Mathematics Guide

From Class 9 to 12 — everything you need, explained simply.

SSC Board Exam Question Bank

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1. Number Systems

Core Idea

What are Real Numbers?

Every number you've ever used falls into the real number family. Here's how they're organized:

Natural Numbers (N): 1, 2, 3, 4, ... (counting numbers)
Whole Numbers (W): 0, 1, 2, 3, ... (natural + zero)
Integers (Z): ..., -2, -1, 0, 1, 2, ... (whole + negatives)
Rational Numbers (Q): Any number written as $\frac{p}{q}$ where $q \neq 0$. Examples: $\frac{3}{4}$, $0.5$, $-7$
Irrational Numbers: Numbers that cannot be written as fractions. Examples: $\sqrt{2}$, $\pi$, $e$

Think of it like this: Rational numbers have decimal expansions that either terminate ($0.25$) or repeat ($0.333...$). Irrational numbers go on forever without repeating ($3.14159...$).

Key Rule

Rationalizing the Denominator

To remove a square root from the bottom of a fraction, multiply top and bottom by the conjugate:

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

Problem: Rationalize $\frac{1}{\sqrt{5} + \sqrt{3}}$

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Multiply numerator and denominator by the conjugate $(\sqrt{5} - \sqrt{3})$

$$\frac{1}{\sqrt{5}+\sqrt{3}} \times \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}} = \frac{\sqrt{5}-\sqrt{3}}{(\sqrt{5})^2 - (\sqrt{3})^2}$$

Simplify: $\frac{\sqrt{5}-\sqrt{3}}{5-3} = \frac{\sqrt{5}-\sqrt{3}}{2}$

2. Polynomials

Definition

What is a Polynomial?

An expression with variables and coefficients using only addition, subtraction, multiplication, and non-negative integer exponents.

$$p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$

Degree = highest power of $x$. A polynomial of degree 1 is linear, degree 2 is quadratic, degree 3 is cubic.

Essential Identity

Algebraic Identities You Must Know

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

Remainder Theorem: When $p(x)$ is divided by $(x - a)$, the remainder is $p(a)$. If $p(a) = 0$, then $(x-a)$ is a factor!

Problem: Factorize $x^3 - 3x^2 - 9x - 5$

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Try $x = -1$: $p(-1) = -1 - 3 + 9 - 5 = 0$ ✓ So $(x+1)$ is a factor.

Divide: $x^3 - 3x^2 - 9x - 5 = (x+1)(x^2 - 4x - 5)$

Factor the quadratic: $x^2 - 4x - 5 = (x-5)(x+1)$

Answer: $(x+1)^2(x-5)$

3. Coordinate Geometry

The Basics

The Cartesian Plane

Two perpendicular number lines (x-axis and y-axis) divide the plane into 4 quadrants. Every point is located by an ordered pair $(x, y)$.

X: 3 Y: 2

Move the sliders to plot any point on the Cartesian plane.

4. Linear Equations in Two Variables

Core Concept

Standard Form

A linear equation in two variables has the form:

$$ax + by + c = 0$$

where $a$, $b$, $c$ are real numbers and $a$, $b$ are not both zero. The graph is always a straight line.

5. Triangles

Key Properties

Triangle Angle Sum

The sum of all angles in a triangle is always $180°$.

$$\angle A + \angle B + \angle C = 180°$$

Congruence Rules

When are two triangles congruent?

SSS — All three sides are equal
SAS — Two sides and the included angle are equal
ASA — Two angles and the included side are equal
AAS — Two angles and a non-included side are equal
RHS — Right angle, hypotenuse, and one side are equal

6. Heron's Formula

Formula

Area of Any Triangle

When you know all three sides $a$, $b$, $c$:

$$s = \frac{a+b+c}{2} \qquad \text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$$

Problem: Find the area of a triangle with sides 7, 8, and 9

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$s = \frac{7+8+9}{2} = 12$

Area $= \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12 \times 5 \times 4 \times 3}$

$= \sqrt{720} = 12\sqrt{5} \approx 26.83$ square units

7. Statistics

Measures of Central Tendency

Mean, Median, Mode

$$\text{Mean} = \bar{x} = \frac{\sum x_i}{n}$$ $$\text{Median} = \text{Middle value when data is sorted}$$ $$\text{Mode} = \text{Most frequently occurring value}$$

8. Probability

Basic Definition

Probability of an Event

$$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$$

Probability always lies between 0 and 1: $0 \leq P(E) \leq 1$

1. Real Numbers

Fundamental Theorem

Euclid's Division Lemma

For any two positive integers $a$ and $b$, there exist unique integers $q$ and $r$ such that:

$$a = bq + r \qquad \text{where } 0 \leq r < b$$

This is the basis for finding HCF using the Euclidean algorithm.

