Question(s) from Search: IIT

Using mathematical induction prove that
 

For x ε R,  is equal to

a) e

b)

c)

d)

Find the centre of the circle passing through (0, 0) and (1, 0) and touching the circle .

The number of integral values of k for which the equation  has a solution is

a) 4

b) 8

c) 10

d) 12

Evaluate if sin |x| - |x| is differentiable at x = 0.

a) Yes

b) No

Prove that  Hence or otherwise evaluate the integral .

Fill in the blank

Let A, B, C be three vectors of length 3, 4, 5 respectively. Let A is perpendicular to B + C, B is perpendicular to C + A, and C is perpendicular to A + B then the length of the vector  is equal to  . . . .

The number of common tangents to the circles  and  is

a) 0

b) 1

c) 3

d) 4

Use mathematical induction to show that (25)^{n + 1} – 24n + 5735 is divisible by (24)² for all n = 1, 2, .  .  .

given that  and

a) does not exist

b) is equal to

c) is equal to

d) is equal to 3

If  are given vectors then the vector B satisfying the equation  and  is . . . . .

If the circles  and  intersect orthogonally then k is

a) 2 or

b) – 2  or

c) 2 or

d) – 2 or

In a ΔABC,  then find the other sides and angles

a) A = 60°, B = 60°, c =

b) A = 45°, B = 75°, c =

c) A = 75°, B = 45°, c =

d) A = 15°, B = 105°, c =

(1 + ax)ⁿ = 1 + 8x + 24x² + .  .  . then a = .  . ., n = .  .  .

Given that a = (1, 1, 1), c = (0, 1, −1), a . b = 3,  then b is equal to

If a > 2b > 0 then the positive value of m for which
 is a common tangent to  and is

a)

b)

c)

d)

Find the coordinates of the point of intersection of the curves
y = cosx and y = sin3x if .

a) (((

b) ((

c) (

d) (

If f (x) = cos (lnx) then f (x) f (y) −   has the value of

a) −1

b)

c) −2

d) None of these

Multiple choices

The function

a) continuous at x = 1

b) differentiable at x = 1

c) continuous at x = 3

d) differentiable at x = 3

The value of  is

a) 0

b) 1

c) 2

d) 4

If b and c are any two non-collinear unit vectors and a is any vector then    .  .  .  .  .

Tangent to the curve  at the point P(1, 7) touches the circle  at a point Q then the coordinates of Q are

a)

b)

c)

d)

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