Question(s) from Search: IIT

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)

Multiple choice

Let a and b be two non-collinear unit vectors. If  and  then  is

a) ||

b)

c)

d)

Let F(x) = f (x) g (x) h (x) for all real x, where f (x), g (x) and h (x) are differentiable functions. At some point x₀

 and

 then k is equal to

a) 12

b) 20

c) 24

d) 28

If  then f

a)

b)

c) 3

d) None

If c be a given non-zero scalar and A and B be given non-zero vectors such that A is perpendicular to B, find the vector X which satisfies the equations

The set of all values of x in the interval (0, π) for which
 is

a) (0, )

b) {  }

c) ( , π)

d) (0,   ) U {  } U ( , π)

If G(x) then  is

a)

b)

c)

d)

Show that

If a, b, c are coplanar, show that
  

 is the reflexion of in the line whose equation is . .

Use mathematical induction to prove that  is divisible by 24 for all n > 0.