# SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 chapter 1 - Differentiation [Latest edition]

## Chapter 1: Differentiation

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Q.1

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 DifferentiationQ.1

#### MCQ [1 Mark]

Q.1 | Q 1

Choose the correct alternative:

If y = 1/sqrt(3x^2 - 2x - 1), then ("d"y)/("d"x) = ?

• (-2)/3 (3x - 2) (3x^2 - 2x - 1)^((-3)/2)

• (-3)/2 (3x - 2) (3x^2 - 2x - 1)^((-3)/2)

• (3x - 2) (3x^2 - 2x - 1)^((-3)/2)

• -(3x - 1) (3x^2 - 2x - 1)^((-3)/2)

Q.1 | Q 2

Choose the correct alternative:

If y = root(3)((3x^2 + 8x - 6)^5, then ("d"y)/("d"x) = ?

• 5/3 (6x + 8) (3x^2 + 8x - 6)^(2/3)

• (-5)/3 (6x + 8) (3x^2 + 8x - 6)^(2/3)

• 3/5 (3x + 4) (3x^2 + 8x - 6)^(2/3)

• (-3)/5 (3x + 4) (3x^2 + 8x - 6)^(2/3)

Q.1 | Q 3

Choose the correct alternative:

What is the rate of change of demand (x) of a commodity with respect to its price (y) if y = 10 + x + 25x3.

• 10/(1 + 75x^2)

• 1/(1 + 75x^2)

• 1 + 75x2

• (-1)/(1 + 75x^2)

Q.1 | Q 4

Choose the correct alternative:

What is the rate of change of demand (x) of a commodity with respect to its price (y) if y = (3x + 7)/(2x^2 + 5)

• (2x^2 + 5)^2/((-6x^2 - 38x + 15))

• (2x^2 + 5)^2/((-6x^2 - 28x + 15))

• (2x^2 + 5)^2/((6x^2 - 28x + 15))

• (2x^2 + 5)^2/((6x^2 - 38x + 15))

Q.1 | Q 5

Choose the correct alternative:

If y = x^(sqrt(x)), then ("d"y)/("d"x) = ?

• y/(2sqrt(x))(log x + 2)

• y/sqrt(x)(log x + 2)

• y/(2sqrt(x))(log x - 2)

• y/sqrt(x)(log x - 2)

Q.1 | Q 6

Choose the correct alternative:

If y = (x )x + (10)x, then ("d"y)/("d"x) = ?

• xx(1 + log x) + 10xlog10x

• xx(1 + log x) – 10xlog10

• x(1 + log x) + 10xlog10

• xx(1 + log x) + 10xlog10

Q.1 | Q 7

Choose the correct alternative:

If xm. yn = (x + y )(m + n), then ("d"y)/("d"x) = ?

• y/x

• (-y)/x

• x/y

• (-x)/y

Q.1 | Q 8

Choose the correct alternative:

If xy = 2x – y, then ("d"y)/("d"x) = ?

• (xlog2 - y)/(xlog2x)

• (xlog2 + y)/(xlog2x)

• (xlog2 + x)/(ylog2x)

• (ylog2 - x)/(xlog2x)

Q.1 | Q 9

Choose the correct alternative:

If x = 2am, y = 2am2, where m be the parameter, then ("d"y)/("d"x) = ?

• 2m

• – 2m

• – am

• am

Q.1 | Q 10

Choose the correct alternative:

If x = "a"("t" - 1/"t"), y = "a"("t" + 1/"t"), where t be the parameter, then ("d"y)/("d"x) = ?

• x/y

• (-x)/y

• y/x

• (-y)/x

Q.1 | Q 11

Choose the correct alternative:

If x = at2, y = 2at, then ("d"^2y)/("d"x^2) = ?

