#### Chapters

Chapter 1.2: Matrices

Chapter 1.3: Differentiation

Chapter 1.4: Applications of Derivatives

Chapter 1.5: Integration

Chapter 1.6: Definite Integration

Chapter 1.7: Application of Definite Integration

Chapter 1.8: Differential Equation and Applications

Chapter 2.1: Commission, Brokerage and Discount

Chapter 2.2: Insurance and Annuity

Chapter 2.3: Linear Regression

Chapter 2.4: Time Series

Chapter 2.5: Index Numbers

Chapter 2.6: Linear Programming

Chapter 2.7: Assignment Problem and Sequencing

Chapter 2.8: Probability Distributions

## Chapter 1: Differentiation

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

#### MCQ [1 Mark]

**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)`

**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)`

**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 + 25x^{3}.

`10/(1 + 75x^2)`

`1/(1 + 75x^2)`

1 + 75x

^{2}`(-1)/(1 + 75x^2)`

**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))`

**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)`

**Choose the correct alternative:**

If y = (x )^{x} + (10)^{x}, then `("d"y)/("d"x)` = ?

x

^{x}(1 + log x) + 10^{x}log10^{x}x

^{x}(1 + log x) – 10^{x}log10x(1 + log x) + 10

^{x}log10x

^{x}(1 + log x) + 10^{x}log10

**Choose the correct alternative:**

If x^{m}. y^{n} = (x + y )(m + n), then `("d"y)/("d"x)` = ?

`y/x`

`(-y)/x`

`x/y`

`(-x)/y`

**Choose the correct alternative:**

If x^{y} = 2^{x} – y, then `("d"y)/("d"x)` = ?

`(xlog2 - y)/(xlog2x)`

`(xlog2 + y)/(xlog2x)`

`(xlog2 + x)/(ylog2x)`

`(ylog2 - x)/(xlog2x)`

**Choose the correct alternative:**

If x = 2am, y = 2am^{2}, where m be the parameter, then `("d"y)/("d"x)` = ?

2m

– 2m

– am

am

**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`

**Choose the correct alternative:**

If x = at^{2}, y = 2at, then `("d"^2y)/("d"x^2)` = ?

`1/("at"^3)`

`(-1)/(2"at"^3)`

`(-1)/("at"^2)`

`1/("at"^2)`

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

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

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

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

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

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

If y = x^{10}, then `("d"y)/("d"x)` is ______

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

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

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

If u = e^{x} and v = log_{e }x, then `("du")/("dv")` is ______

If y = x^{2}, then `("d"^2y)/("d"x^2)` is ______

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

#### [1 Mark]

**State whether the following statement is True or False:**

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

True

False

**State whether the following statement is True or False:**

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

True

False

**State whether the following statement is True or False:**

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

True

False

**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

**State whether the following statement is True or False:**

If y = e^{x}, then `("d"y)/("d"x)` = e^{x}_{ }

True

False

**State whether the following statement is True or False:**

If y = 4^{x}, then `("d"y)/("d"x)` = 4^{x}

True

False

**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

**State whether the following statement is True or False:**

If x^{2} + y^{2} = a^{2}, then `("d"y)/("d"x)` = = 2x + 2y = 2a

True

False

**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

**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

**State whether the following statement is True or False:**

If y = e^{x}, then `("d"^2y)/("d"x^2)` = e^{x}

True

False

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

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

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

Find `("d"y)/("d"x)`, if y = (6x^{3} – 3x^{2} – 9x)^{10}

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

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

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

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

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

If x = t.logt, y = t^{t}, then show that `("d"y)/("d"x)` = t^{t}

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

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

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

Find `("d"y)/("d"x)`, if y = (log x)^{x} + (x)log^{x}

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

Find `("d"y)/("d"x)`, if y = x^{x} + (7x – 1)^{x}

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

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

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

If x^{5}.y^{7} = (x + y)^{12}, then show that `("d"y)/("d"x) = y/x`

If x^{a} .y^{b} = (x + y )(a + b), then show that `("d"y)/("d"x) = y/x`

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

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

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

#### Activity: [4 Marks]

y = (6x^{4} – 5x^{3} + 2x + 3)^{6}, find `("d"y)/("d"x)`

**Solution:** Given,

y = (6x^{4} – 5x^{3} + 2x + 3)^{6}

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

∴ y = `"u"^square`

∴ `("d"y)/"du"` = 6u^{6–1}

∴ `("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)`

The rate of change of demand (x) of a commodity with respect to its price (y), if y = 20 + 15x + x^{3}.

**Solution:** Let y = 20 + 15x + x^{3}

Diff. w.r.to x, we get

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

∴ `("d"y)/("d"x)` = 15 + 3x^{2}

∴ 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)`

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 = x^{x}

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 = 20^{x}

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)` = x^{x}(1 + log x) + 20^{x}.log(20)

Find `("d"y)/("d"x)`, if x = e^{m}, y = `"e"^(sqrt("m"))`

**Solution:** Given, x = e^{m} 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 = e^{m}

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

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

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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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