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

Evaluate the following :

Find the limit of the function, if it exists, at x = 1

f(x) = `{(7 - 4x, "for", x < 1),(x^2 + 2, "for", x ≥ 1):}`

Evaluate the following :

Given that 7x ≤ f(x) ≤ 3x² – 6 for all x. Determine the value of `lim_(x -> 3) "f"(x)`

Evaluate the following :

`lim_(x -> 0) [(sqrt(1 - cosx))/x]`

Evaluate the following :

`lim_(x -> 1) [(x + 3x^2 + 5x^3 + ... + (2"n" - 1)x^"n" - "n"^2)/(x - 1)]`

Evaluate the following :

`lim_(x -> 0) {1/x^12 [1 - cos(x^2/2) - cos(x^4/4) + cos(x^2/2) cos(x^4/4)]}`

Examine the continuity of f(x) = x³ + 2x² − x − 2 at x = − 2

Examine the continuity of `"f"(x)  {:(= sin x",",  "for"  x ≤ pi/4), (= cos x",",  "for"  x > pi/4):}}  "at"  x = pi/4`

Examine the continuity of `f(x) = {:((x^2 - 9)/(x  - 3)",",  "for"  x ≠ 3),(=8",",  "for"  x = 3):}}` at x = 3.

Examine whether the function is continuous at the points indicated against them:

f(x)  `{:(= x^3 - 2x + 1",",  "if"  x ≤ 2),(= 3x - 2",",  "if"  x > 2):}}` at x = 2

Examine whether the function is continuous at the points indicated against them :

f(x) `{:( = (x^2 + 18x - 19)/(x - 1)",",  "for"  x ≠ 1),(= 20",",  "for"  x = 1):}}` at x = 1

Examine whether the function is continuous at the points indicated against them :

f(x) `{:(= x/(tan3x) + 2",",   "for"  x < 0),(= 7/3",",  "for"  x ≥ 0):}}  "at"  x = 0`

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):}`