Question(s) from Search: IIT

For every positive integer n, prove that
 
Hence or otherwise prove that
 
Where [ ] denotes greatest integer not exceeding x.

If  are positive real numbers whose product is a fixed number c then the minimum value of
  is

a)

b)

c)

d)

If from each of the three boxes containing 3 white and one black; 2 white and 2 black; 1 white and 3 black balls, one ball is drawn at random then the probability that 2 white and 1 black ball will be drawn is

a)

b)

c)

d)

Let a and b the roots of the equation  and those of  are c and d, then find the value of a + b + c + d when a ≠ b ≠ c ≠ d.

 =

a) True

b) False

The scalar  equals

a) 0

b)

c)

d) None of these

Fill in the blanks

If  is a root of the equation  where p and q are real then (p, q)  …………

If the m^th, n^th and p^th term of an Arithmetic Progression and a Geometric Progression are equal and are x, y, z then prove that
 

A fair die is rolled. The probability that 1 occurs at the even number of trail is

a)

b)

c)

d)

Fill in the blank

If the quadratic equation
  and  have a common root then the numerical value of a + b is …………

Show that   =
 

Let a, b, c be distinct non-negative numbers. If the vectors   lie in a plane then c is

a) Arithmetic mean of a and b

b) Geometric mean of a and b

c) Harmonic mean of a and b

d) Equal to zero

(One or more correct answers)
For two given events A and B, P (A ∩ B) is

a) Not less than P (A) + P (B) − 1

b) Not greater than P (A) + P (B)

c) Equal to P (A) + P (B) − P (A ∪ B)

d) Equal to P (A) + P (B) + P (A ∪ B)

Fill in the blank

The sum of the real roots of the equation
 is ………..

If  are in Arithmetic Progression, determine the value of x.

Let F(x) be an indefinite integral of sin²x
Statement 1: The function F(x) satisfies F(x + π) = F(x) for all real x because
Statement 2: sin²(x + π) = sin²x for all real x

Then which one of the following statements is true?

a) Statement 1 and 2 are true statements and Statement 2 is a correct explanation of Statement 1

b) Statement 1 and 2 are true statements and statement 2 is not a correct explanation of statement 1

c) Statement 1 is true, Statement 2 is false

d) Statement 1 is false, Statement 2 is true

Let are non–coplanar unit vectors such that

 then the angle between a and b is

a)

b)

c)

d) π

The number  is

a) an integer

b) a rational number

c) an irrational number

d) a prime number

The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of four consecutive terms of it, prove that the resulting sum is square of an integer.

Let f : ℝ → ℝ be a differentiable function and f (1) = 4. Then show that the value of   =

A box contains 2 black, 4 white and 3 red balls. One ball is drawn at random from the box and kept aside from the remaining balls in the box. Another ball is drawn at random and kept besides the first. This process is repeated till all the balls are drawn from the box. Find the probability that the balls drawn are in the sequence of 2 black, 4 white and 3 red.

The expression
 
 is equal to

a) 0

b) 1

c) 3

d) sin4α + cosα

If  then  is equal to

a)

b)

c)

d)

The value of  where [.] represents the greatest integer function is

a)

b)

c)

d)

Let the vectors  be such that  . Let P₁ and P₂ be the planes determined by the pairs of vectors a, b and c, d respectively. Then the angle between P₁ and P₂ is

a) 0

b)

c)

d)