Quantitative Aptitude

A Concept & Formula Compendium for GATE · CAT · XAT · GMAT · SNAP · Banking · SSC

Prof. Mithun Mondal · Department of EEE, BITS Pilani · Revision Notes

Section 01

Introduction & Strategy

Why Quantitative Aptitude?

Purpose

Quantitative Aptitude (QA) measures numerical reasoning, problem-solving speed, and logical thinking under time pressure. It is a core section in nearly every competitive examination — GATE (≈15% of marks), CAT / XAT / IIFT / SNAP / NMAT, GMAT / GRE Quant reasoning, and Banking, SSC, RRB, and UPSC CSAT.

Skills Tested

Speed & accuracy · mental arithmetic · pattern recognition · approximation & estimation · logical structuring.

Recommended Study Sequence

Number Systems → Arithmetic → Algebra → Geometry / Mensuration → Modern Math → Data Interpretation

Test-Taking Strategy

Before the Exam

Memorise the formula sheet (final section of these notes).
Drill mental math — squares to 30, cubes to 15, square roots to 25.
Memorise fraction-to-percentage conversions.
Solve ≥ 10 full mock papers under timed conditions.

During the Exam

Skim once, mark easy problems first.
Allocate ≈ 1.5–2 min per QA question.
Approximate before exact computation.
Use answer choices — back-substitute, eliminate.
Revisit skipped questions only if time permits.

Common Pitfalls

Reading errors · unit confusion (km vs m, hr vs min) · off-by-one in counting · misapplied percentages on shifting bases · over-investing time in one tough problem.

Section 02

Number Systems

Classification of Numbers

Hierarchy

$\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$

$\mathbb{N}$: Naturals $\{1,2,3,\dots\}$
$\mathbb{W}$: Whole $\{0,1,2,\dots\}$
$\mathbb{Z}$: Integers $\{\dots,-2,-1,0,1,\dots\}$
$\mathbb{Q}$: Rationals $p/q$, $q\neq 0$
Irrationals: $\sqrt{2},\pi,e$ (non-repeating, non-terminating)
$\mathbb{R}$: Reals $= \mathbb{Q}\cup\text{Irrationals}$
$\mathbb{C}$: Complex $a+bi$, $i^2=-1$

Nested ellipses diagram showing the hierarchy of number sets: Natural numbers inside Whole inside Integer inside Rational inside Real inside Complex, with representative examples in each region. — Nested-set diagram illustrating the inclusion hierarchy of number systems from natural numbers ($\mathbb{N}$) through complex numbers ($\mathbb{C}$), with representative elements placed in each unique region.

Divisibility Rules

Divisor	Rule
2	Last digit even
3	Sum of digits divisible by 3
4	Last two digits divisible by 4
5	Last digit 0 or 5
6	Divisible by both 2 and 3
7	Double the last digit, subtract from rest; repeat until result divisible by 7
8	Last three digits divisible by 8
9	Sum of digits divisible by 9
10	Last digit 0
11	Alternating sum of digits divisible by 11
13	Multiply last digit by 4, add to rest; repeat

Example — Divisibility by 9

Is 4,725 divisible by 9? Digit sum $= 4+7+2+5 = 18$, and $18/9 = 2$. Yes.

Example — Divisibility by 11

Test 73,953 for divisibility by 11: alternating sum $= 3-5+9-3+7 = 11$. Divisible by 11. ✓

Shortcut

For divisibility by $2^k$, only the last $k$ digits matter. For divisibility by 3 or 9, only the digit sum matters.

HCF and LCM

Definitions

HCF (GCD): largest integer dividing all given numbers. LCM: smallest positive integer divisible by all given numbers.

\[\text{HCF}(a,b)\times \text{LCM}(a,b) \;=\; a\times b\]

Methods

Prime factorisation: HCF = product of common primes (lowest power); LCM = all primes (highest power).
Euclidean algorithm: $\gcd(a,b)=\gcd(b,\, a\bmod b)$.

Example — HCF and LCM of 36 and 60

$36=2^2\cdot 3^2,\quad 60=2^2\cdot 3\cdot 5$.
HCF $=2^2\cdot 3=12$; LCM $=2^2\cdot 3^2\cdot 5=180$. Check: $12\times 180=2160=36\times 60$. ✓

Remainders & Modular Arithmetic

Key Identities

If $a\equiv r_1\pmod{n}$ and $b\equiv r_2\pmod{n}$: \[(a+b)\equiv(r_1+r_2),\quad (a\cdot b)\equiv(r_1\cdot r_2),\quad a^k\equiv r_1^k \pmod{n}\]

Fermat's little theorem: $a^{p-1}\equiv 1 \pmod{p}$, where $p$ is prime and $\gcd(a,p)=1$.

Example — Remainder of $7^{100}$ divided by 5

$7\equiv 2\pmod{5}$, so $7^{100}\equiv 2^{100}\pmod{5}$. Since $2^4=16\equiv 1$, we get $2^{100}=(2^4)^{25}\equiv 1\pmod{5}$. Remainder $=$ 1.

Unit Digit (Cyclicity)

$a\bmod 10$	Cycle Length	Cycle of Last Digit (powers 1, 2, 3, …)
0, 1, 5, 6	1	$\{0\},\ \{1\},\ \{5\},\ \{6\}$
4, 9	2	$\{4,6\},\ \{9,1\}$
2, 3, 7, 8	4	$\{2,4,8,6\},\ \{3,9,7,1\},\ \{7,9,3,1\},\ \{8,4,2,6\}$

Example — Unit digit of $7^{83}$

Cycle of 7 has length 4: $\{7,9,3,1\}$. Compute $83\bmod 4 = 3$. The 3rd entry is 3.

