Engineering Mathematics is a foundational, formula-dense section common to ESE Paper I, GATE, and SSC JE across all engineering branches. This module spans Linear Algebra (matrices, eigenvalues), Calculus (limits, differentiation, integration, multivariable calculus), Differential Equations (first-order and second-order ODEs, Laplace transforms), Complex Variables (analytic functions, Cauchy's theorems, residues), Probability & Statistics (distributions, hypothesis testing), and Numerical Methods (root finding, integration, ODEs) — with every formula, identity, and standard result carried over, plus worked examples and diagrams for each topic.
After studying this chapter you will be able to:
Engineering Mathematics underpins nearly every other technical subject — matrix methods feed into Theory of Structures, differential equations into Structural Dynamics, and numerical methods into virtually every design calculation. Once you've worked through the chapters below, head to the Engineering Mathematics hub page to generate practice tests, or explore Study Material for other subjects.
| Term | Definition / Property |
|---|---|
| Rank of a matrix | Number of linearly independent rows (or columns); = number of non-zero rows in row echelon form |
| Singular matrix | \(\det(A) = 0\); rank < n; system \(Ax = b\) may have no or infinite solutions |
| Symmetric | \(A = A^T\) |
| Skew-symmetric | \(A = -A^T\); diagonal elements \(= 0\) |
| Orthogonal | \(AA^T = I\); \(A^T = A^{-1}\); columns are orthonormal |
| Idempotent | \(A^2 = A\) |
| Nilpotent | \(A^k = 0\) for some integer \(k\) |
\(Ax = b\); augmented matrix \([A|b]\). \(\rho(A)\) = rank of \(A\); \(\rho([A|b])\) = rank of augmented matrix.
\(Ax = \lambda x\); characteristic equation: \(\det(A - \lambda I) = 0\).
Properties of eigenvalues: Sum of eigenvalues \(= \text{trace}(A) = \Sigma a_{ii}\). Product of eigenvalues \(= \det(A)\). Eigenvalues of \(A^T\) \(=\) eigenvalues of \(A\). Eigenvalues of \(A^{-1} = \dfrac{1}{\lambda}\) (if \(A\) invertible). Eigenvalues of \(A^k = \lambda^k\). Symmetric matrix → all eigenvalues are real. Skew-symmetric → eigenvalues are zero or purely imaginary. Orthogonal matrix → \(|\lambda| = 1\).
\(\Sigma\lambda = \text{trace}(A)\); \(\Pi\lambda = \det(A)\)
\(\rho(A) \ne \rho([A|b])\) → no solution; \(= n\) → unique; \(< n\) → infinite solutions
\(\text{rank}(A) + \text{nullity}(A) = n\)
\(A^2 = (\text{trace})A - (\det)I\)
Given: A system \(Ax = b\) has a 3×3 coefficient matrix \(A\) with rank 2, and the augmented matrix \([A|b]\) also has rank 2. What can be said about the solution?
Solution: Since \(\rho(A) = \rho([A|b]) = 2 < n = 3\), the system is consistent but has infinitely many solutions, with \(n - \rho = 3 - 2 = 1\) free variable.
Answer: Infinitely many solutions, with 1 free variable.
Given: A 3×3 matrix has trace 9 and determinant 24. If two of its eigenvalues are 2 and 3, find the third.
Solution: Sum of eigenvalues = trace = \(9\), so \(\lambda_3 = 9 - 2 - 3 = 4\). Checking via the product: \(2 \times 3 \times 4 = 24 = \det(A)\) ✓, confirming consistency.
Answer: \(\lambda_3 = 4\).
Given: A 2×2 matrix \(A\) has trace 5 and determinant 6. Express \(A^2\) in terms of \(A\) and \(I\) using the Cayley–Hamilton theorem.
Solution: By Cayley–Hamilton for a 2×2 matrix, \(A^2 = (\text{trace})A - (\det)I = 5A - 6I\).
Answer: \(A^2 = 5A - 6I\).
