Question(s) from Search: IIT

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

Solve for x

The rational number which equals the numbers with recurring decimals is

a)

b)

c)

d)

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

Let  be roots of the equations and  respectively. If the system of equations  and  have non-trivial solutions then prove that

If  are in Arithmetic Progression
then

a) a, b, c are in Arithmetic Progression

b)  are in  Arithmetic Progression

c) a, b, c are in Geometric Progression

d) a, b, c are in Harmonic Progression

The harmonic means of the roots of the equation
 is

a) 2

b) 4

c) 6

d) 8

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  . . . . .

Suppose p(x) =
If  prove that
 

The sum of the first 2n terms of the Arithmetic Progression 2, 5, 8, . . . .  is equal to the sum of the first n terms of the Arithmetic Progression 57, 59, 61, . . . . then n equals

a) 100

b) 12

c) 11

d) 13

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

a)

b)

c)

d)

 where a, b ε R then find the value of a for which equation has unequal roots for all values of b.

If α, β are roots of  and  are in Geometric Progression and  then

a)

b)

c)

d)

Two vertices of an equilateral triangle are (- 1, 0) and (1, 0) and its third vertex lies above the X–axis, the equation of circumcircle is . . .

If x and y are positive real numbers and m and n are any positive integers then

a) True

b) False

If x, y, z are in Harmonic Progression then show that
 

Fill in the blank

The solution of the equation
 is …………..

If  are in Arithmetic Progression where  for all i, show that
 

The equation of the directrix of the parabola y² + 4y + 4x +2 = 0 is

a) x = − 1

b) x = 1

c)

d)

Fill in the blank

If x < 0, y < 0,  and  then x  ……….. and y  ………..

The sum of the squares of three distinct real numbers which are in Geometric Progression is . If their sum is , show that
 

The focal chord of  is tangent to  then the possible value of the slope of this chord are

a)   

b)  

c)   

d)   

If then x lies in the interval

a) (2, ∞)

b) (1, 2)

c) (−2, −1)

d) None of these