Number of Factors & Sum of Factors

If $N=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$ is the prime factorisation:

\[\tau(N) = \prod_{i=1}^k (a_i+1) \qquad \sigma(N)=\prod_{i=1}^{k}\frac{p_i^{a_i+1}-1}{p_i-1}\]

Example — $N = 72 = 2^3\cdot 3^2$

$\tau(N)=(3+1)(2+1)=12$; $\sigma(N)=\dfrac{2^{4}-1}{2-1}\cdot\dfrac{3^{3}-1}{3-1}=15\cdot 13=$195.

Highest Power of a Prime in $n!$ — Legendre's Formula

\[\nu_p(n!) \;=\; \sum_{k=1}^{\infty}\left\lfloor \frac{n}{p^k}\right\rfloor\]

Example — Trailing zeros in 100!

Trailing zeros = power of 5 in 100! (since 2's are more abundant).
$\left\lfloor \tfrac{100}{5}\right\rfloor+\left\lfloor \tfrac{100}{25}\right\rfloor+\left\lfloor \tfrac{100}{125}\right\rfloor=20+4+0=$24.

Sum Formulas

\[\sum_{k=1}^{n} k = \frac{n(n+1)}{2}\qquad \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}\qquad \sum_{k=1}^{n} k^3 = \left[\frac{n(n+1)}{2}\right]^2\]

Useful Corollaries

Sum of first $n$ odd numbers $= n^2$.
Sum of first $n$ even numbers $= n(n+1)$.
$\displaystyle \sum_{k=1}^{n}(2k-1)^2 = \frac{n(2n-1)(2n+1)}{3}$.

Surds, Indices & Rationalisation

Laws of Indices

$a^m\!\cdot a^n = a^{m+n};\quad \dfrac{a^m}{a^n}=a^{m-n};\quad (a^m)^n=a^{mn};\quad a^0=1;\quad a^{-n}=\dfrac{1}{a^n}$

Surds

$\sqrt[n]{a}=a^{1/n};\quad \sqrt{a}\sqrt{b}=\sqrt{ab};\quad \dfrac{\sqrt{a}}{\sqrt{b}}=\sqrt{\tfrac{a}{b}}$

Rationalising the Denominator

\[\frac{1}{\sqrt{3}-\sqrt{2}}\cdot\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}}=\]$\sqrt{3}+\sqrt{2}$.

Section 03

Arithmetic

Percentages

\[x\%\text{ of }y \;=\; \frac{x}{100}\cdot y \qquad \text{\% change} = \frac{\text{Final}-\text{Initial}}{\text{Initial}}\times 100\%\]

Successive Percentage Change

A change of $a\%$ followed by $b\%$ yields a net change of $\left(a + b + \dfrac{ab}{100}\right)\%$.

Example — Salary up 10%, then down 10%

Net $= 10 + (-10) + \dfrac{(10)(-10)}{100} = -1\%$, i.e. a 1% loss, not zero!

Standard Fraction-to-Percentage Conversions

Fraction	%	Fraction	%	Fraction	%
$1/2$	50	$1/3$	33.33	$1/4$	25
$1/5$	20	$1/6$	16.67	$1/7$	14.29
$1/8$	12.5	$1/9$	11.11	$1/10$	10
$1/11$	9.09	$1/12$	8.33	$1/16$	6.25

Profit and Loss

\[\text{Profit\%}=\frac{\text{SP}-\text{CP}}{\text{CP}}\times 100\qquad \text{Loss\%}=\frac{\text{CP}-\text{SP}}{\text{CP}}\times 100\] \[\text{SP}=\text{CP}\!\left(1\pm\frac{P\%}{100}\right);\qquad \text{Discount}=\text{MP}-\text{SP}\]

Example — Marked 25% above CP, Discount 10%

Let CP $=100$. Then MP $=125$, SP $=125\times 0.9=112.5$. Profit $=$12.5%.

Special Cases

False weight: Gain% $= \dfrac{W-w}{w}\times 100$ where $w$ is weight used, $W$ true weight.
Same SP, equal %: If one item sold at gain $x\%$ and another at loss $x\%$ with same SP, Net loss $= \dfrac{x^2}{100}\%$.

Watch Out

Always compute profit% on CP, never on SP. "Discount" is on MP; "profit" is on CP.

Simple Interest & Compound Interest

\[\text{SI} \;=\; \frac{P\cdot R\cdot T}{100};\qquad A = P + \text{SI}\]

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

With compounding frequency $n$ per year: $A = P\!\left(1+\dfrac{R}{100\,n}\right)^{\!nT}$.

CI − SI Shortcuts

For 2 years: $\text{CI}-\text{SI} = P\!\left(\dfrac{R}{100}\right)^{\!2}$
For 3 years: $\text{CI}-\text{SI} = P\!\left(\dfrac{R}{100}\right)^{\!2}\!\left(3+\dfrac{R}{100}\right)$

Example — P = 10,000, R = 10%, 2 yr

CI $-$ SI $=10000\times 0.01=$\$100. (CI $=\$2100$, SI $=\$2000$.)

Ratio, Proportion & Variation

Key Properties

Componendo: $\dfrac{a}{b}=\dfrac{c}{d}\Rightarrow \dfrac{a+b}{b}=\dfrac{c+d}{d}$
Dividendo: $\dfrac{a-b}{b}=\dfrac{c-d}{d}$
Componendo & Dividendo: $\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}$
Mean Proportional of $a,b$: $x=\sqrt{ab}$
Direct variation: $y=kx$; Inverse: $xy=k$

Partnership

Profit Sharing

Equal time: ratio of capitals. Unequal time: ratio of $C_i\cdot t_i$ products.

