Question(s) from Search: IIT

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

If  and , then constants A and B are

a)

b)

c)

d)

If the vectors  form sides BC, CA and AB respectively of a triangle ABC then

a)

b)

c)

d)

Cards are drawn one by one at random from a well shuffled pack of 52 playing cards until 2 aces are drawn for the first time. If N is the number of cards required to be drawn show that
 where 2 < n ≤ 50

If the lengths of the sides of a triangle are 3, 5, 7 then the largest angle of the triangle is

a)

b)

c)

d)

If y = y (x) and it follows the relation xcosy + ycosx = π then  is

a) – 1

b) π

c) – π

d) 1

Let f be a positive function. Let
 
 where
2k – 1 > 0 then  is

a) 2

b) k

c)

d) 1

Let
,
then  depends on

a) Only x

b) Only y

c) Neither x nor y

d) Both x and y

In a multiple choice question there are four alternative answers out of which one or more is correct. A candidate will get full marks in the question only if he ticks the correct answers. If he is allowed up to three chances to answer the question, find the probability that he will get marks in the question?

Let the Harmonic Mean and Geometric Mean of two positive numbers be in the ratio of 4:5. Then the two numbers are in the ratio .  .  .  .  .

If f (x) = , find  from first principle.

a)

b)

c)

d)