Esc (direct). Ctrl+Alt+Del only opens the security menu first. 'Direct' is what SSC tests.

File systems and file management

⚡ FAT32's four-wall. FAT32 cannot hold a single file bigger than 4 GB - the classic exam line. NTFS breaks that wall and adds security permissions + journaling.

⚡ Wildcards. In Windows search, ? masks exactly one character and * masks any number: *.docx finds all Word files, s?t finds sat/set/sit.…

MS Word

Word interface, views and basics

⚡ Views ladder. Default view = Print Layout (what you print is what you see). Reading = book mode; Web = browser look; Outline = headings hierarchy; Draft = bare text. If a question says 'default', Print Layout is the answer.

⚡ Backstage tab. New, Open, Save, Print, Options live under the File tab (Backstage view), not the Home ribbon. Any 'where do you find Print/Options' question -> File tab.

Formatting text, paragraphs and pages

⚡ Alignment letters. The shortcut letter sits inside the word: ctrE? No - simply: Left=Ctrl+L, CEnter=Ctrl+E, Right=Ctrl+R, Justify=Ctrl+J. Only Centre breaks the pattern with E.

⚡ Spacing numbers. Line-spacing shortcuts ARE the spacing: Ctrl+1 single, Ctrl+2 double, Ctrl+5 1.5. (Five looks like S for single-and-a-half.)

⚡ Case cycle. Shift+F3 cycles case: start at 'hello world' -> HELLO WORLD -> Hello World. Press it repeatedly to see all five styles; no menu needed.

Mail merge, references, review and macros

⚡ No style, no TOC. An automatic Table of Contents is built from Heading styles. If the headings were bolded by hand (not styled), the TOC comes out empty - the classic trap.

⚡ Footnote = F below, Endnote = D for 'enD'. Alt+Ctrl+F = Footnote (bottom of page); Alt+Ctrl+D = enDnote (end of document/section).

⚡ Merge field marks. Mail-merge placeholders appear inside guillemets - «Name». Data comes from the data source; the letter layout lives in the main document. Two parts, always.

File types and defaults

⚡ X = XML = 2007+. Across Office: docx/xlsx/pptx = 2007 and later (Office Open XML, zip-compressed); the 3-letter .doc/.xls/.ppt = 97-2003. If a question mentions the X, it means XML.

⚡ Font eras. Word 2003 default = Times New Roman 12. Word 2007-2021 = Calibri 11. Office 365 (mid-2023) & Office 2024 = Aptos 12. Exams still ask Calibri 11 most often.

MS Excel

Workbook, cells and references

⚡ Dollar locks. A $ before the letter locks the column; before the number locks the row. $A$1 = both locked ('dollar = padlock'). F4 taps through the four states.…

⚡ Rows x Cols. 2007+: 2^20 rows (10,48,576) x 2^14 columns (16,384); last column XFD. Old Excel: 65,536 x 256 ('256 = IV roman-ish, XFD = 16384').

Functions you must know

⚡ COUNT counts Numbers, COUNTA counts Anything. COUNT = digits only; COUNTA = everything non-empty (text too); COUNTBLANK = the gaps. Test any MCQ by tagging each cell in the range N (number), T (text), E (empty).

⚡ V of VLOOKUP = Vertical. VLOOKUP hunts down the first column; HLOOKUP along the first row. The 4th argument FALSE = exact match ('F for Full match').

⚡ TODAY vs NOW. TODAY() = date only; NOW() = date + time. Both refresh on recalculation (unlike Ctrl+; which stamps a fixed date).

Formula errors, charts and data tools

⚡ ###### is not an error. Widen the column - done. Real errors start with #: DIV/0 (zero), NAME (spelling), VALUE (type), REF (deleted), N/A (not found). Match the cause table above.

⚡ Chart chooser. Pie = Parts, Line = Line of time (trend), Column = Compare, Scatter = relationship. 'Trend over months' questions always end at Line.

⚡ Goal Seek direction. Goal Seek: you give the answer, it finds the input. Pivot Table: you give the data, it gives summaries.

Excel shortcuts and file facts

⚡ Date stamp duo. Ctrl+; = date, Ctrl+Shift+; = time - the Shift makes it work-o'clock (time). 'Current date in a cell' questions answer Ctrl+;.

⚡ F-key row for Excel. F2 edit, F4 lock, F7 spell, F9 recalc, F11 chart sheet, F12 Save As - the F-row reads like Excel's toolbar.

⚡ Alt+= = AutoSum. Alt with the plus key writes =SUM() over the adjacent numbers instantly - the fastest marks in the section.

MS PowerPoint

Slides, placeholders and file types

⚡ X-family again. pptx/docx/xlsx = 2007+; ppt/doc/xls = 97-2003; the extra M (pptm/xlsm/docm) = macros; ppsx = the self-running show ('s for show).

⚡ M for More slides. Ctrl+M = More slide (new slide). Ctrl+D = Duplicate the current slide. Remember M-then-D: More, then Duplicate.

