Question(s) from Search: IIT

The circles each of radius 5 units touch each other at (1, 2). If the equation of the common tangent is , find the equation of the circles.

If , then find the values of n and r

The function  increases if

a)

b)

c)

d)

a) True

b) False

The triangle formed by the tangent to the curve

  at (1, 1) and the coordinate axes, lies in the first quadrant if its area is 2. Then the value of b is

a) – 1

b) 3

c) – 3

d) 1

Consider a curve and a point P not on the curve. A line drawn from the point P intersects the curve at points Q and R. If PQ.QR is independent of the slope of the line then show that the curve is a circle.

Let

Determine a and b so that f is continuous at x = 0.

a)

b)

c)

d)

A box contains 2 white balls, 3 black balls and 4 red balls. In how many ways can three balls be drawn from a box if at least one black ball is to be included in the draw?

Multiple choices
y = f ( x ) =  then

a) x = f (y)

b) f (1) = 3

c) y is increasing with x for x < 1

d) f is a rational function of x

A committee of 12 is to be formed from 9 women and 8 men. In how many ways this can be if at least five women have to be in the committee? In how many ways in these committees (i) The women are in majority, (ii)The men are in majority

The area enclosed between y = ax² and x = ay² (a > 0)

is one square unit. Then the value of a is

a)

b)

c) 1

d)

Let f (x + y) = f (x) f (y) for all x, y. Suppose that f (5) = 2 and  (0) = 3. Find f (5).

a) 1

b) 2

c) 3

d) 6

If a function f : is an odd function such that  for x ε [a, 2a] and the left hand derivative at

x = a is 0 then find the left hand derivative at x =  

a) 0

b) 1

c) a

d) 2a

A country produces 90% of its food diet. The population grows continuously at a rate of 3% per year. Its annual food production every year is 4% more than that of last year. Assuming that the average food requirement per person remains constant, prove that the country will become self sufficient in food after n years, where n is the smallest integer bigger than or equal to

If f(x) is a polynomial of degree less than or equal to 2 and S be the set of all such polynomials so that

P(0) = 0

P(1) = 1, and

Then

a) S = ɸ

b) S = ax + (1 – a) x² ⩝ a ε (0, 2)

c) S = ax + (1 – a) x² ⩝ a ε (0, ∞)

d) S = ax + (1 – a) x² ⩝ a ε (0, 1)

The line  is a diameter of the circle

a) True

b) False

One or more correct answers
In a triangle PQR, sin P, sin Q, sin R are in arithmetic progression then

a) Altitudes are in arithmetic progression

b) Altitudes are in harmonic progression

c) Medians are in geometric progression

d) Medians are in arithmetic progression

f(x) is a function such that  and the tangent at any point passes through (1, 2). Find the equation of the tangent.

a) x = 2

b) y = 2

c) x + y = 2

d) x – y = 2

The lines  and  are tangents to the same circle. The radius of this circle is . . . . .

The external radii  of ΔABC are in harmonic progression then prove that a, b, c are in arithmetic progression

a) True

b) False

True / False

If f (x) = ( a – xⁿ )^1/n  where a > 0 and n is a positive integer then f ( f ( x ) ) = x.

a) True

b) False

If then equals

a)

b)

c)

d)

The area of the triangle formed by the tangents from the point (4, 3) to the circle  and the line joining their point of contact is .

L =  = .  .  .  .

a) – 1

b) 0

c) 1

d) 2

Let  then the value of  is

a) 3ω

b) 3ω(ω – 1)

c) 3ω²

d) 3ω(1 – ω)