Question(s) from Search: IIT

Find the integral solutions of the following system of inequality
 

a) x = 1

b) x = 2

c) x = 3

d) x = 4

Area bounded by  and

mn squares of equal size are arranged to form a rectangle of dimension m by n, where m and n are natural numbers. Two squares will be called neighbours if they have exactly one common side. A natural number is written in each square such that the number written in any square is the arithmetic mean of the numbers written in the neighbouring squares. Show that this is possible only if all the numbers used are equal.

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
 

If A, B, C are such that |B| = |C|. Prove that

Let u and v be unit vectors. If w is a vector such that , then prove that  and that equality holds if and only if  is perpendicular to

Let n be an odd integer. If sin nθ =  for every value of θ, then

a) = 1, = 3

b) = 0, = n

c) = −1, = n

d) = 1, =

The points with position vectors  and  are collinear for all real values of k.

a) True

b) False