RD Sharma Solutions Class 12 Mathematics Chapter 10 FBQ

Q: Are the RD Sharma class 12 solutions chapter 10 FBQ considered the best review material for exam preparations?

RD Sharma class 12th solutions exercise FBQ, are arranged as PDFs. RD Sharma class 12th exercise FBQ Additionally, indicating these solutions can fundamentally support students' certainty in noting the troublesome and long-answer type questions effectively.

Q: What is the best approach to prepare for the public exams using the RD Sharma Solutions Class 12 Maths Chapter 10 FBQ book?

The students must get to know the chapters and practice every exercise to get a proper practice to face the public exams. The RD Sharma Class 12 Solutions Chapter 10 FBQ makes this process easier.

Q: How can I practice for the mathematics chapter 10 FBQ of class 12?

Try answering the FBQ by yourself and use the RD Sharma class 12th exercise FBQ material to recheck your answers. Change your mistakes and learn the ways to solve the sum.

Q: At what cost is the RD Sharma class 12 exercise FBQ Chapter 10 available at the Career 360 website?

The Class 12 RD Sharma chapter 10 exercise FBQ material is easily available at the Career 360 website for free of cost. No kind of payment is charged to utilize this resource material.

Q: How many questions are answered in the RD Sharma Solutions Class 12 Maths Chapter 10 FBQ material?

The solutions for all the 26 questions asked in the textbook can be found at the Class 12 RD Sharma chapter 10 exercise FBQ material.

RD Sharma Solutions Class 12 Mathematics Chapter 10 FBQ

Edited By Satyajeet Kumar | Updated on Jan 20, 2022 06:34 PM IST

The RD Sharma Class 12 chapter 10 exercise FBQ arrangement – Our experts plan Differentiation to help ensure students in understanding the thoughts solicited in this chapter and procedures to deal with issues in a more restricted period.

In this RD Sharma class 12 solutions FBQ Chapter 10, Differentiation is a process in Maths, where we find the quick rate of function change based on one of its variables. Class 12 RD Sharma chapter 10 exercise FBQ solution consists of various topics that play a major role in finding the rate of displacement. At Careers360, RD Sharma class 12 chapter 10 exercise FBQ Solutions help students who seek to get a decent scholarly score in the exam.

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RD Sharma Class 12 Solutions Chapter 10 FBQ Differentiation - Other Exercise
Differentiation Excercise: FBQ
RD Sharma Chapter-wise Solutions

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RD Sharma Class 12 Solutions Chapter 10 FBQ Differentiation - Other Exercise

Differentiation Excercise: FBQ

Differentiation exercise Fill in the blanks question 1

Answer: ${(\frac{d y}{d u})}_{u = - 1} = 2$
Hint: $y = (\begin{array}{cc} u^{2} & u \geq 0 \\ - u^{2} & u < 0 \end{array})$
Given: $y = u | u |$
Solution:
$\begin{aligned} y = (\begin{array}{cc} u^{2} & u \geq 0 \\ - u^{2} & u < 0 \end{array}) \\ \frac{d y}{d u} = (\begin{array}{cc} 2 u^{2} & u \geq 0 \\ - 2 u^{2} & u < 0 \end{array}) \end{aligned}$
$\begin{aligned} {(\frac{d y}{d u})}_{- 1} = (- 2 (- 1) \end{aligned}$
$= 2$

Differentiation exercise Fill in the blanks question 2

Answer: ${(\frac{d y}{d x})}_{- 1} = 1 and {(\frac{d y}{d x})}_{x = 1} = 3$
Hint: $y = (\begin{array}{cc} 3 x & x \geq 0 \\ x & x < 0 \end{array})$
Given: $y = 2 x + | x |$
Solution:

\begin{aligned} y = (\begin{array}{ll} 2 x + x & x \geq 0 \\ 2 x - x & x < 0 \end{array}) \\ y = (\begin{array}{cc} 3 x & x \geq 0 \\ x & x < 0 \end{array}) \\ \frac{d y}{d x} = (\begin{array}{ll} 3 & x > 0 \\ 1 & x < 0 \end{array}) \end{aligned}

