Math Formula Sheet : All Classes

Arithmetic & Number

9 formulas

Percentages, ratios, interest, GCD/LCM : the everyday-money formulas.

Percentage

VI–VIII

p\% \text{ of } n = \dfrac{p}{100} \times n

Percentage change

VII

\% \text{ change} = \dfrac{\text{new} - \text{old}}{\text{old}} \times 100

Simple interest

VII–VIII

SI = \dfrac{P \, R \, T}{100}

Compound interest

VIII

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

Discount

VIII

\text{SP} = \text{MP} \left(1 - \dfrac{d}{100}\right)

Profit %

VII–VIII

\text{Profit\%} = \dfrac{\text{SP} - \text{CP}}{\text{CP}} \times 100

EMI

\text{EMI} = \dfrac{P \, r (1 + r)^n}{(1 + r)^n - 1}

r = monthly rate, n = months.

GCD × LCM identity

VI

\gcd(a,b) \cdot \operatorname{lcm}(a,b) = a \cdot b

Ratio split

VI–VII

x : y \Rightarrow \text{shares} = \dfrac{x}{x+y}T,\; \dfrac{y}{x+y}T

Algebraic Identities

11 formulas

The expansions you reach for inside every algebra step.

(a+b)²

VIII

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

(a−b)²

VIII

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

a²−b²

VIII

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

(a+b)³

IX

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

(a−b)³

IX

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

a³+b³

IX

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

a³−b³

IX

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

Sum a³+b³+c³ (special)

IX

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

Quadratic formula

X

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

Discriminant

X

\Delta = b^2 - 4ac

Sum & product of roots

X–XI

\alpha + \beta = -\dfrac{b}{a},\quad \alpha\beta = \dfrac{c}{a}

Exponents & Logarithms

11 formulas

Laws of indices and logs : the workhorse moves in algebra.

Product rule

VII–VIII

a^m \cdot a^n = a^{m+n}

Quotient rule

VII–VIII

\dfrac{a^m}{a^n} = a^{m-n}

Power of a power

VIII

(a^m)^n = a^{mn}

Power of a product

VIII

(ab)^n = a^n b^n

Negative index

VIII

a^{-n} = \dfrac{1}{a^n}

Zero index

VIII

a^0 = 1\quad (a \ne 0)

Fractional index

IX

a^{1/n} = \sqrt[n]{a}

Log product

XI

\log_b (xy) = \log_b x + \log_b y

Log quotient

XI

\log_b \dfrac{x}{y} = \log_b x - \log_b y

Log power

XI

\log_b (x^k) = k \, \log_b x

Change of base

XI

\log_b x = \dfrac{\log_c x}{\log_c b}

Sequences & Series

8 formulas

Arithmetic and geometric patterns.

AP nth term

X

a_n = a + (n-1) d

AP sum

X

S_n = \dfrac{n}{2}\bigl(2a + (n-1)d\bigr) = \dfrac{n}{2}(a + l)

GP nth term

XI

a_n = a r^{n-1}

GP sum (r≠1)

XI

S_n = \dfrac{a(r^n - 1)}{r - 1}

Infinite GP (|r|<1)

XI

S_\infty = \dfrac{a}{1 - r}

Sum of first n naturals

XI

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

Sum of first n squares

XI

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

Sum of first n cubes

XI

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

Geometry (2-D)

14 formulas

Lengths, areas, perimeters of plane figures.

Square area

V–VII

A = s^2

Rectangle area

V–VII

A = l \cdot w

Triangle area

VI

A = \dfrac{1}{2} \cdot \text{base} \cdot \text{height}

Heron's formula

IX

A = \sqrt{s(s-a)(s-b)(s-c)},\; s = \tfrac{a+b+c}{2}

Parallelogram

VII

A = b \cdot h

Trapezium

VIII

A = \dfrac{1}{2}(a + b) \cdot h

Rhombus (diagonals)

VIII

A = \dfrac{1}{2} d_1 d_2

Circle area

VII

A = \pi r^2

Circle circumference

VII

C = 2 \pi r

Sector area

X

A = \dfrac{\theta}{360^\circ} \, \pi r^2

Arc length

X

\ell = \dfrac{\theta}{360^\circ} \, 2\pi r

Sum of polygon angles

VIII

(n - 2) \times 180^\circ

Each interior angle (regular)

VIII

\dfrac{(n-2) \cdot 180^\circ}{n}

Pythagoras

VII–IX

a^2 + b^2 = c^2

Mensuration (3-D)

14 formulas

Volumes and surface areas of solids.

