HSC Science (General) इयत्ता ११ वी - Maharashtra State Board Question Bank Solutions

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?

Discuss the continuity of f on its domain, where f(x) `{:(= |x + 1|",", "for"  -3 ≤ x ≤ 2),(= |x - 5|",", "for"  2 < x ≤ 7):}`.

Discuss the continuity of f(x) at x = `pi/4` where, 

f(x) `{:(= ((sinx + cosx)^3 - 2sqrt(2))/(sin 2x - 1)",", "for"  x ≠ pi/4),(= 3/sqrt(2)",", "for"  x = pi/4):}`

Determine the values of p and q such that the following function is continuous on the entire real number line.

f(x) `{:(= x + 1",", "for"   1 < x < 3),(= x^2 + "p"x + "q"",", "for"  |x - 2| ≥ 1):}`

Show that there is a root for the equation 2x³ − x − 16 = 0 between 2 and 3.

Show that there is a root for the equation x³ − 3x = 0 between 1 and 2.