Example

A invests \$5,000 for 12 mo; B invests \$8,000 for 6 mo.
Ratio $=5000\cdot 12 : 8000\cdot 6 = 60000:48000=$5:4.

Mixtures & Alligation

\[\frac{\text{Quantity of cheaper}}{\text{Quantity of dearer}} = \frac{\text{Dearer price}-\text{Mean price}}{\text{Mean price}-\text{Cheaper price}}\]

Repeated Dilution

After $n$ replacements of $r$ litres from $V$ litres of pure liquid: \[\text{Pure left} = V\!\left(1-\frac{r}{V}\right)^{\!n}\]

Averages & Ages

\[\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i \qquad \bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}\ (\text{weighted})\]

Effect of New Entry

If average of $n$ items is $A$, and a new item $x$ joins: New avg $= \dfrac{nA+x}{n+1}$.

Example — Father and Son Ages

Five years ago the ratio of father's age to son's was 7:2; ten years hence it will be 2:1.
Let current ages be $F$ and $S$: $\dfrac{F-5}{S-5}=\dfrac{7}{2}$ and $\dfrac{F+10}{S+10}=2$.
Solving: Father 40, Son 15.

Section 04

Algebra

Algebraic Identities

Standard Identities

\[\begin{align} (a+b)^2 &= a^2+2ab+b^2 & (a-b)^2 &= a^2-2ab+b^2\\ a^2-b^2 &= (a+b)(a-b) & (a+b)^3 &= a^3+3a^2b+3ab^2+b^3\\ a^3+b^3 &= (a+b)(a^2-ab+b^2) & a^3-b^3 &= (a-b)(a^2+ab+b^2) \end{align}\]

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

Useful Corollary

If $a+b+c=0$, then $a^3+b^3+c^3=3abc$.

Binomial Theorem

\[(a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{\,n-k}b^{\,k}\] General term: $T_{k+1}=\dbinom{n}{k}a^{\,n-k}b^{\,k}$ (the $(k+1)$-th term, $k=0,1,\dots,n$).

Example — Coefficient of $x^4$ in $(2x+3)^7$

For $x^4$: $7-k=4\Rightarrow k=3$. Coefficient $=\binom{7}{3}\cdot 2^{4}\cdot 3^{3}=35\cdot 16\cdot 27=$15120.

Linear & Quadratic Equations

Two-Variable System

For $a_1x+b_1y=c_1,\ a_2x+b_2y=c_2$:

Unique solution: $\dfrac{a_1}{a_2}\ne\dfrac{b_1}{b_2}$
Infinitely many: $\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}$
No solution: $\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\ne \dfrac{c_1}{c_2}$

Quadratic formula for $ax^2+bx+c=0$:
\[x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\] Vieta's: $\alpha+\beta=-\dfrac{b}{a},\quad \alpha\beta=\dfrac{c}{a}$

Discriminant $\Delta = b^2-4ac$

$\Delta>0$: two distinct real roots. $\quad\Delta=0$: equal real roots. $\quad\Delta<0$: complex-conjugate roots.

Quadratic Roots — Common Transformations

Required Roots	Sum, Product	New Equation
Reciprocals $(1/\alpha,\ 1/\beta)$	$\tfrac{\alpha+\beta}{\alpha\beta},\ \tfrac{1}{\alpha\beta}$	$cx^2+bx+a=0$
Negatives $(-\alpha,\ -\beta)$	$-(\alpha+\beta),\ \alpha\beta$	$ax^2-bx+c=0$
Squares $(\alpha^2,\ \beta^2)$	$(\alpha+\beta)^2-2\alpha\beta,\ (\alpha\beta)^2$	$a^2x^2-(b^2-2ac)x+c^2=0$
Roots shifted by $h$	—	Substitute $x\to x-h$

Polynomials — Remainder & Factor Theorems

Key Theorems

Remainder theorem: When $p(x)$ is divided by $(x-a)$, the remainder is $p(a)$.
Factor theorem: $(x-a)$ is a factor of $p(x)$ iff $p(a)=0$.

Inequalities

AM–GM–HM Inequality (Non-negative Reals)

\[\text{AM}\geq \text{GM}\geq \text{HM}:\qquad\frac{a+b}{2}\geq \sqrt{ab}\geq \frac{2ab}{a+b}\] Equality if and only if $a=b$.

Sequences & Series

Arithmetic Progression (AP)

\[T_n = a+(n-1)d \qquad S_n = \frac{n}{2}\bigl(2a+(n-1)d\bigr) = \frac{n}{2}(a+\ell)\]

Geometric Progression (GP)

\[T_n=ar^{\,n-1} \qquad S_n=\frac{a(r^n-1)}{r-1},\ r\ne 1;\qquad S_\infty=\frac{a}{1-r},\ |r|<1\]

Harmonic Progression (HP) & Means

Means Relationship

$\text{HM}=\dfrac{2ab}{a+b}$; $\text{GM}^2=\text{AM}\cdot \text{HM}$.

Logarithms

Laws of Logarithms

\[\begin{align} \log_b(xy) &= \log_b x + \log_b y & \log_b(x/y) &= \log_b x - \log_b y\\ \log_b(x^k) &= k\log_b x & \log_b x &= \frac{\log_a x}{\log_a b}\ (\text{change of base}) \end{align}\] \[\log_b 1 = 0,\qquad \log_b b = 1,\qquad b^{\log_b x}=x\]

Memorise

$\log_{10} 2\approx 0.3010,\quad \log_{10} 3\approx 0.4771,\quad \log_{10} 7\approx 0.8451$.

