Question(s) from Search: IIT

Multiple choices

If x + |y| = 2y, then y as a function of x is

a) Defined for all real x

b) Continuous at x = 0

c) Differentiable for all x

d) Such that  for x < 0

The value of the integral  is equal to a

a) True

b) False

A unit vector coplanar with  and  and perpendicular to  is . . . . .

The centre of the circle inscribed in the square formed by the lines  and

a) (4, 7)

b) (7, 4)

c) (9, 4)

d) (4, 9)

Find the number of solutions of  

a) 0

b) 1

c) 2

d) Infinitely many

The domain of definition of the function
y =  +

a) (−3, −2) excluding −2.5

b) [0, 1] excluding 0.5

c) [−2, 1) excluding 0

d) None of these

Multiple choices

Let g(x) be a function defined on  If the area of the equilateral triangle with two of its vertices at (0, 0) and (x, g (x)) is   then the function g (x) is

a)

b)

c)

d)

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 =

The real roots of the equation x +  = 1 in the interval (−π, π) are …...........

a) x = 0

b) x = ±  

c) x = 0 , x = ±  

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