{(\frac{d y}{d x})}_{- 1} = 1 and {(\frac{d y}{d x})}_{1} = 3

Differentiation exercise Fill in the blanks question 3

Answer: $f^{1} (2) = 3$
Hint: $f (x) = (\begin{array}{ll} x - x^{2} & x^{2} - x \leq 0 \\ x^{2} - x & x^{2} - x \geq 0 \end{array})$
Given: $f (x) = | x^{2} - x |$
Solution: $f (x) = | x^{2} - x |$

f (x) = (\begin{array}{cc} - (x^{2} - x) & x^{2} - x < 0 \\ x^{2} - x & x^{2} - x \geq 0 \end{array})

\begin{aligned} f (x) = (\begin{array}{cc} (x - x^{2}) & x^{2} - x < 0 \\ x^{2} - x & x^{2} - x \geq 0 \end{array}) \\ f^{^{'}} (x) = (\begin{array}{cc} (1 - 2 x) & x^{2} - x < 0 \\ 2 x - 1 & x^{2} - x \geq 0 \end{array}) \end{aligned}

\begin{aligned} f^{^{'}} (2) = 2 x - 1 \\ = 2 (2) - 1 \\ = 3 \end{aligned}

Differentiation exercise Fill in the blanks question 4

Answer: $k = 1$
Hint: $\frac{d y}{d x} = \cos x^{0}$
Given: $y = \sin x^{0} and \frac{d y}{d x} = k \cos x^{0}$
Solution: $y = \sin x^{0}$ $∵ x^{0} = \frac{π}{180} x$
Differentiating y.u.1 to x

\begin{aligned} \frac{d y}{d x} = \frac{π}{180} \cos x^{0} \dots \dots \dots (i) \\ And \frac{d y}{d x} = k \cos x^{0} \dots \dots (i i) \end{aligned}

By (i) and (ii)

k = \frac{π}{180}

Differentiation exercise Fill in the blanks question 5

Answer: $f^{1} (0) = 0$
Hint: $f^{1} (x) = g^{1} (x) > g^{1} (x)$
Solution: $f (x) = e^{x} g (x)$
Differentiating

f (x)

we get

\begin{aligned} f^{1} (x) = e^{x} g^{1} (x) + e^{x} g (x) \\ f^{1} (0) = e^{0} g^{1} (0) + e^{0} g (0) \end{aligned}

\begin{aligned} = 1 (1) + 1 (2) \\ = 1 + 2 \\ = 3 \end{aligned}

Differentiation exercise Fill in the blanks question 6

Answer: $f (- 3) = 3$
Hint:

\begin{aligned} | x + 2 | = (\begin{array}{ll} (x + 2) & x \geq - 2 \\ - x > 2 & x < 2 \end{array}) \\ = & f (x) = 3 | x + 2 | \end{aligned}

Solution: $f (x) = 3 | x + 2 |$

\begin{aligned} f (x) = (\begin{array}{cc} 3 (x + 2) & x \geq - 2 \\ - 3 (x + 2) & x < - 2 \end{array}) \\ f^{1} (x) = (\begin{array}{cc} 3 & x \geq - 2 \\ - 3 & x < - 2 \end{array}) \\ f^{1} (- 3) = - 3 \end{aligned}

Differentiation exercise Fill in the blanks question 7

Answer: 2
Hint: using chain rule
Given: $f (1) = 3$
Solution: $\frac{d}{d x} = \ln f (e^{x} + 2 x)$

\begin{aligned} = \frac{1}{f (e^{x} + 2 x)} f^{^{'}} (e^{x} + 2 x) \times (e^{x} + 2) \\ = (e^{x} + 2) \frac{f^{^{'}} (e^{0} + 2 x)}{f (e^{0} + 2 x)} \end{aligned}

\begin{aligned} = 3 \frac{f^{^{'}} (1)}{f (1)} \\ = 3 \times \frac{2}{3} = 2 \end{aligned}