Example — Number of Digits in $2^{50}$

$\log_{10}(2^{50})=50\log_{10}2 \approx 50(0.3010)=15.05$. Digits $=\lfloor 15.05\rfloor+1=$16.

Section 05

Geometry

Lines, Angles & Polygons

Key Facts

Sum of angles on a straight line $=180^\circ$.
Vertically opposite angles are equal.
Parallel lines cut by a transversal: corresponding and alternate angles equal; co-interior angles sum to $180^\circ$.

Regular polygon with $n$ sides: Interior angle $= \dfrac{(n-2)\cdot 180^\circ}{n}$; Exterior $=\dfrac{360^\circ}{n}$.

Triangle Properties

\[\text{Area} = \tfrac{1}{2}\cdot b\cdot h = \tfrac{1}{2}ab\sin C\] Heron's formula: $\text{Area}=\sqrt{s(s-a)(s-b)(s-c)},\ \ s=\dfrac{a+b+c}{2}$

Special Triangles

Equilateral (side $a$): Area $=\tfrac{\sqrt{3}}{4}a^2$; $r=\tfrac{a}{2\sqrt{3}}$; $R=\tfrac{a}{\sqrt{3}}$
Right triangle: $a^2+b^2=c^2$; triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25)
Isosceles: two equal sides, base angles equal

Classical Centres

Centroid $G$: medians; divides each 2:1 from vertex
Incenter $I$: angle bisectors; inscribed circle radius $r$
Circumcenter $O$: perpendicular bisectors; circumscribed radius $R$
Orthocenter $H$: altitudes; $O$, $G$, $H$ collinear ($OG:GH=1:2$)

Triangle ABC with all three medians drawn as dashed lines meeting at the centroid G, clearly showing the 2:1 division from each vertex to the midpoint of the opposite side. — Triangle with all three medians (dashed) intersecting at centroid $G$. Each median is divided by $G$ in the ratio 2:1 measured from the vertex.

Similarity & Congruence

Key Results

Congruence (SSS, SAS, ASA, RHS): identical shape and size.
Similarity (AA, SSS, SAS): same shape; corresponding sides in ratio $k$. Area ratio $=k^2$; volume ratio $=k^3$.
Thales' theorem: A line parallel to one side of a triangle cuts the other two sides proportionally.

Circles

Key Theorems

Circumference $=2\pi r$; Area $=\pi r^2$.
Angle at centre $=2\times$ angle at circumference (same arc).
Angle in a semicircle $=90^\circ$.
Tangent $\perp$ radius at point of contact.
Two tangents from an external point are equal in length.

Power of a point: $PA\cdot PB = PC\cdot PD = PT^2$

Sectors and Segments

In radians ($\theta$): Arc $= r\theta$; Area $= \tfrac{1}{2}r^2\theta$
In degrees: Arc $= \dfrac{\theta}{360}(2\pi r)$; Area $= \dfrac{\theta}{360}\,\pi r^2$
Segment area $= \tfrac{1}{2}r^2(\theta-\sin\theta)$ ($\theta$ in radians)

Coordinate Geometry

Distance: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$; Section formula: $P=\!\left(\dfrac{mx_2+nx_1}{m+n},\ \dfrac{my_2+ny_1}{m+n}\right)$

Area of triangle: $\tfrac{1}{2}\bigl|\,x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\,\bigr|$

Straight Line Forms

Form	Equation
Slope-intercept	$y=mx+c$
Point-slope	$y-y_1 = m(x-x_1)$
Two-point	$\dfrac{y-y_1}{y_2-y_1}=\dfrac{x-x_1}{x_2-x_1}$
Intercept	$\dfrac{x}{a}+\dfrac{y}{b}=1$
Perpendicular lines	$m_1 m_2 = -1$
Distance from point to line	$\dfrac{\|Ax_0+By_0+C\|}{\sqrt{A^2+B^2}}$

Conic Sections

Equations and Properties

Circle: $(x-h)^2+(y-k)^2=r^2$
Parabola: $y^2=4ax$; focus $(a,0)$; directrix $x=-a$
Ellipse: $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$; $e=\sqrt{1-b^2/a^2}$
Hyperbola: $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$; $e=\sqrt{1+b^2/a^2}$; asymptotes $y=\pm \tfrac{b}{a}x$

Four-panel diagram showing the standard graphs of circle, ellipse, parabola, and hyperbola on coordinate axes with equations, foci, and key parameters labelled. — Standard forms of the four conic sections — circle, ellipse, parabola, and hyperbola — shown on coordinate axes with foci and defining parameters.

