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SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 chapter 2 - Differentiation [Latest edition]

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12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 - Shaalaa.com
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Chapter 2: Differentiation

MCQVery Short AnswersShort Answers IShort Answers IILong Answers III:: Theorems ::
MCQ

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 Chapter 2 Differentiation MCQ

2 Marks each

MCQ | Q 1

If y = sec (tan−1x) then `("d"y)/("d"x)` at x = 1 is ______

  • `1/2`

  • 1

  • `1/sqrt(2)`

  • `sqrt(2)`

MCQ | Q 2

If f(x) = logx (log x) then f'(e) is ______

  • 1

  • e

  • `1/"e"`

  • 0

MCQ | Q 3

If y = `25^(log_5sin_x) + 16^(log_4cos_x)` then `("d"y)/("d"x)` = ______

  • 1

  • 0

  • 9

  • cos x – sin x

MCQ | Q 4

If f'(4) = 5, f(4) = 3, g'(6) = 7 and R(x) = g[3 + f(x)] then R'(4) = ______

  • 35

  • 12

  • `7/5`

  • 105

MCQ | Q 5

If y = `tan^-1((2x)/(1 - x^2))`, x ∈ (1, 1) then `("d"y)/("d"x)` = ______

  • `(-2)/(1 + x^2)`

  • 1

  • `(2)/(1 + x^2)`

  • `1/(1 + x^2)`

MCQ | Q 6

If g is the inverse of f and f'(x) = `1/(1 + x^4)` then g'(x) = ______

  • `1/(1 + ["g"(x)]^4`

  • `(4x^3)/(1 + x^4)`

  • `1/(1 + ["g"(x)]^3`

  • `1 + ["g"(x)]^4`

MCQ | Q 7

If sin−1(x3 + y3) = a then `("d"y)/("d"x)` = ______

  • `(-x)/(cos"a")`

  • `(-x^2)/(y^2)`

  • `(y^2)/(x^2)`

  • `sin"a"/y`

MCQ | Q 8

If x = cos−1(t), y = `sqrt(1 - "t"^2)` then `("d"y)/("d"x)` = ______

  • t

  • – t

  • `(-1)/"t"`

  • `1/"t"`

MCQ | Q 9

If x2 + y2 = 1 then `("d"^2x)/("d"y^2)` = ______

  • x3

  • y3

  • – y3 

  • `(-1)/x^3`

MCQ | Q 10

If x2 + y2 = t + `1/"t"` and x4 + y4 = t2 + `1/"t"^2` then `("d"y)/("d"x)` = ______

  • `x/(2y)`

  • `(-y)/x`

  • `(-x)/(2y)`

  • `y/x`

MCQ | Q 11

If x = a t4 y = 2a t2 then `("d"y)/("d"x)` = ______

  • `1/"t"`

  • `(-1)/"t"`

  • `1/"t"^2`

  • `(-1)/"t"^2`

Very Short Answers

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 Chapter 2 Differentiation Very Short Answers

1 Mark each

Very Short Answers | Q 1

Differentiate y = `sqrt(x^2 + 5)` w.r. to x

Very Short Answers | Q 2

Differentiate y = etanx w.r. to x

Very Short Answers | Q 3

If y = sin−1 (2x), find `("d"y)/(""d"x)` 

Very Short Answers | Q 4

If f(x) is odd and differentiable, then f′(x) is

Very Short Answers | Q 5

If y = `"e"^(1 + logx)` then find `("d"y)/("d"x)` 

Short Answers I

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 Chapter 2 Differentiation Short Answers I

2 mark each

Short Answers I | Q 1

If y = log [cos(x5)] then find `("d"y)/("d"x)`

Short Answers I | Q 2

If y = `sqrt(tansqrt(x)`, find `("d"y)/("d"x)`

Short Answers I | Q 3

Find the derivative of the inverse of function y = 2x3 – 6x and calculate its value at x = −2

Short Answers I | Q 4

Let f(x) = x5 + 2x – 3 find (f−1)'(-3)

Short Answers I | Q 5

If y = cos−1 [sin (4x)], find `("d"y)/("d"x)`

Short Answers I | Q 6

If y = `tan^-1[sqrt(1 + cos x)/(1 - cos x)]`, find `("d"y)/("d"x)`

Short Answers I | Q 7

If x = sin θ, y = tan θ, then find `("d"y)/("d"x)`

Short Answers I | Q 8

Differentiate sin2 (sin−1(x2)) w.r. to x

Short Answers II

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 Chapter 2 Differentiation Short Answers II

