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

## Chapter 2: Differentiation

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

### 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)

### 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

### 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)

### 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"))

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

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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.

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