Section 06

Mensuration

2D Mensuration

Shape	Area	Perimeter
Square (side $a$)	$a^2$	$4a$
Rectangle ($l, b$)	$lb$	$2(l+b)$
Triangle	$\tfrac{1}{2}bh$	$a+b+c$
Equilateral (side $a$)	$\tfrac{\sqrt{3}}{4}a^2$	$3a$
Circle (radius $r$)	$\pi r^2$	$2\pi r$
Semicircle	$\tfrac{1}{2}\pi r^2$	$\pi r+2r$
Trapezium ($a\parallel b$, ht $h$)	$\tfrac{1}{2}(a+b)h$	sum of sides
Regular hexagon (side $a$)	$\tfrac{3\sqrt{3}}{2}a^2$	$6a$

3D Mensuration

Solid	Volume	TSA	LSA / CSA
Cube (side $a$)	$a^3$	$6a^2$	$4a^2$
Cuboid ($l,b,h$)	$lbh$	$2(lb+bh+lh)$	$2h(l+b)$
Cylinder ($r,h$)	$\pi r^2 h$	$2\pi r(r+h)$	$2\pi r h$
Cone ($r,h,\ell$)	$\tfrac{1}{3}\pi r^2 h$	$\pi r(r+\ell)$	$\pi r \ell$
Sphere ($r$)	$\tfrac{4}{3}\pi r^3$	$4\pi r^2$	—
Hemisphere ($r$)	$\tfrac{2}{3}\pi r^3$	$3\pi r^2$	$2\pi r^2$

Frustum, Prism & Pyramid

Frustum (radii $r_1, r_2$, ht $h$): $V = \tfrac{1}{3}\pi h(r_1^2 + r_2^2 + r_1 r_2)$; Slant ht $\ell = \sqrt{h^2+(r_1-r_2)^2}$
Prism: $V = \text{Base area}\times \text{Height}$
Pyramid: $V = \tfrac{1}{3}\cdot \text{Base area}\cdot \text{Height}$

Diagonals

Square: $a\sqrt{2}$; Rectangle: $\sqrt{l^2+b^2}$; Cube: $a\sqrt{3}$; Cuboid: $\sqrt{l^2+b^2+h^2}$

Section 07

Trigonometry

Trigonometric Ratios & Standard Values

\[\sin\theta=\frac{\text{Opp}}{\text{Hyp}},\quad \cos\theta=\frac{\text{Adj}}{\text{Hyp}},\quad \tan\theta=\frac{\text{Opp}}{\text{Adj}}\] \[\csc\theta=\frac{1}{\sin\theta},\quad \sec\theta=\frac{1}{\cos\theta},\quad \cot\theta=\frac{1}{\tan\theta}\]

$\theta$	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$
$\sin$	0	$\tfrac{1}{2}$	$\tfrac{1}{\sqrt{2}}$	$\tfrac{\sqrt{3}}{2}$	1
$\cos$	1	$\tfrac{\sqrt{3}}{2}$	$\tfrac{1}{\sqrt{2}}$	$\tfrac{1}{2}$	0
$\tan$	0	$\tfrac{1}{\sqrt{3}}$	1	$\sqrt{3}$	undefined

Trigonometric Identities

Pythagorean Identities

\[\sin^2\theta+\cos^2\theta = 1\qquad 1+\tan^2\theta=\sec^2\theta\qquad 1+\cot^2\theta=\csc^2\theta\]

Sum / Difference Formulas

\[\begin{align} \sin(A\pm B) &= \sin A\cos B \pm \cos A \sin B\\ \cos(A\pm B) &= \cos A \cos B \mp \sin A \sin B\\ \tan(A\pm B) &= \frac{\tan A\pm \tan B}{1\mp \tan A\tan B} \end{align}\]

\[\sin 2A=2\sin A\cos A;\quad \cos 2A=\cos^2 A-\sin^2 A=1-2\sin^2 A=2\cos^2A-1\]

Sine & Cosine Rules

Sine rule: $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=2R$; Cosine rule: $c^2 = a^2+b^2 -2ab\cos C$

Heights and Distances

Tower Problem

A pole casts a shadow of 30 m when the sun's elevation is $30^\circ$.
Height $=30\tan 30^\circ = 30/\sqrt{3}=10\sqrt{3}\approx$17.3 m.

Two-angle setup (angles $\alpha <\beta$ from points $a$ apart on same side):\quad $h = \dfrac{a\,\tan\alpha\,\tan\beta}{\tan\beta-\tan\alpha}$

Section 08

Permutations & Combinations

\[n! = n\cdot(n-1)\cdots 2\cdot 1;\qquad 0!=1\] \[\nPr{n}{r} = \frac{n!}{(n-r)!}\qquad \nCr{n}{r} = \binom{n}{r} = \frac{n!}{r!\,(n-r)!}\]

Permutation Variants

Repetitions allowed: $n^r$
Circular (distinct items): $(n-1)!$
Necklace/bracelet: $(n-1)!/2$
Indistinguishable items ($p$ of one kind, $q$ of another): $\dfrac{n!}{p!\,q!}$

Combination Properties

$\nCr{n}{r}=\nCr{n}{n-r}$
$\nCr{n}{0}=\nCr{n}{n}=1$
Pascal's rule: $\nCr{n}{r} = \nCr{n-1}{r-1}+\nCr{n-1}{r}$
Sum: $\displaystyle\sum_{r=0}^n \nCr{n}{r} = 2^n$

Example — Letters of MISSISSIPPI

11 letters: M×1, I×4, S×4, P×2.
$\dfrac{11!}{4!\,4!\,2!\,1!}=$34,650.

