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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Q. Newton-Raphson method is used to compute a root of the equation x^2-13=0 with 3.5 as the initial value. The approximation after one iteration is

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Q. What is the value of Limn->∞(1-1/n)^2n ?

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