Commerce (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

Show that the function defined by  g(x) = x = [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x.

Find the points of discontinuity of f, where `f (x) = {(sinx/x, if x<0),(x + 1, if x >= 0):}`

Determine if f defined by `f(x) = {(x^2 sin  1/x, "," if x != 0),(0, "," if x = 0):}` is a continuous function?

Examine the continuity of f, where f is defined by `f(x) = {(sin x - cos x, if x != 0),(-1, "," if x = 0):}`

Using mathematical induction prove that  `d/(dx) (x^n) = nx^(n -1)` for all positive integers n.

Using the method of integration find the area of the region bounded by lines: 2x + y = 4, 3x – 2y = 6 and x – 3y + 5 = 0

Area bounded by the curve y = x³, the x-axis and the ordinates x = –2 and x = 1 is ______.

The area bounded by the curve y = x | x|, x-axis and the ordinates x = –1 and x = 1 is given by ______.

[Hint: y = x² if x > 0 and y = –x² if x < 0]

Determine the value of the constant 'k' so that function f(x) `{((kx)/|x|, ","if  x < 0),(3"," , if x >= 0):}` is continuous at x = 0

`sin^-1  1/2-2sin^-1  1/sqrt2`

`sin^-1{cos(sin^-1  sqrt3/2)}`

Find the domain of the following function:

`f(x)=sin^-1x^2`

Find the domain of the following function:

`f(x) = sin^-1x + sinx`

Find the domain of the following function:

`f(x)sin^-1sqrt(x^2-1)`

Find the domain of the following function:

`f(x)=sin^-1x+sin^-1 2x`

If `sin^-1 x + sin^-1 y+sin^-1 z+sin^-1 t=2pi` , then find the value of x² + y² + z² + t² 

Evaluate the following:

`tan^-1 1+cos^-1 (-1/2)+sin^-1(-1/2)`

Evaluate the following:

`tan^-1(-1/sqrt3)+tan^-1(-sqrt3)+tan^-1(sin(-pi/2))`