Q. Two n bit binary strings, S1 and S2, are chosen randomly with uniform probability. The probability that the Hamming distance between these strings (the number of bit positions where the two strings differ) is equal to d is

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Q. A point is randomly selected with uniform probability in the X-Y plane within the rectangle with corners at (0,0), (1,0), (1,2) and (0,2). If p is the length of the position vector of the point, the expected value of p^2 is

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Q. Let G1 = (V, E1) and G2 = (V, E2) be connected graphs on the same vertex set V with more than two vertices. If G1 ∩ G2 = (V, E1 ∩ E2) is not a connected graph, then the graph G1 U G2 = (V, E1 U E2)

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Q. For the composition table of a cyclic group shown below:

* a b c d

a a b c d

b b a d c

c c d b a

d d c a b

Which one of the following choices is correct?

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Q. The product of the non-zero eigenvalues of the matrix

1 0 0 0 1
0 1 1 1 0
0 1 1 1 0
0 1 1 1 0
1 0 0 0 1

is ______

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Q. Which one of the following statements is TRUE about every n×n matrix with only real eigenvalues?

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Q. Consider the following system of equations in three real variables x1, x2 and x3:

2x1 - x2 + 3x3 = 1
3x1- 2x2 + 5x3 = 2
-x1 + 4x2 + x3 = 3

This system of equations has

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Q. Let A, B, C, D be n × n matrices, each with non-­zero determinant. If ABCD = 1, then B^-1 is

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Q. In the LU decomposition of the matrix,

| 2 2 |
| 4 9 |

if the diagonal elements of U are both 1, then the lower diagonal entry l22 of L is

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Q. Consider the following 2 × 2 matrix A where two elements are unknown and are marked by a and b. The eigenvalues of this matrix are –1 and 7. What are the values of a and b?

A = | 1 4 |
| b a |

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Q. The larger of the two eigenvalues of the matrix

⎡ 4 5 ⎤
⎣ 2 1 ⎦

is ____

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Q. Perform the following operations on the matrix:

⎡ 3 4 45⎤
⎢ 7 8 105⎥
⎣13 2 195⎦

1. Add the third row to the second row.
2. Subtract the third column from the first column.

The determinant of the resultant matrix is _____________.

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Q. If the following system has non-trivial solution,

px + qy + rz = 0
qx + ry + pz = 0
rx + py + qz = 0

then which one of the following options is True?

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Q. Let a(x, y), b(x, y,) and c(x, y) be three statements with variables x and y chosen from some universe. Consider the following statement:

(∃x)(∀y)[(a(x, y) ∧ b(x, y)) ∧ ¬c(x, y)]

Which one of the following is its equivalent?  

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Q. Two eigenvalues of a 3 x 3 real matrix P are (2 + √ -1) and 3. The determinant of P is _____  

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Q. Let f (x) be a polynomial and g(x) = f (x) be its derivative. If the degree of (f(x) + f(−x)) is 10, then the degree of (g(x) − g(−x)) is _______________ .

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Q. Consider the systems, each consisting of m linear equations in n variables.

I. If m < n, then all such systems have a solution
II. If m > n, then none of these systems has a solution
III. If m = n, then there exists a system which has a solution

Which one of the following is CORRECT?

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Q. Suppose that the eigenvalues of matrix A are 1, 2, 4. The determinant of (A^−1)^T is _________

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Q. Consider the function f(x) = sin(x) in the interval [π/4, 7π/4]. The number and location(s) of the local minima of this function are

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Q. The bisection method is applied to compute a zero of the function f(x) = x^4 – x^3 – x^2 – 4 in the interval [1,9]. The method converges to a solution after _______ iterations.

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