Question(s) from Search: IIT

The minimum value of the expression  where  are real numbers satisfying  is

a) Positive

b) Zero

c) Negative

d) –3

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)

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.

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.

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

For the primitive differential equation
 

then  is

a) 3

b) 5

c) 1

d) 2

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

If  P =  , A =  and Q = PAP^T

then P^T (Q²⁰⁰⁵) P is equal to

a)

b)

c)

d)

Show that the system of equations
3x – y + 4z = 3
x + 2y − 3z = −2
6x + 5y + λz = −3
has at least one solution for any real number λ ≠ −5. Find the set of solutions if λ = −5

a)

b)

c)

d)

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

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

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.

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

a) True

b) False

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  , 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

The area of the equilateral triangle which contains three coins of unit radius is

a)  square units

b)  square units

c)  square units

d)  square units

a) True

b) False

a) True

b) False