3 marks each

Short Answers II | Q 1

If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`

Short Answers II | Q 2

If y = `log[4^(2x)((x^2 + 5)/sqrt(2x^3 - 4))^(3/2)]`, find `("d"y)/("d"x)`

Short Answers II | Q 3

Differentiate `cot^-1((cos x)/(1 + sinx))` w.r. to x

Short Answers II | Q 4

Differentiate `sin^-1((2cosx + 3sinx)/sqrt(13))` w.r. to x

Short Answers II | Q 5

Differentiate `tan^-1((8x)/(1 - 15x^2))` w.r. to x

Short Answers II | Q 6

If log5 `((x^4 + y^4)/(x^4 - y^4))` = 2, show that `("d"y)/("d"x) = (12x^2)/(13y^3)`

Short Answers II | Q 7

If y = `sqrt(cos x + sqrt(cos x + sqrt(cos x + ...... ∞)`, show that `("d"y)/("d"x) = (sin x)/(1 - 2y)`

Short Answers II | Q 8

Find the derivative of cos−1x w.r. to `sqrt(1 - x^2)`

Short Answers II | Q 9

If x sin(a + y) + sin a cos(a + y) = 0 then show that `("d"y)/("d"x) = (sin^2("a" + y))/(sin"a")`

Short Answers II | Q 10

If y = 5x. x5. xx. 55 , find `("d"y)/("d"x)`

Long Answers III

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 Chapter 2 Differentiation Long Answers III

4 Marks each

Long Answers III | Q 1

If y = `"e"^("m"tan^(-1)x`, show that `(1 + x^2) ("d"^2y)/("d"x^2) + (2x - "m")("d"y)/("d"x)` = 0

Long Answers III | Q 2

If x7 . y5 = (x + y)12, show that `("d"y)/("d"x) = y/x`

Long Answers III | Q 3

Differentiate `tan^-1[(sqrt(1 + x^2) - 1)/x]` w.r. to `tan^-1[(2x sqrt(1 - x^2))/(1 - 2x^2)]`

Long Answers III | Q 4

If y = `sin^-1[("a"cosx - "b"sinx)/sqrt("a"^2 + "b"^2)]`, then find `("d"y)/("d"x)`

Long Answers III | Q 5

If y = cos(m cos–1x), then show that `(1 - x^2) ("d"^2y)/("d"x^2) - x("d"y)/("d"x) + "m"^2y` = 0

:: Theorems ::

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 Chapter 2 Differentiation :: Theorems ::

:: Theorems :: | Q 1

If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x such that the composite function y = f[g(x)] is a differentiable function of x, then `("d"y)/("d"x) = ("d"y)/("d"u)*("d"u)/("d"x)`. Hence find `("d"y)/("d"x)` if y = sin2x

:: Theorems :: | Q 2

Suppose y = f(x) is a differentiable function of x on an interval I and y is one – one, onto and `("d"y)/("d"x)` ≠ 0 on I. Also if f–1(y) is differentiable on f(I), then `("d"x)/("d"y) = 1/(("d"y)/("d"x)), ("d"y)/("d"x)` ≠ 0

:: Theorems :: | Q 3

If x = f(t) and y = g(t) are differentiable functions of t so that y is a differentiable function of x and it `("d"x)/("dt")` ≠ 0 then `("d"y)/("d"x) = (("d"y)/("dt"))/(("d"x)/("dt"))`

Chapter 2: Differentiation

MCQVery Short AnswersShort Answers IShort Answers IILong Answers III:: Theorems ::
12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 - Shaalaa.com

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 chapter 2 - Differentiation

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 chapter 2 (Differentiation) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 chapter 2 Differentiation are Differentiation, Derivatives of Composite Functions - Chain Rule, Geometrical Meaning of Derivative, Derivatives of Inverse Functions, Logarithmic Differentiation, Derivatives of Implicit Functions, Derivatives of Parametric Functions, Higher Order Derivatives.

Using SCERT Maharashtra Question Bank 12th Board Exam solutions Differentiation exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in SCERT Maharashtra Question Bank Solutions are important questions that can be asked in the final exam. Maximum students of Maharashtra State Board 12th Board Exam prefer SCERT Maharashtra Question Bank Textbook Solutions to score more in exam.

Get the free view of chapter 2 Differentiation 12th Board Exam extra questions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 and can use Shaalaa.com to keep it handy for your exam preparation

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