Distributing $n$ identical balls into $r$ distinct boxes (no restriction): $\nCr{n+r-1}{r-1}$ (stars-and-bars).
Derangements: $D_n = n!\sum_{k=0}^{n}\dfrac{(-1)^k}{k!}\approx \dfrac{n!}{e}$

Section 09

Probability

\[P(A) = \frac{\text{favourable outcomes}}{\text{total equally-likely outcomes}}\qquad 0\leq P(A)\leq 1;\quad P(\bar{A})=1-P(A)\] \[P(A\cup B) = P(A) + P(B) - P(A\cap B)\] \[P(A\mid B) = \frac{P(A\cap B)}{P(B)},\qquad P(A\cap B) = P(A\mid B)\cdot P(B)\]

Bayes' theorem: $\displaystyle P(A\mid B) = \frac{P(B\mid A)\,P(A)}{P(B)}$

Standard Distributions

Binomial Distribution

$n$ independent Bernoulli trials; probability of exactly $k$ successes: \[P(X=k) = \binom{n}{k}p^k (1-p)^{n-k}\qquad \mu=np;\quad \sigma^2=np(1-p)\]

Geometric Distribution

Number of trials until first success:\quad $P(X=k)=(1-p)^{k-1}p,\quad E[X]=\tfrac{1}{p}$

Poisson Distribution (Rare Events)

$P(X=k)=\dfrac{\lambda^k e^{-\lambda}}{k!},\quad E[X]=\mathrm{Var}(X)=\lambda$

Expectation: $E[X]=\displaystyle\sum_i x_i\,P(X=x_i);\qquad \mathrm{Var}(X)=E[X^2]-\bigl(E[X]\bigr)^2$

Section 10

Set Theory

\[|A\cup B| = |A|+|B|-|A\cap B|\] \[|A\cup B\cup C| = |A|+|B|+|C|-|A\cap B|-|B\cap C|-|A\cap C|+|A\cap B\cap C|\]

De Morgan's Laws & Other Properties

$(A\cup B)^c = A^c\cap B^c$; $(A\cap B)^c = A^c\cup B^c$; $|\mathcal{P}(A)| = 2^{|A|}$; $|A\times B| = |A|\cdot|B|$

Example — Tea / Coffee Preferences

In 100 students: 60 like tea, 50 like coffee, 30 like both.
$|T\cup C|=60+50-30=80$. 20 like neither.

Section 11

Statistics

Measures of Central Tendency

Mean: $\bar{x} = \dfrac{\sum x_i}{n}$ (raw); $\bar{x}=\dfrac{\sum f_i x_i}{\sum f_i}$ (grouped).

Median (grouped data): \[\text{Median}=L+\frac{\tfrac{n}{2}-F}{f}\cdot h\] where $L$ = lower limit of median class, $F$ = cumulative frequency before, $f$ = class frequency, $h$ = class width.

Empirical relation (Karl Pearson): $\text{Mode}\approx 3\,\text{Median} - 2\,\text{Mean}$.

Measures of Dispersion

\[\sigma^2 = \frac{1}{n}\sum_{i=1}^{n} (x_i-\bar{x})^2;\qquad \sigma=\sqrt{\sigma^2}\] Sample variance: $s^2 = \dfrac{1}{n-1}\sum (x_i-\bar{x})^2$; CV $= \dfrac{\sigma}{\bar{x}}\times 100\%$

Pearson correlation: $r = \dfrac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum(x_i-\bar{x})^2}\,\sqrt{\sum(y_i-\bar{y})^2}}$; $-1\leq r\leq 1$

Linear Regression $y = a + bx$

\[b = \frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x})^2};\qquad a=\bar{y} - b\bar{x}\]

Section 12

Time, Speed & Distance

\[D = S\times T\qquad 1\,\text{km/h}=\tfrac{5}{18}\,\text{m/s};\qquad 1\,\text{m/s}=\tfrac{18}{5}\,\text{km/h}\] Average speed (equal distances at speeds $u, v$): $\bar{v} = \dfrac{2uv}{u+v}$ (harmonic mean)

Trains

Passing Distances

Pole / standing person: distance = length of train
Platform / another train: distance = sum of lengths

Relative Speed

Same direction: $|v_1-v_2|$; Opposite direction: $v_1+v_2$

Boats and Streams

Downstream speed $= b+s$; Upstream speed $= b-s$
$b = \tfrac{1}{2}(\text{down}+\text{up});\qquad s=\tfrac{1}{2}(\text{down}-\text{up})$

Round trip time: $\dfrac{d}{b-s}+\dfrac{d}{b+s}=\dfrac{2bd}{b^2-s^2}$

Races and Clocks

Clock Angle Formula

Hour hand: $30^\circ$/hr $= 0.5^\circ$/min. Minute hand: $6^\circ$/min.
Angle between hands at $H:M$: $\bigl|30H - 5.5M\bigr|\pmod{360^\circ}$.
Hands coincide every $\dfrac{720}{11}\,\text{min} \approx 65\,\text{min}\,27\,\text{s}$, i.e. 11 times in 12 hours.

Section 13

Time and Work

Combined rate $=\dfrac{1}{a}+\dfrac{1}{b}$; Time together $=\dfrac{ab}{a+b}$

Work conservation: $\dfrac{M_1 D_1 H_1}{W_1} = \dfrac{M_2 D_2 H_2}{W_2}$

Example — Two Workers

A finishes in 12 days, B in 18 days. Together: $\dfrac{12\cdot 18}{12+18}=\dfrac{216}{30}=$7.2 days.

Example — Pipes and Cisterns

Pipe A fills in 6 h; pipe B empties in 8 h. Net rate $= \tfrac{1}{6}-\tfrac{1}{8}=\tfrac{1}{24}$. Tank fills in 24 hours.

Watch the Sign

Always check whether each pipe fills or empties before adding rates. An overlooked outlet flips the answer.

Section 14

Data Interpretation

Common Chart Types

Tables: rows/columns of values.
Bar graphs: categorical comparison; vertical or horizontal.
Line graphs: trends over time.
Pie charts: proportions of a whole; total $=100\%$ or $360^\circ$.
Stacked/grouped bars: composition with comparison.
Caselets: paragraph-form data.

