Question(s) from Search: IIT

The equation  has

a) No solution

b) One solution

c) More than one real solution

d) Cannot be said

The value of  is   

a) 1

b) – 1

c) 0

d) None of these

Evaluate

a)

b)

c)

d)

The number of solutions of the equation

a) 0

b) 1

c) 2

d) Infinitely many

a) exists and equals

b) exists and equals

c) does not exist because x – 1 → 0

d) does not exist because the left hand limit is not equal to the right hand limit.

The number of values of x in the interval (0, 5π) satisfying the equation  is

a) 0

b) 5

c) 6

d) 10

Find the natural number a for which
 
where the function f satisfies the relation f (x + y) = f (x).f(y)for all natural numbers x and y and further f (1) = 2

The lines  and  are diameters of a circle of area 154 square units. Then the equation of the circle is

a)

b)

c)

d)

If α + β =  and β + γ = α, then tanα equals

a) 2(tanβ + tanγ)

b) tanβ + tanγ

c) tanβ + 2tanγ

d) 2tanβ + tanγ

Let n be a positive integer and
(1 + x + x²)ⁿ = a₀ + a₁x + a₂x + a₂x² +  .  .  . + a_2nx²ⁿ then prove that
 

Evaluate

a) πln2

b)

c)

d)

The angle between a pair of tangents drawn from a point P to the circle  is 2α. Then the locus of P is

a)

b)

c)

d)

If  where n is a non–zero real number, then a is equal to

a) 0

b)

c) n

d)

If f (x) is an even function then prove that
 .

If A, B, C are three non-coplanar vectors then
 

The triangle PQR is inscribed in the circle. If Q and R  have coordinates (3, 4) and (-4, 3) respectively, then the ∠QPR is equal to

a)

b)

c)

d)

The larger of 99⁵⁰ + 100⁵⁰ and 101⁵⁰  is

The projection of a vector a along and perpendicular to a non-zero vector  are . . . . . and . . . . . respectively.

If the tangent at the point P on the circle  meets the straight line  at a point Q on the Y-axis, then the length of PˆQ is

a) 4

b)

c) 5

d)

The integral  dx where [ ] denotes the greatest integer function equals .  .   .

a)

b)  + 1

c)

d)

A non-zero vector a is parallel to the line of intersection of the plane determined by the vectors  and the plane determined by the vectors . The angle between a and  is .  .  .  .  .

A circle is given by , another circle C touches it externally and also the X-axis, then the locus of the centre of C is

a)

b)

c)

d)

Find all solutions of
 in

a)

b)

c)

d)

Let f (x) = sin x and g (x) = ln|x|. If the range of the composition

functions fog and gof are R₁ and R₂ respectively, then

a) R₁ = [ u : −1 ≤ u < 1], R₂ = [ v : − < v < 0 ]

b) R₁ = [ u : − < u < 0 ], R₂ = [ v : −1 ≤ v ≤ 0]

c) R₁ = [ u : −1 < u < 1], R₂ = [ v : − < v < 0 ]

d) R₁ = [ u : −1 ≤ u ≤ 1], R₂ = [ v : − < v ≤ 0 ]

Let f (x) =   then for all x

a)

b) f is differentiable

c)  is continuous

d) f is continuous