Differentiation exercise Fill in the blanks question 8

Answer: $f^{^{'}} (x) = (\begin{array}{cc} 2 x & x \geq 0 \\ - 2 x & x^{2} \leq 0 \end{array})$
Hint: $x | x | = (\begin{array}{cc} x^{2} & x \geq 0 \\ - x^{2} & x < 0 \end{array})$
Given: $f | x | = x | x |$
Solution:
$f | x | = x | x |$

\begin{aligned} f (x) = (\begin{array}{cc} x^{2} & x \geq 0 \\ - x^{2} & x < 0 \end{array}) \\ f^{1} (x) = (\begin{array}{cc} 2 x & x \geq 0 \\ - 2 x & x < 0 \end{array}) \end{aligned}

Differentiation exercise Fill in the blanks question 9

Answer: $f^{1} (2) = 0$
Hint: the mod function
Given: $f (x) = | x - 1 | + | x - 3 |$
Solution:
$\begin{aligned} | x - 1 | = (\begin{array}{cc} u - 1 & u > 1 \\ - (u - 1) & u < 1 \end{array}) \\ | x - 3 | = (\begin{array}{cc} x - 3 & x > 3 \\ - (x - 3) & x \leq 3 \end{array}) \end{aligned}$
$\begin{aligned} Case- 1 : x < 1 \\ \begin{aligned} f (x) = & | x - 1 | + | x - 3 | \\ = - x + 1 - x > 3 \\ = - 2 x > 4 \end{aligned} \end{aligned}$

$\begin{aligned} Case- 2 : 1 \leq x < 3 \\ \begin{aligned} f (x) = & | x - 1 | - (x - 3) \\ = x - 1 - x + 3 = 2 \end{aligned} \end{aligned}$

$\begin{aligned} Case. 3 \\ f (x) = x - 1 > x + 3 = 2 x - 4 \\ f (x) = (\begin{array}{cc} - 2 x + 4 & x < 1 \\ 2 & 1 \leq x < 3 \\ - (x - 3) & x \leq 3 \end{array}) \end{aligned}$

\begin{aligned} f^{^{'}} (x) = (\begin{array}{cc} - 2 & x < 1 \\ 0 & 1 \leq x < 3 \\ 2 & x \geq 3 \end{array}) \\ f^{^{'}} (x) = 0 \end{aligned}

Differentiation exercise Fill in the blanks question 10

Answer: $\frac{1 + \sqrt{3}}{2}$
Hint: $\cos x < \sin x where 0 \leq x \leq \frac{π}{4}$
Given: $f (x) = | \cos x - \sin x |$
Solution: $f (x) = | \cos x - \sin x |$
$\begin{array}{ll} f (x) = \cos x - \sin x & 0 \leq x \leq π / 4 \\ f (x) = - (\cos x - \sin x) & \frac{π}{4} \leq x \leq \frac{π}{2} \end{array}$
$\begin{aligned} f (x) = - \cos x - \sin x 0 \leq x \leq π / 4 \\ f (x) = \cos x + \sin x \frac{π}{4} \leq x \leq \frac{π}{2} \end{aligned}$
$\begin{aligned} = \frac{1}{\sqrt{2}} + \frac{\sqrt{3}}{2} \\ = \frac{1 + \sqrt{3}}{2} \end{aligned}$

Differentiation exercise Fill in the blanks question 11

Answer: $f^{^{'}} (\frac{π}{x}) = (\frac{1}{\sqrt{2}})$
Hint: $f (x) = (\begin{array}{cc} \cos x & 0 \leq x \leq \frac{π}{2} \\ - \cos x & \frac{π}{2} < x < π \end{array})$
Given: $f (x) = | \cos x |$
Solution:
$f (x) = | \cos x |$
$f (x) = (\begin{array}{cc} \cos x & 0 \leq u \leq \frac{π}{2} \\ - \cos x & \frac{π}{2} < u < π \end{array})$

\begin{aligned} f^{^{'}} (x) & = (\begin{array}{cc} - \sin x & 0 \leq x \leq \frac{π}{2} \\ \sin x & \frac{π}{2} < x < π \end{array}) \\ f^{1} (\frac{π}{2}) & = - \sin \frac{π}{2} \\ = - \frac{1}{\sqrt{2}} \end{aligned}