Compound Annual Growth Rate (CAGR)

\[\text{CAGR} = \left(\frac{V_T}{V_0}\right)^{\!1/T} - 1\qquad \text{Simple YoY growth: }g = \frac{V_T - V_{T-1}}{V_{T-1}}\times 100\%\]

Approximation & Estimation

Common Shortcuts

Round to nearest 10 / 100 before multiplying.
Convert % to fractions: $33\%=1/3$, $25\%=1/4$.
Newton step for square root: $\sqrt{n}\approx\tfrac{1}{2}(g+\tfrac{n}{g})$.
Difference of squares: $a^2-b^2=(a-b)(a+b)$.

DI Rule of Thumb

Identify the question type first: ratio, growth, total, or relative comparison. Then select only the data point you need — don't read everything. Only approximate when answer choices are well-separated; if two options differ by less than 2%, compute exactly.

Side-by-side illustration of a bar chart showing quarterly revenue and a line chart showing annual sales trends, with labelled axes and data points. — Representative data interpretation charts: bar chart for categorical comparison of quarterly revenue (left) and line chart for tracking year-on-year sales trends (right).

Section 15

Summary & Universal Cheat Sheet

★

Exam Mantra

Score ∝ Accuracy × Speed × Pattern Recognition

Master fundamentals first, not shortcuts. Build a personal formula sheet. Practice with a timer from week one. Review mistakes more than new problems.

Arithmetic Formulas

$\text{SI}=\dfrac{PRT}{100}$; $\text{CI}=P\!\left(1+\tfrac{R}{100}\right)^{T}\!-P$
Profit% $=\dfrac{\text{SP}-\text{CP}}{\text{CP}}\times 100$
Successive %: $a+b+\dfrac{ab}{100}$
Avg speed (equal dist.): $\dfrac{2xy}{x+y}$
Alligation: $\dfrac{Q_{\text{cheap}}}{Q_{\text{dear}}}=\dfrac{D-M}{M-C}$

Algebra Formulas

$(a\pm b)^2=a^2\pm 2ab+b^2$
$a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)$
Quadratic: $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
AP: $S_n=\tfrac{n}{2}(2a+(n-1)d)$
GP: $S_n=\dfrac{a(r^n-1)}{r-1}$
Binomial: $T_{k+1}=\binom{n}{k}a^{n-k}b^k$

Geometry / Mensuration

Triangle area $=\tfrac{1}{2}bh$; Heron's $\sqrt{s(s-a)(s-b)(s-c)}$
Circle: $A=\pi r^2$, $C=2\pi r$
Sphere: $V=\tfrac{4}{3}\pi r^3$, $S=4\pi r^2$
Cone: $V=\tfrac{1}{3}\pi r^2 h$, $S=\pi r(r+\ell)$
Cylinder: $V=\pi r^2 h$, $S=2\pi r(r+h)$

Trigonometry

$\sin^2\theta+\cos^2\theta=1$
$1+\tan^2\theta=\sec^2\theta$
$\sin(A\pm B)=\sin A\cos B\pm \cos A\sin B$
Sine rule: $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=2R$
Cosine rule: $c^2=a^2+b^2-2ab\cos C$

Counting & Probability

${}^nP_r=\dfrac{n!}{(n-r)!}$; ${}^nC_r=\dfrac{n!}{r!(n-r)!}$
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
Bayes: $P(A\mid B)=\dfrac{P(B\mid A)P(A)}{P(B)}$
Binomial: $E[X]=np$, $\mathrm{Var}=np(1-p)$

Statistics & TSD & Work

Mean: $\bar{x}=\dfrac{\sum x_i}{n}$; Var: $\sigma^2=\dfrac{\sum(x_i-\bar{x})^2}{n}$
Empirical: Mode $\approx 3\,\text{Median} - 2\,\text{Mean}$
$D=ST$; km/h → m/s: $\times\tfrac{5}{18}$
Combined work: $\tfrac{1}{T}=\tfrac{1}{a}+\tfrac{1}{b}$
Logs: $\log(ab)=\log a+\log b$; $\log_b a=\dfrac{\log a}{\log b}$
Sets: $|A\cup B|=|A|+|B|-|A\cap B|$

Common Pitfalls — Don't Lose Easy Marks

Calculation Traps

Confusing % change with % point change.
Forgetting base year in CAGR / growth.
Mixing units (km vs m, hr vs s, % vs decimal).
Squaring negatives: $(-3)^2=9$, not $-9$.
Treating $\sqrt{a^2+b^2}$ as $a+b$.

Algebra Traps

Cancelling a variable that may be zero.
Squaring an inequality with a negative side.
Forgetting domain restrictions in logs ($x>0$).

Geometry Traps

Assuming a figure is to scale.
Confusing radius with diameter in surface area.
Forgetting Pythagoras condition ($c$ = hypotenuse).
Sector vs segment area.

Probability & DI Traps

Assuming independence without proof.
Reading a pie chart in degrees vs %.
Comparing absolute vs relative values.
Trusting visual proportions over actual numbers.

