Question(s) from Search: IIT

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

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.

Three circles of radii 3, 4 and 5 units touch each other externally and tangents drawn at the points of contact intersect at P. Find the distance between P and the point of contact.

In ΔABC, D is the midpoint of BC. If AD is perpendicular to AC then .

a) True

b) False

A function f : R  R where R is the set of real numbers is defined by f (x) = . Find the interval of values of α for which f is onto. Is the function one to one for α = 3? Justify your answer.

a) 2 ≤ α ≤ 14

b) α ≥ 2

c) α ≤ 14

d) none of the above

If f₁ ( x ) and f₂ ( x ) are defined by domains D₁ and D₂ respectively, then f₁ ( x ) + f₂ ( x ) is defined as on D₁ D₂.

a) True

b) False