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WBJEE Mathematics Key Formulas

Essential coordinate geometry, calculus, vector algebra, and algebra identities for WBJEE mathematics preparation.

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Distance Between Parallel Lines

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Distance Between Parallel Lines

d = \frac{∣ c _{1} - c _{2} ∣}{a ^{2} + b ^{2}}

for lines

a x + b y + c_{1} = 0

and

a x + b y + c_{2} = 0

Condition of Tangency to Circle

x^{2} + y^{2} = r^{2}

The line

y = m x + c

is tangent if

c^{2} = r^{2} (1 + m^{2})

Ellipse Eccentricity (

a > b

)

e = 1 - \frac{b ^{2}}{a ^{2}}

for standard ellipse

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

Latus Rectum of Parabola

y^{2} = 4 a x

Length of latus rectum is

4 a

, with focus at

(a, 0)

and directrix

x = - a

Asymptotes of Hyperbola

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

Equations are

y = \pm \frac{b}{a} x

, passing through the center

(0, 0)

Derivative of

tan^{- 1} (x)

\frac{d}{d x} [tan^{- 1} (x)] = \frac{1}{1 + x ^{2}}

for all real

x

Standard Integral

\int \frac{d x}{x ^{2} + a ^{2}}

\int \frac{1}{x ^{2} + a ^{2}} d x = \frac{1}{a} tan^{- 1} (\frac{x}{a}) + C

Integration by Parts Formula

\int u v^{'} d x = uv - \int u^{'} v d x

, ordered by ILATE rule.

Leibniz Differentiation of Definite Integral

\frac{d}{d x} \int_{u (x)}^{v (x)} f (t) d t = f (v (x)) v^{'} (x) - f (u (x)) u^{'} (x)

Condition for Coplanar Vectors

Vectors

a, b, c

are coplanar if scalar triple product

[a b c] = 0

Shortest Distance Between Skew Lines

d = \frac{∣ ( a _{2} - a _{1} ) \cdot ( b _{1} \times b _{2} ) ∣}{∣ b _{1} \times b _{2} ∣}

Angle Between Two Planes

cos θ = \frac{∣ A _{1} A _{2} + B _{1} B _{2} + C _{1} C _{2} ∣}{A _{1}^{2} + B _{1}^{2} + C _{1}^{2} A _{2}^{2} + B _{2}^{2} + C _{2}^{2}}

Roots of Cubic Equation

a x^{3} + b x^{2} + c x + d = 0

\sum α = - \frac{b}{a}

\sum α β = \frac{c}{a}

α β γ = - \frac{d}{a}

Cube Roots of Unity Properties

1 + ω + ω^{2} = 0

and

ω^{3} = 1

, where

ω = e^{i 2 π /3}

De Moivre's Theorem

(cos θ + i sin θ)^{n} = cos (n θ) + i sin (n θ)

for integer

n

Sum of Infinite Geometric Series (

∣ r ∣ < 1

)

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

, with common ratio

r

Sum of Binomial Coefficients

\sum_{r = 0}^{n} (r n) = 2^{n}

, and

\sum_{r = 0}^{n} (- 1)^{r} (r n) = 0

Rolle's Theorem Conditions

Continuous on

[a, b]

, differentiable on

(a, b)

, and

f (a) = f (b) ⟹ \exists c \in (a, b)

with

f^{'} (c) = 0

Lagrange Mean Value Theorem

f^{'} (c) = \frac{f ( b ) - f ( a )}{b - a}

for some point

c \in (a, b)

Derivative of

sin^{- 1} (x)

\frac{d}{d x} [sin^{- 1} (x)] = \frac{1}{1 - x ^{2}}

for

∣ x ∣ < 1

Standard Trigonometric Limit

lim_{x \to 0} \frac{s i n x}{x} = 1

, where

x

is measured in radians.

Addition Theorem of Probability

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

Bayes' Theorem Formula

P (A_{i} ∣ B) = \frac{P ( B ∣ A _{i} ) P ( A _{i} )}{\sum _{j} P ( B ∣ A _{j} ) P ( A _{j} )}

Binomial Distribution Mean and Variance

Mean

μ = n p

and Variance

σ^{2} = n p (1 - p) = n pq

Determinant Identity with Identical Rows

If any two rows or columns of a square matrix are identical,

det (A) = 0

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