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GATE CE Engineering Mathematics Previous Year Questions

60 solved GATE CE questions on Engineering Mathematics, drawn from 4 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 hypotheses before applying a formula

State the domain, continuity, or convergence condition a formula requires before substituting values. These selected foundations connect linear algebra, calculus, differential equations, complex analysis, probability, and numerical methods as they appear in the GATE CE syllabus. Each topic needs 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

  • Single-variable derivatives, definite integrals, and basic limit evaluation.
  • Matrix operations, determinants, and systems of linear equations.

Concepts to revise before solving

Linear algebra: eigenvalues and rank

Eigenvalues satisfy det(A − λI) = 0. The trace equals the eigenvalue sum and the determinant equals the eigenvalue product for square matrices. Rank is the number of linearly independent rows (or columns); a system Ax = b is consistent when rank(A) equals rank([A|b]).

Check yourself: Have you checked whether the determinant is zero before inverting a matrix?

Calculus: multivariable and vector

A partial derivative holds other variables constant. The gradient points in the steepest-ascent direction. Green's theorem connects a line integral around a simple closed curve to a double integral over the enclosed region, assuming continuously differentiable components.

Check yourself: Are the partial derivatives continuous in the region where you apply the theorem?

Ordinary differential equations

A first-order linear ODE dy/dx + P(x)y = Q(x) has integrating factor e^(∫P dx). For second-order constant-coefficient ODEs, the characteristic equation determines the general solution form: distinct real roots, repeated roots, or complex conjugate roots each produce a different basis.

Check yourself: Does your particular solution satisfy the original equation, not just the homogeneous part?

Probability and statistics

For mutually exclusive events, P(A ∪ B) = P(A) + P(B). For independent events, P(A ∩ B) = P(A)P(B). The mean of a continuous distribution is ∫x f(x) dx over the support. Variance is E(X²) − [E(X)]², which is always non-negative.

Check yourself: Are the events mutually exclusive, independent, or neither?

Numerical methods

Newton-Raphson iteration x_{n+1} = x_n − f(x_n)/f'(x_n) requires f'(x_n) ≠ 0 and a starting point sufficiently close to the root. The trapezoidal rule approximates ∫f dx using linear segments; Simpson's 1/3 rule uses quadratic segments and requires an even number of subintervals.

Check yourself: Is the derivative non-zero at the current iterate, and is the interval count compatible with the rule?

Mistakes to avoid

Inverting a singular matrix without checking the determinant.
Compute the determinant first; if it is zero, the matrix has no inverse and the system may have no solution or infinitely many.
Applying the trapezoidal rule formula with Simpson's 1/3 rule coefficients.
Trapezoidal uses weights (1, 2, 2, …, 2, 1) × h/2; Simpson's 1/3 uses (1, 4, 2, 4, …, 4, 1) × h/3. Verify the formula before substituting.
Confusing independent events with mutually exclusive events.
Mutually exclusive events cannot occur together, so P(A ∩ B) = 0. Independent events satisfy P(A ∩ B) = P(A)P(B), which is generally nonzero.

Original teaching example · not a PYQ

Work through the reasoning

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

  1. The characteristic equation is det(A − λI) = (4 − λ)(3 − λ) − (1)(2) = λ² − 7λ + 10 = 0.
  2. Factoring: (λ − 5)(λ − 2) = 0, giving eigenvalues λ₁ = 5 and λ₂ = 2.
  3. Trace check: 4 + 3 = 7, and 5 + 2 = 7. ✓
  4. Determinant check: (4)(3) − (1)(2) = 10, and (5)(2) = 10. ✓

Eigenvalues are λ = 5 and λ = 2.

Try it before reading the answer

Apply one iteration of Newton-Raphson to f(x) = x² − 9 starting from x₀ = 4. What is x₁?

Show answer and reasoning

x₁ = 3.125.

f(4) = 16 − 9 = 7. f'(x) = 2x, so f'(4) = 8. Then x₁ = 4 − 7/8 = 4 − 0.875 = 3.125. The true root is 3, so the iteration moved closer.

Go deeper with free learning resources

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

  • MIT OpenCourseWare: Linear Algebra, Spring 2010

    Free supplemental lectures and problem sets by Gilbert Strang covering matrix operations, eigenvalues, and linear systems. Use the relevant lectures alongside your ODE and calculus references.

  • NPTEL: Engineering Mathematics

    Free video lectures covering ODEs, PDEs, complex analysis, and numerical methods aligned with the GATE engineering mathematics syllabus.

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 CE papers. Questions can carry more than one subject tag; counts are not marks weightage.

