Question(s) from Search: IIT

If a, b, c, d are distinct vectors satisfying relation  and . Prove that

If two circles  and  intersect in two distinct points, then

a) 2 < r < 8

b) r < 2

c) r = 2

d) r > 2

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

Eight chairs are numbered one to eight. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs marked 1 to 4 then the men select the chairs from amongst the remaining. The number of possible arrangements is

a)

b)

c)

d) None of these

AB is a diameter of a circle and C is any point on the circumference of the circle. Then

a) The area of △ABC is maximum if it is isosceles

b) The area of △ABC is minimum if it is isosceles

c) The perimeter of △ABC is minimum when it is isosceles

d) None of these

The abscissas of two points A and B are the roots of the equation  and their ordinates are the roots of the equation . Find the equation of the circle on AB as diameter.

Find  if f(x) =

a) 0

b)

c)

d)

An n digit number is a positive number with exactly n–digits. Nine hundred distinct n–digit numbers are to be formed with only the three digits 2, 5 and 7. The smallest value of n for which this is possible is

a) 6

b) 7

c) 8

d) 9