GATE EC Previous Year Solved Paper

Transcription

1 GATE EC Previous Year Solved Paper GATE 2017 is just round the corner. Engineering Mathematics is a highly scoring portion in GATE. A basic understanding of the concepts and a thorough practice is enough to boost your score in Engineering Mathematics section. Solving Previous Year Papers is of vital importance since it provides you an idea of the type of questions anticipated in the exam as well as helps you gauge your preparation level. With the exam just approaching, we have brought forward solved Engineering Mathematics section of GATE 2016 paper. Solved Engineering Mathematics Questions of GATE 2016 Q1. The second moment of a Poisson-distributed random variable is 2. Compute the mean of the random variable? Answer: λ = 1 E(x 2 ) = 2 V(X) = E(X 2 ) (E(X)) 2 Let x denote the mean of the Poisson variable x = 2 x 2

2 x 2 + x 2 = 0 x = 1, 2 Therefore, Mean is λ = 1 Q2. For what value of x does the matrix A = Has zero has Eigen value? Correct Answer: x = 1 Explanation For Eigen value of A to be 0, Determinant (A) = 0 3 { ( x) + 52 } 2 { ( x) + 78 } + 4 { } = 0 Therefore, x = 1 Q3. Which of the following describes the behavior of the function f(x) = x 3 3x as x varies from 1 to 3?

3 A. f(x) increases monotonically B. f(x) increases, then decreases and increases again C. f(x) decreases, then increases and decreases again D. f(x) increases and then decreases Correct Answer: Option B Since f(-1) = -3, f(0) = 1, f(1) = -1, f(2) = -3 and f(3) = 1 Q4. The Ordinary Differential Equation dx = -3x + 2 with x(0) = 1 is to be solved using the Forward Euler method. What is the largest time step which can be used to solve the equation without making the numerical solution unstable? $% $& = -3y + 2, y(0) = 1 If 1 3h < 1, then solution of differential equation is stable -1 <1 <-3h < 1-2 < -3h < 0 0 < 3h < 2 dt

4 0 < h < ' ( Therefore, if 0 < h < 0.66, then we get a stable solution Q5. The Matrix A has Det(A) = 100 and trace(a) = 14 Compute the value of a b? Correct Answer: 3 Given Trace(A) = 14 a + b + 7 = 14 a + b = 7 Given Det (A) = ab = 100 ab = 10

5 Therefore, a = 5, b = 2 or a = 2, b = 5 a b = 3 Q6. If the vectors e 1 = (1, 0, 2), e 2 = (0, 1, 0) and e 3 = (-2, 0, 1) form an orthogonal basis of the 3-Dimensional real space R 3, then the vector u = (4, 3, -3) R 3 can be expressed as: A. u = )2 5 e 1 3e e 3 B. u = )2 5 e 1 3e e 3 C. u = )2 5 e 1+ 3e e 3 D. u = )2 5 e 1+ 3e e 3 Correct Answer: Option (D) u = x 1 e 1 + x 2 e 2 + x 3 e 3 (4, 3, -3) = x 1 (1, 0, 2) + x 2 (0, 1, 0) + x 3 (-2, 0, 1) x 1-2x 3 = 4. (1) x 2 = 3 (2) 2x 1 + x 3 = -3 (3)

6 Solving these equations, we get x 1 = )' - x 2 = 3 x 3 = ).. - Therefore, u = )' - e 1+ 3e e 3 Q7. The Discrete Fourier Transform (DFT) of the 4-point sequence X[n] = {x[0], x[1], x[2], x[3]} = {3, 2, 3, 4} is X[k] = {X[0], X[1], X[2], X[3]} = {12, 2j, 0, 2j} If X 1 [k] is the DFT of the 12-point sequence x 1 [n] = {3, 0, 0, 2, 0, 0, 3, 0, 0, 4, 0, 0}, Then, compute the value of Correct Answer: 6 Interpolation in time domain is equal to replication in frequency domain

7 x 1 (n) = x( / ( ) X 1 (k) = [12, 2j, 0, 2j, 12, 2j, 0, 2j, 12, 2j, 0, 2j] X 1 (8) = 12, X 1 (11) = 2j The value of is equal to = 6 Q8. Consider the Complex valued function f(z) = 2z 3 + b z 3 where z denotes a complex variable. Compute the value of b for which the function f(z) is analytic? Correct Answer: 0 f(z) = 2z 3 + b z 3 For b = 0, f(z) becomes polynomial Therefore, it is analytic everywhere only when b = 0

8 Q9. Let M 4 = I (where I denotes the identity Matrix) and M I, M 2 I, M 3 I Then compute the value of M -1 for any natural number k? A. M 4k+1 B. M 4k+2 C. M 4k+3 D. M 4k Correct Answer: Option C M 4 = I M 8 = M 4 = I M 7 =M -1 M 12 = M 8 = I M 11 = M -1 M 16 = M 12 = I M 15 = M -1 Therefore, M -1 = M 4k+3 where k is a natural number

9

10 Q10. The probability of getting a head in a single toss of a biased coin is 0.3. The coin is tossed repeatedly till a head is obtained. In case tosses are independent, compute the probability of getting head for the first time in the 5 th toss? Correct Answer: P = (0.7) 4 (0.3) = Q11. Consider the following statements with regard to the function f: R R. Select the correct option: P: If f(x) is continuous at x = x 0, then it is also differentiable at x = x 0 Q: If f(x) is continuous at x = x 0, then it may not be differentiable at x = x 0 R: If f(x) is continuous at x = x 0, then it is also different at x = x 0 A. P is true, Q is false, R is false B. P is false, Q is true, R is true C. P is false, Q is true, R is false D. P is true, Q is false, R is true

11 Correct Answer: Option B Option B is correct since continuous function may not be differentiable. But differentiable function is always continuous. Hope that this post will aid you in your GATE 2017 preparation. More Updates will follow Stay Tuned ǃǃ