Engineering Mathematics carries about 13 marks in GATE CSE, and many of those questions are quick wins if your basics are sharp. This set focuses on topics that our earlier practice sets touched only briefly: vector spaces and linear independence, the rank-nullity theorem, continuity versus differentiability, and continuous random variables, including probability density functions, the uniform distribution and joint distributions. Each answer comes with a short worked solution.
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
- Linear independence, span and dimension
- Subspaces and basis of matrix and polynomial spaces
- Rank-nullity theorem and homogeneous systems
- Continuity vs differentiability
- PDFs, CDFs, expectation and the uniform distribution
- Joint and marginal distributions, independence
Engineering Mathematics MCQs with solutions
Q1. What is the dimension of the subspace of R^3 spanned by the vectors (1, 2, 3), (2, 4, 6) and (1, 0, 1)?
a) 1
b) 2
c) 3
d) 0
Q2. A is a 4 × 6 matrix with rank 3. What is the nullity of A?
a) 1
b) 2
c) 3
d) 6
Q3. A is a 3 × 5 matrix of rank 2. How many linearly independent solutions does the homogeneous system Ax = 0 have?
a) 3
b) 2
c) 5
d) 0
Q4. A is an n × n matrix with det(A) ≠ 0. What can you say about the solutions of Ax = 0?
a) Infinitely many solutions
b) No solution
c) Exactly n solutions
d) Only the trivial solution x = 0
Q5. What is the dimension of the vector space of all real 3 × 3 symmetric matrices?
a) 3
b) 9
c) 6
d) 5
Q6. Which of these sets is a subspace of R^3?
a) {(x, y, z) : x + y + z = 0}
b) {(x, y, z) : x + y + z = 1}
c) {(x, y, z) : x ≥ 0}
d) {(x, y, z) : xyz = 0}
Q7. What is the dimension of the vector space of all real polynomials of degree at most 3?
a) 3
b) 4
c) Infinite
d) 6
Q8. Consider f(x) = |x| at x = 0. Which statement is true?
a) f is neither continuous nor differentiable at 0
b) f is differentiable but not continuous at 0
c) f is continuous and differentiable at 0
d) f is continuous but not differentiable at 0
Q9. Consider f(x) = x|x| at x = 0. Which statement is true?
a) f is differentiable at 0 and f'(0) = 0
b) f is not continuous at 0
c) f is continuous but not differentiable at 0
d) f'(0) = 1
Q10. X is uniformly distributed on [2, 6]. What is Var(X)?
a) 2
b) 16/12 = 4/3
c) 4
d) 1/4
Q11. X has probability density f(x) = kx for 0 ≤ x ≤ 2, and 0 elsewhere. What is k?
a) 1
b) 2
c) 1/2
d) 1/4
Q12. For the same density f(x) = x/2 on [0, 2], what is E[X]?
a) 1
b) 4/3
c) 2/3
d) 3/2
Q13. X is a continuous random variable with a valid probability density function. What is P(X = 2)?
a) f(2)
b) F(2)
c) 1 − F(2)
d) 0
Q14. The joint density of X and Y is f(x, y) = 4xy for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. Which statement is correct?
a) X and Y are independent, with marginal densities f_X(x) = 2x and f_Y(y) = 2y
b) X and Y are dependent because the density contains the product xy
c) The marginal density of X is 4x
d) f(x, y) is not a valid joint density
Q15. X and Y are independent and each uniformly distributed on [0, 1]. What is P(X + Y ≤ 1)?
a) 1/4
b) 1/3
c) 1/2
d) 2/3
Quick revision notes
- Rank-nullity: rank(A) + nullity(A) = number of columns; Ax = 0 has n − rank independent solutions.
- det(A) ≠ 0 ⇔ full rank ⇔ only the trivial solution. A subspace must contain 0 and be closed under + and scalar ×.
- Dimensions: n × n symmetric matrices = n(n + 1)/2; polynomials of degree ≤ n = n + 1.
- Differentiable ⇒ continuous, not the reverse: |x| is continuous but not differentiable at 0; x|x| is differentiable there.
- Uniform U(a, b): mean (a + b)/2, variance (b − a)²/12. For continuous X, P(X = c) = 0; joint density that factors on a rectangle ⇒ independent.
Frequently asked questions
How many marks does Engineering Mathematics carry in GATE CSE?
Engineering Mathematics, including discrete mathematics, usually carries around 13 marks, in addition to the 15 marks of General Aptitude. Linear algebra, probability and calculus questions are often short and are good scoring areas.
Is the rank-nullity theorem asked directly in GATE?
It is often used inside questions about the number of solutions of a linear system or the dimension of a null space. Remember that it uses the number of columns, not rows.
Which continuous distributions should I revise for GATE CSE?
Focus on the uniform, exponential and normal distributions: their density functions, mean, variance and how to find probabilities by integration. Also practise joint, marginal and conditional distributions.
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