Question(s) from Search: IIT

Let  and f = R – [R] where [ ] denotes the greatest integer function. Prove that Rf = 4^{2n + 4}

One or more than one correct option

Circle(s) touching X – axis at a distance 3 from the origin and having an intercept of length $2\sqrt{7}$

on Y – axis is/are

a) x² + y² – 6x + 8y + 9 = 0

b) x² + y² – 6x + 7y + 9 = 0

c) x² + y² – 6x − 8y + 9 = 0

d) x² + y² – 6x − 7y + 9 = 0

Using induction or otherwise, prove that for any non-negative integers m, n, r and k
 

Let V be the volume of the parallelepiped formed by the vectors  and . If a_r, b_r, c_r where r = 1, 2, 3 are non-negative real numbers and , show that V ≤ L³

One or more than one correct option

A circle S passes through the point (0, 1) and is orthogonal to the circles (x – 1)² + y² = 16 and x² + y² = 1, then

a) Radius of S is 8

b) Radius of S is 7

c) Centre of S is (−7, 1)

d) Centre of S is (−8, 1)

The locus of the midpoint of a chord of the circle  which subtend a right angle at the origin is

a)

b)

c)

d)

If n is a positive integer and 0 ≤ v < π then show that

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

f(x) is twice differentiable polynomial function such that f (1) = 1, f (2) = 4, f (3) = 9, then

a) there exists at least one x  (1, 2) such that

b) there exists at least one x  (2, 3) such that

c)

d) there exists at least one x  (1, 3) such that

The radius of a circle having minimum area which touches the curve y = 4 – x² and the line y = |x| is

a) $2\sqrt{2}$

b) $2\left(\sqrt{2}-1\right)$

c) $4\left(\sqrt{2}-1\right)$

d) $4\left(\sqrt{2}+1\right)$

Let AB be a chord of the circle subtending a right angle at the centre then the locus of the centroid of the triangle PAB as P moves on the circle is

a) A parabola

b) A circle

c) An ellipse

d) A pairing straight line

Given a circle 2x² + 2y² = 5 and a parabola $y^2=4\sqrt{5}x$

Statement 1: An equation of a common tangent to the curves is $y=x+\sqrt{5}$ Statement 2: If the line $y=\mathrm{mx}+\frac{\sqrt{5}}{m}(m\ne 0)$ is the common tangent then m satisfies m⁴ – 3m² + 2 = 0

a) Statement 1 is correct. Statement 2 is correct. Statement 2 is a correct explanation for statement 1

b) Statement 1 is correct. Statement 2 is correct. Statement 2 is not a correct explanation for statement 1

c) Statement 1 is correct. Statement 2 is incorrect.

d) Statement 1 is incorrect. Statement 2 is correct.

One or more than one correct option

Let L be a normal to the parabola y² = 4x. If L passes through the point (9, 6) then L is given by

a) y – x + 3 = 0

b) y + 3x – 33 = 0

c) y + x – 15 = 0

d) y – 2x + 12 = 0

Let ABCD be a quadrilateral with area 18 with side AB parallel to CD and AB = 2CD. Let AD be perpendicular to AB and CD. A circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius is

a) 3

b) 2

c)

d) 1

Multiple choices

The function f (x) = max  is

a) continuous at all points

b) differentiable at all points

c) differentiable at all points except x = 1 and x =

d) continuous at all points except at x=1 and x=-1 where it is discontinuous

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