HSC Arts (English Medium) ११ वीं कक्षा - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Find all the point of discontinuities of f(x) = [x] on the interval (− 3, 2).

Discuss the continuity of the function f(x) = |2x + 3|, at x = `(−3)/(2)`

Test the continuity of the following function at the point or interval indicated against them :

f(x)  `{:(= (sqrt(x - 1) - (x - 1)^(1/3))/(x - 2)",",  "for"  x ≠ 2),(= 1/5",",  "for"  x = 2):}}`at x = 2

Test the continuity of the following function at the point or interval indicated against them :

f(x)  `{:(= (x^3 - 8)/(sqrt(x + 2) - sqrt(3x - 2))",",  "for"  x ≠ 2),(= -24",",  "for"  x = 2):}}` at x = 2

Test the continuity of the following function at the point or interval indicated against them :

f(x) `{:(= 4x + 1",",  "for"  x ≤  8/3),(= (59 - 9x)/3 ",",  "for"  x > 8/3):}}  "at"  x = 8/3`

Test the continuity of the following function at the point or interval indicated against them :

f(x) `{:(= ((27 - 2x)^(1/3) - 3)/(9 - 3(243 + 5x)^(1/5))",",  "for"  x ≠ 0),(= 2",",  "for"  x = 0):}}` at x = 0.

Test the continuity of the following function at the point or interval indicated against them:

f(x) `{:( =(x^2 + 8x - 20)/(2x^2 - 9x + 10)",",  "for"  0 < x < 3","  x ≠ 2),(= 12",",  "for"  x = 2),(= (2 - 2x - x^2)/(x - 4)",",  "for"  3 ≤ x < 4):}}` at x = 2

Identify discontinuities for the following function as either a jump or a removable discontinuity :

f(x) = `(x^2 - 10x + 21)/(x - 7)`

Identify the discontinuity for the following function as either a jump or a removable discontinuity.

f(x) `{:(= x^2 + 3x - 2",",  "for"  x ≤ 4),(= 5x + 3",",  "for"  x > 4):}`

Identify discontinuities for the following function as either a jump or a removable discontinuity :

f(x) `{:(= x^2 - 3x - 2",",  "for"  x < -3),(= 3 + 8x",",  "for"  x > -3):}`

Identify discontinuities for the following function as either a jump or a removable discontinuity :

f(x) `{:(= 4 + sin x",",  "for"  x < pi),(= 3 - cos x",",  "for"  x > pi):}`

Show that following function have continuous extension to the point where f(x) is not defined. Also find the extension :

f(x) = `(1 - cos2x)/sinx`, for x ≠ 0

Show that following function have continuous extension to the point where f(x) is not defined. Also find the extension :

f(x) = `(3sin^2 x + 2cos x(1 - cos 2x))/(2(1 - cos^2x)`, for x ≠ 0

Show that following function have continuous extension to the point where f(x) is not defined. Also find the extension :

f(x) = `(x^2 - 1)/(x^3 + 1)` for x ≠ – 1

Discuss the continuity of the following function at the point indicated against them :

f(x) = `{:(=( sqrt(3) - tanx)/(pi - 3x)",", x ≠ pi/3),(= 3/4",", x = pi/3):}}  "at"  x = pi/3`

Discuss the continuity of the following function at the point indicated against them :

f(x)  `{:(=("e"^(1/x) - 1)/("e"^(1/x) + 1)",",  "for"  x ≠ 0),(= 1",", "for"  x = 0):}}` at x = 0

Discuss the continuity of the following function at the point indicated against them :

f(x)  `{:(=(4^x - 2^(x + 1) + 1)/(1 - cos 2x)",",  "for"  x ≠ 0),(= (log 2)^2/2",",  "for"  x = 0):}}` at x = 0.

The following function has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it becomes continuous :

f(x) `{:(=("e"^(5sinx) - "e"^(2x))/(5tanx - 3x)",",   "for"  x ≠ 0),(= 3/4",",   "for"  x = 0):}}` at x = 0

The following function has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it become continuous :

f(x) `{:(= log_((1 + 3x)) (1 + 5x)",", "for"  x > 0),(=(32^x - 1)/(8^x - 1)",",  "for"  x < 0):}}` at x = 0

The following function has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it become continuous :

f(x) = `((3 - 8x)/(3 - 2x))^(1/x)`, for x ≠ 0