Differentiation exercise Fill in the blanks question 12

Answer: $\frac{2}{3 x}$
Hint: $take u = x^{2} and v = x^{3}$
Given:The derivative of

x^{2}

and

x^{3}

Solution:

\begin{aligned} Let u = x^{2} \\ \frac{d u}{d x} = 2 x \\ And let v = x^{3} \end{aligned}

\begin{aligned} \frac{d v}{d x} = 3 x^{2} \\ \frac{d u}{d v} = \frac{d u / d x}{d v / d x} = \frac{2 x}{3 x^{2}} = \frac{2}{3 x} \end{aligned}

Differentiation exercise Fill in the blanks question 14

Answer: $f^{1} (\frac{π}{4}) = (\frac{1}{\sqrt{2}})$
Hint: $f^{1} (x) = (\begin{array}{cc} \sin x & 0 \leq x \leq \frac{π}{2} \\ - \sin x & \frac{π}{2} < x < π \end{array})$
Given: $f (x) = | \sin x |$
Solution:

\begin{aligned} f (x) = (\begin{array}{cc} \sin x & 0 \leq x \leq \frac{π}{2} \\ - \sin x & \frac{π}{2} < x < π \end{array}) \\ f^{^{'}} (x) = (\begin{array}{cc} \cos x & 0 \leq x \leq \frac{π}{2} \\ - \cos x & \frac{π}{2} < x < π \end{array}) \end{aligned}

\begin{aligned} f^{^{'}} \frac{π}{2} = & \cos \frac{π}{4} \\ = \frac{1}{\sqrt{2}} \end{aligned}

Differentiation exercise Fill in the blanks question 15

Answer: $f^{1} (\frac{π}{6}) = \frac{1 + \sqrt{3}}{2}$
Hint:
Given: $f (x) = | \sin x - \cos x |$
Solution: $f (x) = | \sin x - \cos x |$

\begin{aligned} f (x) = (\begin{array}{cc} \sin x - \cos x & 0 \leq x \leq \frac{π}{4} \\ - (\sin x - \cos x) & \frac{π}{4} < x \leq \frac{π}{4} \end{array}) \\ f^{^{'}} (x) = (\begin{array}{cc} \sin x + \cos x & 0 \leq x \leq \frac{π}{4} \\ - (\sin x + \cos x) & \frac{π}{4} < x \leq \frac{π}{4} \end{array}) \end{aligned}

\begin{aligned} f^{^{'}} (\frac{π}{6}) = & \sin \frac{π}{6} + \cos \frac{π}{6} \\ = \frac{1}{2} + \frac{\sqrt{3}}{2} \end{aligned}

$= \frac{\sqrt{3} + 1}{2}$

Differentiation exercise Fill in the blanks question 16

Answer: 2
Hint: $\frac{d (\tan x)}{d x} = \sec^{2} x$
Given: $y = \tan x^{0}$
$∵ x^{0} = \frac{π}{180} x ∴ \frac{d y}{d x} = \frac{π}{180} \cdot \sec^{2} \frac{π}{180} x = \frac{π}{180} \sec^{2} x^{0}$

Differentiation exercise Fill in the blanks question 17

Answer: $\frac{d y}{d x} = 0$
Hint: $\frac{d}{d x} = (\sin^{- 1} x) = \frac{1}{\sqrt{1 - x^{2}}}$
Given: $y = \sin^{- 1} (e^{x}) + \cos^{- 1} (e^{x})$
Solution: $y = \sin^{- 1} (e^{x}) + \cos^{- 1} (e^{x})$
Differentiating (i) we get;

\begin{aligned} \frac{d y}{d x} & = \frac{1}{\sqrt{1 + e^{2 x}}} e^{x} - \frac{1}{\sqrt{1 + e^{2 x}}} e^{x} \\ = 0 \end{aligned}