GATE CE 202620 questions

  1. Set 1 Q3At how many points will the curves y=x2y = x^2 and y=−x2−2x−1y = -x^2 - 2x - 1 intersect in the real (x,y)(x, y) plane?MCQ · +1 marks · Easy
  2. Set 1 Q11Matrix PP is given as P=[101010101]P = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix} The TRUE option isMCQ · +1 marks · Easy
  3. Set 1 Q12Given: [111102]{x1x2x3}={00}\begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{Bmatrix} x_1 \\ x_2 \\ x_3 \end{Bmatrix} = \begin{Bmatrix} 0 \\ 0 \end{Bmatrix} The…MCQ · +1 marks · Medium
  4. Set 1 Q26Bag I contains 4 white and 6 black balls. Bag II contains 4 white and 3 black balls. One ball is drawn at random from any one of the two bags and it is found…NAT · +1 marks · Medium
  5. Set 1 Q27A matrix is given as: [9151550]\begin{bmatrix} 9 & 15 \\ 15 & 50 \end{bmatrix} By performing Cholesky decomposition, ∣l22∣|l_{22}| of the lower triangular matrix is…NAT · +1 marks · Medium
  6. Set 1 Q36Let f(x)f(x) be a continuous function defined in [0,2]→R[0,2] \rightarrow \mathbb{R} and satisfying the equation ∫02f(x)[x−f(x)]dx=23.\int_{0}^{2} f(x)[x - f(x)]dx = \frac{2}{3}. The…MCQ · +2 marks · Hard
  7. Set 1 Q37An ordinary differential equation is given below. x2d2ydx2=6yx^2 \frac{d^2y}{dx^2} = 6y Considering aa and bb as arbitrary constants, the general solution of the…MCQ · +2 marks · Easy
  8. Set 1 Q48Starting with the first approximation as x=0.5x = 0.5, the second approximation for the root of the following function by the Newton-Raphson method is ______…NAT · +2 marks · Medium
  9. Set 1 Q49Values of yy for different values of xx are tabulated below. | xx | −2-2 | 11 | 22 | |---|---|---|---| | yy | 2828 | 44 | 1616 | If a second-degree…NAT · +2 marks · Medium
  10. Set 1 Q55A bridge with an expected life of 50 years is designed for a flood of 10000 m3/s10000\text{ m}^3/\text{s} corresponding to the return period of 100 years. The risk…NAT · +2 marks · Medium
  11. Set 2 Q11Matrix AA has the eigenvalues 1, 2, and 3. The Trace of A2A^2 isMCQ · +1 marks · Easy
  12. Set 2 Q12A fifth-degree polynomial in xx is defined for x>0x > 0. All coefficients of the polynomial are positive. The first derivative of the polynomial is obtained…MCQ · +1 marks · Medium
  13. Set 2 Q13Periodic function f(x)f(x) is given below.…MCQ · +1 marks · Easy
  14. Set 2 Q30A partial differential equation is given below. ∂2u∂x2−∂2u∂y2=0\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2} = 0 Possible solution(s) is/are:MSQ · +1 marks · Medium
  15. Set 2 Q31The eigenvalues of [A]=[2−3.563.552818.5][A] = \begin{bmatrix} 2 & -3.5 & 6 \\ 3.5 & 5 & 2 \\ 8 & 1 & 8.5 \end{bmatrix} are λ1=−1.547\lambda_1 = -1.547, λ2=12.330\lambda_2 = 12.330, and…NAT · +1 marks · Easy
  16. Set 2 Q32The probability (in %) that a storm having return period of 15 years may occur in the next 10 years is ______ (rounded off to two decimal places).NAT · +1 marks · Medium
  17. Set 2 Q36Let f(x)=∣x3sin⁡xcos⁡x6−10pp2p3∣f(x) = \begin{vmatrix} x^3 & \sin x & \cos x \\ 6 & -1 & 0 \\ p & p^2 & p^3 \end{vmatrix} where pp is a constant. The value of d3dx3f(x)\frac{d^3}{dx^3} f(x)…MCQ · +2 marks · Medium
  18. Set 2 Q37Vector field V⃗\vec{V} is defined as V⃗=3x2yzi^−5xyj^+6yz2k^\vec{V} = 3x^2yz \hat{i} - 5xy \hat{j} + 6yz^2 \hat{k} The curl of V⃗\vec{V} at point (2,−1,1)(2, -1, 1) isMCQ · +2 marks · Medium
  19. Set 2 Q46Consider differential equation dydx+xy=x\frac{dy}{dx} + xy = x with the condition as y=0y = 0 at x=0x = 0. The value of yy at x=1.0x = 1.0 is ______ (rounded off to two…NAT · +2 marks · Easy
  20. Set 2 Q47The age (in years) of a population is normally distributed with a mean of 36 and standard deviation of 12. The height (in cm) of the same population is also…NAT · +2 marks · Medium

