Find the area of the greatest rectangle that can be inscribed in the ellipse   +  = 1

A right circular cylinder is inscribed in a right circular cone. Show that the curved surface of the cylinder will be maximum when the diameter of the cylinder is equal to the radius of the base of the cone.

If α and β be the roots of ax² + bx + c = 0 then

 is

a) ln a(α – β)

b)

c)

d) None of these

The point representing complex number z for which
 |z – 3| = |z – 5| lies on the locus given by

a) an ellipse

b) a circle

c) a straight line

d) none of these

If a + b + c = 0 then the equation 3ax² + 2bx + c = 0 has

a) At least one root in (0, 1)

b) One positive and one negative root

c) No real roots

d) None of these

Let C be the curve  . If H is the set of points on the curve C when the tangent is horizontal and v be the set of all points on the curve C when the tangent is vertical then H  =  . . . . .  and v = . . . . .

The centre of the circle passing through (0, 1) and touching the curve  at (2, 4) is

a)

b)

c)

d) None of these

A point P is given on the circumference of a circle of radius r. Chord QR is parallel to the tangent at P. Determine the maximum possible area of ΔPQR.

Let a, b, c be non-zero real numbers such that
 
 
Then the quadratic function  has

a) no root in (0, 2)

b) at least one root in (1, 2)

c) a double root in (0, 2)

d) two imaginary roots

The area bounded by the curve y = f(x), the X–axis and the ordinate x = 1 and x = b is (b – 1) sin (3b + 4). Then f(x) is

a) (x – 1) cos (3x + 4)

b) sin(3x + 4)

c) sin(3x + 4) + 3(x – 1) cos (3x + 4)

d) none of these

AB is a diameter of a circle and C is any point on the circumference of the circle. Then

a) The area of △ABC is maximum if it is isosceles

b) The area of △ABC is minimum if it is isosceles

c) The perimeter of △ABC is minimum when it is isosceles

d) None of these

If  has its extremum value at x = 1 and x = 2, then

a) a = 2, b = 1

b) a = 2,

c) a =  2,

d) None of these

Let  be a polynomial in a real variable x with 0< then the function p(x) has

a) neither maximum nor minimum

b) only one maximum

c) only one minimum

d) only one maximum and only one minimum

e) none of these

Which of the following curves cut the parabola  at right angles?

a)

b)

c)

d)

The function defined by  is

a) Decreasing for all x

b) Decreasing in  and increasing in

c) Increasing for all x

d) Decreasing in  and increasing in  

The function  is

a) Increasing on (0, ∞)

b) Decreasing on (0, ∞)

c) Increasing on  and decreasing on  

d) Increasing on  and decreasing on

Let P(asecθ, btanθ) and Q(asecɸ, btanɸ) where θ + ɸ =  be two points on the hyperbola . If (h, k) be the point of intersection of the normals at P and Q then k is equal to

a)

b)

c)

d)

The function  increases if

a)

b)

c)

d)

Consider the following Statement (S) and Reason (R)

S: Both sinx, cosx are decreasing functions in the interval

R: If a differentiable function decreases in an interval (a, b) then the derivative also decreases in (a, b)

Which of the following is true?

a) Both S and R are wrong

b) Both S and R are correct but R is not the correct explanation of S

c) S is correct and R is the correct explanation of S

d) S is correct and R is wrong

Let

Then at x = 0, f has

a) A local maximum

b) No local maximum

c) A local minimum

d) No extremum

The triangle formed by the tangent to the curve

  at (1, 1) and the coordinate axes, lies in the first quadrant if its area is 2. Then the value of b is

a) – 1

b) 3

c) – 3

d) 1

If  then f(x) is

a) Increasing on

b) Decreasing on ℝ

c) Increasing on ℝ

d) Decreasing on

Tangent is drawn to the ellipse  at  where  . Then the value of θ such that the sum of intercept on the axes made by the tangents is minimum is

a)

b)

c)

d)