Views, slide master, transitions vs animations

⚡ Transition = TRansfer. Transition transfers you between slides; animation acts on atoms (objects) inside one slide. If the question says 'between slides', the answer is transition.

⚡ Sorter = order desk. Slide Sorter shows every slide as a thumbnail - the view for rearranging, duplicating and deleting slides quickly. Normal edits content, Sorter edits order.

Slideshow and editing shortcuts

⚡ F5 = Full show. F5 from the First slide; Shift+F5 from where you Stand. Esc = Exit. The F-key is PowerPoint's signature - no other app uses F5 to present.

⚡ B/W = Blackout/Washout. Mid-show, B blanks the screen to black and W washes it to white - for pauses/questions. Pressing the same key again brings the slide back.

Internet, Browsers & E-mail

Internet, WWW, URL and DNS

⚡ Internet vs WWW. The Internet = hardware network (cables, routers, servers); the WWW = content service (hyperlinked pages) riding on it. Invented later (1989) by ONE man - Berners-Lee; the Internet is older (1969 ARPANET).

⚡ DNS = phonebook. DNS translates names to numbers - never 'assigns IPs' (that is DHCP) and never 'sends mail' (that is SMTP). Any question phrased 'converts domain names into IP addresses' = DNS.

Browsers, search engines, downloading and uploading

⚡ Browser vs engine. Browser = the car; search engine = the GPS you ask inside it. Chrome/Edge/Firefox/Safari/Opera = cars. Google/Bing/Yahoo/DuckDuckGo = GPS. If it is a program you install, browser; if it is a website you visit, engine…

⚡ Ctrl-letter browser map. Tab new, Wipe (close) tab, Drop a bookmark, History, Junk... no - Jetisoned files (Downloads), Normal window, N+Shift No-trace (incognito). F5 = freshen.

⚡ Up vs Down. Upload = YOU send up (to the server); Download = server sends down to you. The direction is always relative to YOUR machine.

E-mail: structure, To/CC/BCC and protocols

⚡ CC vs BCC. CC = Carbon Copy - Everyone sees. BCC = Blind - eyes covered, so To/CC recipients cannot see the blind list. 'Who was secretly copied?' - nobody but the sender knows.

⚡ SMTP Sends, POP Picks, IMAP In-sync. SMTP = Send (outgoing). POP3 = Pick up on one device (downloads, server emptied). IMAP = In All devices (server copy stays, everything syncs).

⚡ @ = at. In ray@gmail.com the @ separates 'who' from 'where' - user at domain. Options without @, or with a space/comma, are dead giveaways.

e-Banking and digital payments

⚡ NEFT vs RTGS. RTGS = Really To Go, in batches of one, 2 lakh+ (real-time, gross = one-by-one, big amounts). NEFT = Nice Easy Batches (settlement in half-hourly batches, any amount).

⚡ IFSC 11. IFSC = 11 characters (4 bank code + 0 + 6 branch code), used on NEFT/RTGS/UPI branch routing. MICR is the 9-digit cheque strip.

⚡ 2FA two keys. Two-factor = password (know) + OTP/biometric (have/are). One stolen password is not enough - that is the whole point.

Networking: Devices, Topologies & Protocols

Network types and topologies

⚡ Mesh half-row sum. Full-mesh links for n nodes = n(n-1)/2: 6 nodes -> 6x5/2 = 15. Half of n rows of (n-1) - compute, don't memorise per-n.

⚡ PLMW span ladder. PAN < LAN < MAN < WAN - Personal, Local, Metropolitan, Wide. The Internet is the WAN of WANs.

⚡ Star = spine. Modern offices run star topology around a switch. If a question says 'central device fails, whole network down', it is describing star's weakness.

Network devices

⚡ Hub-Switch-Router ladder. Hub hears and shouts to all (layer 1); Switch = Selective, by MAC (layer 2, same network); Router = Routes between networks by IP (layer 3). Exam verbs: 'broadcasts to all' -> hub; 'MAC address table'…

⚡ Modem = modulator + demodulator. The full form itself is the answer: digital computer data is modulated onto an analog carrier and demodulated back. 'Digital to analog conversion' in a question = modem.

OSI and TCP/IP reference models

⚡ OSI mnemonics. Bottom-up: "Please Do Not Throw Sausage Pizza Away" (Physical, Data link, Network, Transport, Session, Presentation, Application). Top-down: "All People Seem To Need Da…

⚡ Layer-device pairs. Router = layer 3 (both R words rhyme with 'routing/IP'); Switch/Bridge = layer 2 (frames/MAC); Hub/Repeater = layer 1 (bits). Any 'works at which layer' question resolves by the device's address type: IP …

IP addressing: IPv4 classes and IPv6

⚡ Class ruler. Remember three cut-points: 1 - 126 - 128 - 191 - 192 - 223. A starts at 1, B at 128, C at 192, D (multicast) at 224. 127 is the loopback no-man's land between A and B.

⚡ Loopback = 127. 127.0.0.1 = localhost - pinging it tests your own TCP/IP stack, never the cable. Any question with 127.x wants 'loopback'.

