Question(s) from Search: IIT

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

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