Problem: Find HCF of 135 and 225

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$225 = 135 \times 1 + 90$

$135 = 90 \times 1 + 45$

$90 = 45 \times 2 + 0$

Remainder is 0, so HCF = 45

Fundamental Theorem of Arithmetic

Unique Prime Factorization

Every integer greater than 1 can be expressed as a product of primes in exactly one way (ignoring order).

$$\text{LCM}(a,b) \times \text{HCF}(a,b) = a \times b$$

2. Quadratic Equations

The Most Important Formula

Quadratic Formula

For $ax^2 + bx + c = 0$:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

The discriminant $D = b^2 - 4ac$ tells you what kind of roots you get:

$D > 0$ → Two distinct real roots
$D = 0$ → Two equal real roots
$D < 0$ → No real roots

a: 1 b: -2 c: -3

Adjust a, b, c to see how the parabola changes and where it crosses the x-axis.

Problem: Solve $2x^2 - 7x + 3 = 0$

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Here $a=2$, $b=-7$, $c=3$

$D = (-7)^2 - 4(2)(3) = 49 - 24 = 25$

$x = \frac{7 \pm \sqrt{25}}{4} = \frac{7 \pm 5}{4}$

$x = \frac{12}{4} = 3$ or $x = \frac{2}{4} = \frac{1}{2}$

3. Arithmetic Progressions (AP)

Formulas

AP Essentials

An AP is a sequence where the difference between consecutive terms is constant ($d$).

$$a_n = a + (n-1)d \qquad \text{(nth term)}$$ $$S_n = \frac{n}{2}[2a + (n-1)d] \qquad \text{(sum of n terms)}$$

Problem: Find the sum of first 20 terms of AP: 5, 8, 11, ...

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$a = 5$, $d = 8 - 5 = 3$, $n = 20$

$S_{20} = \frac{20}{2}[2(5) + (20-1)(3)]$

$= 10[10 + 57] = 10 \times 67 = 670$

4. Similar Triangles

BPT (Thales' Theorem)

Basic Proportionality Theorem

If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally:

$$\text{If } DE \parallel BC, \text{ then } \frac{AD}{DB} = \frac{AE}{EC}$$

Similarity Criteria

AA, SSS, SAS Similarity

AA: Two angles of one triangle equal two angles of another
SSS: All three pairs of sides are proportional
SAS: Two pairs of sides proportional + included angle equal

5. Trigonometry

The Big Three

Trigonometric Ratios

In a right triangle with angle $\theta$:

$$\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \qquad \cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \qquad \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}$$

Must Memorize

Standard Values

$\theta$	$0°$	$30°$	$45°$	$60°$	$90°$
$\sin\theta$	$0$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	$1$
$\cos\theta$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	$0$
$\tan\theta$	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	$\infty$

Identity

Pythagorean Trigonometric Identity

$$\sin^2\theta + \cos^2\theta = 1$$

6. Circles

Key Theorem

Tangent to a Circle

A tangent to a circle is perpendicular to the radius at the point of contact.

If two tangents are drawn from an external point, they are equal in length.

7. Surface Area & Volume

Formulas

3D Shapes

$$\text{Cylinder: } V = \pi r^2 h, \quad SA = 2\pi r(r+h)$$ $$\text{Cone: } V = \frac{1}{3}\pi r^2 h, \quad SA = \pi r(r+l)$$ $$\text{Sphere: } V = \frac{4}{3}\pi r^3, \quad SA = 4\pi r^2$$

1. Sets

Foundation

Set Operations

A set is a well-defined collection of distinct objects.

$$A \cup B = \{x : x \in A \text{ or } x \in B\} \quad \text{(Union)}$$ $$A \cap B = \{x : x \in A \text{ and } x \in B\} \quad \text{(Intersection)}$$ $$A - B = \{x : x \in A \text{ and } x \notin B\} \quad \text{(Difference)}$$

Important Formula

Cardinality

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

2. Relations & Functions

Core Concept

What is a Function?

