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GATE: Engineering Mathematics Key Results

Flashcards on linear algebra, calculus, probability, differential equations and discrete mathematics for GATE. Original study material.

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Trace and determinant in terms of eigenvalues

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Trace and determinant in terms of eigenvalues

Trace = sum of eigenvalues; determinant = product of eigenvalues.

Rank–nullity relation

Number of free parameters = number of unknowns − rank.

Laplace transform of

t^{n}

\frac{n !}{s ^{n + 1}}

Laplace transform of

e^{a t}

\frac{1}{s - a}

Divergence of

F = (F_{1}, F_{2}, F_{3})

\frac{\partial F _{1}}{\partial x} + \frac{\partial F _{2}}{\partial y} + \frac{\partial F _{3}}{\partial z}

Newton–Raphson step

x_{n + 1} = x_{n} - \frac{f ( x _{n} )}{f ^{'} ( x _{n} )}

Taylor series of

e^{x}

\sum \frac{x ^{n}}{n !}

Limit of

sin (a x) / x

x \to 0

a

Conditional probability

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}

Expected value

E [X] = \sum x p (x)

Binomial probability

(k n) p^{k} (1 - p)^{n - k}

Solution of

d y / d x = - a y

y = y_{0} e^{- a x}

Edges in the complete graph

K_{n}

\frac{n ( n - 1 )}{2}

Edges in a tree with n vertices

n - 1

Cayley's formula

n^{n - 2}

labelled trees on n vertices.

Relations on an n-element set

2^{n^{2}}

Reflexive relations on an n-element set

2^{n^{2} - n}

Functions from an m-set to an n-set

n^{m}

Probability of at least one of two independent events

1 - (1 - p_{1}) (1 - p_{2})

Complement rule

P (A^{'}) = 1 - P (A)

Determinant of a triangular matrix

The product of the diagonal entries.

Sum of 1 to n

\frac{n ( n + 1 )}{2}

Chain rule

\frac{d y}{d x} = \frac{d y}{d u} \frac{d u}{d x}

Integration by parts

\int u d v = uv - \int v d u

Double integral over a rectangle

Integrate one variable at a time with its own limits.

Variance

E [X^{2}] - (E [X])^{2}

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