⚡ IPv4 vs IPv6 size. IPv4 = 32-bit dotted decimal; IPv6 = 128-bit colon-hex. 'Runs out of addresses' questions -> IPv6 as the fix.

Protocols and port numbers; TCP vs UDP

⚡ Port anchors. Memorise six: 80 HTTP, 443 HTTPS, 25 SMTP, 53 DNS, 110 POP3, 21 FTP - these cover most CKT options. Then '22 SSH, 23 Telnet' (Telnet = the insecure 23).

⚡ TCP = Tracked, UDP = Untracked. TCP Tracks every packet (reliable, slower - web/mail/files); UDP is Unreliable but quick (streaming, gaming, DNS). 'Guaranteed delivery' in the question = TCP.

Cyber Security

Malware types

⚡ V-W-T triangle. Virus needs a Vehicle (host file); Worm Wanders alone; Trojan Tricks you into installing. Replication + host = virus; replication without host = worm; disguise without replication = trojan.

⚡ Ransom = Ransomware. Encrypts your files, demands money -> ransomware (WannaCry). 'Keystrokes recorded' -> keylogger. 'Hides with admin rights' -> rootkit. 'Zombie army' -> botnet.

Cyber attacks and social engineering

⚡ The -ishing family. Phishing = fake mail/site; Vishing = Voice; SMishing = SMS; Pharming = fake site with NO click (DNS poisoned). If the victim 'entered a real-looking site without clicking any link', think pharming.

⚡ MITM = postman reading letters. Man-in-the-Middle: both parties think they talk to each other; the attacker relays (and reads/edits) everything - classic on open Wi-Fi. Pure listening without relaying = sniffing.

Preventive measures and security tools

⚡ AV detects, FW decides. Antivirus = detective (scans files for malware already arriving); Firewall = gatekeeper (allows/blocks traffic by rules). 'Filters network traffic' -> firewall; 'removes virus from a file' -> antivirus.

⚡ 2FA two doors. Two-factor = knowledge (password) + possession/inherence (OTP, token, fingerprint). One stolen password opens only the first door.

IT Act 2000, offences and authorities

⚡ Section story-line. 43 = damage fine, 65 = source code, 66 = computer crime, 66C = identity (C for Credential theft), 66D = cheating by personation (D for Donning a false identity), 66F = terrorism (F for Fearsome), …

⚡ 2000-17-10. IT Act passed and effective in 2000 (in force 17 October 2000), UNCITRAL-based, amended 2008. CERT-In follows in 2004.

⚡ Helpline 1930. Money lost to online fraud? Dial 1930 fast and report at cybercrime.gov.in - the hotline aims to freeze the siphoned money in the banking chain.

Keyboard Shortcuts

Windows 10/11 and File Explorer shortcuts

⚡ Win + first letter. Most Win shortcuts use the first letter of the action: Explorer, Lock, Run, I (settIngs), Search, Desktop, Minimise, Project, V (paste history, like Ctrl+V). Learn the letter and you k…

⚡ Alt = Active window. Alt shortcuts act on the window in front of you: Alt+Tab switches it, Alt+F4 closes it, Alt+PrtScn captures only it, Alt+Enter shows the selected item's properties.

Universal Ctrl keys and MS Word shortcuts

⚡ L-E-R-J alignment. Left, cEntre, Right, Justify. C is already taken by Copy, so centre borrows the E from cEntre.

⚡ Spacing = the number itself. Ctrl+1 gives single spacing, Ctrl+2 gives double, and Ctrl+5 gives 1.5. Five is the odd one out: read it as 1.5.

MS Excel and PowerPoint shortcuts

⚡ Same key, different app. Shift+F3: change case in Word, Insert Function in Excel. Ctrl+D: Font dialog in Word, fill down in Excel, duplicate in PowerPoint, bookmark in a browser. F5: Go To in Word/Excel, slide show in PowerPoint, ref…

⚡ Page keys move pages (sheets). In Excel, Ctrl+PageDown moves to the next sheet and Ctrl+PageUp to the previous one, as if turning pages of the workbook.

Web browser shortcuts (Chrome, Edge, Firefox)

⚡ Shift = undo the tab. Ctrl+T opens a tab and Ctrl+Shift+T brings back the one you closed. Shift adds 'the opposite' or 'the stronger version': Ctrl+Shift+N is a stronger (private) new window, and Ctrl+Shift+Delete is a stronger delete (al…

⚡ Alt+arrows = time travel. Alt+Left goes back in history and Alt+Right goes forward. Backspace no longer goes back in Chrome.

Function keys F1-F12

⚡ 1-2-5-7-12 ladder. F1 = help (first thing you need), F2 = rename (give it a 2nd name), F5 = refresh/show (5 = S for Show), F7 = spell check (7 letters in 'SPELLER'), F12 = Save As (end of the row, you save at the end).

⚡ Shift+F10 = right-click. Shift+F10 opens the context menu, the same one you get with a right mouse click. The Menu key next to right Ctrl does the same.