Fig. 1.1 — The three consistency cases of the Rouché–Capelli theorem: inconsistent, unique, and infinite solutions.
L'Hôpital's rule: if \(\lim f/g \to 0/0\) or \(\infty/\infty\), then \(\lim f/g = \lim f'/g'\). Continuity at \(x=a\): \(\lim_{x\to a^-} f = \lim_{x\to a^+} f = f(a)\). Differentiability \(\Longrightarrow\) Continuity (the converse is not true).
Chain rule: \(\dfrac{d}{dx}[f(g(x))] = f'(g(x))\cdot g'(x)\). Product rule: \((uv)' = u'v + uv'\). Quotient rule: \(\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}\). Leibniz rule (nth derivative of product): \((uv)^{(n)} = \Sigma C(n,k)\,u^{(k)} v^{(n-k)}\).
Integration by parts: \(\int u\, dv = uv - \int v\, du\) (ILATE order: Inverse, Log, Algebraic, Trig, Exp).
Total differential: \(df = (\partial \dfrac{f}{\partial} x)dx + (\partial \dfrac{f}{\partial} y)dy\). Euler's theorem for homogeneous functions of degree \(n\): \(x(\partial \dfrac{f}{\partial} x) + y(\partial \dfrac{f}{\partial} y) = nf\).
Maxima/minima test (two variables): \(D = f_{xx}\cdot f_{yy} - f_{xy}^2\). \(D > 0, f_{xx} > 0\) → minimum; \(D > 0, f_{xx} < 0\) → maximum; \(D < 0\) → saddle point.
Jacobian: \(J = |\dfrac{\partial(u, v)}{\partial(x, y)}|\) (used in change of variables for double integrals).
For \(0/0\) or \(\infty/\infty\): \(\lim f/g = \lim f'/g'\)
\(D = f_{xx}f_{yy} - f_{xy}^2\); \(D>0,f_{xx}>0\)→min; \(D>0,f_{xx}<0\)→max; \(D<0\)→saddle
\(\Gamma(n) = (n-1)!\) for integer \(n\); \(\Gamma(\dfrac{1}{2}) = \sqrt\pi\); \(\Gamma(n+1) = n\Gamma(n)\)
\(x\,\partial \dfrac{f}{\partial} x + y\,\partial \dfrac{f}{\partial} y = nf\) for degree-\(n\) homogeneous \(f\)
Given: At a critical point of \(f(x,y)\), \(f_{xx} = 4\), \(f_{yy} = 9\), and \(f_{xy} = 2\). Classify the point.
Solution: \(D = f_{xx}f_{yy} - f_{xy}^2 = (4)(9) - (2)^2 = 36 - 4 = 32 > 0\). Since \(D > 0\) and \(f_{xx} = 4 > 0\), this is a local minimum.
Answer: Local minimum.
Given: Evaluate \(\Gamma(5)\) using the factorial relation for integers.
Solution: \(\Gamma(n) = (n-1)!\) for integer \(n\), so \(\Gamma(5) = 4! = 24\).
Answer: \(\Gamma(5) = 24\).
Given: For \(f(x,y) = x^2y + xy^2\) (a homogeneous function of degree 3), verify Euler's theorem.
Solution: \(\partial \dfrac{f}{\partial} x = 2xy + y^2\), \(\partial \dfrac{f}{\partial} y = x^2 + 2xy\). Then \(x(2xy+y^2) + y(x^2+2xy) = 2x^2y + xy^2 + x^2y + 2xy^2 = 3x^2y + 3xy^2 = 3(x^2y+xy^2) = 3f\), confirming \(n=3\).
Answer: Verified — the result equals \(3f\), matching the degree \(n=3\).
Fig. 2.1 — The second-derivative (D) test: minimum, maximum, and saddle point classifications.
