SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 1.3 - Differentiation [Latest edition]

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SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 1.3 - Differentiation - Shaalaa.com
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Solutions for Chapter 1.3: Differentiation

Below listed, you can find solutions for Chapter 1.3 of Maharashtra State Board SCERT Maharashtra Question Bank for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022.


Q.1Q.2Q.3Q.4Q.5Q.6
Q.1

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 1.3 Differentiation Q.1

MCQ [1 Mark]

Q.1 | Q 1

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 - 1) (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) + 10xlog10

  • 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 `("dy")/("dx")` = ?

  • `"y"/"x"`

  • `(-"y")/"x"`

  • `"x"/"y"`

  • `(-"x")/"y"`

Q.1 | Q 8

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

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 2022 Chapter 1.3 Differentiation Q.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 2022 Chapter 1.3 Differentiation Q.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 2022 Chapter 1.3 Differentiation Q.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 `(dy)/(dx)`, 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 2022 Chapter 1.3 Differentiation Q.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 + 8x + 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

Solve the following:

If `"x"^5 * "y"^7 = ("x + y")^12` then show that, `"dy"/"dx" = "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 2022 Chapter 1.3 Differentiation Q.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^x`

∴ 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"m))/square`

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

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

Solutions for Chapter 1.3: Differentiation

Q.1Q.2Q.3Q.4Q.5Q.6
SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 1.3 - Differentiation - Shaalaa.com

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 1.3 - Differentiation

Shaalaa.com has the Maharashtra State Board Mathematics 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. SCERT Maharashtra Question Bank solutions for Mathematics 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Maharashtra State Board 1.3 (Differentiation) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.

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Concepts covered in 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 1.3 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.

Using SCERT Maharashtra Question Bank 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 solutions Differentiation exercise by students is an easy way to prepare for the exams, as they involve solutions arranged chapter-wise and also page-wise. The questions involved in SCERT Maharashtra Question Bank Solutions are essential questions that can be asked in the final exam. Maximum Maharashtra State Board 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 students prefer SCERT Maharashtra Question Bank Textbook Solutions to score more in exams.

Get the free view of Chapter 1.3, Differentiation 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 additional questions for Mathematics 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Maharashtra State Board, and you can use Shaalaa.com to keep it handy for your exam preparation.

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