Question(s) from Search: IIT

A tangent PT is drawn to the circle x² + y² = 4 at the point $P\left(\sqrt{3},1\right)$

. A straight line L, perpendicular to PT is tangent to the circle (x – 3)² + y² = 1A possible equation of L is

a) $x-\sqrt{3}y=1$

b) $x+\sqrt{3}y=1$

c) $x-\sqrt{3}y=-1$

d) $x+\sqrt{3}y=5$

Let 0 < A_i < π for i = 1, 2, .  .  . n. Use mathematical induction to prove that
 
where n ≥ 1 is a natural number.

The centre of those circles which touch the circle x² + y² – 8x – 8y = 0, externally and also touch the X- axis, lie on

a) A circle

b) An ellipse which is not a circle

c) A hyperbola

d) A parabola

Solve

 for every 0 < α, β < 2.

Let (x, y) be any point on the parabola y² = 4x. Let P be the point that divides the line segment from (0, 0) to (x, y) in the ratio of 1 : 3. Then the locus of P is

a) x² = y

b) y² = 2x

c) y² = x

d) x² = 2y

The value of  where x > 0 is

a) 0

b) – 1

c) 1

d) 2

The value of

a) 5050

b) 5051

c) 100

d) 101

Let the curve C be the mirror image of the parabola y² = 4x with respect to the line x + y + 4 = 0. If A and B are points of intersection of C with the line y = −5 then the distance between A and B is . . .?

Consider the parabola y² = 8x. Let △₁ be the area of the triangle formed by the end points of its latus rectum and the point $P\left(\frac{1}{2},2\right)$

on the parabola and △₂ be the area of the triangle formed by drawing tangent at P and the end points of the latus rectum. Then $\frac{{△}_1}{{△}_2}$ is

Multiple choices

Let g (x) = x f (x), where   at x = 0

a) g is  but  is not continuous

b) g is  while f is not

c) f and g are both differentiable

d) g is  and  is continuous

Find the equation of the circle passing through ( 4, 3) and touching the lines x + y = 4 and .

A circle touches the line y = x at a point P such that  , where O is the origin. The circle contains the point  in its interior and the length of its chord on the line  is  . Determine its equation.

a)

b)

c)

d)

 equals

a)

b)

c)

d)

Let g (x) be a polynomial of degree one and f (x) be defined by

Find the continuous function f (x) satisfying

a)

b)  

c)

d) None of the above

In how many ways can a pack of 52 cards be divided equally amongst 4 players in order?

Find the interval in which ‘a’ lies for which the line y + x = 0 bisects the chord drawn from the point  to the circle

The points on the curve   where the tangent is vertical, is (are)

a)

b)

c)

d)

Let T₁, T₂ be two tangents drawn from (−2, 0) onto the circle C: x² + y² = 1. Determine the circle touching C and having T₁, T₂ as their pair of tangents. Further find the equation of all possible common tangents to the circles, when taken two at a time.

Let  for all real x and y. If   exists and  then find f(2)

a) – 1

b) 0

c) 1

d) 2

Let  and  where  are continuous functions. If A(t) and B(t) are non-zero vectors for all t and

A(0) =

then A(t) and b(t) are parallel for some t.

a) True

b) False

Let n be any positive integer. Prove that
For each non negative integer m ≤ n

Find the centre and radius of the circle formed by all the points represented by  satisfying the relation  where α and β are complex numbers given by
 

Using permutation or otherwise prove that    is an integer, where n is a positive integer.