Question(s) from Search: IIT

Find the sum of the series
 

The set of all points where the function  is differentiable is

a)

b) [0, ∞)

c)  

d)  (0, ∞)

e)  None of these

Given a function f (x) such that
i) it is integrable over every interval on the real axis and
ii) f (t + x) = f (x) for every x and a real t, then show that the integral  is independent of a.

If the algebraic sum of the perpendicular distance from the point
(2, 0), (0, 2) and (1, 1) to a variable straight line be zero then the line passes through a fixed point whose coordinates are

The general solution of
 is

a)

b)

c)

d)

The function f(x) =  denotes the greatest integer function is discontinuous at

a) All x

b) All integer points

c) No x

d) x which is not an integer

If f (x) and g (x) are continuous functions on (0, a) satisfying f (x) = f (a – x) and g (x) + g (a – x) = 2 then show that

The equation of the circles through (1, 1) and the point of intersection of
 
is

a)

b)

c)

d) None of these

The general value of θ satisfying the equation
 is

a)

b)

c)

d)

A cubic f (x) vanishes at x = −2 and has a relative minimum/maximum at x = −1 and . If , find the cube f (x).

a) x³ + x² + x + 1

b) x³ + x² − x + 1

c) x³ − x² + x + 2

d) x³ + x² − x + 2

If a circle passes through the points (a, b) and cuts the circle  orthogonally, then the equation of the locus of its centre is

a)

b)

c)

d)

In ΔPQR, angle R . If tan  and tan  are roots of the equation

a)

b)

c)

d)

Prove that
where  and n is an even integer.

 equals

a) – π

b) π

c)

d) 1

Evaluate

The locus of the centre of circles which touches externally  and which touches the Y-axis is given by the equation

a)

b)

c)

d)

The values of θ ε (0, 2π) for which  are

a)

b)

c)

d)

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 = …..