Question(s) from Search: IIT

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)

For n > 0,  is

a)

b) π

c)

d)

The value of the definite integral  is

a) – 1

b) 2

c)

d)

Let A be the centre of the circle . Suppose the tangents at the points B (1, 7) and D (4, 2) on the circle meet at the point C, find the area of the quadrilateral ABCD.

Find all the values of θ in the interval  satisfying the equation .

a)

b)

c)

d)

If f (x) = 3x – 5 then f ^-1 (x)

a) is given by

b) is given by

c)

d)

Evaluate

a) 0

b)

c)

d) 1

If  has its extremum value at x = 1 and x = 2, then

a) a = 2, b = 1

b) a = 2,

c) a =  2,

d) None of these

Lines  and  touch a circle C₁ of diameter 6. If the centre of C₁ lies in the first quadrant, find the equation of the circle C₂ which is concentric with C₁ and cuts intercepts of length 8 on these lines.