Question(s) from Search: IIT

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)

Let C₁ , C₂ be two circles with C₂ lying inside C₁. A circle C lying inside C₁ touches C₁ internally and C₂ externally. Identify the locus of the center of C .

The value of the determinant  is …………

a) 0

b) 1

c) a² + b² + c² – abc

d) a² + b² + c² – 3abc

If f(x) is differentiable and strictly increasing function then the value of  is

a) 1

b) 0

c) – 1

d) 2

Find all the real values of x which satisfy  and  .

The area bounded by the curves

y = (x + 1)²

y = (x – 1)²

and the line  is

a)

b)

c)

d)

For a ≤ 0, determine all real roots of the equation
 

If a, b, c, d and p are distinct real numbers such that
 
then a, b, c, d

a) Are in Arithmetic Progression

b) Are in Geometric Progression

c) Are in Harmonic Progression

d) Satisfy ab = cd

e) Satisfy none of these

Multiple choice

The function  

has local minimum at x =

a) 0

b) 1

c) 2

d) 3

Let  be a circle. A pair of tangents from (4, 5) and a pair of radii form a quadrilateral of area . . . . .

Solve  

If  are in Geometric Progression then
 are in

a) Arithmetic Progression

b) Geometric Progression

c) Harmonic Progression

d) None of these

A polygon of nine sides, each of length 2, is inscribed in a circle. The radius of the circle is . . . . .

Let T_n denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If  then n equals

a) 5

b) 7

c) 6

d) 4

A circle passes through the point of intersection of the coordinate axes with the lines  and x , then λ = . . . . .

Let a, b, c be real numbers with a ≠ 0 and let α, β be roots of the equation . Express the roots of  in terms of α, β.

Suppose a, b, c are in Arithmetic Progression and  are in Geometric Progression. If  then the value of a is

a)

b)

c)

d)

Show that   for all x ≥ 0.

For each natural number k, let C_k denote the circle with radius k centimeters and center at the origin. On the circle C_k, a particle moves k centimeters in the counterclockwise direction. After completing its motion on C_k the particle moves to C_{k + 1} in the radial direction. The motion of the particle continues in this manner. The particle starts at ( 1, 0 ). If the particle crosses the positive direction of the X–axis for the first time on the circle C_n then n = . . . . .

The equation  has an irrational root.

a) False

b) True

Multiple Choice

For  if
 , then

a)  

b)  

c)  

d)  

Fill in the blank

If the product of the roots of the equation
  is 7
Then the roots are real for  ………….

If  is a normal to  then k is

a) 3

b) 9

c) – 9

d) – 3

Fill in the blank

There are exactly two distinct linear functions ………. and ………. which map {−1, 1} onto {0, 2}.

Find three numbers a, b, c between 2 and 18 such that (i) their sum is 25 (ii) 2, a, b are consecutive terms of an Arithmetic Progression and (iii) the numbers b, c, 18 are consecutive terms of a Geometric Progression