Question(s) from Search: IIT

The maximum value of cos₁ cos₂ cos_{3 ……} cos_nunder the restriction 0  _{1 , 2} , ₃ …. , _n   and cot₁ cot₂ cot_{3 ……} cot_n= 1 is

a)

b)

c)

d)

The left hand derivative of f (x) = [x] sinπx at k, k an integer is

a) (k – 1)π

b) (k – 1)π

c) kπ

d)  kπ

Determine the value of

a)

b)

c)

d)

Let f : ℝ → ℝ be such that f (1) = 3 and  then

 equals

a) 1

b)

c)

d)

For x > 0, let  find the function  and show that . Here .

a)

b)

c)

d)

If two distinct chords drawn from the point (p, q) on the circle  (where pq ≠ 0) are bisected by the X-axis then

a)

b)

c)

d)

Let   are the perpendiculars from the vertices of a triangle to the opposite sides, then  

a) True

b) False

The coefficient of x⁹⁹ in the polynomial
(x – 1) (x – 2) .  .  . (x – 100) is

Evaluate

The sum of the rational terms in the expansion of  is

A unit vector perpendicular to the plane determined by the points P (1, -1, 2), Q (2, 0, -1) and R (0, 2, 1) is .  .  .  .  .

If one of the diameters of the circle  is a chord to the circle with centre (2, 1) then the radius of the circle is

a)

b)

c) 3

d) 2

Which of the following functions is periodic?

a) f(x) = x – [x] where [x] denotes the greatest integer less than or equal to the real number x

b) f(x) = sin  x ≠ 0, f(0) = 0

c) f(x) = x cos x

d) None of these

a)

b)

c) 1

d) 2

Let a, b and c be three vectors having magnitudes 1, 1 and 2 respectively. If  then the acute angle between a and c is  .  .  .  .  .

The equation of the tangents drawn from the origin to the circle  are

a) x= 6

b) y = 0

c)

d)

Let f (x) be defined for all x > 0 and be continuous. If f (x) satisfies
f  = f (x) – f (y) for all x and y and f (e) = 1 then

a) f (x) is bounded

b) f  → 0 as x → 0

c) x f  → 0 as x → 0

d) f (x) = lnx

The value of  is equal to

a)

b)

c)

d) None of these

The area bounded by the curve y = f(x), the X–axis and the ordinate x = 1 and x = b is (b – 1) sin (3b + 4). Then f(x) is

a) (x – 1) cos (3x + 4)

b) sin(3x + 4)

c) sin(3x + 4) + 3(x – 1) cos (3x + 4)

d) none of these

Through a fixed point (h, k) secants are drawn to the circle  . Show that the locus of the mid points of the secant intercepted by the circle is

There exists a solution of θ between 0 and 2π that satisfies the equation .

a) True

b) False

The number of values of x where the function
f (x) = cos x + cos () attains the maximum is

a) 0

b) 1

c) 2

d) Infinite

Evaluate

a) 0

b)

c) 1

d) 2

The circle  is inscribed in a triangle which has two of its sides along the co-ordinate axes. The locus of the circum centre of the triangle is  find k.

The domain of definition of the function f (x) given by the equation

2^x + 2^y = 2 is

a) 0 < x ≤ 1

b) 0 ≤ x ≤ 1

c)  < x ≤ 0

d)  < x ≤ 1