Separable: \(f(y)\,dy = g(x)\,dx\) → integrate both sides. Linear: \(\dfrac{dy}{dx} + P(x)y = Q(x)\) → integrating factor \(\mu = e^{\int P\, dx}\) → solution: \(y\mu = \int Q\mu\, dx\).
Exact: \(M\,dx + N\,dy = 0\) is exact if \(\partial \dfrac{M}{\partial} y = \partial \dfrac{N}{\partial} x\) → solution \(F(x, y) = c\) where \(\partial \dfrac{F}{\partial} x = M, \partial \dfrac{F}{\partial} y = N\). Bernoulli: \(\dfrac{dy}{dx} + P(x)y = Q(x)y^n\) → substitute \(v = y^{1-n}\) → reduces to linear.
\(ay'' + by' + cy = f(x)\); auxiliary equation: \(am^2 + bm + c = 0\).
| \(f(t)\) | \(F(s) = \mathcal{L}\{f(t)\}\) |
|---|---|
| \(1\) | \(\dfrac{1}{s}\) |
| \(t\) | \(\dfrac{1}{s^2}\) |
| \(t^n\) | \(n!/s^{n+1}\) |
| \(e^{at}\) | \(\dfrac{1}{s-a}\) |
| \(\sin(at)\) | \(\dfrac{a}{s^2+a^2}\) |
| \(\cos(at)\) | \(s/(s^2+a^2)\) |
| \(e^{at}f(t)\) | \(F(s-a)\) [First shifting theorem] |
| \(f(t-a)u(t-a)\) | \(e^{-as}F(s)\) [Second shifting theorem] |
| \(f'(t)\) | \(sF(s) - f(0)\) |
| \(f''(t)\) | \(s^2 F(s) - sf(0) - f'(0)\) |
\(\mu = e^{\int P\, dx}\); solution \(y\mu = \int Q\mu\, dx\)
Real distinct: \(C_1e^{m_1x}+C_2e^{m_2x}\). Repeated: \((C_1+C_2x)e^{mx}\). Complex: \(e^{\alpha x}[C_1\cos\beta x+C_2\sin\beta x]\)
\(e^{at}\to \dfrac{1}{s-a}\); \(\sin(at)\to \dfrac{a}{s^2+a^2}\); \(\cos(at)\to s/(s^2+a^2)\)
\(\mathcal{L}\{f'(t)\} = sF(s)-f(0)\); \(\mathcal{L}\{f''(t)\} = s^2F(s)-sf(0)-f'(0)\)
Given: Solve the auxiliary equation for \(y'' - 5y' + 6y = 0\) and write the complementary solution.
Solution: Auxiliary equation: \(m^2 - 5m + 6 = 0 \Rightarrow (m-2)(m-3) = 0 \Rightarrow m = 2, 3\) (real, distinct roots).
Answer: \(y_c = C_1e^{2x} + C_2e^{3x}\).
Given: Find the Laplace transform of \(f(t) = e^{3t}\sin(2t)\).
Solution: Using the first shifting theorem, \(\mathcal{L}\{e^{at}f(t)\} = F(s-a)\). Since \(\mathcal{L}\{\sin(2t)\} = \dfrac{2}{s^2+4}\), substituting \(s \to s-3\) gives \(\mathcal{L}\{e^{3t}\sin(2t)\} = \dfrac{2}{(s-3)^2+4}\).
Answer: \(\dfrac{2}{(s-3)^2+4}\).
Given: Find the integrating factor for \(\dfrac{dy}{dx} + 2y = e^{-2x}\).
Solution: Here \(P(x) = 2\), so \(\mu = e^{\int 2\, dx} = e^{2x}\).
Answer: \(\mu = e^{2x}\).
Fig. 3.1 — Complementary solution shapes for second-order ODEs: real distinct roots (sum of exponentials), real repeated roots, and complex conjugate roots (oscillating envelope).
\(f(z) = u(x, y) + iv(x, y)\); \(z = x + iy\).
Cauchy's theorem: if \(f(z)\) is analytic inside and on closed contour \(C\), then \(\oint_C f(z)\, dz = 0\).
