Question(s) from Search: IIT

If the m^th, n^th and p^th term of an Arithmetic Progression and a Geometric Progression are equal and are x, y, z then prove that
 

Fill in the blank

If the quadratic equation
  and  have a common root then the numerical value of a + b is …………

The equation of the common tangent touching the circle
 and the parabola , above X–axis is

a)

b)

c)

d)

Fill in the blank

The sum of the real roots of the equation
 is ………..

If  are in Arithmetic Progression, determine the value of x.

The angle between the tangents drawn from the point (1, 4) to the parabola  is

a)

b)

c)

d)

The number  is

a) an integer

b) a rational number

c) an irrational number

d) a prime number

The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of four consecutive terms of it, prove that the resulting sum is square of an integer.

A is a point on the parabola . The normal at A cuts the parabola again at B. If AB subtends a right angle at the vertex of the parabola, find the slope of AB.

 is equal to

a)

b)

c)

d)

If a, b, c are positive real numbers then prove that
 

Show that the locus of a point that divides a chord of slope 2 of the parabola  internally in the ratio 1:2 is a parabola. Find its vertex.

If the lengths of the sides of a triangle are 3, 5, 7 then the largest angle of the triangle is

a)

b)

c)

d)

Three normals with slopes  are drawn from a point P not on the axis of the parabola . If  results in the locus of P being a part of the parabola, find the value of α.

Let the Harmonic Mean and Geometric Mean of two positive numbers be in the ratio of 4:5. Then the two numbers are in the ratio .  .  .  .  .

The order of the differential equation whose general solution is given by  is

a) 5

b) 4

c) 3

d) 2

If  and  then  equals

a)

b)

c)

d) 1

Let , where f is such that  and  then g(2) satisfies the inequality

a)

b)

c)

d)

If y =  

Prove that  

The value of  is

a) π

b) aπ

c)

d) 2π

Let F(x) = f (x) g (x) h (x) for all real x, where f (x), g (x) and h (x) are differentiable functions. At some point x₀

 and

 then k is equal to

a) 12

b) 20

c) 24

d) 28

If  then f

a)

b)

c) 3

d) None

If G(x) then  is

a)

b)

c)

d)

Show that

If

Where [x] denotes the greatest integer less than or equal to x then  equals

a) 1

b) 0

c) – 1

d) None of these