Cube volume

VIII

V = a^3

Cube TSA

VIII

S = 6a^2

Cuboid volume

VIII

V = l \cdot w \cdot h

Cuboid TSA

VIII

S = 2(lw + wh + lh)

Cylinder volume

IX

V = \pi r^2 h

Cylinder CSA

IX

S = 2 \pi r h

Cylinder TSA

IX

S = 2\pi r (r + h)

Cone volume

IX

V = \dfrac{1}{3} \pi r^2 h

Cone CSA

IX

S = \pi r l,\; l = \sqrt{r^2 + h^2}

Sphere volume

IX

V = \dfrac{4}{3} \pi r^3

Sphere SA

IX

S = 4 \pi r^2

Hemisphere volume

IX

V = \dfrac{2}{3} \pi r^3

Hemisphere TSA

IX

S = 3 \pi r^2

Frustum of cone V

X

V = \dfrac{\pi h}{3}\!\bigl(r_1^2 + r_2^2 + r_1 r_2\bigr)

Coordinate Geometry

13 formulas

Distances, lines and conics in the plane.

Distance between points

IX–X

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Mid-point

X

M = \left(\dfrac{x_1 + x_2}{2},\; \dfrac{y_1 + y_2}{2}\right)

Section formula (internal)

X

P = \left(\dfrac{m x_2 + n x_1}{m + n},\; \dfrac{m y_2 + n y_1}{m + n}\right)

Area of a triangle (coordinates)

X

A = \tfrac{1}{2}\bigl|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\bigr|

Slope between points

X–XI

m = \dfrac{y_2 - y_1}{x_2 - x_1}

Slope–intercept

X–XI

y = mx + c

Point–slope

XI

y - y_1 = m(x - x_1)

General line

XI

ax + by + c = 0

Distance from a point to a line

XI

d = \dfrac{|a x_0 + b y_0 + c|}{\sqrt{a^2 + b^2}}

Circle (centre h, k)

XI

(x - h)^2 + (y - k)^2 = r^2

Ellipse

XI

\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1

Parabola

XI

y^2 = 4 a x

Hyperbola

XI

\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1

Trigonometry

14 formulas

Ratios, identities, addition and multiple-angle formulas.

Reciprocal pair

X

\csc\theta = \dfrac{1}{\sin\theta},\; \sec\theta = \dfrac{1}{\cos\theta},\; \cot\theta = \dfrac{1}{\tan\theta}

Pythagorean identity

X

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

Identity 2

X

1 + \tan^2\theta = \sec^2\theta

Identity 3

X

1 + \cot^2\theta = \csc^2\theta

Sin(A+B)

XI

\sin(A + B) = \sin A \cos B + \cos A \sin B

Sin(A−B)

XI

\sin(A - B) = \sin A \cos B - \cos A \sin B

Cos(A+B)

XI

\cos(A + B) = \cos A \cos B - \sin A \sin B

Cos(A−B)

XI

\cos(A - B) = \cos A \cos B + \sin A \sin B

Tan(A+B)

XI

\tan(A + B) = \dfrac{\tan A + \tan B}{1 - \tan A \tan B}

Sin 2A

XI

\sin 2A = 2 \sin A \cos A

Cos 2A

XI

\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A

Tan 2A

XI

\tan 2A = \dfrac{2 \tan A}{1 - \tan^2 A}

Sine rule

XI

\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} = 2R

Cosine rule

XI

c^2 = a^2 + b^2 - 2ab \cos C

Calculus

15 formulas

Limits, derivatives, integrals.

Limit (sin x)/x

XI

\lim_{x \to 0} \dfrac{\sin x}{x} = 1

Limit (1 − cos x)/x²

XI

\lim_{x \to 0} \dfrac{1 - \cos x}{x^2} = \dfrac{1}{2}

Definition of derivative

XI

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

Power rule

XI–XII

\dfrac{d}{dx} x^n = n x^{n-1}

Sin / Cos / Tan derivatives

XI–XII

(\sin x)' = \cos x,\; (\cos x)' = -\sin x,\; (\tan x)' = \sec^2 x

Exponential / log

XII

(e^x)' = e^x,\; (\ln x)' = \dfrac{1}{x}

Product rule

XI

(uv)' = u' v + u v'

Quotient rule

XII

\left(\dfrac{u}{v}\right)' = \dfrac{u' v - u v'}{v^2}

Chain rule

XII

\dfrac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)

Integral x^n

XII

\int x^n \, dx = \dfrac{x^{n+1}}{n+1} + C \quad (n \ne -1)

Integral 1/x

XII

\int \dfrac{1}{x}\, dx = \ln |x| + C

Integral sin / cos

XII

\int \sin x \, dx = -\cos x + C,\; \int \cos x \, dx = \sin x + C

Integral e^x

XII

\int e^x \, dx = e^x + C

Definite integral (FTC)

XII

\int_a^b f(x) \, dx = F(b) - F(a)

Integration by parts

XII

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

Vectors & 3-D Geometry

10 formulas

Dot product, cross product and lines/planes in space.