Differentiation exercise Fill in the blanks question 18

Answer: $\frac{3}{\sqrt{1 - x^{2}}}$
Hint: $put x = \sin θ$
Given: $y = \sin^{- 1} (3 x - 4 x^{3})$
Solution: By putting

x = \sin θ in y

\begin{aligned} y = \sin^{- 1} (3 \sin θ - 4 \sin θ) \\ y = \sin^{- 1} (\sin 3 θ) \end{aligned}

\begin{aligned} = 3 θ \\ = 3 \sin^{- 1} x \end{aligned}

\frac{d y}{d x} = \frac{3}{\sqrt{1 - x^{2}}}

Differentiation exercise Fill in the blanks question 19

Answer: 0
Hint: $\cos^{- 1} x + \sin^{- 1} x = \frac{π}{2}$
Given: $y = \sec^{- 1} (\frac{\sqrt{x} + 1}{\sqrt{x} - 1}) + \sin^{- 1} (\frac{\sqrt{x} - 1}{\sqrt{x} + 1})$
Solution:
$y = \cos^{- 1} (\frac{\sqrt{x} - 1}{\sqrt{x} + 1}) + \sin^{- 1} (\frac{\sqrt{x} - 1}{\sqrt{x} + 1})$
$\begin{aligned} = \frac{π}{2} \\ = 0 \end{aligned}$

Differentiation exercise Fill in the blanks question 20

Answer: $tan u$
Hint: $put u = \cos x and v = \sin x$
Given: The derivative of

\cos x

w.r.t

\sin x

Solution:

\begin{aligned} Let u = \cos x and v = \sin x \\ \frac{d u}{d v} = - \sin x & \frac{d v}{d u} = \cos x \end{aligned}

Therefore:

\begin{aligned} \frac{d u}{d v} = \frac{d u ∣ d v}{d v ∣ d u} = \frac{- \sin x}{\cos x} \\ = - \tan x \end{aligned}

Differentiation exercise Fill in the blanks question 21

Answer: $\frac{\log_{10} e}{x}$
Hint: $use: [\frac{d}{d x} \log_{e} x = \frac{1}{x}]$
Given: $If y = \log_{10} x$
Solution: $\log_{10} x$

\begin{aligned} y = \log_{10} e \cdot \log_{e} x \\ \frac{d y}{d x} = \log_{10} e \times \frac{1}{x} \end{aligned}

So the answer is $\frac{\log_{10} e}{x}$

Differentiation exercise Fill in the blanks question 22

Answer: the correct answer is

\frac{3 x^{2}}{1 + x^{6}}

Hint: integrate the function.
Given: $\frac{d}{dx} f (x) = \frac{1}{1 + x^{2}}$
Solution: integrating we get

\begin{aligned} \int \frac{d}{d x} (f (x)) d x = \int [\frac{1}{1 + x^{2}}] d x \\ f (x) = \tan^{- 1} x + e \end{aligned}

\begin{aligned} f (x^{3}) = \tan^{- 1} (x^{3}) + c \\ f^{1} (x^{3}) = \frac{3 x^{2}}{1 + x^{6}} \end{aligned}

So the answer is

\frac{3 x^{2}}{1 + x^{6}}

Differentiation exercise Fill in the blanks question 23

Answer: the correct answer is ='0'
Hint: $y = \cos (\sin x^{2}) \frac{d}{d x} \sin x = - \cos x$
$\frac{d y}{d x} = - \sin (\sin x^{2}) - \cos x^{2} \cdot 2 x$
Given: $y = \cos (\sin x^{2})$
Solution: using

\frac{d y}{d x}

to get the value at

\frac{λ}{2}

Now at

x = \sqrt{π / 2}

\frac{d}{d x} = - \sin (\sin \sqrt{\frac{π}{2}}) \cdot \cos \sqrt{\frac{π}{2}} \cdot 2 \sqrt{\frac{π}{2}} = 0

So the correct answer is ‘0'

Differentiation exercise Fill in the blanks question 24

Answer: The correct answer is

\frac{1}{x}

Hint: Differentiate the function but u should not be equal to zero
Given: $y = \log_{e} | x |$
Solution:

y = \log | x | = (\begin{array}{cc} \log (- x), & x < 0 \\ \log (x), & x > 0 \end{array})

\frac{d y}{d x} = (\begin{array}{cc} \frac{- 1}{- x} = \frac{1}{x},, & x < 0 \\ \frac{1}{x}, & x > 0 \end{array})

