Question(s) from Search: IIT

Evaluate   where n is a positive integer and t is a parameter independent of x.

a)

b)

c)

d)

Let OABC be a parallelogram with O as the origin and OC a diagonal. Let D be the midpoint of OA. Using vector method, prove that BD and CO intersect in the same ratio.

For positive integers n₁ and n₂ the value of the expression  where  is real if and only if

a)

b)

c)

d)

 is equal to

a) 0

b)

c)

d) None of these

Find the area bounded by the X - axis, part of the curve  and the ordinate at x = 2 and x = 4. If the ordinate at x = a divide the area into two equal parts, find a,

a)

b)

c)

d)

Determine the value of c so that for all real x the vector cx and  make an obtuse angle with each other.

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)