Question(s) from Search: IIT

Show that  =

Let p and q be the position vectors of P and Q respectively with respect to O and . The points R and S divide PQ internally and externally in the ratio 2:3 respectively. If OR and OS are perpendicular then

a)

b)

c)

d)

Let u, v and w be vectors such that  . If  then  is equal to

a) 47

b) –25

c) 0

d) 25

(One or more correct answers)
There are four machines and it is known that exactly two of them are faulty. They are tested one by one, in a random order till both the faulty machines are identified. Then the probability that only two tests are needed

a)

b)

c)

d)

(Subjective problem)

Solve  
where a > 0, b = a²x.

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)

If A, B, C be events such that P(A) = 0.3, P(B) = 0.4, P(C) = 0.8, P(AB) = 0.08, P(AC) = 0.28, P(ABC) = 0.09 and P(A ∪ B ∪ C) ≥ 0.75, then show that P(BC) lies in the interval [0.23, 0.48].

If ω be the cube root of unity then the value of
 is

a)

b)

c)

d)

If  then the value of f(1) is

a)

b) 0

c) 1

d)

The value of a so that the volume of parallelopiped formed by  becomes minimum is

a)  

b)  3

c)  

d)  

 is true if

a) x + y= 0

b) x = y, x ≠ 0

c) x = y

d) x ≠ 0, y ≠ 0

Given  find

Let , where f is such that  and  then g(2) satisfies the inequality

a)

b)

c)

d)

Suppose the probability for A winning a game against B is 0.4. If A has an option of playing either a best of 3 games or best of 5 games match against B, which option should he choose so that the probability of his winning the match is higher.

Coefficient of x⁴ in  is

a)

b)

c)

d) None of these

If y =  

Prove that  

The value of  is

a) π

b) aπ

c)

d) 2π

Let  be unit vectors such that . Which one of the following is correct

a)

b)

c)

d)  are mutually perpendicular

The complex number z = x + iy which satisfies the equation  lies on

a) The real axis

b) The straight line y = 5

c) Circle passing through origin

d) None of these

In the binomial expansion of  the sum of the 5^th term and 6^th term is zero, then  equals

a)

b)

c)

d)