Question(s) from Search: IIT

The area enclosed between y = ax² and x = ay² (a > 0)

is one square unit. Then the value of a is

a)

b)

c) 1

d)

Let f (x + y) = f (x) f (y) for all x, y. Suppose that f (5) = 2 and  (0) = 3. Find f (5).

a) 1

b) 2

c) 3

d) 6

If a function f : is an odd function such that  for x ε [a, 2a] and the left hand derivative at

x = a is 0 then find the left hand derivative at x =  

a) 0

b) 1

c) a

d) 2a

A country produces 90% of its food diet. The population grows continuously at a rate of 3% per year. Its annual food production every year is 4% more than that of last year. Assuming that the average food requirement per person remains constant, prove that the country will become self sufficient in food after n years, where n is the smallest integer bigger than or equal to

If f(x) is a polynomial of degree less than or equal to 2 and S be the set of all such polynomials so that

P(0) = 0

P(1) = 1, and

Then

a) S = ɸ

b) S = ax + (1 – a) x² ⩝ a ε (0, 2)

c) S = ax + (1 – a) x² ⩝ a ε (0, ∞)

d) S = ax + (1 – a) x² ⩝ a ε (0, 1)

The line  is a diameter of the circle

a) True

b) False

One or more correct answers
In a triangle PQR, sin P, sin Q, sin R are in arithmetic progression then

a) Altitudes are in arithmetic progression

b) Altitudes are in harmonic progression

c) Medians are in geometric progression

d) Medians are in arithmetic progression

f(x) is a function such that  and the tangent at any point passes through (1, 2). Find the equation of the tangent.

a) x = 2

b) y = 2

c) x + y = 2

d) x – y = 2

The lines  and  are tangents to the same circle. The radius of this circle is . . . . .

The external radii  of ΔABC are in harmonic progression then prove that a, b, c are in arithmetic progression

a) True

b) False

True / False

If f (x) = ( a – xⁿ )^1/n  where a > 0 and n is a positive integer then f ( f ( x ) ) = x.

a) True

b) False

Let f(x) =

If f is continuous for all x, then k is equal to

a) 3

b) 5

c) 7

d) 9

Fill in the blank

The domain of the function f (x) =  is

a) [− 2, − 1]

b) [1, 2]

c) [− 2, − 1] ⋃ [1, 2]

d) None of the above

 

 

Then

a) 0

b) 1

c) 2

d) 4

The complex numbers  satisfying  are the vertices of the triangle which is

a) of zero area

b) right angle isosceles

c) equilateral

d) obtuse angled isosceles

Let x and y be two real variables such that x > 0 and xy = 1. Find the minimum value of x + y.

a) 1

b) 2

c) 3

d) 4

ABC is an isosceles triangle in a circle of radius r. If AB = AC and h is the altitude from A to BC then the triangle ABC has perimeter , area A = . . . . .

Also  . . . . .

Let f(x) = x|x|. The set of points where f(x) is twice differentiable is .  .  .  .

a) ℝ

b) 0

c) ℝ − {0, 1}

Find the shortest distance of the point (0, c) from the parabola
y = x², where 0 ≤ c ≤ 5.

a)

b)

c)

d)

Intercepts on the line y = x by the circle  is AB. Equation of the circle with AB as diameter is  . . . . .

The number of real solutions of the equation | x |² – 3 | x | + 2 = 0 is

a) 4

b) 1

c) 3

d) 2

Match the following

Let the function defined in column 1 has domain

a) i) → A, ii) → B

b) i) → A, ii) → C

c) i) → C, ii) → A

d) i) → B, ii) → C

A man walks a distance of three units from the origin towards north-east (N direction. From there he walks a distance of 4 units towards north–west (N direction to reach a point P. Then the position of P in the argand plane is

a)

b)

c)

d)

Find the coordinates of the point on the curve  where the tangent to the curve has the greatest slope.

a) (0, 0)

b)

c)

d)

If p, q, r are any real numbers, then

a) Max ( p, q ) < max ( p, q, r )

b) Min ( p, q ) =  

c) Max ( p, q ) < min ( p, q, r )

d) none of these