The domain of the function  is

If  then the domain of f(x) is

If  and  then (gof)(x) is equal to

If f is an even function defined on (−5, 5) then the four real values of x satisfying the equation  are

Let a, b, c be real numbers. Then the following system of equations in x, y, z

  + −  = 1

  − +  = 1

−  + +  = 1  has

a) No solution

b) Unique solution

c) Infinitely many solutions

d) Finitely many solutions

Let A and B be square matrices of equal degree, then which one is correct amongst the following

a) A + B = B + A

b) A + B = A – B

c) A – B = B – A

d) AB = BA

The parameter on which the value of the determinant
Δ =
does not depend upon is

a) a

b) p

c) d

d) x

If  P =  , A =  and Q = PAP^T

then P^T (Q²⁰⁰⁵) P is equal to

a)

b)

c)

d)

For what value of m does the system of equations 3x + my = m, 2x − 5y = 20 have a solution satisfying the condition x > 0, y > 0.

a) m  (−∞, ∞)

b) m  (−∞, −15) ∪ (30, ∞)

c)  

d)  

Show that the system of equations
3x – y + 4z = 3
x + 2y − 3z = −2
6x + 5y + λz = −3
has at least one solution for any real number λ ≠ −5. Find the set of solutions if λ = −5

a)

b)

c)

d)

If α is a repeated root of a quadratic equation f(x) = 0 and A(x), B(x), C(x) be polynomials of degree 3, 4, 5 respectively, Then show that
 

is divisible by f(x) where prime denotes the derivatives.

Show that  =

For all A, B, C, P, Q, R show that
 = 0

Prove that for all values of θ
 = 0

Let a, b, c be real numbers with a² + b² + c² = 1. Show that the equation represents a straight line
 = 0

If M is a 3 x 3 matrix where det (M) = 1 and MM^T = I, then prove that det (M – I) = 0.

A = , B = , U = , V =

If AX = U has infinitely many solutions, prove that BX = V has no unique solution. Also prove that if afd ≠ 0 then BX = V has no solution. X is a vector.

If  
then the two triangles with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃), and (a₁, b₁), (a₂, b₂), (a₃, b₃) must be congruent.

a) True

b) False

Let A =
 
AU₁ =  , AU₂ =  and AU₃ =
 

a) 3

b) −3

c)  

d) 2

Let A =

 
AU₁ =  , AU₂ =  and AU₃ =

a) −1

b) 0

c) 1

d) 3

Multiple choices

If the first and  term of an Arithmetic Progression, a Geometric Progression and a Harmonic Progression are equal and their n^th term are a, b, c respectively then

a)

b)

c)

d)

The value of . Given that a, x, y, z, b are in Arithmetic Progression while the value of . If a, x, y, z, b are in Harmonic Progression then find a and b.

The interior angles of a polygon are in Arithmetic Progression. The smallest angle is 120° and the common difference is 5. Find the number of sides of the polygon.