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GATE EE Engineering Mathematics Previous Year Questions

60 solved GATE EE questions on Engineering Mathematics, drawn from 6 exam years and grouped by year. Every question shows the official answer and a step-by-step solution.

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Engineering Mathematics: verify conditions before applying a formula

State the domain, convergence condition, or regularity requirement a theorem needs before substituting. These selected foundations connect linear algebra, calculus, differential equations, complex analysis, and probability as they appear in the GATE EE syllabus. Each topic requires its own deeper study.

Our study notes and original examples support the PYQs below; they are not official exam questions or a replacement for the current syllabus.

Before you start

  • Limits, derivatives, and definite integrals of single-variable functions.
  • Matrix operations, determinants, and basic set and function notation.

Concepts to revise before solving

Eigenvalues and matrix properties

Find eigenvalues by solving det(A − λI) = 0. For an n×n matrix, eigenvalue sum equals the trace and eigenvalue product equals the determinant. A real symmetric matrix has all real eigenvalues. Diagonalisation requires n linearly independent eigenvectors.

Check yourself: Does eigenvalue sum match the trace and eigenvalue product match the determinant?

Multivariable calculus and vector calculus

Gradient gives the direction of steepest ascent. Divergence of a vector field measures net outward flux per unit volume; Stokes' theorem relates a surface integral of curl to a line integral around its boundary. Confirm orientation and surface normals before applying integral theorems.

Check yourself: Are the surface normal and boundary traversal consistently oriented?

Ordinary differential equations

For the linear first-order ODE dy/dx + P(x)y = Q(x), the integrating factor is e^(∫P dx). Second-order constant-coefficient ODEs reduce to a characteristic equation; the form of the particular solution depends on whether the forcing term duplicates a homogeneous solution.

Check yourself: Does the candidate solution satisfy both the equation and the initial conditions?

Complex analysis

A function is analytic where the Cauchy–Riemann equations hold and partial derivatives are continuous. Residues at isolated singularities determine contour integrals via the residue theorem: the integral equals 2πj times the sum of enclosed residues. Identify singularity type before computing residues.

Check yourself: Is the singularity a simple pole, a higher-order pole, or an essential singularity?

Probability and random variables

For equally likely outcomes, probability is favourable over total. Use conditional probability P(A|B) = P(A ∩ B)/P(B). Expected value is linear: E(X + Y) = E(X) + E(Y) regardless of dependence. Variance of independent variables adds: Var(X + Y) = Var(X) + Var(Y).

Check yourself: Are the events independent, or do you need a joint distribution?

Mistakes to avoid

Applying a diagonalisation formula without checking for n independent eigenvectors.
A repeated eigenvalue may or may not supply enough eigenvectors. Verify the eigenspace dimension.
Confusing the integrating factor with the complementary solution.
The integrating factor e^(∫P dx) multiplies the entire ODE to make the left side an exact derivative; it is not itself a solution.
Summing probabilities of non-mutually-exclusive events without subtracting the overlap.
Use P(A ∪ B) = P(A) + P(B) − P(A ∩ B); direct addition overcounts the intersection.

Original teaching example · not a PYQ

Work through the reasoning

Original mini-example: find the eigenvalues of the matrix A = [[4, 2], [1, 3]].

  1. Form A − λI = [[4−λ, 2], [1, 3−λ]]. The characteristic polynomial is (4−λ)(3−λ) − 2 = λ² − 7λ + 10.
  2. Factor: λ² − 7λ + 10 = (λ − 5)(λ − 2) = 0. Alternatively, discriminant = 49 − 40 = 9, giving λ = (7 ± 3)/2.
  3. Eigenvalues are λ₁ = 5 and λ₂ = 2.
  4. Cross-check: sum 5 + 2 = 7 equals trace 4 + 3 = 7; product 5 × 2 = 10 equals determinant 4 × 3 − 2 × 1 = 10.

Eigenvalues are 5 and 2.

Try it before reading the answer

Solve the first-order linear ODE dy/dx + 2y = 6 with initial condition y(0) = 1.

Show answer and reasoning

y(x) = 3 − 2e^(−2x).

