Engineering Mathematics

Sheet 1

Linear Algebra & Calculus

10 formulas

Core concepts

• Triangular/diagonal matrix: eigenvalues are exactly the diagonal entries
• Cayley-Hamilton: every square matrix satisfies its own characteristic equation

Key formulas

Rank of a Matrix ☆

\[ \text{Rank}(A) = \text{Rank}(A^T) \]

= largest order of a non-zero minor

Characteristic Equation ☆

\[ \det(A - \lambda I) = 0 \]

roots are the eigenvalues

Eigenvalue Sum & Product ☆

\[ \sum \lambda_i = \text{trace}(A) \qquad \prod \lambda_i = \det(A) \]

Orthogonal Matrix ☆

\[ A^{-1} = A^T \]

Mean Value Theorem ☆

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

L'Hopital's Rule ☆

\[ \lim \frac{f}{g} = \lim \frac{f'}{g'} \]

for 0/0 or ∞/∞ forms

Maxima / Minima Test ☆

\[ f'(x)=0,\ f''(x)>0 \Rightarrow \text{min} \qquad f'(x)=0,\ f''(x)<0 \Rightarrow \text{max} \]

Continuity at a Point ☆

\[ \lim_{x \to a} f(x) = f(a) \]

Differentiability at a Point ☆

\[ f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h} \]

LHD = RHD for Differentiability ☆

\[ \lim_{h \to 0} \frac{f(a)-f(a-h)}{h} \ = \ \lim_{h \to 0} \frac{f(a+h)-f(a)}{h} \]

left-hand derivative must equal right-hand derivative

Test your recall

Can you recall the rank of a matrix formula?

Reveal formula

\[ \text{Rank}(A) = \text{Rank}(A^T) \]

GATE traps 6 traps

Singular means a zero eigenvalue

A singular matrix (det=0) has at least one zero eigenvalue -- a matrix with all non-zero eigenvalues is always invertible.

Inverse eigenvalues are reciprocals

Eigenvalues of A^-1 are reciprocals of eigenvalues of A -- only valid when A is invertible.

L'Hopital only fits 0/0 or ∞/∞

L'Hopital only applies to 0/0 or infinity/infinity indeterminate forms -- applying it to other forms is a classic trap.

Continuity doesn't imply differentiability

Differentiability implies continuity, but continuity does NOT imply differentiability -- the reverse direction is a frequently tested trick option.

|x| is the classic counterexample

f(x)=|x| is continuous everywhere (including at x=0) but NOT differentiable at x=0 -- its LHD=-1 and RHD=+1 disagree there. The single most common example GATE uses to test the continuity-vs-differentiability distinction.

Repeated eigenvalues aren't automatically bad

Repeated eigenvalues don't automatically mean a matrix is non-diagonalizable -- check whether geometric multiplicity equals algebraic multiplicity before concluding either way.

Sheet 2

Probability & Statistics

7 formulas

Key formulas

Union of Events ☆

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

Conditional Probability ☆

\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \]

Bayes' Theorem ☆

\[ P(A \mid B) = \frac{P(B \mid A)\,P(A)}{P(B)} \]

Expectation & Variance ☆

\[ E[X] = \sum x\,P(x) \qquad \text{Var}(X) = E[X^2] - (E[X])^2 \]

Binomial Distribution ☆

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

mean = np, variance = np(1−p)

Poisson Distribution ☆

\[ P(X=k) = \frac{e^{-\lambda}\lambda^k}{k!} \]

mean = variance = λ

Standard Deviation ☆

\[ \sigma = \sqrt{\text{Var}(X)} \]

Test your recall

Can you recall the bayes' theorem formula?

Reveal formula

\[ P(A \mid B) = \frac{P(B \mid A)\,P(A)}{P(B)} \]

GATE traps 4 traps

Mutually exclusive ≠ independent

Mutually exclusive events are NEVER independent (unless one has probability 0) -- one of the most common probability traps on GATE. Independence requires P(A intersect B) = P(A)P(B); mutual exclusivity gives P(A intersect B) = 0, which only equals P(A)P(B) if one probability is 0.

Variance isn't linear in constants

Variance is NOT linear in the additive constant: Var(aX+b) = a^2 . Var(X), the +b vanishes.

E[XY]=E[X]E[Y] needs independence

E[XY] = E[X] . E[Y] only holds when X and Y are independent -- not in general.

'At least one' = 1 minus 'none'

'At least one success' = 1 - P(zero successes) -- almost always faster than summing every case directly, and the shortcut GATE numericals are built around.

