Question(s) from Search: IIT

The axis of the parabola is along the line  and the distance of the vertex and focus from origin are  and  respectively. If vertex and focus both lie in the first quadrant, then the equation of the parabola is

a)

b)

c)

d)

(Subjective problem)

Solve  
where a > 0, b = a²x.

Sketch the region bounded by the curves y = x² and  . Find the area.

a)

b)

c)

d)

Find the equation of the normal to the curve  which passes through the point (1, 2).

The expression
 
 is equal to

a) 0

b) 1

c) 3

d) sin4α + cosα

If ω be the cube root of unity then the value of
 is

a)

b)

c)

d)

At any point P on the parabola  , a tangent is drawn which meets the directrix at Q. Find the locus of the point R which divides QP externally in the ratio  .

 is true if

a) x + y= 0

b) x = y, x ≠ 0

c) x = y

d) x ≠ 0, y ≠ 0

Find the value of the expression

1.(2−ω)(2−+ 2.(3−ω)(3−+ … (n−1).(n−ω)(n−

where ω is an imaginary cube root of unity.

a) n(n−1)(+3n+4)

b) n(n+1)(+3n+4)

c) n(n−1)(+n+1)

d) n(n+1)(+n+1)

A solution of the differential equation  is

a)  y = 2

b)  y = 2x

c)  

d)  2

A spherical rain drop evaporates at a rate proportional to its surface area at any instant. The differential equation giving the rate of change of the radius vector of the rain drop is . . . . .

The value of  is

a) 0

b) 1

c) 2

d) 4

Let y =  Find

a)

b)

c)

d) 0

If

Then  =

a) 0

b) 1

c) 2

d) 3

The derivative of an even function is always an odd function.

a) False

b) True

The derivative of  with respect to  at x =  is

a) 0

b) 1

c) 2

d) 4

If f (x) is differentiable and  , then  equals

a)

b)

c)

d)

 equals

a)

b)

c)

d) 4 f (2)

The function  is not defined at x = 0. The value which should be assigned to f at x = 0 so that it is continuous at x = 0 is

a) a – b

b) a + b

c) lna – lnb

d) None of these

Find the value of

a)

b)

c)

d)

The set of all points where the function  is differentiable is

a)

b) [0, ∞)

c)  

d)  (0, ∞)

e)  None of these

Given a function f (x) such that
i) it is integrable over every interval on the real axis and
ii) f (t + x) = f (x) for every x and a real t, then show that the integral  is independent of a.

The function f(x) =  denotes the greatest integer function is discontinuous at

a) All x

b) All integer points

c) No x

d) x which is not an integer

If f (x) and g (x) are continuous functions on (0, a) satisfying f (x) = f (a – x) and g (x) + g (a – x) = 2 then show that

A cubic f (x) vanishes at x = −2 and has a relative minimum/maximum at x = −1 and . If , find the cube f (x).

a) x³ + x² + x + 1

b) x³ + x² − x + 1

c) x³ − x² + x + 2

d) x³ + x² − x + 2