Question(s) from Search: IIT

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 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)

Let

Then at x = 0, f has

a) A local maximum

b) No local maximum

c) A local minimum

d) No extremum

Let C be any circle with centre (0, . Prove that at the most two rational points can be there on C (A rational point is a point both of whose coordinates are rational numbers).

Let   then the real roots of the equation

 are

a) ± 1

b)

c)

d) 0 and 1

Consider a family of circles . If in the first quadrant, the common tangent to a circle of the family and the ellipse  meet the coordinate axes at A and B, then find the locus of the mid-point of AB.

The area bounded by the curves  

and the X–axis in the first quadrant is

a) 9

b)

c) 36

d) 18

Find the point on   which is nearest to the line

If f(x) = x^a lnx and f(0) = 0 then the value of a for which Rolle’s theorem can be applied in [0, 1] is

a) – 2

b) – 1

c) 0

d)

The points of intersection of the line  and the circle  is . . . . . 

Multiple choice

For which value of m, is the area of the region bounded by the curve y = x –x² and the line y = mx equal to

a) – 4

b) – 2

c) 2

d) 4

The equation of the line passing through the points of intersection of the circles
 and
 is . . . . .

If the triangle  another circle C₂ of radius 5 in such a manner that the common chord is of maximum length and a slope equal to  , then the coordinates of the centre of C₂ are . . . . .

The equation of the locus of the midpoints of the chord of the circle  that subtends an angle of  at the centre is  . . . . .

Find the area bounded by the X–axis, part of the curve  and the ordinates at x = 2 and x = 4. If the ordinate x = a divides the area in two equal parts, find a.

a)

b)

c)

d)

The chord of contact of the pair of tangents drawn from each point on the line  to the circle  passes through the point . . . . .

Find the tangents to the curve
y = cos(x + y), − 2π ≤ x ≤ 2π
that are parallel to the line x + 2y = 0

Evaluate   where n is a positive integer and t is a parameter independent of x.

a)

b)

c)

d)

 is equal to

a) 0

b)

c)

d) None of these

Find the area bounded by the X - axis, part of the curve  and the ordinate at x = 2 and x = 4. If the ordinate at x = a divide the area into two equal parts, find a,

a)

b)

c)

d)

The value of  is   

a) 1

b) – 1

c) 0

d) None of these

Evaluate

a)

b)

c)

d)

a) exists and equals

b) exists and equals

c) does not exist because x – 1 → 0

d) does not exist because the left hand limit is not equal to the right hand limit.

Evaluate

a) πln2

b)

c)

d)

If  where n is a non–zero real number, then a is equal to

a) 0

b)

c) n

d)