Question(s) from Search: IIT

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

Determine the values of a, b, c for which the function

is continuous at x = 0

a)

b)

c)

d)

If

a)

b) [2, ∞)

c)

d)

The function  is

a) Increasing on (0, ∞)

b) Decreasing on (0, ∞)

c) Increasing on  and decreasing on  

d) Increasing on  and decreasing on

A point P is given on the circumference of a circle of radius r. Chord QR is parallel to the tangent at P. Determine the maximum possible area of ΔPQR.

If we consider only the principal values of the inverse trigonometric functions then the value of

 is

a)

b)

c)

d)

Let g (x) = 1 + x – [ x ] and f (x) =  then for all x,
f (g (x)) is equal to

a) x

b) 1

c) f ( x )

d) g ( x )

Let P(asecθ, btanθ) and Q(asecɸ, btanɸ) where θ + ɸ =  be two points on the hyperbola . If (h, k) be the point of intersection of the normals at P and Q then k is equal to

a)

b)

c)

d)

Find the value of  at  where
.

a) 1

b)

c)

d)

Let ℝ be the set of real numbers and f : ℝ → ℝ such that for all x and y in ℝ, . Then f (x) is a constant.

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