Magnitude

XII

|\vec a| = \sqrt{a_1^2 + a_2^2 + a_3^2}

Unit vector

XII

\hat a = \dfrac{\vec a}{|\vec a|}

Dot product

XII

\vec a \cdot \vec b = a_1 b_1 + a_2 b_2 + a_3 b_3 = |\vec a||\vec b| \cos\theta

Cross product magnitude

XII

|\vec a \times \vec b| = |\vec a||\vec b| \sin\theta

Projection of a on b

XII

\text{proj}_{\vec b}\vec a = \dfrac{\vec a \cdot \vec b}{|\vec b|^2}\, \vec b

Distance between two points

XII

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

Direction cosines

XII

l^2 + m^2 + n^2 = 1

Line vector form

XII

\vec r = \vec a + t \vec b

Plane equation (normal form)

XII

\vec r \cdot \hat n = d

Distance from point to plane

XII

d = \dfrac{|a x_0 + b y_0 + c z_0 + d_0|}{\sqrt{a^2 + b^2 + c^2}}

Statistics & Probability

12 formulas

Central tendency, dispersion and probability rules.

Mean (raw)

VII–IX

\bar x = \dfrac{1}{n}\sum_{i=1}^{n} x_i

Mean (frequency)

IX–X

\bar x = \dfrac{\sum f_i x_i}{\sum f_i}

Median (odd n)

VIII–IX

M = x_{(n+1)/2}

Median (grouped)

X

M = L + \left(\dfrac{\tfrac{n}{2} - F}{f}\right) h

Mode (grouped)

X

Mo = L + \left(\dfrac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) h

Variance

XI

\sigma^2 = \dfrac{1}{n}\sum (x_i - \bar x)^2

Std. deviation

XI

\sigma = \sqrt{\sigma^2}

Probability of an event

IX–XI

P(E) = \dfrac{\text{favourable}}{\text{total}}

Addition rule

XI

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

Conditional probability

XII

P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}

Bayes' theorem

XII

P(A_i \mid B) = \dfrac{P(B \mid A_i) P(A_i)}{\sum_j P(B \mid A_j) P(A_j)}

Binomial probability

XII

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

Matrices & Determinants

8 formulas

Common operations and 2×2 / 3×3 shortcuts.

2×2 determinant

XII

\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc

2×2 inverse

XII

M^{-1} = \dfrac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

Adjoint identity

XII

A \cdot \operatorname{adj}(A) = \det(A) \, I

Cramer's rule (2×2)

XII

x = \dfrac{D_x}{D},\quad y = \dfrac{D_y}{D}

Property: row swap

XII

\det \text{ changes sign on row swap}

Property: scalar multiply row

XII

\det \to k \cdot \det \text{ if a row is multiplied by } k

Transpose product

XII

(AB)^{T} = B^{T} A^{T}

Inverse product

XII

(AB)^{-1} = B^{-1} A^{-1}

Permutations, Combinations & Binomial

7 formulas

Counting arrangements and the binomial theorem.

Factorial

XI

n! = n \cdot (n - 1) \cdots 2 \cdot 1,\; 0! = 1

Permutation P(n,r)

XI

{}^nP_r = \dfrac{n!}{(n - r)!}

Combination C(n,r)

XI

\binom{n}{r} = \dfrac{n!}{r!\, (n - r)!}

Pascal's identity

XI

\binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1}

Binomial theorem

XI

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

Sum of binomial coeffs

XI

\sum_{r=0}^{n} \binom{n}{r} = 2^n

Middle term (even n)

XI

T_{n/2 + 1} = \binom{n}{n/2}\, a^{n/2} b^{n/2}

Math Formula Sheet

Arithmetic & Number

Algebraic Identities

Exponents & Logarithms

Sequences & Series

Geometry (2-D)

Mensuration (3-D)

Coordinate Geometry

Trigonometry

Calculus

Vectors & 3-D Geometry

Statistics & Probability

Matrices & Determinants

Permutations, Combinations & Binomial

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