Integrating factor e^(2x) gives d(ye^(2x))/dx = 6e^(2x). Integrating: ye^(2x) = 3e^(2x) + C, so y = 3 + Ce^(−2x). Applying y(0) = 1 yields C = −2. Verification: y′ + 2y = 4e^(−2x) + 6 − 4e^(−2x) = 6.

Go deeper with free learning resources

Supplemental reading, not an official GATE reading list or an endorsement of these notes.

Apply this to the previous-year questions

Previous-year questions by year

This page shows 60 recent questions from the released archive, newest first. For older questions and complete papers, browse all GATE EE papers. Questions can carry more than one subject tag; counts are not marks weightage.

GATE EE 202612 questions

  1. Set 1 Q14Consider the following differential equation: t2d2ydt2+7tdydt+8ty=10sin⁡(t)t^2 \frac{d^2y}{dt^2} + 7t \frac{dy}{dt} + 8ty = 10 \sin(t) Which one of the following options is correct?MCQ · +1 marks · Easy
  2. Set 1 Q30Two n×nn \times n matrices AA and BB have a common eigenvalue 22, and the same corresponding nonzero eigenvector. Which of the following options is/are…MSQ · +1 marks · Medium
  3. Set 1 Q32Given that F⃗(x,y,z)=sin⁡(y)x^+cos⁡(x)y^+5z^\vec{F}(x, y, z) = \sin(y) \hat{x} + \cos(x) \hat{y} + 5 \hat{z}, the integral ∯SF⃗(x,y,z)⋅ds⃗\oiint_S \vec{F}(x, y, z) \cdot d\vec{s} over the unit sphere…NAT · +1 marks · Easy
  4. Set 1 Q35AA is an m×mm \times m skew-symmetric matrix with real-valued entries, and xx is an mm-dimensional column vector with real-valued entries such that…NAT · +1 marks · Easy
  5. Set 1 Q36A time-limited waveform g(x)g(x) is specified as follows:…MCQ · +2 marks · Medium
  6. Set 1 Q49Which one of the following statements is ALWAYS correct about a collection of pp column vectors, each having nn real-valued entries?MCQ · +2 marks · Easy
  7. Set 1 Q50Consider the second-order differential equation d2ydx2+dydx+y=0\frac{d^2y}{dx^2} + \frac{dy}{dx} + y = 0 with initial conditions…MCQ · +2 marks · Easy
  8. Set 1 Q55Consider an n×nn \times n orthogonal matrix AA with real entries and each column having unit Euclidean norm. Which of the following statements is/are correct?MSQ · +2 marks · Medium
  9. Set 1 Q56Consider the system of linear equations: Ax=bA\mathbf{x} = \mathbf{b}, where AA is an n×nn \times n matrix, and x\mathbf{x} and b\mathbf{b} are nn-dimensional…MSQ · +2 marks · Medium
  10. Set 1 Q62The magnitude of the contour integral ∮C((z+1)2(z−i)(z−2))dz\oint_C \left( \frac{(z+1)^2}{(z-i)(z-2)} \right) dz over the contour C:∣z−2−i∣=3/2C: |z - 2 - i| = 3/2 is ______ (Round off to…NAT · +2 marks · Medium
  11. Set 1 Q64Let XX and YY be two real-valued random variables with E(X)=1,E(Y)=2,E(X2)=4,E(Y2)=9E(X) = 1, E(Y) = 2, E(X^2) = 4, E(Y^2) = 9, and E(XY)=0.9E(XY) = 0.9, where EE denotes the expectation…NAT · +2 marks · Medium
  12. Set 1 Q65The integral 1π∫0∞x2026(1+x2026)(1+x2)dx\frac{1}{\pi} \int_{0}^{\infty} \frac{x^{2026}}{(1 + x^{2026})(1 + x^2)} dx evaluates to _________ (Round off to two decimal places)NAT · +2 marks · Hard

