GATE Engineering Mathematics Formula Sheet for ECE, EE, ME and CE

Engineering Mathematics is the one section the ECE, EE, ME and Civil papers share, and it mostly tests whether you know a few dozen results exactly. This sheet gives them group by group, each with a formula table, a short worked example and the trap that costs marks. If you write the CSE paper, the CSE engineering mathematics formula sheet covers the discrete mathematics that paper adds.

Key takeaways

• Eigenvalues add up to the trace and multiply to the determinant, and those two checks settle most matrix questions.
• A zero determinant means "no unique solution", which still splits into no solution or infinitely many; compare ranks to tell which.
• L'Hôpital's rule works only on a $\displaystyle \frac{0}{0}$ or $\displaystyle \frac{\infty}{\infty}$ form, so check the form before you differentiate.
• When the right-hand side $e^{ax}$ has $a$ as a root of the auxiliary equation, the particular integral picks up a factor of $x$.
• In a contour integral, only the poles strictly inside the curve contribute residues.
• Simpson's 1/3 rule needs an even number of sub-intervals, and its error falls as $h^{4}$ against $h^{2}$ for the trapezoidal rule.

The words this sheet uses

• The rank of a matrix is its number of independent rows, which equals its number of independent columns.
• An eigenvalue $\lambda$ of a square matrix $A$ satisfies $A\mathbf{x} = \lambda\mathbf{x}$ for some non-zero vector $\mathbf{x}$, its eigenvector. The trace is the sum of the diagonal entries.
• A system $A\mathbf{x} = \mathbf{b}$ is consistent if it has at least one solution.
• A complex function is analytic at a point if it is differentiable throughout a small disc around it. A pole is a point where it blows up like a power of $\displaystyle \frac{1}{z - a}$. The residue is the coefficient of $\displaystyle \frac{1}{z - a}$ in its expansion there.
• A particular integral is any one solution of a non-homogeneous differential equation. The complementary function solves the same equation with the right-hand side set to zero.
• The order of error of a numerical rule says how the error shrinks with the step $h$: order $h^{2}$ falls to a quarter when $h$ halves.

How many marks Engineering Mathematics carries in each paper

Each branch book counted every question of its papers against the official answer keys, placing a borderline question where it leans. The totals are:

So 13 marks is the settled figure for ME and CE, EE has run higher, and ECE has swung either side. These are past counts, not a forecast; check the current notification. For the branch subjects themselves, use the ECE, EE, ME and Civil formula sheets.

Remember: The four syllabi differ in detail. EE files vector calculus under Calculus, ECE lists Vector Analysis on its own, and only ME and CE name numerical methods. Read Section 1 of your own paper's syllabus before you revise.

Linear algebra: determinants, rank and eigenvalues

Most matrix questions need two eigenvalue checks and one rank comparison. In the table, $A$ is $n \times n$.

Worked example. Take $A = \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix}$. Its trace is 6 and its determinant is 8, so $\lambda^{2} - 6\lambda + 8 = 0$ and the eigenvalues are 2 and 4. By Cayley–Hamilton, $A^{2} - 6A + 8I = 0$. Multiply through by $A^{-1}$:

Trap: A zero determinant only tells you the solution is not unique. The system may then have no solution or infinitely many, and a question with a parameter usually wants one of the two. Compare $\operatorname{rank} A$ with the rank of the augmented matrix to decide.

Calculus: limits, series, extrema and integrals

Know these results by heart and the working stays short.

A Taylor series often settles a limit faster than repeated L'Hôpital. Since $\displaystyle x - \sin x = \frac{x^{3}}{6} - \cdots$, the ratio $\displaystyle \frac{x - \sin x}{x^{3}}$ tends to $\displaystyle \frac{1}{6}$. The Leibniz rule is just as quick: $\displaystyle \frac{d}{dx}\int_0^{x^{2}} \sin t\,dt = 2x\sin(x^{2})$.

Worked example. Find the extrema of $f = x^{3} + y^{3} - 3xy$. Setting $f_x = 3x^{2} - 3y = 0$ and $f_y = 3y^{2} - 3x = 0$ gives the real points $(0, 0)$ and $(1, 1)$. Then:

Trap. Check the form before using L'Hôpital. As $x \to 0$, $\displaystyle \frac{1 + \cos x}{1 + x}$ is simply $\displaystyle \frac{2}{1} = 2$. Differentiating top and bottom gives $\displaystyle \frac{-\sin x}{1} \to 0$, which is wrong, because the limit was never $\displaystyle \frac{0}{0}$.

Vector calculus: gradient, divergence, curl and the three theorems

The three integral theorems trade a hard integral for an easier one, given the right orientation and a closed boundary.

