Question(s) from Search: IIT

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)

Sketch the region bounded by the curves
 and y = |x – 1|
and find its area.

a)

b)

c)

d) 5π + 2

Tangents are drawn from the point (17, 7) to the circle .
Statement 1 – The tangents are mutually perpendicular, because

Statement 2 – The locus of points from which mutually perpendicular tangents are drawn to the given circle is .

The question contains statement – 1 (assertion) and statement 2 (reason). Of these statements mark correct choice if

a) Statement 1 and 2 are true. Statement 2 is a correct explanation for statement 1.

b) Statement 1 and 2 are true. Statement 2 is not a correct explanation for statement 1.

c) Statement 1 is true. Statement 2 is false.

d) Statement 1 is false. Statement 2 is true

Let  be the vertices of the triangle. A parallelogram AFDE is drawn with the vertices D, E and F on the line segments BC, CA and AB respectively. Using calculus find the area of the parallelogram.

a)  

b)  

c)  

d)  

Two rays in the first quadrant x + y = |a| and ax – y = 1 intersect each other in the interval a ε (a₀, ∞). The value of a₀ is

Find the area of the region bounded by the curve C: y = tanx, tangent drawn to C at  and the X–axis.

a) ln2 – 1

b)

c)

d)

then tan t =

Sketch the curves and identify the region bounded by
 

Consider the following linear equations
ax + by + cz = 0
bx + cy + az = 0
cx + ay + bz = 0
Match the statements/expressions in column 1 with column 2

The domain of the function y(x) given by the equation  is

a) 0 < x ≤ 1

b) 0 ≤ x ≤ 1

c)  < x ≤ 0

d)  < x < 1

If A = , 6A^-1 = A² + cA + dI

then (c, d ) is

a) (−11, 6)

b) (−6, 11)

c)  (6, 11 )

d)  (11, 6 )

Prove that