Question(s) from Search: IIT

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

Let A =

 
AU₁ =  , AU₂ =  and AU₃ =

a) −1

b) 0

c) 1

d) 3

If f : [1, ∞) → [2, ∞) is given by  then  equals

a)

b)

c)

d)

For the primitive differential equation
 

then  is

a) 3

b) 5

c) 1

d) 2

Consider the system of linear equations
 
 
 
Find the value of θ for which the systems of equations have non-trivial solutions.

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

Let a, b, c, d be real numbers in geometric progression. If u, v, w satisfy the system of equations

 
 
 
Then show that the roots of the equation
 
 
and  are reciprocal of each other.

Subjective Problems
Let f (x + y) = f (x) . f (y) for all x, y. Suppose f (5) = 2 and  = 3. Find f (5).

Find the natural number a for which  where the function f satisfies the relation f(x + y) = f(x) f(y) for all natural numbers x and y and further f(1) = 2.

The interior angles of a polygon are in Arithmetic Progression. The smallest angle is 120° and the common difference is 5. Find the number of sides of the polygon.

If where a > 0 and n is a positive integer then f(f(x)) = x.

a) True

b) False

A vector a has components 2p and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If with respect to new system a has components p + 1 and 1 then

a) p ≠ 0

b) p = 1 or p =

c) p = −1 or

d) p = 1 or p = −1

e) None of these

The domain of the function  is

If f is an even function defined on (−5, 5) then the four real values of x satisfying the equation  are

Let a₁, a₂, … a_n be positive real numbers in Geometric Progression. For each n let A_n, G_n, H_n be respectively the arithmetic mean, geometric mean and harmonic mean of a₁, a₂, .  .  .  ., a_n. Find the expressions for the Geometric mean of G₁, G₂, .  .  .  .G_n in terms of A₁, A₂, .  .  .  .,A_n; H₁, H₂, .  .  .  .H_n

Let  , 0 < x < 2 are integers m ≠ 0, n > 0 and let p be the left hand derivative of |x − 1| at x = 1. If , then

a) n = −1, m = 1

b) n = 1, m = −1

c) n = 2, m = 2

d) n > 2, n = m

For three vectors  which of the following expressions is not equal to any of the remaining three

a)

b)

c)

d)

If total number of runs scored in n matches is
 where n > 1 and the runs scored in the k^th match are given by k.2^{n + 1 – k}  where 1 ≤ k ≤ n. Find n.