Question(s) from Search: IIT

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

a) ln2

b) ln3

c) ln6

d) ln2 ln3

For all complex numbers satisfying  = 5, the minimum value of

a) 0

b) 2

c) 7

d) 17

Use the function  , x > 0 to determine the bigger of the numbers e^π and π^e.

a) e^π

b) π^e

In a triangle ABC, D and E are points on  and  respectively such that  and . Let P be the point of intersection of  and . Find  using vector method.

a)

b)

c)

d) 2

The minimum value of  where a, b c are all not equal integers and ω(≠1) a cube root of unity is

a) 1

b) 0

c)

d)

Match the following
Let the functions defined in column 1 have domain

a) i) → A, ii) → B

b) i) → A, ii) → C

c) i) → C, ii) → A

d) i) → B, ii) → C

Find the area of the region bounded by the X–axis and the curve defined by
 
 

a) ln2

b) 2ln2

c)

d)

Let ABCD be a square with side of length 2 units. C₂ is the circle through the vertices A, B, C, D and C₁ is the circle touching all the sides of the square ABCD. L is a line through A.

A circle touching the line L and the circle C₁ externally such that both the circles are on the same side of the line, then the locus of the centre of circle is

a) Ellipse

b) Hyperbola

c) Parabola

d) Pair of straight lines

Find three dimensional vectors u₁, u₂, u₃ satisfying
u₁.u₁ = 4; u₁.u₂ = −2; u₁.u₃ = 6; u₂.u₂  = 2; u₂.u₃ = −5; u₃.u₃ = 29

If a continuous function f defined on the real line ℝ, assumes positive and negative values in ℝ then the equation f(x) = 0 has a root in ℝ. For example, it is known that if a continuous function f on ℝ is positive at some points and its minimum value is negative then the equation f(x) = 0 has a root in ℝ. Consider the function f(x) =  for all real x where k is a real constant.

For k > 0, the set of all values of k for which  has two distinct roots is

a)

b)

c)

d) (0, 1)

Let f(x) = x³ – x² + x + 1 and
 
Discuss the continuity and differentiability of f(x) in the interval (0, 2)

a) Continuous and differentiable in (0, 2)

b) Continuous and differentiable in (0, 2)except x = 1

c) Continuous in (0, 2). Differentiable in (0, 2) except x = 1

d) None of the above

A relation R on the set of complex numbers is defined by iff  is real. Show that R is an equivalence relation.

Find the point on the curve 4x² + a²y² = 4a², 4 < a² < 8 that is farthest from the point (0, −2).

a) (a, 0)

b)

c)

d) (0, - 2)

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y² = 4ax is another parabola with directrix

a) x = −a

b)

c)

d)