Cauchy's integral formula: \(f(a) = \dfrac{1}{2\pi i} \oint_C \dfrac{f(z)}{z-a}\, dz\) (\(a\) inside \(C\)); \(f^{(n)}(a) = \dfrac{n!}{2\pi i} \oint_C \dfrac{f(z)}{(z-a)^{n+1}}\, dz\).
Pole of order \(m\) at \(z = a\): \((z-a)^m f(z)\) remains finite and non-zero as \(z\to a\).
\(\partial \dfrac{u}{\partial} x = \partial \dfrac{v}{\partial} y\); \(\partial \dfrac{u}{\partial} y = -\partial \dfrac{v}{\partial} x\)
If \(f(z)\) analytic in/on \(C\): \(\oint_C f(z)\,dz = 0\)
\(\operatorname{Res}[f,a] = \lim_{z\to a}(z-a)f(z)\)
\(\oint_C f(z)\,dz = 2\pi i \times \Sigma(\text{residues inside }C)\)
Given: For \(f(z) = z^2 = (x+iy)^2\), verify the Cauchy–Riemann equations.
Solution: \(u = x^2-y^2\), \(v = 2xy\). Then \(\partial \dfrac{u}{\partial} x = 2x\), \(\partial \dfrac{v}{\partial} y = 2x\) ✓ (equal); \(\partial \dfrac{u}{\partial} y = -2y\), \(-\partial \dfrac{v}{\partial} x = -2y\) ✓ (equal). Both C–R equations hold.
Answer: C–R equations are satisfied, so \(f(z)=z^2\) is analytic everywhere.
Given: Find the residue of \(f(z) = \dfrac{1}{z-3}\) at \(z=3\).
Solution: This is a simple pole at \(z=3\). \(\operatorname{Res}[f,3] = \lim_{z\to 3}(z-3)\cdot\dfrac{1}{z-3} = 1\).
Answer: Residue \(= 1\).
Given: Evaluate \(\oint_C \dfrac{1}{z-3}\, dz\) where \(C\) is a contour enclosing \(z=3\).
Solution: By the residue theorem, \(\oint_C f(z)\,dz = 2\pi i \times \Sigma(\text{residues})\). With residue \(=1\) at the enclosed pole, the integral equals \(2\pi i \times 1 = 2\pi i\).
Answer: \(2\pi i\).
Fig. 4.1 — A closed contour C enclosing a single pole at z = a; the contour integral equals 2πi times the residue at that pole.
Total probability theorem: if \(B_1, B_2, \ldots, B_n\) partition the sample space, \(P(A) = \sum_i P(A|B_i)P(B_i)\). Two events are independent if \(P(A \cap B) = P(A)P(B)\); mutually exclusive if \(P(A \cap B) = 0\).
| Distribution | PMF / PDF | Mean | Variance |
|---|---|---|---|
| Binomial \(B(n,p)\) | \(P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}\) | \(np\) | \(np(1-p)\) |
| Poisson \(P(\lambda)\) | \(P(X=k) = \dfrac{e^{-\lambda}\lambda^k}{k!}\) | \(\lambda\) | \(\lambda\) |
| Normal \(N(\mu,\sigma^2)\) | \(f(x) = \dfrac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/2\sigma^2}\) | \(\mu\) | \(\sigma^2\) |
| Exponential \(Exp(\lambda)\) | \(f(x) = \lambda e^{-\lambda x},\ x\ge 0\) | \(\dfrac{1}{\lambda}\) | \(\dfrac{1}{\lambda^2}\) |
| Uniform \(U(a,b)\) | \(f(x) = \dfrac{1}{b-a},\ a\le x\le b\) | \(\dfrac{a+b}{2}\) | \(\dfrac{(b-a)^2}{12}\) |
Sample mean: \(\bar{x} = \dfrac{1}{n}\sum_i x_i\). Sample variance: \(s^2 = \dfrac{1}{n-1}\sum_i (x_i-\bar{x})^2\). Standard error: \(SE = \dfrac{\sigma}{\sqrt{n}}\).
