GATE · Engineering Mathematics
Linear algebra, calculus and probability
74 practice questions, each with the answer and an explanation.
Practise Linear algebra, calculus and probability now
Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.
All 74 questions
- What is the rank of the matrix with rows (3, 6, 9); (6, 12, 18); (9, 18, 27)?
- If P(A and B) = 0.05 and P(B) = 0.5, what is P(A given B)?
- Find the divergence of F = (x², y², z²) at the point (1, 2, 2).
- The solution of dy/dx = −6y with y(0) = 6 is
- How many labelled spanning trees does the complete graph K7 have?
- How many functions are there from a set of 5 elements to a set of 6 elements?
- A 2×2 matrix has eigenvalues 4 and -4. What are the trace and determinant?
- A system of 3 linear equations in 3 unknowns has a coefficient matrix of rank 1. How many free parameters does its general solution have (if consistent)?
- A random variable takes the values 3, 4 and 1 with probabilities 1/2, 1/3 and 1/6 respectively. What is its expected value?
- The Laplace transform of e^(2t) is
- The solution of dy/dx = −3y with y(0) = 1 is
- A 2×2 matrix has eigenvalues -2 and 5. What are the trace and determinant?
- What is the rank of the matrix with rows (2, 4, 6); (3, 6, 9); (5, 10, 15)?
- The Laplace transform of e^(4t) is
- The solution of dy/dx = −2y with y(0) = 4 is
- How many edges does a tree with 8 vertices have?
- The Taylor series of e^(6x) about x = 0 has the coefficient of x³ equal to
- A 2×2 matrix has eigenvalues 3 and -3. What are the trace and determinant?
- If P(A and B) = 0.05 and P(B) = 0.6, what is P(A given B)?
- Find the divergence of F = (x², y², z²) at the point (1, 1, 3).
- The solution of dy/dx = −2y with y(0) = 1 is
- How many labelled spanning trees does the complete graph K5 have?
- How many relations are there on a set with 5 elements?
- What is the limit of (sin 4x)/x as x tends to 0?
- A 3×3 matrix has eigenvalues -2, 2 and 6. What is its determinant?
- A system of 5 linear equations in 5 unknowns has a coefficient matrix of rank 3. How many free parameters does its general solution have (if consistent)?
- A random variable takes the values 2, 6 and 1 with probabilities 1/2, 1/3 and 1/6 respectively. What is its expected value?
- The Laplace transform of t^5 is
- The solution of dy/dx = −6y with y(0) = 2 is
- The Taylor series of e^(4x) about x = 0 has the coefficient of x³ equal to
- A 3×3 matrix has eigenvalues 7, 3 and 4. What is its determinant?
- A system of 6 linear equations in 6 unknowns has a coefficient matrix of rank 4. How many free parameters does its general solution have (if consistent)?
- The Laplace transform of e^(6t) is
- The solution of dy/dx = −3y with y(0) = 6 is
- How many edges does the complete graph K6 have?
- A 3×3 matrix has eigenvalues 5, 7 and 5. What is its determinant?
- What is the limit of (sin 2x)/x as x tends to 0?
- The eigenvalues of the matrix with rows (-1, -1); (0, 5) are
- What is the rank of the matrix with rows (2, 4, 6); (4, 8, 12); (6, 12, 18)?
- Two independent events have probabilities 0.5 and 0.375. What is the probability that at least one occurs?
- The solution of dy/dx = −5y with y(0) = 2 is
- How many labelled spanning trees does the complete graph K6 have?
- The Taylor series of e^(2x) about x = 0 has the coefficient of x³ equal to
- A 3×3 matrix has eigenvalues 0, -3 and 2. What is its determinant?
- How many functions are there from a set of 3 elements to a set of 4 elements?
- The eigenvalues of the matrix with rows (4, 3); (0, -1) are
- What is the rank of the matrix with rows (2, 1, 3); (3, 6, 9); (5, 7, 12)?
- Evaluate the double integral of (x + y) over the rectangle 0 ≤ x ≤ 1, 0 ≤ y ≤ 2.
- The solution of dy/dx = −4y with y(0) = 2 is
- A 3×3 matrix has eigenvalues 6, -2 and 3. What is its determinant?
- A 3×3 matrix has eigenvalues -3, 5 and 6. What is its determinant?
- What is the rank of the matrix with rows (3, 2, 5); (4, 8, 12); (7, 10, 17)?
- A random variable takes the values 4, 2 and 3 with probabilities 1/2, 1/3 and 1/6 respectively. What is its expected value?
- The solution of dy/dx = −4y with y(0) = 1 is
- How many reflexive relations are there on a set with 4 elements?
- The eigenvalues of the matrix with rows (4, -1); (0, -2) are
- Find the divergence of F = (x², y², z²) at the point (4, 3, 3).
- A 2×2 matrix has eigenvalues 7 and -3. What are the trace and determinant?
- What is the rank of the matrix with rows (3, 4, 7); (1, 2, 3); (4, 6, 10)?
- Two independent events have probabilities 0.5 and 0.2. What is the probability that at least one occurs?
- The solution of dy/dx = −3y with y(0) = 2 is
- A 3×3 matrix has eigenvalues 3, -3 and 4. What is its determinant?
- The solution of dy/dx = −6y with y(0) = 3 is
- How many edges does a tree with 9 vertices have?
- A 2×2 matrix has eigenvalues -1 and 5. What are the trace and determinant?
- What is the rank of the matrix with rows (4, 8, 12); (8, 16, 24); (12, 24, 36)?
- Find the divergence of F = (x², y², z²) at the point (4, 1, 4).
- A 3×3 matrix has eigenvalues 6, 5 and 4. What is its determinant?
- The solution of dy/dx = −2y with y(0) = 2 is
- How many edges does the complete graph K5 have?
- The eigenvalues of the matrix with rows (-2, -1); (0, -4) are
- What is the rank of the matrix with rows (4, 2, 6); (3, 6, 9); (7, 8, 15)?
- If P(A and B) = 0.1 and P(B) = 0.5, what is P(A given B)?
- Find the divergence of F = (x², y², z²) at the point (3, 4, 1).