Class 10 Mathematics — Complete Formula Guide & Master Repository

Complete 100% exhaustive formula guide for Class 10 Mathematics (NCERT, CBSE & State Board Syllabus). Covers Real Numbers, Polynomials, Pair of Linear Equations, Quadratic Equations, Arithmetic Progressions, Trigonometry & Applications, Circles, Constructions, Areas Related to Circles, Surface Areas & Volumes, Statistics, and Probability with clear KaTeX formulas, tables, worked examples, and animated SVG diagrams.

sin²θ + cos²θ = 1

a² + b² = c²

a = bq + r

π = 22/7

aₙ = a + (n-1)d

x = (-b ± √D)/2a

1. Real Numbers

1.1 Euclid's Division Lemma & Algorithm

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

$a = bq + r, \quad \text{where } 0 \le r < b$

Fundamental Theorem of Arithmetic: Every composite number can be expressed (factored) as a product of primes uniquely, apart from the order in which prime factors occur.

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

2. Polynomials

2.1 Zeros and Coefficients Relationships

For quadratic polynomial $p(x) = ax^2 + bx + c$ with zeros $\alpha, \beta$:

$\text{Sum of Zeros } (\alpha + \beta) = -\frac{b}{a} = -\frac{\text{Coefficient of } x}{\text{Coefficient of } x^2}$ $\text{Product of Zeros } (\alpha \cdot \beta) = \frac{c}{a} = \frac{\text{Constant term}}{\text{Coefficient of } x^2}$

Forming Quadratic Polynomial with given zeros $\alpha, \beta$:

$p(x) = k \left[ x^2 - (\alpha + \beta)x + \alpha\beta \right]$

3. Pair of Linear Equations in Two Variables

3.1 Solvability Conditions

Equations: $a_1 x + b_1 y + c_1 = 0$ and $a_2 x + b_2 y + c_2 = 0$

Ratio Comparison	Graphical Representation	Algebraic Solution	Consistency
$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$	Intersecting Lines	Exactly One Unique Solution	Consistent
$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$	Coincident Lines	Infinitely Many Solutions	Dependent / Consistent
$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$	Parallel Lines	No Solution	Inconsistent

4. Quadratic Equations

4.1 Quadratic Formula & Discriminant

Standard Form: $ax^2 + bx + c = 0 \quad (a \neq 0)$

Discriminant: $D = b^2 - 4ac$

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

Nature of Roots:

If $D > 0$: Two distinct real roots
If $D = 0$: Two equal real roots ($x = -\frac{b}{2a}$)
If $D < 0$: No real roots

5. Arithmetic Progressions (AP)

5.1 AP Essential Formulas

Standard AP: $a, a+d, a+2d, a+3d, \dots$

First term: $a$, Common Difference: $d = a_n - a_{n-1}$
$n$-th Term (General Term): $a_n = a + (n - 1)d$
$n$-th Term from End: $a_n' = l - (n - 1)d$ (where $l$ is the last term)
Sum of First $n$ Terms: $S_n = \frac{n}{2} \left[ 2a + (n - 1)d \right] \quad \text{or} \quad S_n = \frac{n}{2} (a + l)$
Number of Terms Formula: $n = \frac{l - a}{d} + 1$

6. Introduction to Trigonometry & Applications

6.1 Trigonometric Ratios

$\sin\theta = \frac{\text{Perpendicular (P)}}{\text{Hypotenuse (H)}}, \quad \csc\theta = \frac{\text{Hypotenuse (H)}}{\text{Perpendicular (P)}}$

$\cos\theta = \frac{\text{Base (B)}}{\text{Hypotenuse (H)}}, \quad \sec\theta = \frac{\text{Hypotenuse (H)}}{\text{Base (B)}}$

$\tan\theta = \frac{\text{Perpendicular (P)}}{\text{Base (B)}}, \quad \cot\theta = \frac{\text{Base (B)}}{\text{Perpendicular (P)}}$

6.2 Standard Angles Table

Ratio / Angle	0°	30°	45°	60°	90°
$\sin\theta$	0	1/2	$1/\sqrt{2}$	$\sqrt{3}/2$	1
$\cos\theta$	1	$\sqrt{3}/2$	$1/\sqrt{2}$	1/2	0
$\tan\theta$	0	$1/\sqrt{3}$	1	$\sqrt{3}$	Undefined (∞)
$\cot\theta$	∞	$\sqrt{3}$	1	$1/\sqrt{3}$	0
$\sec\theta$	1	$2/\sqrt{3}$	$\sqrt{2}$	2	∞
$\csc\theta$	∞	2	$\sqrt{2}$	$2/\sqrt{3}$	1

6.3 Fundamental Identities

$\sin^2\theta + \cos^2\theta = 1$ $1 + \tan^2\theta = \sec^2\theta \implies \sec^2\theta - \tan^2\theta = 1$ $1 + \cot^2\theta = \csc^2\theta \implies \csc^2\theta - \cot^2\theta = 1$

7. Circles

7.1 Circle Theorems with SVG Diagrams

Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

$OP \perp \text{Tangent Line}$

Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal.