Hypothesis testing: null hypothesis \(H_0\) vs. alternative \(H_1\). Type I error (\(\alpha\)) = rejecting \(H_0\) when true; Type II error (\(\beta\)) = failing to reject \(H_0\) when false; Power \(= 1-\beta\).
Z-test (large \(n\), known \(\sigma\)): \(Z = \dfrac{\bar{x}-\mu_0}{\sigma/\sqrt{n}}\). t-test (small \(n\), unknown \(\sigma\)): \(t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}\), with \(n-1\) degrees of freedom.
\(P(A_i|B) = \dfrac{P(B|A_i)P(A_i)}{\sum_j P(B|A_j)P(A_j)}\)
\(P(X=k) = \dfrac{e^{-\lambda}\lambda^k}{k!}\); mean = variance = \(\lambda\)
\(Z = \dfrac{\bar{x}-\mu_0}{\sigma/\sqrt{n}}\)
\(t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}\), df \(= n-1\)
Given: Machine A produces 60% of items with 2% defective; Machine B produces 40% with 5% defective. An item is found defective. Find \(P(\text{from A}|\text{defective})\).
Solution: \(P(D) = 0.6(0.02) + 0.4(0.05) = 0.012+0.020 = 0.032\). \(P(A|D) = \dfrac{0.6 \times 0.02}{0.032} = \dfrac{0.012}{0.032} = 0.375\).
Answer: \(P(A|D) = 0.375\) (37.5%).
Given: A call center receives calls with mean rate \(\lambda = 3\) per minute. Find \(P(X=2)\).
Solution: \(P(X=2) = \dfrac{e^{-3}3^2}{2!} = \dfrac{e^{-3}\times 9}{2} \approx \dfrac{0.0498\times9}{2} \approx 0.224\).
Answer: \(P(X=2) \approx 0.224\).
Given: Population mean \(\mu_0=50\), \(\sigma=10\), sample of \(n=25\) has \(\bar{x}=54\). Test \(H_0: \mu=50\).
Solution: \(Z = \dfrac{54-50}{10/\sqrt{25}} = \dfrac{4}{2} = 2\). Since \(|Z|=2 > 1.96\) (5% two-tailed critical value), reject \(H_0\).
Answer: \(Z=2\); reject \(H_0\) at 5% significance.
Fig. 5.1 — Probability tree for the Bayes' theorem worked example: branching by machine, then by defect status.
Bisection method: repeatedly halves an interval \([a,b]\) where \(f(a)\) and \(f(b)\) have opposite signs. Guaranteed to converge, but linear convergence (slow). Secant method: uses a linear approximation through two points; super-linear convergence, no derivative needed.
Trapezoidal rule: \(\int_a^b f(x)\,dx \approx \dfrac{h}{2}\big[f(x_0)+2\sum f(x_i)+f(x_n)\big]\); error \(O(h^2)\). Works with any number of intervals.
Euler's method: \(y_{n+1} = y_n + h\,f(x_n,y_n)\); error \(O(h)\) — simple but low accuracy.
\(x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}\); \(O(h^2)\) quadratic
\(\dfrac{h}{3}[f_0+4\Sigma f_{odd}+2\Sigma f_{even}+f_n]\); \(n\) even
\(\dfrac{h}{2}[f_0+2\Sigma f_i+f_n]\); error \(O(h^2)\)
\(y_{n+1}=y_n+\dfrac{h}{6}(k_1+2k_2+2k_3+k_4)\)
Given: Find a root of \(f(x)=x^2-2\) starting from \(x_0=1.5\) (one iteration).
Solution: \(f'(x)=2x\). \(x_1 = 1.5 - \dfrac{1.5^2-2}{2(1.5)} = 1.5 - \dfrac{0.25}{3} = 1.5-0.0833 = 1.4167\).