Sheet 3

Discrete Math (Sets, Relations, Graphs, Counting)

8 formulas

Key formulas

2-Set Inclusion-Exclusion ☆

\[ |A \cup B| = |A|+|B|-|A \cap B| \]

3-Set Inclusion-Exclusion ☆

\[ |A\cup B\cup C| = |A|{+}|B|{+}|C|-|A{\cap}B|-|B{\cap}C|-|A{\cap}C|+|A{\cap}B{\cap}C| \]

Relations on a Set ☆

\[ \text{Relations on an } n\text{-set} = 2^{n^2} \]

Reflexive & Symmetric Relations ☆

\[ \text{Reflexive} = 2^{n^2-n} \qquad \text{Symmetric} = 2^{n(n+1)/2} \]

Counting Functions ☆

\[ |B|^{|A|} \text{ functions} \qquad \frac{|B|!}{(|B|-|A|)!} \text{ one-one} \]

one-one needs |A| ≤ |B|

Handshaking Lemma ☆

\[ \sum \deg(v) = 2|E| \]

Permutations & Combinations ☆

\[ {}^{n}P_r = \frac{n!}{(n-r)!} \qquad {}^{n}C_r = \frac{n!}{r!(n-r)!} \]

Pigeonhole Principle ☆

\[ n{>}m \Rightarrow \text{some container has} \ge \lceil n/m \rceil \]

n items, m containers

Test your recall

Can you recall the permutations & combinations formula?

Reveal formula

\[ {}^{n}P_r = \frac{n!}{(n-r)!} \qquad {}^{n}C_r = \frac{n!}{r!(n-r)!} \]

GATE traps 5 traps

Reflexive needs every element

A relation is reflexive only if (a,a) is present for EVERY element a -- missing even one breaks it, and GATE tests this with small explicit sets where it's easy to miss one pair.

Equivalence vs. partial order

Equivalence relation = reflexive + symmetric + transitive. Partial order = reflexive + antisymmetric + transitive. These get mixed up constantly under time pressure.

n-1 edges alone isn't a tree

A connected graph with n-1 edges is a tree, but n-1 edges alone doesn't guarantee connectivity -- both conditions (connected AND n-1 edges, equivalently connected AND acyclic) must be checked.

Don't drop the triple-overlap term

In 3-set inclusion-exclusion, forgetting the final +|A intersect B intersect C| term is the single most common mistake.

Permutation vs. combination wording

Permutation = order matters (arrangements); combination = order doesn't (selections) -- GATE word problems often disguise which one actually applies.

Sheet 4

Numerical Methods

4 formulas

Core concepts

• Newton-Raphson has quadratic convergence once close to the root; Bisection has only linear convergence but always converges once a root is bracketed
• Simpson's 1/3 rule needs an EVEN number of sub-intervals; the trapezoidal rule works for any number

Key formulas

Newton-Raphson Method ☆

\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]

Bisection Method ☆

\[ c = \frac{a+b}{2} \]

valid when f(a)·f(b) < 0

Trapezoidal Rule ☆

\[ \int_a^b f(x)\,dx \approx \frac{h}{2}\Big[f(x_0)+2\sum f(x_i)+f(x_n)\Big] \]

h = (b−a)/n

Simpson's 1/3 Rule ☆

\[ \int_a^b f(x)\,dx \approx \frac{h}{3}\Big[f(x_0)+4\sum_{odd} f(x_i)+2\sum_{even} f(x_i)+f(x_n)\Big] \]

needs an even number of intervals

Test your recall

Can you recall the newton-raphson method formula?

Reveal formula

\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]

GATE traps 4 traps

Newton-Raphson isn't guaranteed to converge

Newton-Raphson can fail to converge (or converge to the wrong root) if f'(x_n) is zero or the initial guess is far from the root -- unlike bisection, it has no guaranteed convergence.

Simpson's 1/3 needs an even interval count

Simpson's 1/3 rule needs an even number of sub-intervals -- an odd count means it can't be applied directly, a common numerical-methods setup error.

Bisection iterations follow a clean log formula

Bisection halves the interval every iteration -- the number of iterations needed for a given accuracy eps is ceil(log2((b-a)/eps)), a frequently tested numerical.

Quadratic convergence kicks in near the root

Higher-order convergence (Newton-Raphson) doesn't mean fewer total iterations from the very first step -- it means the error shrinks quadratically ONCE you're already close to the root.