A function $f: A \rightarrow B$ is a rule that assigns exactly one element in $B$ to every element of $A$.

Domain: The set of all input values
Range: The set of all output values
Codomain: The set $B$ (range $\subseteq$ codomain)

3. Trigonometric Functions

Extended Identities

Compound Angle Formulas

$$\sin(A+B) = \sin A \cos B + \cos A \sin B$$ $$\cos(A+B) = \cos A \cos B - \sin A \sin B$$ $$\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$$

Radian Measure: $180° = \pi$ radians. To convert: multiply degrees by $\frac{\pi}{180}$.

4. Complex Numbers

New Number

The Imaginary Unit

We define $i = \sqrt{-1}$, so $i^2 = -1$. A complex number is $z = a + bi$.

$$|z| = \sqrt{a^2 + b^2} \qquad \text{(modulus)}$$ $$\bar{z} = a - bi \qquad \text{(conjugate)}$$ $$z \cdot \bar{z} = |z|^2 = a^2 + b^2$$

5. Linear Inequalities

Rules

Solving Inequalities

Same rules as equations, with one critical difference:

When you multiply or divide by a negative number, FLIP the inequality sign!

$$\text{If } -2x > 6, \text{ then } x < -3$$

6. Permutations & Combinations

Counting

When Order Matters vs. When It Doesn't

$$^nP_r = \frac{n!}{(n-r)!} \quad \text{(order matters)}$$ $$^nC_r = \frac{n!}{r!(n-r)!} \quad \text{(order doesn't matter)}$$

Problem: How many 3-letter words from A, B, C, D, E (no repetition)?

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Order matters (ABC $\neq$ BAC), so this is a permutation.

$^5P_3 = \frac{5!}{(5-3)!} = \frac{120}{2} = 60$

7. Binomial Theorem

Expansion

Binomial Theorem

$$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$

General term: $T_{r+1} = \binom{n}{r} a^{n-r} b^r$

8. Sequences & Series

Geometric Progression

GP Formulas

$$a_n = ar^{n-1} \qquad S_n = \frac{a(r^n - 1)}{r - 1} \text{ (if } r \neq 1\text{)}$$ $$S_\infty = \frac{a}{1-r} \text{ (if } |r| < 1\text{)}$$

9. Straight Lines

Forms

Equation of a Line

$$y - y_1 = m(x - x_1) \quad \text{(point-slope)}$$ $$y = mx + c \quad \text{(slope-intercept)}$$ $$\frac{x}{a} + \frac{y}{b} = 1 \quad \text{(intercept form)}$$

Distance from point $(x_1, y_1)$ to line $ax + by + c = 0$:

$$d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$$

10. Limits & Derivatives

The Gateway to Calculus

What is a Limit?

$\lim_{x \to a} f(x) = L$ means as $x$ gets closer and closer to $a$, $f(x)$ gets closer to $L$.

$$\lim_{x \to a} \frac{x^n - a^n}{x - a} = na^{n-1}$$ $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$

Derivative

First Principles

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Quick rules: $(x^n)' = nx^{n-1}$, $(\sin x)' = \cos x$, $(\cos x)' = -\sin x$

11. Statistics

Measures of Dispersion

Variance & Standard Deviation

$$\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} \qquad \sigma = \sqrt{\sigma^2}$$

12. Probability

Set-Based Probability

Addition Rule

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

For mutually exclusive events: $P(A \cap B) = 0$, so $P(A \cup B) = P(A) + P(B)$.