GATE CE 202520 questions

  1. Set 1 Q3The sum of the following infinite series is: 11!+12!+13!+14!+15!+…\frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \dotsMCQ · +1 marks · Easy
  2. Set 1 Q11Suppose λ\lambda is an eigenvalue of matrix AA and xx is the corresponding eigenvector. Let xx also be an eigenvector of the matrix B=A−2IB = A - 2I, where II…MCQ · +1 marks · Easy
  3. Set 1 Q12Let A=[1113−2−3]A = \begin{bmatrix} 1 & 1 \\ 1 & 3 \\ -2 & -3 \end{bmatrix} and b=[b1b2b3]b = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}. For Ax=bAx = b to be solvable, which…MCQ · +1 marks · Medium
  4. Set 1 Q13Which one of the following options is the correct Fourier series of the periodic function f(x)f(x) described below:…MCQ · +1 marks · Medium
  5. Set 1 Q14XX is a random variable that can take any one of the values 0, 1, 7, 11, and 12. The probability mass function for XX is…MCQ · +1 marks · Medium
  6. Set 1 Q28Which of the following equations belong/belongs to the class of second-order, linear, homogeneous partial differential equations:MSQ · +1 marks · Medium
  7. Set 1 Q36The value of lim⁡x→∞(x−x2+x)\lim_{x \to \infty} (x - \sqrt{x^2 + x}) is equal toMCQ · +2 marks · Easy
  8. Set 1 Q47Let yy be the solution of the initial value problem y′′+0.8y′+0.16y=0y'' + 0.8y' + 0.16y = 0, where y(0)=3y(0) = 3 and y′(0)=4.5y'(0) = 4.5. Then, y(1)y(1) is equal to…NAT · +2 marks · Medium
  9. Set 1 Q48The maximum value of the function h(x)=−x3+2x2h(x) = -x^3 + 2x^2 in the interval [−1,1.5][-1, 1.5] is equal to _________________________ (rounded off to 1 decimal place).NAT · +2 marks · Medium
  10. Set 1 Q49Consider the differential equation given below. Using the Euler method with the step size (hh) of 0.5, the value of yy at x=1.0x = 1.0 is equal to ____________…NAT · +2 marks · Medium
  11. Set 1 Q63A one-way, single lane road has traffic that consists of 30% trucks and 70% cars. The speed of trucks (in km/h) is a uniform random variable on the interval…NAT · +2 marks · Medium
  12. Set 2 Q11For the matrix [A][A] given below, the transpose is ______. [A]=[234145432][A] = \begin{bmatrix} 2 & 3 & 4 \\ 1 & 4 & 5 \\ 4 & 3 & 2 \end{bmatrix}MCQ · +1 marks · Easy
  13. Set 2 Q12Integration of ln⁡(x)\ln(x) with xx i.e., ∫ln⁡(x)dx=______.\int \ln(x) dx = \_\_\_\_\_\_.MCQ · +1 marks · Easy
  14. Set 2 Q21Consider a velocity vector, V⃗\vec{V} in (x,y,z)(x, y, z) coordinates given below. Pick one or more CORRECT statements(s) from the choices given below.…MSQ · +1 marks · Easy
  15. Set 2 Q22Given that AA and BB are not null sets, which of the following statements regarding probability is/are CORRECT?MSQ · +1 marks · Medium
  16. Set 2 Q29The “order” of the following ordinary differential equation is ______.…NAT · +1 marks · Easy
  17. Set 2 Q36Pick the CORRECT solution for the following differential equation dydx=ex−y\frac{dy}{dx} = e^{x-y}MCQ · +2 marks · Easy
  18. Set 2 Q44Consider the function given below and pick one or more CORRECT statement(s) from the following choices. f(x)=x3−152x2+18x+20f(x) = x^3 - \frac{15}{2}x^2 + 18x + 20MSQ · +2 marks · Medium
  19. Set 2 Q45Pick the CORRECT eigen value(s) of the matrix [A][A] from the following choices. [A]=[6842][A] = \begin{bmatrix} 6 & 8 \\ 4 & 2 \end{bmatrix}MSQ · +2 marks · Easy
  20. Set 2 Q51Consider a discrete random variable XX whose probabilities are given below. The standard deviation of the random variable is ______ (round off to one decimal…NAT · +2 marks · Medium

