HSC Arts (English Medium) इयत्ता ११ वी - Maharashtra State Board Question Bank Solutions

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

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

f(x) `{:(= 3x + 2",",  "for"  -4 ≤ x ≤-2),(= 2x - 3";",  "for"  -2 < x ≤ 6):}`

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

f(x) `{:(= (x^3 - 8)/(x^2 - 4)",",  "for"  x > 2),(= 3",",  "for"  x = 2),(= ("e"^(3(x - 2)^2 - 1))/(2(x - 2)^2) ",",  "for"  x < 2):}`

If f(x) = `(sqrt(2 + sin x) - sqrt(3))/(cos^2x), "for"  x ≠ pi/2`, is continuous at x = `pi/2` then find `"f"(pi/2)`

If f(x) = `(cos^2 x - sin^2 x - 1)/(sqrt(3x^2 + 1) - 1)` for x ≠ 0, is continuous at x = 0 then find f(0)

If f(x) = `(4^(x - π) + 4^(π - x) - 2)/(x - π)^2` for x ≠ π, is continuous at x = π, then find f(π).

If f(x) `{:(= (24^x - 8^x - 3^x + 1)/(12^x - 4^x - 3^x + 1)",",  "for"  x ≠ 0), (= "k"",",  "for"  x = 0):}}` is continuous at x = 0, find k

If f(x)  `{:(= (5^x + 5^(-x) - 2)/(x^2)"," , "for"  x ≠ 0),(= k",",  "for"  x = 0):}}` is continuous at x = 0, find k

If f(x) `{:(= (sin2x)/(5x) - "a"",", "for"  x > 0),(= 4 ",", "for"  x = 0),(= x^2 + "b" - 3",", "for"  x < 0):}}` is continuous at x = 0, find a and b

For what values of a and b is the function

f(x) `{:(= "a"x + 2"b" + 18",",  "for"  x ≤ 0),(= x^2 + 3"a" - "b"",",  "for"  0 < x ≤ 2),(= 8x - 2",",  "for"  x > 2):}}` continuous for every x?

For what values of a and b is the function

f(x) `{:(= (x^2 - 4)/(x - 2)",", "for"  x < 2),(= "a"x^2 - "b"x + 3",", "for"  2 ≤ x < 3),(= 2x - "a" + "b"",", "for"  x ≥ 3):}}` continuous for every x on R?