Question(s) from Search: IIT

The value of  is

Ten different letters of an alphabet are given. Words with five letters are formed from the given letters. Then the number of words which have at least one letter repeated is

a) 69760

b) 30240

c) 99748

d) None of these

Let a, b, c be non-zero real numbers such that
 
 
Then the quadratic function  has

a) no root in (0, 2)

b) at least one root in (1, 2)

c) a double root in (0, 2)

d) two imaginary roots

Prove that the value of the function  do not lie between  and 3 for any real x.

a) True

b) False

If g (f (x)) = |sin x| and f (g (x)) = (sin)², then

a) f (x) = sin² x, g (x) =

b) f (x) = sin x, g (x) =

c) f (x) = x², g (x) = sin

d) f and g cannot be determined

Evaluate

a) 0

b)

c)

d) 1

If   then  equals

a)

b)

c)

d) None of these

Let  be a polynomial in a real variable x with 0< then the function p(x) has

a) neither maximum nor minimum

b) only one maximum

c) only one minimum

d) only one maximum and only one minimum

e) none of these

Let a given line L₁ intersect the X-axis and Y-axis at P and Q respectively. Let another line L₂ perpendicular to L₁ cut the X and Y axis at R and S respectively. Show that the locus of the point of intersection of the lines PS and QR is a circle passing through the origin.

Fill in the blank
General values of θ satisfying the equation  are

a) θ = nπ

b)

c)

d) θ = nπ or θ =

If f (x + y) = f (x) + f (y) for all x and y. If the function f is continuous at x = 0 then f is continuous for all x.

a) True

b) False

How many different 9 digit numbers can be formed from the numbers 223355888 by rearranging its digits so that the odd digits occupy even positions

a) 16

b) 36

c) 60

d) 180

The function defined by  is

a) Decreasing for all x

b) Decreasing in  and increasing in

c) Increasing for all x

d) Decreasing in  and increasing in  

The principal value of is

a)

b)

c)

d)

e) None of these

Let f(x) =

Discuss the continuity of  on [0, 2]

a)  is continuous for all x  ℝ

b)  is continuous for all x  ℝ except at x = 1

c)  is continuous for all x  ℝ except at x = 1 and x = 2

d)  is continuous for all x  ℝ except at x = 0, x = 1 and x = 2

Let a circle be given by . Find the condition on a and b if two chords each bisected by the X–axis can be drawn from .

The value of x for which  is

a)

b) 1

c) 0

d)

The domain of the derivative of the function
f (x) =

a) R  { 0 }

b) R

c) R

d) R

The greater of the two angles
 and  is

a) A

b) B

c) Both are equal

If f (x) = sinx + cosx, g (x) = x² – 1 then g ( f (x)) is invertible in the domain

a)

b)

c)

d)

One or more correct answers
In a triangle the length of the two larger sides are 10 and 9 respectively. If the angles are in arithmetic progression then the length of the third side can be

a)

b)

c) 5

d)

e) None of these

Let f (x) = Ax² + Bx + C where A, B , C are real numbers. Prove that if f (x) is an integer then the numbers 2A, A + B and C are all integers. Conversely prove that if the numbers 2A, A + B and C are all integers then f ( x ) is an integer whenever x is an integer.

A ladder rests against a wall at an angle α to the horizontal. If its foot is pulled away from the wall through a distance a, so that it slides a distance b down the wall making an angle β with the horizontal, then .

a) True

b) False

Let be the vertices of an n sided regular polygon such that   . Then find n.

a) 5

b) 6

c) 7

d) 8

A variable plane at a distance of one unit from the origin cuts the coordinate axes at A, B and C. If the centroid D(x, y, z) of triangle ABC satisfies the relation  then the value of k is

a) 9

b)

c) 1

d) 3