Question(s) from Search: IIT

Let A =
 
AU₁ =  , AU₂ =  and AU₃ =
 

a) 3

b) −3

c)  

d) 2

The domain of definition of  is

a)  

b)  

c)  

d)  

Let f : ℝ → ℝ be defined by f(x) = 2x + sinx for all x  ℝ. Then f is

a) One to one and onto

b) One to one but not onto

c) Onto but not one to one

d) Neither one to one nor onto

Range of    ;   x  ℝ is

a) (1, ∞)

b)

c)

d)

Let a, b, c, ε R and α, β be roots of  such that  and  then show that .

If  where
. Given F(5) = 5, then f(10) is equal to

a) 5

b) 10

c) 0

d) 15

Subjective problems
Let .  Find all real values of x for which y takes real values.

a) [− 1, 2)

b)  [3, ∞)

c) [− 1, 2) ∪ [3, ∞)

d) None of the above

Let R be the set of real numbers and f : R → R be such that for all x and y in R, . Prove that f(x) is constant.

If f₁(x) and f₂(x) are defined by domains D₁ and D₂ respectively then f₁(x) + f₂(x) is defined as on D₁ ⋂ D₂

a) True

b) False

If  then the domain of f(x) is

The real numbers x₁, x₂, x₃ satisfying the equation x³ – x² + βx + γ = 0 are in Arithmetic Progression. Find the interval in which β and γ lie.

Let p, q, r be three mutually perpendicular vectors of the same magnitude. If x satisfies the equation p  ((x – q)  p) + q ((x – r)  q) + r  ((x – p)  r) = 0 then x is given by

a)

b)

c)

d)

Let f(x) be a non constant differentiable function defined on (−∞, ∞) such that f(x) = f(1 – x) and  then

a)  vanishes at twice an (0, 1)

b)

c)

d)

Let and a unit vector c be coplanar. If c is perpendicular to a then c is equal to

a)

b)

c)

d)

Number of solutions of  lying in the interval  is

a) 0

b) 1

c) 2

d) 3

If three complex numbers are in Arithmetic Progression, then they lie on a circle in a complex plane.

a) True

b) False

Multiple choice

The vector  is

a) A unit vector

b) Makes an angle  with the vector

c) Parallel to vector

d) Perpendicular to the vector

A₁, A₂, …… , A_n are the vertices of  a regular polygon with n sides and O is the centre. Show that
 

Let  be non–coplanar unit vectors equally inclined to one another at an angle θ. If find p, q, r in terms of θ

If  is the unit vector along the incident ray,  is a unit vector along the reflected ray and is a unit vector along the outward drawn normal to the plane mirror at the point of incidence. Find  in terms of  and

True / False

For any three vectors a, b and c
 

a) True

b) False

Multiple choices
For a positive integer n, let
 
.  .  . then

a)

b)

c)

d)

For all ,

a) True

b) False

Let f (x) = |x – 1| then

a) f (x²) = |f (x)|²

b) f (x + y) = f (x) + f (y)

c) f () = |f (x)|

d) None of these

Let the vectors represent the edges of a regular hexagon

Statement 1 -  because

Statement 2 -

a) Statement 1 and 2 are true and Statement 2 is a correct explanation of statement 1.

b) Statement 1 and 2 are true and Statement 2 is not a correct explanation of statement 1.

c) Statement 1 is true. Statement 2 is false.

d) Statement 1 is false. Statement 2 is true.