Worked example. Find the outward flux of $\mathbf{F} = (x^{2}, y^{2}, z^{2})$ through the surface of the unit cube $0 \le x, y, z \le 1$. Instead of six face integrals, use Gauss. Here $\nabla \cdot \mathbf{F} = 2x + 2y + 2z$, and each of $x$, $y$, $z$ averages $\displaystyle \frac{1}{2}$ over the cube:

Trap. Make the direction a unit vector first. For $\phi = x^{2}y + z$ at $(1, 2, 0)$, $\nabla\phi = (4, 1, 1)$. Along $(1, 2, 2)$ the dot product is 8, but the unit vector is $\displaystyle \frac{1}{3}(1, 2, 2)$, so the directional derivative is $\displaystyle \frac{8}{3}$.

Differential equations: the solution forms to know

Recognise the standard form, and the solution follows a fixed recipe.

Worked example. Solve $y'' - 3y' + 2y = e^{x}$. Here $f(D) = D^{2} - 3D + 2$, with roots 1 and 2. Since $f(1) = 0$, the plain rule fails and you use the derivative form:

Trap. A repeated root needs its extra factor. For the Cauchy–Euler equation $x^{2}y'' - 3xy' + 4y = 0$, the auxiliary equation $m^{2} - 4m + 4 = 0$ gives $m = 2$ twice. The solution is $(C_1 + C_2 \ln x)\,x^{2}$: the factor is $\ln x$, not $x$.

Each branch book's last-minute sheet carries the full Engineering Mathematics table for its own syllabus, including the Laplace transforms, PDE classification and Fourier series this sheet leaves out. It comes with the GATE 2027 books.

Complex variables: Cauchy–Riemann, Cauchy's formula and residues

ECE and EE write $j$ for $\sqrt{-1}$; the results are the same with $i$. Contours run anticlockwise unless the question says otherwise.

Worked example. Evaluate the integral of $\displaystyle f(z) = \frac{z^{2} + 1}{z(z - 3)}$ round $\vert z \vert = 2$. The poles are $z = 0$, inside the circle, and $z = 3$, outside it:

Trap: The pole at $z = 3$ has residue $\displaystyle \frac{10}{3}$, and adding it gives $6\pi i$, the wrong answer. A pole counts only when it lies strictly inside the contour, and a clockwise contour flips the sign.

Probability and statistics: distributions and Bayes

Most probability questions name a distribution and ask for a mean, a variance or one probability.

Two rules sit beside the table: $\text{Var}(X) = E[X^{2}] - (E[X])^{2}$ and $\text{Var}(aX + b) = a^{2}\,\text{Var}(X)$. Bayes' theorem reverses a conditional probability, with the total probability of the evidence below the line:

Worked example. Supplier $S_1$ makes 70% of the parts, 1% of them defective. Supplier $S_2$ makes 30%, 4% defective. For a defective part:

Trap. Rescale a Poisson rate to the interval asked. Four arrivals an hour means $\lambda = 2$ over 30 minutes, so $P(X \ge 1) = 1 - e^{-2} \approx 0.865$. Using $\lambda = 4$ gives the wrong answer.

Numerical methods: Newton–Raphson, trapezoidal and Simpson's rules

The ME and CE syllabi name numerical methods; the 2027 EC and EE syllabi do not. Questions test each rule's order as often as its value.

Worked example. Find $\sqrt{3}$ with Newton–Raphson on $f(x) = x^{2} - 3$ from $x_0 = 2$, then integrate $x^{3}$ from 0 to 2 with $h = 1$, whose ordinates are 0, 1 and 8:

Two Newton steps match $\sqrt{3} \approx 1.732051$ to four decimal places, and Simpson's 4 is exact because the rule is exact for cubics.

In one line: Halving $h$ cuts the trapezoidal error to about a quarter and Simpson's error to about a sixteenth, and Simpson's 1/3 rule works only with an even number of sub-intervals.

Quick revision

1. Eigenvalues: sum equals the trace, product equals the determinant; $\det(kA) = k^{n}\det A$.
2. Unique solution only when $\operatorname{rank} A = \operatorname{rank}[A \mid \mathbf{b}] = n$; a zero determinant means none or infinitely many.
3. L'Hôpital only on $\displaystyle \frac{0}{0}$ or $\displaystyle \frac{\infty}{\infty}$; for two variables, the sign of $rt - s^{2}$ decides.
4. Leibniz: $\displaystyle \frac{d}{dx}\int_{a(x)}^{b(x)} f(t)\,dt = f(b)\,b' - f(a)\,a'$.
5. Gauss turns a closed-surface flux into $\iiint \nabla \cdot \mathbf{F}\,dV$; a directional derivative needs a unit vector.
6. The particular integral for $e^{ax}$ picks up a factor $x$ when $f(a) = 0$; a repeated Cauchy–Euler root brings $\ln x$.
7. Only poles strictly inside the contour count; at a simple pole the residue is $\displaystyle \frac{p(a)}{q'(a)}$.
8. Binomial variance $np(1 - p)$, Poisson mean and variance $\lambda$, exponential $\displaystyle \frac{1}{\lambda}$ and $\displaystyle \frac{1}{\lambda^{2}}$; trapezoidal error $h^{2}$, Simpson's $h^{4}$.