2
###
Let k be an integer such that the triangle with vertices (k, –3k), (5, k) and (–k, 2)

Let k be an integer such that the triangle with vertices (k, –3k), (5, k) and (–k, 2) has area 28 sq. units. Then the orthocentre of this triangle is at the point

- 2, -1/2
- 1, 3/4
- 1, -3/4
- 2, 1/2

3
###
If S is the set of distinct values of 'b' for which

If S is the set of distinct values of 'b' for which the following system of linear equations

x + y + z = 1

x + ay + z = 1

ax + by + z = 0

has no solution, then S is:

- an empty set
- an infinite set
- a finite set containing two or more elements
- a singleton

4
###
If the system of linear equations x+2ay+az=0

If the system of linear equations x + 2ay + az = 0, x + 3by + bz = 0, x + 4cy + cz = 0 has a non-zero solution, then a, b, c

- satisfy a + 2b + 3c
- are in G.P
- are in A.P
- are in H.P

5
###
If A and B are square matrices of size n × n such that

If A and B are square matrices of size n × n such that A^{2} – B^{2} = (A – B)(A + B), then which of the following will be always true?

- either A or B is an identity matrix
- either A or B is a zero matrix
- AB = BA
- A = B

6
###
Let A be a square matrix all of whose entries are integers

Let A be a square matrix all of whose entries are integers. Then which one of the following is true?

- If det A = ± 1, then A
^{-1}exists and all its entries are integers - If det A = ± 1, then A
^{-1}exists and all its entries are non-integers - If det A = ± 1, then A
^{-1}exists and all its entries are not necessarily integers - If det A = ± 1, then A
^{-1}need not exist

7
###
Let P and Q be 3 × 3 matrices with P ≠ Q

Let P and Q be 3 × 3 matrices with P ≠ Q. If P^{3} = Q^{3} and P^{2}Q = Q^{2}P, then determinant of (P^{2} + Q^{2}) is equal to

- 0
- 1
- -1
- -2

8
###
If A^2 – A + I = 0, then the inverse of A is

If A^{2} – A + I = 0, then the inverse of A is

- A - I
- A
- I + A
- I - A

9
###
If A and B given, then what is determinant of AB

If \( A = \begin{bmatrix}1 & 2 \\2 & 3 \end{bmatrix} \) and \( B = \begin{bmatrix}1 & 0 \\1 & 0 \end{bmatrix} \) then what is determinant of AB?

- 0
- 1
- 10
- 20

10
###
A and B are two matrices such that AB = A and BA = B then what is B^2 equal to

A and B are two matrices such that AB = A and BA = B then what is B^{2} equal to?

- B
- A
- I
- -I

11
###
The system of linear equations has a non-trivial solution for

The system of linear equations

x + λy – z = 0

λx – y – z = 0

x + y – λz = 0

has a non-trivial solution for:

- exactly one value of λ.
- exactly two values of λ.
- exactly three values of λ.
- infinitely many values of λ.