Question(s) from Search: IIT

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

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

A curve passes through  and slope at the point  is

. Find the equation of the curve and the area between the

curve and the X-axis in the fourth quadrant.

Find the integral solutions of the following system of inequality
 

a) Ø

b) x = 1

c) x = 2

d) x = 3

Cosine of angle of intersection of curve y = 3^{x – 1}lnx and y = x^x – 1 is