Question(s) from Search: IIT

The value of the integral
 is

a) sin⁻¹ x – 6tan⁻¹(sin⁻¹ x) + c

b) sin⁻¹x – 2(sinx)⁻¹ + c

c) sin⁻¹x – 2(sinx)⁻¹ − 6tan⁻¹(sin⁻¹x) + c

d) sin⁻¹x – 2(sinx)⁻¹ + 5tan⁻¹(sin⁻¹x) + c

Three identical dice are rolled. The probability that the same number will appear on each of them is

a)

b)

c)

d)

Let  be in Arithmetic Progression and
 be in Harmonic Progression. If  and
 then  is

a) 2

b) 3

c) 5

d) 6

Integrate

a)

b)

c)

d)

An unbiased die with faces marked 1, 2, 3, 4, 5 and 6 is rolled 4 times. Out of four face values obtained, the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5 is

a) 16/81

b) 1/81

c) 80/81

d) 65/81

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