Question(s) from Search: IIT

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

If for non-zero x, where a ≠ b thenis equal to

a)

b)

c)

d)

Find the equation of the circle which passes through the point (2, 0) and whose centre is the limit of the point of intersection of the lines .