GATE EE 202511 questions

  1. Set 1 Q11Consider the set SS of points (x,y)∈R2(x, y) \in \mathbb{R}^2 which minimize the real valued function f(x,y)=(x+y−1)2+(x+y)2f(x, y) = (x + y - 1)^2 + (x + y)^2 Which of the following…MCQ · +1 marks · Medium
  2. Set 1 Q12Let v1\mathbf{v_1} and v2\mathbf{v_2} be the two eigenvectors corresponding to distinct eigenvalues of a 3×33 \times 3 real symmetric matrix. Which one of the…MCQ · +1 marks · Easy
  3. Set 1 Q13Let A=[111−1−1−101−1]A = \begin{bmatrix} 1 & 1 & 1 \\ -1 & -1 & -1 \\ 0 & 1 & -1 \end{bmatrix}, and b=[1/3−1/30]b = \begin{bmatrix} 1/3 \\ -1/3 \\ 0 \end{bmatrix}. Then, the system of…MCQ · +1 marks · Easy
  4. Set 1 Q14Let P=[210−100001]P = \begin{bmatrix} 2 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} and let II be the identity matrix. Then P2P^2 is equal toMCQ · +1 marks · Easy
  5. Set 1 Q15Consider discrete random variables XX and YY with probabilities as follows: P(X=0 and Y=0)=14P(X = 0 \text{ and } Y = 0) = \frac{1}{4}…MCQ · +1 marks · Easy
  6. Set 1 Q36Let XX and YY be continuous random variables with probability density functions PX(x)P_X(x) and PY(y)P_Y(y), respectively. Further, let Y=X2Y = X^2 and…MCQ · +2 marks · Medium
  7. Set 1 Q51Let aR\mathbf{a}_R be the unit radial vector in the spherical co-ordinate system. For which of the following value(s) of nn, the divergence of the radial…MSQ · +2 marks · Medium
  8. Set 1 Q53Let (x,y)∈ℜ2(x, y) \in \Re^2. The rate of change of the real valued function, V(x,y)=x2+x+y2+1V(x, y) = x^2 + x + y^2 + 1 at the origin in the direction of the point (1,2)(1, 2) is…NAT · +2 marks · Medium
  9. Set 1 Q54Consider ordinary differential equations given by x˙1(t)=2x2(t)\dot{x}_1(t) = 2x_2(t) x˙2(t)=r(t)\dot{x}_2(t) = r(t) with initial conditions x1(0)=1x_1(0) = 1 and x2(0)=0x_2(0) = 0. If…NAT · +2 marks · Easy
  10. Set 1 Q55Let CC be a clockwise oriented closed curve in the complex plane defined by ∣z∣=1|z| = 1. Further, let f(z)=jzf(z) = jz be a complex function, where j=−1j = \sqrt{-1}.…NAT · +2 marks · Easy
  11. Set 1 Q62Consider the state-space model x˙(t)=Ax(t)+Br(t),\dot{\mathbf{x}}(t) = A\mathbf{x}(t) + Br(t), y(t)=Cx(t)y(t) = C\mathbf{x}(t) where x(t),r(t),y(t)\mathbf{x}(t), r(t), y(t) are the state,…NAT · +2 marks · Easy

GATE EE 202410 questions

  1. Set 1 Q11Which one of the following matrices has an inverse?MCQ · +1 marks · Easy
  2. Set 1 Q29Let XX be a discrete random variable that is uniformly distributed over the set {−10,−9,…,0,…,9,10}\{-10, -9, \dots, 0, \dots, 9, 10\}. Which of the following random variables…MSQ · +1 marks · Medium
  3. Set 1 Q30Which of the following complex functions is/are analytic on the complex plane?MSQ · +1 marks · Easy
  4. Set 1 Q31Consider the complex function f(z)=cos⁡z+ez2f(z) = \cos z + e^{z^2}. The coefficient of z5z^5 in the Taylor series expansion of f(z)f(z) about the origin is ______ (rounded…NAT · +1 marks · Easy
  5. Set 1 Q32The sum of the eigenvalues of the matrix A=[1234]2A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}^2 is ______ (rounded off to the nearest integer).NAT · +1 marks · Medium
  6. Set 1 Q36Consider a vector uˉ=2x^+y^+2z^\bar{u} = 2\hat{x} + \hat{y} + 2\hat{z}, where x^,y^,z^\hat{x}, \hat{y}, \hat{z} represent unit vectors along the coordinate axes x,y,zx, y, z…MCQ · +2 marks · Medium
  7. Set 1 Q43Let f(t)f(t) be a real-valued function whose second derivative is positive for −∞<t<∞-\infty < t < \infty. Which of the following statements is/are always true?MSQ · +2 marks · Medium
  8. Set 1 Q44Consider the function f(t)=(max⁡(0,t))2f(t) = (\max(0,t))^2 for −∞<t<∞-\infty < t < \infty, where max⁡(a,b)\max(a,b) denotes the maximum of aa and bb. Which of the following…MSQ · +2 marks · Medium
  9. Set 1 Q45Which of the following differential equations is/are nonlinear?MSQ · +2 marks · Medium
  10. Set 1 Q52The given equation represents a magnetic field strength H⃗(r,θ,ϕ)\vec{H}(r, \theta, \phi) in the spherical coordinate system, in free space. Here, r^\hat{r} and…NAT · +2 marks · Hard