• 1/("at"^3)

• (-1)/(2"at"^3)

• (-1)/("at"^2)

• 1/("at"^2)

Q.2

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 DifferentiationQ.2

#### Fill in the blanks: [1 Mark]

Q.2 | Q 1

If y = (5x3 – 4x2 – 8x)9, then ("d"y)/("d"x) is ______

Q.2 | Q 2

If y = "a"^((1 + log"x")), then ("d"y)/("d"x) is ______

Q.2 | Q 3

The rate of change of demand (x) of a commodity with respect to its price (y) is ______ if y = 5 + x2e–x + 2x

Q.2 | Q 4

The rate of change of demand (x) of a commodity with respect to its price (y) is ______ if y = xe–x + 7

Q.2 | Q 5

If y = x10, then ("d"y)/("d"x) is ______

Q.2 | Q 6

If y = ("e")^((2x + 5)), then ("d"y)/("d"x) is ______

Q.2 | Q 7

If sqrt(x) + sqrt(y) = sqrt("a"), then ("d"y)/("d"x) is ______

Q.2 | Q 8

If u = 5x and v = log x, then ("du")/("dv") is ______

Q.2 | Q 9

If u = ex and v = loge x, then ("du")/("dv") is ______

Q.2 | Q 10

If y = x2, then ("d"^2y)/("d"x^2) is ______

Q.3

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 DifferentiationQ.3

#### [1 Mark]

Q.3 | Q 1

State whether the following statement is True or False:

If y = log(log x), then ("d"y)/("d"x) = logx

• True

• False

Q.3 | Q 2

State whether the following statement is True or False:

If y = 10x + 1, then ("d"y)/("d"x) = 10x.log10

• True

• False

Q.3 | Q 3

State whether the following statement is True or False:

If y = x2, then the rate of change of demand (x) of a commodity with respect to its price (y) is 1/(2x)

• True

• False

Q.3 | Q 4

State whether the following statement is True or False:

If y = 7x + 1, then the rate of change of demand (x) of a commodity with respect to its price (y) is 7

• True

• False

Q.3 | Q 5

State whether the following statement is True or False:

If y = ex, then ("d"y)/("d"x) = ex

• True

• False

Q.3 | Q 6

State whether the following statement is True or False:

If y = 4x, then ("d"y)/("d"x) = 4x

• True

• False

Q.3 | Q 7

State whether the following statement is True or False:

If sqrt(x), + sqrt(y) = sqrt("a"), then ("d"y)/("d"x) = 1/(2sqrt(x)) + 1/(2sqrt(y)) = 1/(2sqrt("a"))

• True

• False

Q.3 | Q 8

State whether the following statement is True or False:

If x2 + y2 = a2, then ("d"y)/("d"x) = = 2x + 2y = 2a

• True

• False

Q.3 | Q 9

State whether the following statement is True or False:

If x = 2at, y = 2a, where t is parameter, then ("d"y)/("d"x) = 1/"t"

• True

• False

Q.3 | Q 10

State whether the following statement is True or False:

If x = 5m, y = m, where m is parameter, then ("d"y)/("d"x) = 1/5

• True

• False

Q.3 | Q 11

State whether the following statement is True or False:

If y = ex, then ("d"^2y)/("d"x^2) = ex

• True

• False

Q.4

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 DifferentiationQ.4

#### Solve the following: [3 Marks]

Q.4 | Q 1

Find ("d"y)/("d"x), if y = [log(log(logx))]2

Q.4 | Q 2

Find ("d"y)/("d"x), if y = (6x3 – 3x2 – 9x)10

Q.4 | Q 3

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = (5x + 7)/(2x - 13)

Q.4 | Q 4

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 5 + x2e–x + 2x

Q.4 | Q 5

Find ("d"y)/("d"x), if xy = yx

Q.4 | Q 6

Find ("d"y)/("d"x), if xy = log(xy)

Q.4 | Q 7

Find ("d"y)/("d"x), if x = sqrt(1 + "u"^2), y = log(1 +u2)

Q.4 | Q 8

If x = t.logt, y = tt, then show that ("d"y)/("d"x) = tt

Q.4 | Q 9

Find ("d"^2y)/("d"x^2), if y = "e"^((2x + 1))

Q.5

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 DifferentiationQ.5

#### Solve the following : [4 Marks]

Q.5 | Q 1

Find ("d"y)/("d"x), if y = (log x)x + (x)logx

Q.5 | Q 2

Find ("d"y)/("d"x), if y = root(5)((3x^2 + x + 5)^4

Q.5 | Q 3

Find ("d"y)/("d"x), if y = xx + (7x – 1)x

Q.5 | Q 4

Find rate of change of demand (x) of a commodity with respect to its price (y) if y = (3x + 7)/(2x^2 + 5)

Q.5 | Q 5

Find ("d"y)/("d"x), if y = x^(x^x)

