60 solved GATE ME questions on Engineering Mathematics, drawn from 5 exam years and grouped by year. Every question shows the official answer and a step-by-step solution.
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State the domain, convergence condition, or matrix property a method requires before substituting values. These selected foundations connect linear algebra, calculus, differential equations, probability, and numerical methods as they appear in the GATE ME 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.
Eigenvalues satisfy det(A - λI) = 0. For a 2x2 matrix, λ₁ + λ₂ = trace and λ₁λ₂ = determinant. A system Ax = b has a unique solution when det(A) ≠ 0. Rank deficiency introduces free variables or inconsistency depending on the augmented matrix.
Check yourself: Did you verify the determinant condition before claiming a unique solution?
Gradient points in the direction of steepest ascent. Stokes' and divergence theorems convert between line/surface integrals and area/volume integrals. Apply Green's theorem for 2D: ∮(P dx + Q dy) = ∬(∂Q/∂x - ∂P/∂y) dA over a simply connected region.
Check yourself: Is the region simply connected and the field continuously differentiable?
A first-order linear ODE dy/dx + P(x)y = Q(x) has integrating factor e^(∫P dx). For second-order constant-coefficient equations, the characteristic equation determines complementary solutions: distinct real roots give exponentials, repeated roots add a polynomial factor, complex roots give sinusoids.
Check yourself: Have you applied the initial or boundary conditions after finding the general solution?
For independent events, P(A ∩ B) = P(A)P(B). Bayes' theorem inverts conditional probability: P(A|B) = P(B|A)P(A)/P(B). The mean of a continuous distribution is ∫x f(x) dx; variance is E(X²) - [E(X)]². The normal distribution is characterized by mean and variance alone.
Check yourself: Are the events independent, or did you assume independence without justification?
Newton-Raphson iterates x_{n+1} = x_n - f(x_n)/f'(x_n), converging quadratically near a simple root when f'(x_n) ≠ 0. The trapezoidal rule approximates ∫f dx with error proportional to h². Simpson's 1/3 rule requires an even number of intervals and has error proportional to h⁴.
Check yourself: Is f'(x_n) near zero, which would cause divergence or oscillation?
Original teaching example · not a PYQ
Original mini-example: find the eigenvalues of A = [[4, 2], [1, 3]].
Eigenvalues are λ = 5 and λ = 2.
Apply one iteration of Newton-Raphson to f(x) = x² - 9 starting from x₀ = 4.
x₁ = 3.125.
f(4) = 16 - 9 = 7. f'(x) = 2x, so f'(4) = 8. x₁ = 4 - 7/8 = 4 - 0.875 = 3.125. The method converges toward the root x = 3.
Supplemental reading, not an official GATE reading list or an endorsement of these notes.
Free supplemental lectures by Gilbert Strang covering eigenvalues, matrix decompositions, and linear systems. Complement with calculus and ODE resources for full coverage.
This page shows 60 recent questions from the released archive, newest first. For older questions and complete papers, browse all GATE ME papers. Questions can carry more than one subject tag; counts are not marks weightage.
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