Question(s) from Search: IIT

Prove that
 

Evaluate

a)

b)

c)

d)

A, B, C , D are four points in a plane with position vectors a, b, c, d respectively, such that . The point D then is the  . . . . . . .  of the triangle ABC.

If   are altitudes of a triangle from the vertices A, B, C and Δ the area of the triangle then  

a) True

b) False

The sum of the coefficients of the polynomial (1 + x – 3x²)²¹⁶³ is

If  at x = π

a)

b) π

c) 2π

d) 4π

If the vectors
 

are coplanar then the value of  . . . . . .

Let n be a positive integer. If the coefficient of the 2^nd, 3^rd and 4^th terms in the expansion of (1 + x)ⁿ are in arithmetic progression then n = …..

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 =