ICAR AIEEA-PG — Agricultural Engineering & Technology (Major Subject Group 10) · India · National Testing Agency for ICAR
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
ICAR AIEEA-PG — Agricultural Engineering & Technology (Unit I)
Unit I combines mathematics and statistics. Expect single-answer MCQs on matrices, differentiation and integration, differential equations and numerical methods at a slightly lower computational load than GATE, often testing a standard result directly (a Laplace pair, a series sum, a Simpson's rule weight).
How our questions for ICAR AIEEA-PG spread across the units of Engineering Mathematics. Difficulty: 32 easy, 52 medium, 21 hard.
| Eigenvalues, eigenvectors and the Cayley-Hamilton theorem | 8 | |
| Vector calculus: gradient, divergence and curl | 8 | |
| Line, surface and volume integrals: Green, Stokes and Gauss theorems | 8 | |
| Laplace transforms | 8 | |
| Numerical solution of equations: bisection and Newton-Raphson | 8 | |
| Euler's theorem on homogeneous functions | 7 | |
| Higher-order linear ordinary differential equations | 7 | |
| Numerical solution of ODEs: Euler and Runge-Kutta methods | 7 | |
| Sequences, series and convergence | 6 | |
| Taylor, Maclaurin and Fourier series | 6 | |
| First-order ordinary differential equations | 6 | |
| Partial differential equations: Laplace, heat and wave equations | 6 | |
| Numerical integration: trapezoidal and Simpson's rules | 6 | |
| Matrices, determinants and systems of linear equations | 5 | |
| Limits, continuity and partial derivatives | 5 | |
| Maxima and minima of functions of several variables | 4 |
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
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
Which substitution turns the Bernoulli equation dy/dx + y = x y³ into a linear equation?
B. v = y⁻²
For dy/dx + P y = Q yⁿ the substitution is v = y^(1−n); with n = 3 that is y⁻². v = y/x is for homogeneous equations.
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, 105 in the app. The Laisha app has every practice question we have written to ICAR AIEEA-PG 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.