Calculus and linear algebra questions in GATE CSE reward knowing a few standard results well. This GATE-style set covers limits, continuity, the mean value theorems, definite integrals, maxima and minima, special matrices, eigenvalue properties, LU decomposition and the Cayley–Hamilton theorem.
How to practise: read each question and tap the option you think is correct. You will see right away whether you got it, along with a short explanation, and your score is tracked at the bottom of the screen. This set has 15 questions.
Topics covered in this set
- Limits and continuity
- Rolle's and mean value theorems
- Definite integrals and their properties
- Maxima and minima
- Symmetric, orthogonal and idempotent matrices
- Eigenvalues, LU decomposition and Cayley–Hamilton
GATE CSE Engineering Mathematics MCQs with answers and explanations
Q1. What is the value of lim (x → 0) (1 − cos 2x)/x²?
a) 1/2
b) 4
c) 2
d) 1
Q2. The function f(x) = (x² − 9)/(x − 3) for x ≠ 3, and f(3) = k. For which value of k is f continuous at x = 3?
a) 3
b) 6
c) 0
d) 9
Q3. Which statement about f(x) = |x| at x = 0 is correct?
a) It is both continuous and differentiable
b) It is neither continuous nor differentiable
c) It is differentiable but not continuous
d) It is continuous but not differentiable
Q4. For f(x) = x³ on the interval [0, 3], which value of c satisfies the conclusion of Lagrange's mean value theorem?
a) 3/2
b) 1
c) √6
d) √3
Q5. For f(x) = x³ − 3x on the interval [0, √3], which value of c satisfies the conclusion of Rolle's theorem?
a) 1/√3
b) 1
c) √3/2
d) 0
Q6. What is the value of ∫₀^π x sin x dx?
a) 2
b) π
c) 0
d) π/2
Q7. What is the value of ∫₀^(π/2) sin³x/(sin³x + cos³x) dx?
a) π/2
b) 0
c) π/4
d) 1
Q8. What is the absolute maximum value of f(x) = x³ − 3x on the closed interval [−2, 3]?
a) 2
b) 9
c) 18
d) −2
Q9. What is the minimum value of f(x) = x + 4/x for x > 0?
a) 5
b) 4
c) 8
d) 2
Q10. Q is a real orthogonal matrix. Which statement is always true?
a) All eigenvalues of Q are real
b) Q is symmetric
c) Q⁻¹ = Qᵀ and |det(Q)| = 1
d) det(Q) = 1
Q11. A 4 × 4 matrix A is idempotent (A² = A) and its trace is 3. What is the rank of A?
a) 4
b) 1
c) 3
d) Cannot be determined
Q12. The matrix A = [[1, 4], [2, 3]] is written as the sum of a symmetric matrix S and a skew-symmetric matrix K. What is S?
a) [[0, 1], [−1, 0]]
b) [[1, 4], [4, 3]]
c) [[1, 2], [2, 3]]
d) [[1, 3], [3, 3]]
Q13. A 3 × 3 matrix A has eigenvalues 1, 2 and −3. What is the trace of A⁻¹?
a) 6
b) −1/6
c) 7/6
d) 0
Q14. A 2 × 2 real matrix A has trace 7 and determinant 10. Which statement about A − 2I is correct?
a) Its determinant is 10
b) Its trace is 7
c) Its eigenvalues are 2 and 5
d) It is singular, because its eigenvalues are 0 and 3
Q15. For A = [[1, 2], [3, 4]], the Cayley–Hamilton theorem gives A² − 5A − 2I = 0. Which expression equals A⁻¹?
a) (A − 5I)/5
b) (5I − A)/2
c) (A − 5I)/2
d) (A + 5I)/2
Quick revision notes
- lim (1 − cos kx)/x² = k²/2; continuity needs limit = function value; |x| is continuous but not differentiable at 0.
- Lagrange MVT: f'(c) = (f(b) − f(a))/(b − a); Rolle is the case f(a) = f(b).
- King's property: ∫₀^a f(x) dx = ∫₀^a f(a − x) dx; check endpoints for absolute extrema.
- Orthogonal: Q⁻¹ = Qᵀ, det = ±1; idempotent: eigenvalues 0 or 1, rank = trace; A = (A + Aᵀ)/2 + (A − Aᵀ)/2.
- Eigenvalues of A⁻¹ are 1/λ, of A − kI are λ − k; Cayley–Hamilton lets you write A⁻¹ as a polynomial in A.
Frequently asked questions
Are these GATE previous year questions?
No, they are original GATE-style practice questions written to the GATE CSE Engineering Mathematics syllabus.
What is LU decomposition used for in GATE questions?
It writes A as a lower triangular L times an upper triangular U, for example [[2, 3], [4, 7]] = [[1, 0], [2, 1]] × [[2, 3], [0, 1]], so Ax = b can be solved by forward and back substitution. Questions usually ask for one entry of L or U.
Is calculus important for GATE CSE?
It carries fewer marks than linear algebra and probability, but limits, continuity, mean value theorems and maxima questions are quick to solve once the standard results are known.
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