Question(s) from Search: IIT

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

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

If M is a 3 x 3 matrix where det (M) = 1 and MM^T = I, then prove that det (M – I) = 0.

Let f(x) be defined for all x > 0 and be continuous. If f(x) satisfies  for all x, y and f(e)=1 then

a) f(x) is bounded

b)

c) x f(x) → 1 as x → 0

d) f(x) = lnx

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)