Section 16

Required Roots	Sum, Product	New Equation
Reciprocals \((1/\alpha,\ 1/\beta)\)	\(\tfrac{\alpha+\beta}{\alpha\beta},\ \tfrac{1}{\alpha\beta}\)	\(cx^2+bx+a=0\)
Negatives \((-\alpha,\ -\beta)\)	\(-(\alpha+\beta),\ \alpha\beta\)	\(ax^2-bx+c=0\)
Squares \((\alpha^2,\ \beta^2)\)	\((\alpha+\beta)^2-2\alpha\beta,\ (\alpha\beta)^2\)	\(a^2x^2-(b^2-2ac)x+c^2=0\)
Roots shifted by \(h\)	—	Substitute \(x\to x-h\)

Shape	Area	Perimeter
Square (side \(a\))	\(a^2\)	\(4a\)
Rectangle (\(l, b\))	\(lb\)	\(2(l+b)\)
Triangle	\(\tfrac{1}{2}bh\)	\(a+b+c\)
Equilateral (side \(a\))	\(\tfrac{\sqrt{3}}{4}a^2\)	\(3a\)
Circle (radius \(r\))	\(\pi r^2\)	\(2\pi r\)
Semicircle	\(\tfrac{1}{2}\pi r^2\)	\(\pi r+2r\)
Trapezium (\(a\parallel b\), ht \(h\))	\(\tfrac{1}{2}(a+b)h\)	sum of sides
Regular hexagon (side \(a\))	\(\tfrac{3\sqrt{3}}{2}a^2\)	\(6a\)

Solid	Volume	TSA	LSA / CSA
Cube (side \(a\))	\(a^3\)	\(6a^2\)	\(4a^2\)
Cuboid (\(l,b,h\))	\(lbh\)	\(2(lb+bh+lh)\)	\(2h(l+b)\)
Cylinder (\(r,h\))	\(\pi r^2 h\)	\(2\pi r(r+h)\)	\(2\pi r h\)
Cone (\(r,h,\ell\))	\(\tfrac{1}{3}\pi r^2 h\)	\(\pi r(r+\ell)\)	\(\pi r \ell\)
Sphere (\(r\))	\(\tfrac{4}{3}\pi r^3\)	\(4\pi r^2\)	—
Hemisphere (\(r\))	\(\tfrac{2}{3}\pi r^3\)	\(3\pi r^2\)	\(2\pi r^2\)

\(\theta\)	\(0^\circ\)	\(30^\circ\)	\(45^\circ\)	\(60^\circ\)	\(90^\circ\)
\(\sin\)	0	\(\tfrac{1}{2}\)	\(\tfrac{1}{\sqrt{2}}\)	\(\tfrac{\sqrt{3}}{2}\)	1
\(\cos\)	1	\(\tfrac{\sqrt{3}}{2}\)	\(\tfrac{1}{\sqrt{2}}\)	\(\tfrac{1}{2}\)	0
\(\tan\)	0	\(\tfrac{1}{\sqrt{3}}\)	1	\(\sqrt{3}\)	undefined

\(a\bmod 10\)	Cycle Length	Cycle of Last Digit (powers 1, 2, 3, …)
0, 1, 5, 6	1	\(\{0\},\ \{1\},\ \{5\},\ \{6\}\)
4, 9	2	\(\{4,6\},\ \{9,1\}\)
2, 3, 7, 8	4	\(\{2,4,8,6\},\ \{3,9,7,1\},\ \{7,9,3,1\},\ \{8,4,2,6\}\)

Fraction	%	Fraction	%	Fraction	%
\(1/2\)	50	\(1/3\)	33.33	\(1/4\)	25
\(1/5\)	20	\(1/6\)	16.67	\(1/7\)	14.29
\(1/8\)	12.5	\(1/9\)	11.11	\(1/10\)	10
\(1/11\)	9.09	\(1/12\)	8.33	\(1/16\)	6.25

Form	Equation
Slope-intercept	\(y=mx+c\)
Point-slope	\(y-y_1 = m(x-x_1)\)
Two-point	\(\dfrac{y-y_1}{y_2-y_1}=\dfrac{x-x_1}{x_2-x_1}\)
Intercept	\(\dfrac{x}{a}+\dfrac{y}{b}=1\)
Perpendicular lines	\(m_1 m_2 = -1\)
Distance from point to line	\(\dfrac{\|Ax_0+By_0+C\|}{\sqrt{A^2+B^2}}\)

Quantitative Aptitude

Introduction & Strategy

Why Quantitative Aptitude?

Test-Taking Strategy

Number Systems

Classification of Numbers

Divisibility Rules

HCF and LCM

Remainders & Modular Arithmetic

Unit Digit (Cyclicity)

Number of Factors & Sum of Factors

Highest Power of a Prime in \(n!\) — Legendre's Formula

Sum Formulas

Surds, Indices & Rationalisation

Arithmetic

Percentages

Profit and Loss

Simple Interest & Compound Interest

Ratio, Proportion & Variation

Partnership

Mixtures & Alligation

Averages & Ages

Algebra

Algebraic Identities

Binomial Theorem

Linear & Quadratic Equations

Quadratic Roots — Common Transformations

Polynomials — Remainder & Factor Theorems

Inequalities

Sequences & Series

Arithmetic Progression (AP)

Geometric Progression (GP)

Harmonic Progression (HP) & Means

Logarithms

Geometry

Lines, Angles & Polygons

Triangle Properties

Similarity & Congruence

Circles

Sectors and Segments

Coordinate Geometry

Straight Line Forms

Conic Sections

Mensuration

2D Mensuration

3D Mensuration

Trigonometry

Trigonometric Ratios & Standard Values

Trigonometric Identities

Sine & Cosine Rules

Heights and Distances

Permutations & Combinations

Probability

Standard Distributions

Set Theory

Statistics

Time, Speed & Distance

Trains

Boats and Streams

Races and Clocks

Time and Work

Data Interpretation

Approximation & Estimation

Summary & Universal Cheat Sheet

Common Pitfalls — Don't Lose Easy Marks

Further Reading & References