Question(s) from Search: IIT

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

The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Determine the sides of triangle.

a) 3, 4, 5

b) 4, 5, 6

c) 4, 5, 7

d) 5, 6, 7

A plane which is perpendicular to two planes  and  passes through (1, −2, 1). The distance of the plane from the point (1, 2, 2) is

a) 0

b) 1

c)

d)

Two lines having direction ratios (1, 0, −1) and (1, −1, 0) are parallel to a plane passing through (1, 1, 1). This plane cuts the coordinate axes at A, B, C. Find the value of the tetrahedron OABC.

Let a, b, c be real numbers. Then the following system of equations in x, y, z

  + −  = 1

  − +  = 1

−  + +  = 1  has

a) No solution

b) Unique solution

c) Infinitely many solutions

d) Finitely many solutions

Consider the lines

 ;

 
The distance of the point (1, 1, 1) from the plane through the point (−1, −2, −1) and whose normal is perpendicular to both lines L₁ and L₂ is

a)

b)

c)

d)

The domain of definition of the function  is

a)  excluding  

b) [0, 1] excluding 0.5

c)  excluding x = 0

d) None of these

A curve  passes through  and the tangent at  cuts the X-axis and Y-axis at A and B respectively such that then

a) Equation of the curve is

b) Normal at  is

c) Curve passes through

d) Equation of the curve is

Let y = f (x) be a curve passing through (1, 1) such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the first quadrant and has area 2. Find the differential equation and determine all such possible curves.

If  
then the two triangles with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃), and (a₁, b₁), (a₂, b₂), (a₃, b₃) must be congruent.

a) True

b) False

If  then

a)

b)

c)

d) f and g cannot be determined

The number of values of x where the function  attains its maximum is

a) 0

b) 1

c) 2

d) infinite

The domain of the definition of the function y given by the equation  is

a) 0 < x < 1

b) 0 ≤ x ≤ 1

c) ∞ < x ≤ 0

d) ∞ < x ≤ 1

Solution of the differential equation is

Let A =

If U₁, U_2, U₃ are column matrices satisfying
AU₁ =  , AU₂ =  and AU₃ =

and U is a 3 x 3 matrix whose columns are U₁, U_2, U₃  then the value of [ 3  2  0 ] U  is

a)

b)

c)

d)

Let f(x) =   , x ≠  then for what value of α, f(f(x)) = x

a)

b)

c)

d)

If  and  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

If
and
Then f – g is

a) Neither one to one nor onto

b) One to one and onto

c) One to one and into

d) Many one and onto