$TP = TQ$

8. Constructions

8.1 Line Division and Tangents Construction

1. Division of Line Segment in ratio $m:n$: Draw acute ray and mark $m+n$ equal arcs, using parallel line construction.
2. Tangents from External Point: Join center $O$ to external point $T$, draw perpendicular bisector of $OT$ to locate midpoint $M$, draw circle with center $M$ and radius $MO$ intersecting original circle at points $P, Q$.

10. Surface Areas and Volumes

10.1 Master 3D Shapes Formula Table

Shape	Curved Surface Area (CSA)	Total Surface Area (TSA)	Volume
Cube	$4a^2$	$6a^2$	$a^3$
Cuboid	$2h(l + b)$	$2(lb + bh + hl)$	$l \times b \times h$
Cylinder	$2\pi rh$	$2\pi r(h + r)$	$\pi r^2 h$
Cone	$\pi rl \quad (l = \sqrt{r^2 + h^2})$	$\pi r(l + r)$	$\frac{1}{3}\pi r^2 h$
Sphere	$4\pi r^2$	$4\pi r^2$	$\frac{4}{3}\pi r^3$
Hemisphere	$2\pi r^2$	$3\pi r^2$	$\frac{2}{3}\pi r^3$
Frustum of Cone	$\pi (r_1 + r_2)l$	$\pi l(r_1 + r_2) + \pi r_1^2 + \pi r_2^2$	$\frac{1}{3}\pi h (r_1^2 + r_2^2 + r_1 r_2)$

11. Statistics

11.1 Mean, Median and Mode Formulas

1. Mean ($\bar{x}$):
🔹 Direct Method: $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$
🔹 Assumed Mean Method: $\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$
🔹 Step-Deviation Method: $\bar{x} = a + \left( \frac{\sum f_i u_i}{\sum f_i} \right) \times h$

2. Median: $\text{Median} = l + \left( \frac{\frac{N}{2} - CF}{f} \right) \times h$

3. Mode: $\text{Mode} = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$

4. Empirical Relationship: $\mathbf{3 \times \text{Median} = \text{Mode} + 2 \times \text{Mean}}$

12. Probability

12.1 Classical Probability Formulas

$P(E) = \frac{\text{Number of outcomes favorable to } E \quad (n(E))}{\text{Total number of possible outcomes} \quad (n(S))}$ $0 \le P(E) \le 1$ $P(E) + P(\bar{E}) = 1 \implies P(\bar{E}) = 1 - P(E)$ $\text{Probability of Certain Event} = 1, \quad \text{Probability of Impossible Event} = 0$

Comments & Reviews

0.0

★★★★★

0 reviews

Tap stars to rate this page:

★ ★ ★ ★ ★

Loading comments...

Ratio / Angle	0°	30°	45°	60°	90°
\(\sin\theta\)	0	1/2	\(1/\sqrt{2}\)	\(\sqrt{3}/2\)	1
\(\cos\theta\)	1	\(\sqrt{3}/2\)	\(1/\sqrt{2}\)	1/2	0
\(\tan\theta\)	0	\(1/\sqrt{3}\)	1	\(\sqrt{3}\)	Undefined (∞)
\(\cot\theta\)	∞	\(\sqrt{3}\)	1	\(1/\sqrt{3}\)	0
\(\sec\theta\)	1	\(2/\sqrt{3}\)	\(\sqrt{2}\)	2	∞
\(\csc\theta\)	∞	2	\(\sqrt{2}\)	\(2/\sqrt{3}\)	1

Shape	Curved Surface Area (CSA)	Total Surface Area (TSA)	Volume
Cube	\(4a^2\)	\(6a^2\)	\(a^3\)
Cuboid	\(2h(l + b)\)	\(2(lb + bh + hl)\)	\(l \times b \times h\)
Cylinder	\(2\pi rh\)	\(2\pi r(h + r)\)	\(\pi r^2 h\)
Cone	\(\pi rl \quad (l = \sqrt{r^2 + h^2})\)	\(\pi r(l + r)\)	\(\frac{1}{3}\pi r^2 h\)
Sphere	\(4\pi r^2\)	\(4\pi r^2\)	\(\frac{4}{3}\pi r^3\)
Hemisphere	\(2\pi r^2\)	\(3\pi r^2\)	\(\frac{2}{3}\pi r^3\)
Frustum of Cone	\(\pi (r_1 + r_2)l\)	\(\pi l(r_1 + r_2) + \pi r_1^2 + \pi r_2^2\)	\(\frac{1}{3}\pi h (r_1^2 + r_2^2 + r_1 r_2)\)