GATE — Agricultural Engineering (AG) · India · IITs and IISc on rotation (2026: IIT Guwahati)
Original practice questions written to the published syllabus · not past papersThe algebra, calculus, differential equations and numerical methods used in every numerical question of the degree. GATE AG gives it a fixed block of marks every year.
In the exams: GATE AG Section 1, about 12–13 marks a year (2019–26); ICAR AIEEA-PG Unit I; NCEES FE 8–12 questions; allied subject in PRC ABE
GATE — Agricultural Engineering (AG)
A fixed 13-mark Section 1. Items are short and computational: eigenvalues and trace/determinant checks, rank and consistency of a system with a parameter, Cayley-Hamilton, a limit or L'Hôpital, Euler's theorem on a homogeneous function, a two-variable extremum, gradient/divergence/curl or a directional derivative, a Green or Gauss theorem shortcut, the solution of a first- or second-order ODE with initial conditions, a Laplace transform pair, classification of a PDE, and one numerical-methods step (a Newton-Raphson iterate, a trapezoidal or Simpson value, an Euler or RK step). Numerical-answer items need the value to a stated precision, so carry enough decimals. The official syllabus text for Section 1 also lists complex variables and probability and statistics; this manual covers only the sixteen units of this course, so study those two topics separately for GATE.
How our questions for GATE spread across the units of Engineering Mathematics. Difficulty: 39 easy, 57 medium, 29 hard.
| Higher-order linear ordinary differential equations | 9 | |
| Matrices, determinants and systems of linear equations | 8 | |
| Eigenvalues, eigenvectors and the Cayley-Hamilton theorem | 8 | |
| Limits, continuity and partial derivatives | 8 | |
| Sequences, series and convergence | 8 | |
| Vector calculus: gradient, divergence and curl | 8 | |
| Line, surface and volume integrals: Green, Stokes and Gauss theorems | 8 | |
| First-order ordinary differential equations | 8 | |
| Laplace transforms | 8 | |
| Partial differential equations: Laplace, heat and wave equations | 8 | |
| Numerical solution of equations: bisection and Newton-Raphson | 8 | |
| Numerical integration: trapezoidal and Simpson's rules | 8 | |
| Numerical solution of ODEs: Euler and Runge-Kutta methods | 8 | |
| Euler's theorem on homogeneous functions | 7 | |
| Taylor, Maclaurin and Fourier series | 7 | |
| Maxima and minima of functions of several variables | 6 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
A is a 3×3 matrix with det(A) = 5. What is det(2A⁻¹)?
D. 8/5
det(kA) = kⁿ det(A) for an n×n matrix, and det(A⁻¹) = 1/det(A). So det(2A⁻¹) = 2³ × (1/5) = 8/5. Using k instead of k³ gives 2/5; 40 is det(2A), forgetting the inverse; 10 multiplies 2 by 5.
Further reading: Grewal, Higher Engineering Mathematics
By the Cayley-Hamilton theorem, the matrix A = [[1, 2], [3, 4]] satisfies A² = ?
D. 5A + 2I
Characteristic polynomial: λ² − (trace)λ + det = λ² − 5λ − 2 (det = 4 − 6 = −2). Cayley-Hamilton says A² − 5A − 2I = 0, so A² = 5A + 2I. Direct check: A² = [[7, 10], [15, 22]] and 5A + 2I = [[7, 10], [15, 22]]. 5A − 2I comes from writing det as +2 instead of −2.
Further reading: Kreyszig, Advanced Engineering Mathematics
What is the degree of homogeneity of u = (x^(1/3) + y^(1/3))/(x^(1/2) + y^(1/2))?
B. −1/6
u(tx, ty) = t^(1/3)/t^(1/2) · u = t^(−1/6) u, so the degree is 1/3 − 1/2 = −1/6. Subtracting in the wrong order gives +1/6; adding the powers gives 5/6.
Further reading: Grewal, Higher Engineering Mathematics
The series Σ n²/2ⁿ (n = 1 to ∞) converges by the ratio test. What is its sum?
D. 6
Ratio test: a(n+1)/a(n) = [(n + 1)²/n²]/2 → 1/2 < 1, so it converges. Using Σ n xⁿ = x/(1 − x)² and Σ n² xⁿ = x(1 + x)/(1 − x)³ with x = 1/2: (1/2)(3/2)/(1/8) = 6. 2 is Σ n/2ⁿ (the first-power sum); 4 comes from dropping the (1 + x) factor.
Further reading: Kreyszig, Advanced Engineering Mathematics
What is the limit of the sequence aₙ = n/(2n + 1) as n → ∞?
C. 0.5
Divide top and bottom by n: 1/(2 + 1/n) → 1/2. Answering 0 or 1 comes from taking the limits of numerator and denominator separately.
Further reading: Grewal, Higher Engineering Mathematics
For any scalar field φ and vector field F with continuous second derivatives, which pair of identities holds?
C. curl(grad φ) = 0 and div(curl F) = 0
Both follow from equality of mixed partial derivatives. div(grad φ) is the Laplacian ∇²φ, which is not zero in general (it is zero only for harmonic φ).
Further reading: Kreyszig, Advanced Engineering Mathematics
What is the volume of the tetrahedron bounded by the coordinate planes and the plane x + y + z = 1, found by triple integration?
D. 1/6
V = ∫₀¹ ∫₀^(1−x) ∫₀^(1−x−y) dz dy dx = ∫₀¹ (1 − x)²/2 dx = 1/6. 1/2 is the area of the base triangle; 1/3 is (1/3)·base·height with base area 1 instead of 1/2.
Further reading: Kreyszig, Advanced Engineering Mathematics
What is the Laplace transform of t²?
D. 2/s³
L{tⁿ} = n!/s^(n+1). For n = 2: 2!/s³ = 2/s³. 1/s³ forgets the n!; 2/s² uses sⁿ instead of s^(n+1).
Further reading: Kreyszig, Advanced Engineering Mathematics
The PDE u_xx + 4u_xy + 4u_yy = 0 is
A. parabolic
For A u_xx + B u_xy + C u_yy, the discriminant B² − 4AC = 16 − 16 = 0 → parabolic. Negative → elliptic (Laplace), positive → hyperbolic (wave).
Further reading: Kreyszig, Advanced Engineering Mathematics
For what value of k does u = e^(−kt) sin 3x satisfy the heat equation u_t = 2u_xx?
B. 18
u_t = −k e^(−kt) sin 3x; u_xx = −9 e^(−kt) sin 3x. So −k = 2(−9) and k = 18. 6 uses 3 instead of 3²; 9 forgets the factor 2; −18 is a sign slip.
Further reading: Kreyszig, Advanced Engineering Mathematics
Ten here, 125 in the app. The Laisha app has every practice question we have written to GATE in this subject, marked as you go, with timed papers, a jump-to-unanswered list and a certificate when you pass at 70 per cent.
Free for ten days, then one subscription of US$10 a month. We would rather say so here than have you find it later.
An AI operating system for agriculture: free guides, manuals and tools written for every country.