National Champion Test - GATE 2018 CS : Engineering Mathematics for GATE & PSU CS Exams

Question 1

What is the differentiation of 1/x ?

Question 2

The value of

is

Question 3

The value of the integral given below is

Question 4

In the following integral, the contour C enclose the points 2πj and -2πj

The value of the integral is ______

Question 5

What is the value of

Question 6

Suppose you want to move from 0 to 100 on the number line. In each step, you either move right by a unit distance or you take a shortcut. A shortcut is simply a pre-specified pair of integers i, j with i < j. Given a shortcut i, j if you are at position i on the number line, you may directly move to j. Suppose T(k) denotes the smallest number of steps needed to move from k to 100. Suppose further that there is at most 1 shortcut involving any number, and in particular from 9 there is a shortcut to 15. Let y and z be such that T(9) = 1 + min(T(y), T(z)). Then the value of the product yz is _____.

Question 7

If

then the value of k is equal to ______ .

Question 8

equals

Question 9

Let A be real valued square symmetric matrix of rank 2 with

Consider the following statements.
(I) One eigen value must be in [-5, 5]
(II) The eigen value with the largest magnitude must be strictly greater than 5.
Which of the above statements about eigen values of A is/are necessarily CORRECT?

Question 10

Consider the following statements:
S1: The sum of two singular n × n matrices may be non-singular
S2: The sum of two n × n non-singular matrices may be singular.
Which of the following statements is correct?

Question 11

Consider the following system of equations:
3x + 2y = 1
4x + 7z = 1
x + y + z = 3
x – 2y + 7z = 0
The number of solutions for this system is _________

Question 12

The value of the dot product of the eigenvectors corresponding to any pair of different Eigen values of a 4-by-4 symmetric positive definite matrix is ___________.

Question 13

Consider the following matrix

If the eigenvalues of A are 4 and 8, then

Question 14

The following system of equations

has a unique solution. The only possible value(s) for α is/are

Question 15

Suppose we uniformly and randomly select a permutation from the 20! Permutations of 1, 2, 3,……, 20. What is the probability that 2 appears at an earlier position than any other even number in the selected permutation?

Question 16

We are given a set

where

A sample S ⊆X is drawn by selecting each x_iindependently with probability

. The expected value of the smallest number in sample S is:

Question 17

Find the standard deviation of 2,4,6 and 8. (Round upto two decimal places)

Question 18

A loaded dice has following probability distribution of occurrences

If three identical dice as the above are thrown, the probability of occurrence of values 1,5 and 6 on the three dice is

Question 19

Let X be a stationary sequence with zero mean and autocovariance function c(m).Then find the best linear predictor Ẋ_n+1 of X_n+1 given X_n

Question 20

The average marks of boys in class is 52 and that of girls is 42. The average marks of boys and girls combined is 50. The percentage of boys in the class is