HSC Science (General) इयत्ता ११ वी - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Discuss the continuity of the following function at the point(s) or on the interval indicated against them:

f(x) = `(cos4x - cos9x)/(1 - cosx)`, for x ≠ 0

f(0) = `68/15`, at x = 0 on `- pi/2 ≤ x ≤ pi/2`

Discuss the continuity of the following function at the point(s) or on the interval indicated against them:

f(x) `{:( = (sin^2pix)/(3(1 - x)^2) ",", "for"  x ≠ 1),(= (pi^2sin^2((pix)/2))/(3 + 4cos^2 ((pix)/2)) ",", "for"  x = 1):}}` at x = 1

Discuss the continuity of the following function at the point(s) or on the interval indicated against them:

f(x) `{:(= (|x + 1|)/(2x^2 + x - 1)",", "for"  x ≠ -1),(= 0",", "for"  x = -1):}}` at x = – 1

Discuss the continuity of the following function at the point(s) or on the interval indicated against them:

f(x) = [x + 1] for x ∈ [−2, 2)

Where [*] is greatest integer function.

Discuss the continuity of the following function at the point(s) or on the interval indicated against them:

f(x) `{:(= 2x^2 + x + 1",", "for"  |x - 3| ≥ 2),(= x^2 + 3",", "for"  1 < x < 5):}`

Identify discontinuity for the following function as either a jump or a removable discontinuity on their respective domain:

f(x) `{:(= x^2 + x - 3,","  "for"  x ∈ [ -5, -2)),(= x^2 - 5,","  "for"  x ∈ (-2, 5]):}`

Identify discontinuity for the following function as either a jump or a removable discontinuity on their respective domain:

f(x) `{:(= x^2 + 5x + 1"," , "for"  0 ≤ x ≤ 3),(= x^3 + x + 5"," , "for"  3 < x ≤ 6):}`

Identify discontinuity for the following function as either a jump or a removable discontinuity on their respective domain:

f(x) `{:(= (x^2 + x + 1)/(x + 1)"," , "for"  x ∈ [0, 3)),(=(3x +4)/(x^2 - 5)"," , "for"  x ∈ [3, 6]):}`

Discuss the continuity of the following function at the point or on the interval indicated against them. If the function is discontinuous, identify the type of discontinuity and state whether the discontinuity is removable. If it has a removable discontinuity, redefine the function so that it becomes continuous:

f(x) = `((x + 3)(x^2 - 6x + 8))/(x^2 - x - 12)`

Discuss the continuity of the following function at the point or on the interval indicated against them. If the function is discontinuous, identify the type of discontinuity and state whether the discontinuity is removable. If it has a removable discontinuity, redefine the function so that it becomes continuous:

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

Find k if following function is continuous at the point indicated against them:

f(x) `{:(= ((5x - 8)/(8 - 3x))^(3/(2x - 4))",", "for"  x ≠ 2),(= "k"",", "for"  x = 2):}}` at x = 2

Find k if following function is continuous at the point indicated against them:

f(x) `{:(= (45^x - 9^x - 5^x + 1)/(("k"^x - 1)(3^x - 1))",", "for"  x ≠ 0),(= 2/3",", "for"  x = 0):}}` at x = 0

Find a and b if following function is continuous at the point or on the interval indicated against them:

f(x) `{:(= (4tanx + 5sinx)/("a"^x - 1)",", "for"  x < 0),(= (9)/(log2)",", "for"  x = 0),(= (11x + 7x*cosx)/("b"^x - 1)",", "for"  x > 0):}`

Find a and b if following function is continuous at the point or on the interval indicated against them:

f(x) `{:(= "a"x^2 + "b"x + 1",", "for"  |2x - 3| ≥ 2),(= 3x + 2",", "for"  1/2 < x < 5/2):}`

Find f(a), if f is continuous at x = a where,

f(x) = `(1 + cos(pi x))/(pi(1 - x)^2)`, for x ≠ 1 and at a = 1

Find f(a), if f is continuous at x = a where,

f(x) = `(1 - cos[7(x - pi)])/(5(x - pi)^2`, for x ≠ π at a = π

Solve using intermediate value theorem:

Show that 5^x − 6x = 0 has a root in [1, 2]

Solve using intermediate value theorem:

Show that x³ − 5x² + 3x + 6 = 0 has at least two real root between x = 1 and x = 5

Find the derivative of the following w. r. t. x by using method of first principle:

x² + 3x – 1

Find the derivative of the following w. r. t. x by using method of first principle:

sin (3x)