Question(s) from Search: IIT

 and
where α, β ε [π, π]. Values of α, β which satisfy both the equations is/are

a) 0

b) 1

c) 2

d) 4

Given positive integers r > 1, n > 2 and the coefficients of (3r)^th term and (r + 2)^th terms in the binomial expansion of (1 + x)²ⁿ are equal then

a) n = 2r

b) n = 2r + 1

c) n = 3r

d) none of these

Let E be the ellipse  and C be the circle . Let P and Q be the points (1, 2) and (2, 1) respectively. Then

a) Q lies inside C but outside E

b) Q lies outside both C and E

c) P lies inside both C and E

d) P lies inside C but outside E

If , then  

a) True

b) False

If in the expansion of (1 + x)^m (1 – x)ⁿ, the coefficients of x and x² are 3 and –6 respectively. then m is

a) 6

b) 9

c) 12

d) 24

If tangents are drawn to the ellipse  then the locus of the mid-points of the intercepts made by the tangents between the coordinate axes is

a)

b)

c)

d)

Suppose  is an identity in x where  are constants and . Then the value of n = ……….

a) 4

b) 5

c) 6

d) 7

Prove that  is divisible by 25 for any natural number n.

Let P be a variable point on the ellipse  with foci F₁ and F_2. . If A is the area of  then the maximum value of A is  . . . . .

The equation  represents

a) An ellipse

b) A hyperbola

c) A circle

d) None of these

The equation  has

a) No solution

b) One solution

c) More than one real solution

d) Cannot be said

For the hyperbola  which of the following remains constant with change in α

a) Abscissae of vertices

b) Abscissae of focii

c) Eccentricity

d) Directrix

The number of solutions of the equation

a) 0

b) 1

c) 2

d) Infinitely many

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

India played two matches each with Australia and West indies. In any match the probability of India getting the points 0, 1, and 2 are 0.45, 0.05 and 0.50 respectively. Assuming that the outcomes are independent, the probability of India getting at least seven points is

a) 0.8730

b) 0.0875

c) 0.0625

d) 0.0250

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
 

Three of the vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral equals

a)

b)

c)

d)

If  are complementary events E and F respectively and if 0 < p(E) < 1, then

a)

b)

c)

d)

The larger of 99⁵⁰ + 100⁵⁰ and 101⁵⁰  is

The numbers are selected from the set S = {1, 2, 3, 4, 5, 6} without replacement one by one. Probability that the minimum of the two numbers is less than 4 is

a)

b)

c)

d)

One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife 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 ]