Question(s) from Search: IIT

If α, β are roots of  and  are roots of  for some constant δ, then prove that
 

Let the positive numbers a, b, c, d be in Arithmetic Progression. Then
abc, abd, acd, bcd are

a) Not in Arithmetic Progression/Geometric Progression/Harmonic Progression

b) In Arithmetic Progression

c) In Geometric Progression

d) In Harmonic Progression

If f(x) be the interval of  find

a) ½

b) 1

c) 2

d) 4

For the three events A, B, C,

P(exactly one of A or B occurs) = P(exactly one of B or C occurs) = P(exactly one of C or A occurs) = p and P(all the three events occur simultaneously =  where . Then the probability of at least one of A, B, C occurring is

a)

b)

c)

d)

If  is the area of a triangle with sides a, b, c then show that
 .
Also show that equality occurs if a = b = c

An infinite Geometric Progression has first term x and sum 5 then

a)

b)

c)

d)

 =

a)

b)

c)

d)

If a < b < c < d then the roots of the equation
  
are real and distinct.

a) True

b) False

The volume of the parallelopiped whose sides are given by
 

a)

b) 4

c)

d) None of these

If three distinct numbers are chosen randomly from the first 100 natural numbers then the probability that all three of them are divisible by 2 and 3 is

a)

b)

c)

d)

The angles of a triangle are in Arithmetic Progression and let . Find the angle A.

Show that
 =   where y =

The number of unit vectors perpendicular to the vectors  and  is

a) 1

b) 2

c) 3

d) Infinitely many

e) None of these

If α, β, γ are the cube roots of P, P < 0, then for any x, y, z,
………..

If n is a natural number such that  and  are distinct primes, then show that
 

For any natural number m, show that
 

Let α, β, γ be distinct real numbers. The points with position vectors
 

a) Are collinear

b) Form an equilateral triangle

c) Form a scalene triangle

d) Form a right angled triangle

The least value of the expression  for x > 1 is

a) 10

b) 2

c) −0.01

d) none of these

The probability that a student passes in Mathematics, Physics and Chemistry are m, p and c respectively. Of these subjects, the student has 75% chances of passing in at least one, a 50% chance of passing in at least two and 40% chance of passing in exactly two. Which of the following relations is true?

a)

b)

c)

d)

If  satisfies ,
find the value of

Let f : ℝ → ℝ and g : ℝ → ℝ be continuous functions. Then the value of integral  is

a) π

b) 1

c) – 1

d) 0

Let  If c is a vector such that  and the angle between  and c is 30° then  is equal to

a)

b)

c) 2

d) 3

An anti–aircraft gun can take a maximum of 4 shots at an enemy plane moving away from it. The probability of hitting the plane at the first shot, 2^nd, 3^rd and 4^th shots are 0.4, 0.3, 0.2 and 0.1 respectively. What is the probability that the gun hits the plane?

The value of the expression  is equal to

a) 2

b)

c) 4

d)

Let f(x) =  where p is a constant

Then  at x = 0 is

a) p

b)

c)

d) Independent of p