GATE CE 202419 questions

  1. Set 1 Q11The smallest positive root of the equation x5−5x4−10x3+50x2+9x−45=0x^5 - 5x^4 - 10x^3 + 50x^2 + 9x - 45 = 0 lies in the rangeMCQ · +1 marks · Medium
  2. Set 1 Q12The second-order differential equation in an unknown function u:u(x,y)u: u(x, y) is defined as ∂2u∂x2=2\frac{\partial^2 u}{\partial x^2} = 2 Assuming g:g(x)g: g(x),…MCQ · +1 marks · Medium
  3. Set 1 Q13The probability that a student passes only in Mathematics is 13\frac{1}{3}. The probability that the student passes only in English is 49\frac{4}{9}. The…MCQ · +1 marks · Easy
  4. Set 1 Q28For the following partial differential equation, x∂2f∂x2+y∂2f∂y2=x2+y22x \frac{\partial^2 f}{\partial x^2} + y \frac{\partial^2 f}{\partial y^2} = \frac{x^2 + y^2}{2} which of…MSQ · +1 marks · Medium
  5. Set 1 Q32Consider the data of f(x)f(x) given in the table. | ii | 0 | 1 | 2 | |---|---|---|---| | xix_i | 1 | 2 | 3 | | f(xi)f(x_i) | 0 | 0.3010 | 0.4771 | The value of…NAT · +1 marks · Medium
  6. Set 1 Q36What are the eigenvalues of the matrix [211141112]\begin{bmatrix} 2 & 1 & 1 \\ 1 & 4 & 1 \\ 1 & 1 & 2 \end{bmatrix} ?MCQ · +2 marks · Easy
  7. Set 1 Q37A vector field p⃗\vec{p} and a scalar field rr are given by…MCQ · +2 marks · Easy
  8. Set 1 Q42A flow velocity field V⃗:V⃗(x,y)\vec{V}: \vec{V}(x, y) for a fluid is represented by V⃗=3i^+(5x)j^\vec{V} = 3 \hat{i} + (5x) \hat{j} In the context of the fluid and the flow,…MCQ · +2 marks · Easy
  9. Set 1 Q47The return period of a large earthquake for a given region is 200 years. Assuming that earthquake occurrence follows Poisson’s distribution, the probability…NAT · +2 marks · Medium
  10. Set 1 Q48A 2 m×2 m2\text{ m} \times 2\text{ m} tank of 3 m3\text{ m} height has inflow, outflow and stirring mechanisms. Initially, the tank was half-filled with fresh water.…NAT · +2 marks · Hard
  11. Set 2 Q11A partial differential equation ∂2T∂x2+∂2T∂y2=0\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0 is defined for the two-dimensional field…MCQ · +1 marks · Medium
  12. Set 2 Q12The statements P and Q are related to matrices A and B, which are conformable for both addition and multiplication. P:…MCQ · +1 marks · Easy
  13. Set 2 Q13The second derivative of a function ff is computed using the fourth-order Central Divided Difference method with a step length hh. The CORRECT expression for…MCQ · +1 marks · Medium
  14. Set 2 Q14The function f(x)=x3−27x+4,1≤x≤6f(x) = x^3 - 27x + 4, 1 \leq x \leq 6 hasMCQ · +1 marks · Medium
  15. Set 2 Q15Consider two Ordinary Differential Equations (ODEs): P: dydx=x4+3x2y2+2y4x3y\frac{dy}{dx} = \frac{x^4+3x^2y^2+2y^4}{x^3y} Q: dydx=−y2x2\frac{dy}{dx} = \frac{-y^2}{x^2} Which one of the…MCQ · +1 marks · Medium
  16. Set 2 Q36In a sample of 100100 heart patients, each patient has 80%80\% chance of having a heart attack without medicine X. It is clinically known that medicine X reduces…MCQ · +2 marks · Medium
  17. Set 2 Q44Three vectors p⃗\vec{p}, q⃗\vec{q}, and r⃗\vec{r} are given as p⃗=i^+j^+k^\vec{p} = \hat{i} + \hat{j} + \hat{k} q⃗=i^+2j^+3k^\vec{q} = \hat{i} + 2\hat{j} + 3\hat{k}…MSQ · +2 marks · Medium
  18. Set 2 Q47The expression for computing the effective interest rate (ieffi_{eff}) using continuous compounding for a nominal interest rate of 5% is…NAT · +2 marks · Easy
  19. Set 2 Q48Consider two matrices A=[214103]\mathbf{A} = \begin{bmatrix} 2 & 1 & 4 \\ 1 & 0 & 3 \end{bmatrix} and…NAT · +2 marks · Easy

GATE CE 20231 question

  1. Set 1 Q7Which one of the options can be inferred about the mean, median, and mode for the given probability distribution (i.e. probability mass function), P(x)P(x), of a…MCQ · +2 marks · Medium

Other GATE CE topics

Structural Engineering(347)General Aptitude(260)Geotechnical Engineering(231)Water Resources Engineering(217)Environmental Engineering(171)Transportation Engineering(166)Geomatics Engineering(86)

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