Question(s) from Search: IIT

An ellipse has OB as a semi-minor axis. F, F’ are its focii and the angle FBF’ is a right angle. Then the eccentricity of the ellipse is . . . . .

If x = 9 is the chord of contact of the hyperbola  then the equation of the corresponding pair of tangents is

a)

b)

c)

d)

Solve for x

The rational number which equals the numbers with recurring decimals is

a)

b)

c)

d)

(Fill in the blanks) The function y = 2x² – ln|x| is monotonically increasing for values of x (≠0) satisfying the inequalities .  .  .  . and monotonically decreasing for values of x satisfying the inequalities .  .  .  .

a)

b)

c)

d)

Find  

a) 0

b) 1

c) 2

d) 4

The probability that an event A happens in one of the experiments is 0.4 Three independent trials of these experiments are performed. The probability that the event A happens at least once is

a) 0.936

b) 0.784

c) 0.904

d) None of these

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

Let f(x) = ∫e^x (x – 1) (x − 2) dx, then f(x) decreases in the interval

a) (−∞, −2)

b) (−2, −1)

c) (1, 2)

d) (2, ∞)

The harmonic means of the roots of the equation
 is

a) 2

b) 4

c) 6

d) 8

Find the integral of   

a) tan⁻¹x² + c

b)

c)

d)

Consider the two curves  then

a)  touch each other at only one point

b)  touch each other exactly at two points

c)  intersect(but not touch) at exactly two points

d)  neither intersect nor touch each other

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

Show that  =

Seven white balls and three black balls are randomly placed in a row. The possibility that no two black balls are placed adjacently equals

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)

 =

a)

b)

c)

d)

 =

a) True

b) False

The scalar  equals

a) 0

b)

c)

d) None of these

Fill in the blanks

If  is a root of the equation  where p and q are real then (p, q)  …………

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
 

A fair die is rolled. The probability that 1 occurs at the even number of trail is

a)

b)

c)

d)