Let A be a real skew symmetric matrix such that A² + I = [0] then

a) A is an orthogonal matrix

b) A is an orthogonal matrix of even order

c) A is an orthogonal matrix of odd order

d) None of these

If A = [a_ij] is a 4 x 4 matrix and C_ij be the cofactor of the element a_ij in |A| then the expression a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃ + a₁₄C₁₄ is equal to

a) 0

b) -1

c) 1

d) |A|

If A is a symmetric matrix then M’AM is

a) Symmetric

b) Skew-symmetric

c) Orthogonal

d) None of these

The system of equations ax + y + z = 0, x + by + z = 0 and x + y + cz = 0 where a, b, c ≠ 1 has a non–trivial solution then the value of  +  +  is

a) -1

b) 0

c) 1

d) None of these

If a, b, c are positive integers, then the determinant
Δ =  is divisible by

a) x³

b) x²

c) a² + b² + c²

d) none of these

For the matrix A =  which of the following is correct

a) A³ + 3A² – I = 0

b) A³  3A² – I = 0

c) A³ + 2A² – I = 0

d) A³  A² + I = 0

Let f(x) =  then   is equal to

a) 0

b) 1

c) 2

d) -2

If x + y + z = 6, x – y + z = 2, 2x + y – z = 1 then x, y, z are respectively

a) 3, 2, 1

b) 2, 1, 3

c) 1, 2, 3

d) None of these

For a = + , the volume of the parallelepiped formed by the vectors î + aĵ, aî + ĵ + k̂ and ĵ + ak̂ is maximum

a) True

b) False

The number of 3 x 3 non-zero matrices with four entries as 1 and all others as 0 is

a) Less than 4

b) 5

c) 0

d) At least 7

Consider the system of linear equations x₁ + 2x₂ + x₃ = 3, 2x₁ + 3x₂ + x₃ = 3, 3x₁ + 5x₂ + 2x₃ = 1. The system has

a) Infinite number of solutions

b) Exactly 3 solutions

c) A unique solution

d) No solution

Let a, b, c be such that b + c ≠ 0. If
 +  = 0,

then the value of n is

a) Zero

b) An even integer

c) An odd integer

d) An integer

Let A be a 2 x 2 matrix with non–zero entries and let A² = I where I is a 2 x 2 identity matrix. Define tr A = sum of diagonal entries of A, |A| = determinant of matrix A
Statement I  tr A = 0
Statement II  |A| =1
Out of the following four statements choose only one which is correct

a)  Statement I is true. Statement II is true. Statement II  is the correct explanation of Statement I

b) Statement I is true. Statement II is true. Statement II  is not the correct explanation of Statement I

c) Statement I is true. Statement II is false

d) Statement I is false. Statement II is true

Prove for all positive integer n the identity
 

Prove that for any real number x₁, x₂ .  .  . , x_n; n ≥ 1
|sinx₁| + |sinx₂| + .  .  . + |sinx_n| +|cos(x₁ + x₂ + . .  .  +x_n)| ≥ 1

Prove that  for all positive integers.

Let n ≥ 6 be an integer. Show that
 

Let a₁, a₂ .  .  . a_n be real numbers greater than 1 then prove that
 

Let f : ℝ → ℝ be a function satisfying

  for any x₁, x₂.

Prove that
 

Show that if a₁, a₂ .  .  . a_n are non - negative numbers then
 

Let f(x) = |x – 1|, then

a)

b)

c)

d) None of these

The domain of definition of the function  is

a)  excluding  

b) [0, 1] excluding 0.5

c)  excluding x = 0

d) None of these

Let , then the set  is

a)  

b)  

c)  

d)  ϕ