Question(s) from Search: IIT

If  is the unit vector along the incident ray,  is a unit vector along the reflected ray and is a unit vector along the outward drawn normal to the plane mirror at the point of incidence. Find  in terms of  and

True / False

For any three vectors a, b and c
 

a) True

b) False

Multiple choices
For a positive integer n, let
 
.  .  . then

a)

b)

c)

d)

For all ,

a) True

b) False

Let f (x) = |x – 1| then

a) f (x²) = |f (x)|²

b) f (x + y) = f (x) + f (y)

c) f () = |f (x)|

d) None of these

Let the vectors represent the edges of a regular hexagon

Statement 1 -  because

Statement 2 -

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

b) Statement 1 and 2 are true 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.

Find the smallest possible value of p for which the equation
 

a)

b)

c)

d)

If f (x) =  for every real x then the minimum value of f

a) does not exist because f is unbounded

b) is not attained even though f is bounded

c) is equal to 1

d) is equal to −1

Find the larger of cos(lnθ) and ln(cosθ) if  < θ < .

a) cos(lnθ)

b) ln(cosθ)

c) Neither is larger throughout the interval

If the function f : [ 1,  ) → [ 1,  ) is defined by f (x) = 2^{x(x – 1)} then
f ^-1(x) is

a)

b)  ()

c)  ()

d)

If are in harmonic progression then  …………

a) 1

b)

c)

d)

If

then x equals

a)

b) 1

c)

d) –1

Let f ( x ) = , x ≠ 1 then for what value of a is f ( f (x)) = x

a)

b)

c) 1

d) 1

If f : [ 0,  )  [ 0,  ) and f (x) =  then f is

a) one-one and onto

b) one-one but not onto

c) onto but not one-one

d) neither one-one nor onto

Match the following

Let (x, y) be such that

 =

f (x) =
and g (x) =
 

a) neither one-one nor onto

b) one-one and onto

c) one-one and into

d) many one and onto

One angle of an isosceles triangle is 120 and the radius of its incircle = . Then the area of the triangle in square units is

a)

b)

c)

d) 2π

In a triangle of base a the ratio of the other two sides is  r (< 1). Then the altitude of the triangle is less than or equal to  .

a) True

b) False

The value of k such that  lies in the plane
  is

a) 7

b) – 7

c) No real value

d) 4

If ABCD are four points in a space, prove that

If a, b, c are distinct positive numbers then the expression
( b + c – a ) ( c + a – b ) ( a + b – c ) –abc is

a) Positive

b) Negative

c) Non–positive

d) None of these

Let A and B be square matrices of equal degree, then which one is correct amongst the following

a) A + B = B + A

b) A + B = A – B

c) A – B = B – A

d) AB = BA

The edges of a parallelepiped are of unit length and are parallel to non-coplanar unit vectors  such that . Then the volume of the parallelepiped is

a)

b)

c)

d)

If  P =  , A =  and Q = PAP^T

then P^T (Q²⁰⁰⁵) P is equal to

a)

b)

c)

d)

Consider three planes
P₁ : x – y + z = 1

P₂ : x + y – z = −1

P₃  : x – 3y + 3z = 2

Let L₁, L₂, L₃ be lines of intersection of planes P₂ and P₃, P₃ and P_1, and P₁ and P₂ respectively.

Statement 1 – At least two of the lines L₁, L₂, L₃ are non parallel

Statement 2 – The three planes do not have a common point.

a) Statement 1 is true. Statement 2 is true. Statement 2 is a correct explanation of statement 1.

b) Statement 1 is true. Statement 2 is true. 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.