1. Relations & Functions

Types of Relations

Reflexive, Symmetric, Transitive

Reflexive: $(a, a) \in R$ for all $a$
Symmetric: If $(a,b) \in R$, then $(b,a) \in R$
Transitive: If $(a,b) \in R$ and $(b,c) \in R$, then $(a,c) \in R$
Equivalence: All three of the above

Types of Functions

One-One & Onto

One-One (Injective): Different inputs → different outputs
Onto (Surjective): Every element in codomain has a preimage
Bijective: Both one-one and onto (invertible)

2. Inverse Trigonometric Functions

Key Properties

Domains & Ranges

$$\sin^{-1}x : [-1,1] \to \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$ $$\cos^{-1}x : [-1,1] \to [0, \pi]$$ $$\tan^{-1}x : \mathbb{R} \to \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$

Key identity: $\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$

3. Matrices & Determinants

Determinant

2x2 and 3x3 Determinants

$$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$$

A matrix is invertible only if its determinant is non-zero.

$$A^{-1} = \frac{1}{|A|} \text{adj}(A)$$

4. Continuity & Differentiability

Definition

Continuity at a Point

$f(x)$ is continuous at $x = a$ if:

$$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$$

Chain Rule

Differentiating Composite Functions

$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$

Derivatives to Remember

Common Derivatives

$$\frac{d}{dx}(e^x) = e^x \qquad \frac{d}{dx}(\ln x) = \frac{1}{x}$$ $$\frac{d}{dx}(\tan x) = \sec^2 x \qquad \frac{d}{dx}(\sec x) = \sec x \tan x$$

5. Integrals

The Reverse of Differentiation

Basic Integration Rules

$$\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$ $$\int \frac{1}{x} dx = \ln|x| + C$$ $$\int e^x dx = e^x + C$$ $$\int \sin x\, dx = -\cos x + C$$ $$\int \cos x\, dx = \sin x + C$$

Techniques

Integration by Parts

$$\int u\, dv = uv - \int v\, du$$

LIATE Rule for choosing $u$: Logarithmic > Inverse trig > Algebraic > Trig > Exponential

Problem: Evaluate $\int x e^x dx$

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Let $u = x$ (algebraic), $dv = e^x dx$ (exponential). By LIATE, $x$ comes first.

$du = dx$, $v = e^x$

$\int xe^x dx = xe^x - \int e^x dx = xe^x - e^x + C$

$= e^x(x - 1) + C$

6. Definite Integrals

Fundamental Theorem

Evaluating Definite Integrals

$$\int_a^b f(x)\,dx = F(b) - F(a) \quad \text{where } F'(x) = f(x)$$

7. Differential Equations

Separable Equations

Variable Separable Method

If you can write the equation as $f(y)\,dy = g(x)\,dx$, just integrate both sides!

$$\frac{dy}{dx} = \frac{g(x)}{h(y)} \implies \int h(y)\,dy = \int g(x)\,dx + C$$

8. Vectors

Operations

Dot Product & Cross Product

$$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3$$ $$|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$$

Dot product gives a scalar (used to find angles). Cross product gives a vector (used to find area).

9. 3D Geometry

Lines & Planes

Equations

$$\text{Line: } \frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$$ $$\text{Plane: } ax + by + cz = d$$

Distance from point $(x_0,y_0,z_0)$ to plane $ax+by+cz=d$:

$$D = \frac{|ax_0+by_0+cz_0-d|}{\sqrt{a^2+b^2+c^2}}$$

10. Linear Programming

Method

Graphical Method

Write objective function: maximize/minimize $Z = ax + by$
Graph all constraints as inequalities
Find the feasible region (intersection of all constraints)
Evaluate $Z$ at each corner point of the feasible region
The corner point giving max/min $Z$ is your answer

11. Probability (Bayes' Theorem)

Conditional Probability

Bayes' Theorem

$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$

This lets you "reverse" conditional probabilities. Used extensively in real-world decision making.

Problem: A factory has 2 machines. Machine A produces 60% of items (5% defective), Machine B produces 40% (3% defective). If an item is defective, what's the probability it came from Machine A?

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$P(A) = 0.6$, $P(B) = 0.4$, $P(D|A) = 0.05$, $P(D|B) = 0.03$

$P(D) = P(D|A)P(A) + P(D|B)P(B) = 0.05(0.6) + 0.03(0.4) = 0.042$

$P(A|D) = \frac{P(D|A) \cdot P(A)}{P(D)} = \frac{0.05 \times 0.6}{0.042} = \frac{0.03}{0.042} \approx 0.714$

There's about a 71.4% chance the defective item came from Machine A.