Answer: \(x_1 \approx 1.4167\) (converging to \(\sqrt2 \approx 1.4142\)).
Given: Estimate \(\int_0^2 x^2\,dx\) using Simpson's 1/3 rule with \(n=2\) (\(h=1\)).
Solution: \(x_0=0,x_1=1,x_2=2\); \(f_0=0,f_1=1,f_2=4\). \(\int \approx \dfrac{1}{3}[0+4(1)+4] = \dfrac{8}{3} \approx 2.667\).
Answer: \(\approx 2.667\) (exact value \(=\dfrac{8}{3}\) — exact here since Simpson's rule is exact for cubics and below).
Given: Solve \(\dfrac{dy}{dx} = x+y\), \(y(0)=1\), find \(y(0.1)\) using Euler's method with \(h=0.1\).
Solution: \(y_1 = y_0 + h\,f(x_0,y_0) = 1 + 0.1(0+1) = 1.1\).
Answer: \(y(0.1) \approx 1.1\).
Fig. 6.1 — Newton-Raphson geometry: the tangent line at (xₙ, f(xₙ)) crosses the x-axis at the next estimate xₙ₊₁, closer to the true root.
\(\Sigma\lambda_i = \operatorname{trace}(A)\); \(\Pi\lambda_i = \det(A)\)
\(\rho(A) = \rho([A|b])\) — Rouché–Capelli
Every matrix satisfies its own characteristic equation
\(A^{-1} = A^T\)
\(0/0\) or \(\infty/\infty\) form: differentiate numerator & denominator
\(\Gamma(n) = (n-1)!\) for positive integers
\(D = f_{xx}f_{yy}-f_{xy}^2\): \(D>0,f_{xx}>0\) min; \(D>0,f_{xx}<0\) max; \(D<0\) saddle
\(\mathcal{L}\{e^{at}\} = \dfrac{1}{s-a}\)
\(\mathcal{L}\{\sin at\} = \dfrac{a}{s^2+a^2}\)
\(y = (c_1+c_2x)e^{mx}\)
\(u_x = v_y\); \(u_y = -v_x\)
\(\oint_C f(z)\,dz = 2\pi i \Sigma(\text{residues})\)
Mean = Variance = \(\lambda\)
\(P(A_i|B) = \dfrac{P(B|A_i)P(A_i)}{\Sigma_j P(B|A_j)P(A_j)}\)
Z-score: \(Z = \dfrac{x-\mu}{\sigma}\)
\(x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}\); quadratic convergence
Requires even number of intervals; error \(O(h^4)\)
\(\dfrac{k_1+2k_2+2k_3+k_4}{6}\) — the 1:2:2:1 ratio
| Topic | Paper I Focus |
|---|---|
| Linear Algebra | Rank/consistency numericals (Rouché–Capelli); eigenvalue property shortcuts; Cayley-Hamilton applications |
| Calculus | L'Hôpital's rule limits; maxima/minima via D-test; Gamma/Beta function evaluations |
| Differential Equations | Auxiliary-equation root-case identification; Laplace transform pairs and shifting theorems |
| Complex Variables | Cauchy-Riemann verification; residue and contour-integral numericals |
| Probability & Statistics | Bayes' theorem conditional-probability problems; distribution mean/variance recall; Z-test/t-test selection |
| Numerical Methods | Newton-Raphson and Simpson's 1/3 rule numericals; RK4 weighted-average recall |
Q1. A 3×3 matrix has eigenvalues 2, 3, and 5. What are its trace and determinant?
Q2. Using the D-test, if \(D = 25\) and \(f_{xx} = -4\) at a critical point, what type of point is it?
Q3. Find \(\mathcal{L}\{e^{2t}\}\) using the standard Laplace transform table.
Q4. A Poisson process has a sample mean of 4.2 defects per batch. What is your estimate of the variance?
Q5. Using Simpson's 1/3 rule, can you directly use \(n=5\) intervals?