GATE EE 202311 questions

  1. Set 1 Q3Given a fair six-faced dice where the faces are labelled ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, and ‘6’, what is the probability of getting a ‘1’ on the first roll of the…MCQ · +1 marks · Easy
  2. Set 1 Q7Which one of the following options represents the given graph? [figure]MCQ · +2 marks · Easy
  3. Set 1 Q11For a given vector w=[1 2 3]T\mathbf{w} = [1 \ 2 \ 3]^T, the vector normal to the plane defined by wTx=1\mathbf{w}^T\mathbf{x} = 1 isMCQ · +1 marks · Easy
  4. Set 1 Q24In the figure, the vectors u\mathbf{u} and v\mathbf{v} are related as: Au=v\mathbf{Au} = \mathbf{v} by a transformation matrix A\mathbf{A}. The correct choice…MCQ · +1 marks · Medium
  5. Set 1 Q25One million random numbers are generated from a statistically stationary process with a Gaussian distribution with mean zero and standard deviation σo\sigma_o.…MCQ · +1 marks · Medium
  6. Set 1 Q34In the following differential equation, the numerically obtained value of y(t)y(t), at t=1t = 1, is _______________ (Round off to 2 decimal places).…NAT · +1 marks · Medium
  7. Set 1 Q35Three points in the xx-yy plane are (−1,0.8)(-1, 0.8), (0,2.2)(0, 2.2) and (1,2.8)(1, 2.8). The value of the slope of the best fit straight line in the least square sense is…NAT · +1 marks · Medium
  8. Set 1 Q42Consider the following equation in a 2-D real-space. ∣x1∣p+∣x2∣p=1|x_1|^p + |x_2|^p = 1 for p>0p > 0 Which of the following statement(s) is/are true.MSQ · +2 marks · Medium
  9. Set 1 Q45The expected number of trials for first occurrence of a “head” in a biased coin is known to be 4. The probability of first occurrence of a “head” in the second…NAT · +2 marks · Medium
  10. Set 1 Q62The closed curve shown in the figure is described by r=1+cos⁡θr = 1 + \cos\theta, where r=x2+y2r = \sqrt{x^2 + y^2}; x=rcos⁡θ,y=rsin⁡θx = r\cos\theta, y = r\sin\theta The magnitude of…NAT · +2 marks · Medium
  11. Set 1 Q65A quadratic function of two variables is given as f(x1,x2)=x12+2x22+3x1+3x2+x1x2+1f(x_1, x_2) = x_1^2 + 2x_2^2 + 3x_1 + 3x_2 + x_1x_2 + 1 The magnitude of the maximum rate of change of…NAT · +2 marks · Medium

