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

Chapters Chapter 1: Differentiation

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 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

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