Q.5 | Q 6

Find ("d"y)/("d"x), if y = root(3)(((3x - 1))/((2x + 3)(5 - x)^2)

Q.5 | Q 7

If x5.y7 = (x + y)12, then show that ("d"y)/("d"x) = y/x

Q.5 | Q 8

If xa .yb = (x + y )(a + b), then show that ("d"y)/("d"x) = y/x

Q.5 | Q 9

If x = (4"t")/(1 + "t"^2), y = 3((1 - "t"^2)/(1 + "t"^2)), then show that ("d"y)/("d"x) = (-9x)/(4y)

Q.5 | Q 10

If x2 + 6xy + y2 = 10, then show that ("d"^2y)/("d"x^2) = 80/(3x + y)^3

Q.6

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 Chapter 1 DifferentiationQ.6

#### Activity: [4 Marks]

Q.6 | Q 1

y = (6x4 – 5x3 + 2x + 3)6, find ("d"y)/("d"x)

Solution: Given,

y = (6x4 – 5x3 + 2x + 3)6

Let u = [6x^4 - 5x^3 + square + 3]

∴ y = "u"^square

∴ ("d"y)/"du" = 6u6–1

∴ ("d"y)/"du" = 6(  )5

and "du"/("d"x) = 24x^3 - 15(square) + 2

By chain rule,

("d"y)/("d"x) = ("d"y)/square xx square/("d"x)

∴ ("d"y)/("d"x) = 6(6x^4 - 5x^3 + 2x + 3)^square xx (24x^3 - 15x^2 + square)

Q.6 | Q 2

The rate of change of demand (x) of a commodity with respect to its price (y), if y = 20 + 15x + x3.

Solution: Let y = 20 + 15x + x3

Diff. w.r.to x, we get

("d"y)/("d"x) = square + square  + square

∴ ("d"y)/("d"x) = 15 + 3x2

∴ By derivative of the inverse function,

("d"x)/("d"y)  1/square, ("d"y)/("d"x) ≠ 0

∴ Rate of change of demand with respect to price = 1/(square + square)

Q.6 | Q 3

Find ("d"y)/("d"x), if y = x(x) + 20(x)

Solution: Let y = x(x) + 20(x)

Let u = x^square and "v"^square

∴ y = u + v

Diff. w.r.to x, we get

("d"y)/("d"x) = square/("d"x) + "dv"/square   .....(i)

Now, u = xx

Taking log on both sides, we get

log u = x × log x

Diff. w.r.to x,

1/"u"*"du"/("d"x) = x xx 1/square + log x xx square

∴ "du"/("d"x) = u(1 + log x)

∴ "du"/("d"x) = x^x (1 +  square)    .....(ii)

Now, v = 20x

Diff.w.r.to x, we get

"dv"/("d"x") = 20^square*log(20)     .....(iii)

Substituting equations (ii) and (iii) in equation (i), we get

("d"y)/("d"x) = xx(1 + log x) + 20x.log(20)

Q.6 | Q 4

Find ("d"y)/("d"x), if x = em, y = "e"^(sqrt("m"))

Solution: Given, x = em and y = "e"^(sqrt("m"))

Now, y = "e"^(sqrt("m"))

Diff.w.r.to m,

("d"y)/"dm" = "e"^(sqrt("m"))("d"square)/"dm"

∴ ("d"y)/"dm" = "e"^(sqrt("m"))*1/(2sqrt("m"))    .....(i)

Now, x = em

Diff.w.r.to m,

("d"x)/"dm" = square    .....(ii)

Now, ("d"y)/("d"x) = (("d"y)/("d"x))/square

∴ ("d"y)/("d"x) = (("e"sqrt("m"))/square)/("e"^"m")

∴  ("d"y)/("d"x) = ("e"^(sqrt("m")))/(2sqrt("m")*"e"^("m")`

## Chapter 1: Differentiation

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## SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 chapter 1 - Differentiation

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2021 chapter 1 (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 (Commerce) Maharashtra State Board 2021 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 (Commerce) Maharashtra State Board 2021 chapter 1 Differentiation are Derivatives of Composite Functions - Chain Rule, Derivatives of Inverse Functions, Derivatives of Logarithmic Functions, Derivatives of Implicit Functions, Derivatives of Parametric Functions, Second Order Derivative.

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