GATE EE 20229 questions

  1. Set 1 Q20Consider a 3×33 \times 3 matrix AA whose (i,j)(i,j)-th element, ai,j=(i−j)3a_{i,j} = (i - j)^3. Then the matrix AA will beMCQ · +1 marks · Easy
  2. Set 1 Q40eAe^A denotes the exponential of a square matrix AA. Suppose λ\lambda is an eigenvalue and vv is the corresponding eigen-vector of matrix AA. Consider the…MCQ · +2 marks · Medium
  3. Set 1 Q41Let f(x)=∫0xet(t−1)(t−2)dtf(x) = \int_{0}^{x} e^t (t - 1)(t - 2) dt. Then f(x)f(x) decreases in the intervalMCQ · +2 marks · Easy
  4. Set 1 Q42Consider a matrix A=[10004−2011]A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & -2 \\ 0 & 1 & 1 \end{bmatrix}. The matrix AA satisfies the equation 6A−1=A2+cA+dI6A^{-1} = A^2 + cA + dI,…MCQ · +2 marks · Medium
  5. Set 1 Q45If the magnetic field intensity (H\mathbf{H}) in a conducting region is given by the expression,…MCQ · +2 marks · Medium
  6. Set 1 Q50Let, f(x,y,z)=4x2+7xy+3xz2f(x, y, z) = 4x^2 + 7xy + 3xz^2. The direction in which the function f(x,y,z)f(x, y, z) increases most rapidly at point P=(1,0,2)P = (1,0,2) isMCQ · +2 marks · Easy
  7. Set 1 Q51Let RR be a region in the first quadrant of the xyxy plane enclosed by a closed curve CC considered in counter-clockwise direction. Which of the following…MCQ · +2 marks · Medium
  8. Set 1 Q52Let E⃗(x,y,z)=2x2i^+5yj^+3zk^\vec{E}(x, y, z) = 2x^2 \hat{i} + 5y \hat{j} + 3z \hat{k}. The value of ∭V(∇⋅E⃗)dV\iiint_V (\nabla \cdot \vec{E}) dV, where VV is the volume enclosed by the…MCQ · +2 marks · Easy
  9. Set 1 Q54Let the probability density function of a random variable xx be given as f(x)=ae−2∣x∣f(x) = ae^{-2|x|}. The value of 'a' is __________.NAT · +2 marks · Easy

GATE EE 20217 questions

  1. Set 1 Q8Let XX be a continuous random variable denoting the temperature measured. The range of temperature is [0,100][0, 100] degree Celsius and let the probability…MCQ · +2 marks · Easy
  2. Set 1 Q11Let pp and qq be real numbers such that p2+q2=1p^2 + q^2 = 1. The eigenvalues of the matrix [pqq−p]\begin{bmatrix} p & q \\ q & -p \end{bmatrix} areMCQ · +1 marks · Easy
  3. Set 1 Q12Let p(z)=z3+(1+j)z2+(2+j)z+3p(z) = z^3 + (1 + j)z^2 + (2 + j)z + 3, where zz is a complex number. Which one of the following is true?MCQ · +1 marks · Medium
  4. Set 1 Q13Let f(x)f(x) be a real-valued function such that f′(x0)=0f'(x_0) = 0 for some x0∈(0,1)x_0 \in (0, 1), and f′′(x)>0f''(x) > 0 for all x∈(0,1)x \in (0, 1). Then f(x)f(x) hasMCQ · +1 marks · Easy
  5. Set 1 Q15Which one of the following vector functions represents a magnetic field B⃗\vec{B}? (x^\hat{x}, y^\hat{y}, and z^\hat{z} are unit vectors along x-axis, y-axis,…MCQ · +1 marks · Easy
  6. Set 1 Q23Suppose the circles x2+y2=1x^2 + y^2 = 1 and (x−1)2+(y−1)2=r2(x - 1)^2 + (y - 1)^2 = r^2 intersect each other orthogonally at the point (u,v)(u, v). Then u+v=u + v = ______.NAT · +1 marks · Medium
  7. Set 1 Q36In the open interval (0,1)(0, 1), the polynomial p(x)=x4−4x3+2p(x) = x^4 - 4x^3 + 2 hasMCQ · +2 marks · Medium

Other GATE EE topics

General Aptitude(181)Electric Circuits(139)Electrical Machines(129)Analog & Digital Electronics(111)Power Systems(108)Signals & Systems(107)Power Electronics(101)Control Systems(98)Electromagnetic Fields(56)Electrical & Electronic Measurements(36)

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