Question(s) from Search: IIT

The tangent PT and the normal PN of the parabola y² = 4ax at the point P on it meet its axis at the points T and N respectively. The locus of the centroid of the triangle PTM is a parabola whose

a) Vertex is $\left(\frac{2a}{3},0\right)$

b) Directrix is x = 0

c) Latus rectum is $\frac{2a}{3}$

d) Focus is (a, 0)

Let f and g be increasing and decreasing functions, respectively from [0, ∞) to [0, ∞). Let h(x) =f(g(x)). If h(0) = 0 then h(x) – h(t) is

a) Always zero

b) Always negative

c) Always positive

d) Strictly increasing

e) None of these

Let E = {1, 2, 3, 4} and F = {1, 2} then the number of onto functions from E to F is

a) 14

b) 16

c) 12

d) 8

On the interval [0, 1] the function  takes the maximum value at the point

a) 0

b)

c)

d)

Let f (x) be continuous and g (x) be a discontinuous function. Prove that f (x) + g (x) is a discontinuous function.

a) True

b) False

c) Could be continuous or discontinuous

Find the coordinates of the point at which the circles x² + y² – 4x – 2y = – 4 and  x² + y² – 12x – 8y = – 36  touch each other. Also find the equation of the common tangents touching the circles at distinct points.

Draw the graph of the function y = [x] + |1 – x|, – 1 ≤ x ≤ 3. Determine the points, if any, where the function is not differentiable.

a) y is differentiable everywhere

b) y is not differentiable at x = 0

c) y is not differentiable at x = 0, 1, 2

d) y is not differentiable at x = 0, 1, 2 and 3

In how many ways can a pack of 52 cards be divided in 4 sets, three of them having 17 cards each and fourth just one card.

The area bounded by the curves

  and   is

a) 1

b) 2

c)

d) 4

Let ABC be an equilateral triangle inscribed in the circle x² + y² = a². Suppose perpendiculars from A, B, C to the major axis of the ellipse  (a > b) meet the ellipse respectively at P, Q, R so that P, Q, R are on the same side of the major axis. Prove that the normals drawn at the points P, Q and R are concurrent.

Which of the following pieces of data does not uniquely determine an acute angled triangle ABC (R being the radius of the circumcircle).

a) a, sinA, sinB

b) a, b , c

c) a, sinB, R

d) a, sinA, R

Let f(x), x ≥ 0 be a non-negative function and let F(x) = . For some c > 0, f(x) ≤ cF(x) for all x ≥ 0. Then for all x ≥ 0, f(x) =

a) 0

b) 1

c) 2

d) 4

Tangents are drawn from P (6, 8) to the circle  . Find the radius of the circle such that the area of the triangle formed by tangents and chord of contact is maximum.

Find the natural number a for which
  
where the function f satisfies the relation f (x + y) = f (x) . f (y)
for all natural numbers x and y and further f (1) = 2

a) 1

b) 2

c) 3

d) 4

In a certain test  students gave wrong answers to at least i questions where i = 1, 2, …, k. No student gave more than k correct answers. Total number of wrong answers given is .  .  .

Multiple choice

If

a) f(x) is increasing on [– 1, 2]

b) f(x) is continuous on [– 1, 3]

c)  does not exist

d) f(x) has maximum value at x = 2

If arg(z) < 0 then arg(−z) – arg(z) is equal to

a) π

b) –π

c) – π/2

d) π/2

Multiple choice

f(x) is a cubic polynomial with f(2) = 18 and f(1) = − 1. Also f(x) has a local maxima at x = − 1 and  has a local minima at x = 0 then

a) The distance between (− 1, 2) and (a, f(a)), where x = a is the point of local minimum, is

b) f(x) is increasing for

c) f(x) has a local minima at x = 1

d) The value of f(0) = 15

From the point A (0, 3) on the circle , a chord AB is drawn and extended to a point M such that AˆM = 2AˆB. The equation of locus of M is . . . . .

In Δ ABC the median to the side BC is of length  and divides ∠A into 30° and 45°. Then find the length of side BC.

a) 1

b) 2

c)

d)

If f is an even function defined on (−5, 5) then the real values of x satisfying the equation f (x) =  are ……………

a)

b)

c)

d)

If ω(≠1) be a cube root of unity and  then the least positive value of n is

a) 2

b) 3

c) 5

d) 6

A circle is inscribed in an equilateral triangle of side a. The area of any square inscribed in the circle is . . . . .

The number of all possible triplets  such that
 for all x is

a) Zero

b) One

c) Three

d) Infinite

e) None

A swimmer S is in the sea at a distance d km. from the closest point A on a straight shore. The house of the swimmer is on the shore at a distance L km. from A. He can swim at a speed of
u km/hour and walk at a speed of v km/hr (v > u). At what point on the shore should he land so that he reaches his house in the shortest possible time.

a)

b)

c)

d)