Given 2x – y – z = 2, x – 2y + z = − 4, x + y + λz = 4 then the value of λ such that the given system of equations has no solution is

a) 3

b) −2

c) 0

d) −3

(Multiple choices)
The determinant
  is equal to zero if

a) a, b, c are in arithmetic progression

b) a, b, c are in geometric progression

c) a, b, c are in harmonic progression

d) α is a root of the equation ax² + bx + c = 0

e) x – α is a factor of ax² + 2bx + c

Given x = cy + bz, y = az + cx, z = bx + ay where x, y, z are not all zero, prove that  a² + b² + c² + 2abc = 1

For what values of k does the following system of equations possess a non-trivial solution over the set of rationals? Find all the solutions.

x + y – 2z = 0

2x – 3y + z = 0

x – 5y + 4z = k

If a, b, c be positive and not all equal, show that the value of the determinant  is negative.

Without expanding a determinant at any stage show that
 = Ax + B

where A, B are non-zero constants

Let Δa =
Then show that  = c, a constant.

Let the three digit numbers A28, 3B9 and 62C where A, B, C are integers between 0 and 9, be divisible by a fixed number k. Show that the determinant
 

is divisible by k.

If a ≠ p, b ≠ q, c ≠ r and
 = 0

Then find the value of
  +  +

a) 0

b) 1

c) 2

d) 3

For a > 0, d > 0, find the value of the determinant
 

a) 0

b) 1

c)

d)

Find the value of the determinant

where a, b, c are respectively p^th, q^th and r^th term of a harmonic

progression.

a) 0

b) 1

c) ½

d) None of the above

If the matrix A is equal to where a, b, c are real positive numbers, abc = 1 and A^TA = I then find the value of a³ + b³ + c³.

a) 1

b) 2

c) 3

d) 4

The determinants   are.

a) Identical

b) Not identical

c) Identical if a = b = c

d) None of the above

Let pλ⁴ + qλ³ + rλ² + sλ + t =  be an identity in λ where p, q, r, s, t are constants. Find the value of t.

a) 0

b) +1

c) –1

d) ±1

The solution set of equation  = 0 is ……….

a) {0}

b) {1, 2}

c) {−1, 2}

d) {−1, −2}

Given that x = −9 is a root of  = 0

.

a) {2, 7}

b) {−2, −7}

c) {2, 0}

d) {0, 7}

A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the value of the determinant chosen is positive is

a)

b)

c)

d)

The value of the determinant  is …………

a) 0

b) 1

c) a² + b² + c² – abc

d) a² + b² + c² – 3abc

For positive numbers x, y and z the numerical value of the determinant
 is ………..

a) 1

b) –1

c) ±1

d) 0

Find the domain of definition of the function
 

a) x ε (0, 5)

b) x ε [1, 4]

c) x ε (0, 1) ∪ (1, 4)

d) x ε [− 4, 4]

Find the domain of definition of the function
 

a) x ε (−∞, ∞)

b) x ε (−∞, 2) ∪ (2, ∞)

c) x ε (−∞, 2) ∪ (3, ∞)

d) x ε (2, 3)

Find the domain of definition of the function
 

a) x ε (0, ∞)

b) x ε (0, 1) ∪ (1, ∞)

c) x ε (1, ∞)

d) x ε (0, 1]

Find the domain of definition of the function
 

a) x ε [1, 4)

b) x ε (1, 4)

c) x ε [1, 5]

d) x ε (−∞, 4)