Question(s) from Search: IIT

The probability of India winning a test match against West Indies is ½. Assuming independence of outcomes in each match, the probability that in a 5 test match series India’s second win will occur in the third test is

a)

b)

c)

d)

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.

If a  are linearly dependent and |c|  then

a)

b)

c)

d)

Six boys and six girls sit in a row at random. Find the probability that the girls and the boys sit alternately.

 is equal to

a)

b)

c)

d)

If a, b, c are positive real numbers then prove that
 

Let f(x) be a quadratic expression which is positive for all values of x. If g(x) =  then for any real x

a) g (x) < 0

b) g (x) > 0

c) g (x) = 0

d) g (x) ≥ 0