Which means it is equal to

\frac{1}{x}

when

x \neq 0

\frac{d}{d x} \log | x | = \frac{1}{x}, x \neq 0

So the answer is

\frac{1}{x}

Differentiation exercise Fill in the blanks question 25

Answer: the correct answer is =0
Hint: put 1,4,5 in the equation to get the respective value the use

f^{1} (1) + f^{1} (u) - f^{1} (5)

Given: $y = a x^{2} + b x + c$
Solution:
$\begin{aligned} y = a x^{2} + b x + c \\ f^{1} (x) = \frac{d y}{d x} = 2 a x + b \\ f^{1} (1) = 2 a + b and \end{aligned}$
$\begin{aligned} f^{1} (x) = 3 a + 4 b \\ f^{1} (5) = 10 a + 5 b \\ f^{1} (1) + f^{1} (u) - f^{1} (5) \end{aligned}$
$= 2 a + b + 8 a + 4 b - 10 a - 5 b$
So the correct answer is 0

Differentiation exercise Fill in the blanks question 26

Answer: the correct answer is

\frac{1}{\sqrt{2}}

Hint:
Given: $f^{1} (1) = 2 and g^{1} (\sqrt{2}) = 4$
Solution:
$\begin{aligned} y = f (\tan x) and z = g (\sec x) \\ \frac{d y}{d x} = f^{'} (\tan x) \sec^{2} x and \frac{d z}{d x} = g^{'} (\sec x) \cdot \sec x + \tan x \end{aligned}$
$\begin{aligned} \frac{d y}{d z} = \frac{f^{1} (\tan x) \cdot \sec^{2} x}{g^{1} (\sec x) \cdot \sec x \tan x} \\ \frac{d y}{d z} = \frac{f^{'} (1) \cdot (\sqrt{2})^{2}}{g^{'} (\sqrt{2}) \cdot (\sqrt{2})} = \frac{1}{\sqrt{2}} \end{aligned}$
So the answer is

\frac{1}{\sqrt{2}}

This chapter of RD Sharma class 12 exercise FBQ evolves around the possibility of lucidness. Students can download the RD Sharma class 12 solutions FBQ Chapter 10 Differentiation to find out with regards to this topic.

The RD Sharma class 12 solution Differentiation exercise FBQ reference book is the most purchased book by the students. It is not a simple task to answer FBQs that too in a complex chapter like the Differentiation. The students must be clear about all the concepts to give the answers quickly by not wasting time.

Moreover, the Class 12 RD Sharma chapter 10 exercise FBQ solution book has many shortcuts and tricks that can be followed to arrive at the solution effortlessly. This Class 12 RD Sharma chapter 10 exercise FBQ material has around 26 solved questions.

Some important concepts used in chapter 10 FBQ:-

Questions related to Differentiation of a function.
Questions related to the Differentiation of inverse trigonometric functions by the chain rule.
Questions related to Differentiation by using trigonometric substitutions.
Solutions on Differentiation of implicit functions.
Questions related to Logarithmic differentiation.
Differentiation of infinite series.
Differentiation of parametric functions.
Differentiation of determinants.

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Frequently Asked Questions (FAQs)

1. Are the RD Sharma class 12 solutions chapter 10 FBQ considered the best review material for exam preparations?

RD Sharma class 12th solutions exercise FBQ, are arranged as PDFs. RD Sharma class 12th exercise FBQ Additionally, indicating these solutions can fundamentally support students' certainty in noting the troublesome and long-answer type questions effectively.

2. What is the best approach to prepare for the public exams using the RD Sharma Solutions Class 12 Maths Chapter 10 FBQ book?

The students must get to know the chapters and practice every exercise to get a proper practice to face the public exams. The RD Sharma Class 12 Solutions Chapter 10 FBQ makes this process easier.

3. How can I practice for the mathematics chapter 10 FBQ of class 12?

Try answering the FBQ by yourself and use the RD Sharma class 12th exercise FBQ material to recheck your answers. Change your mistakes and learn the ways to solve the sum.

4. At what cost is the RD Sharma class 12 exercise FBQ Chapter 10 available at the Career 360 website?

The Class 12 RD Sharma chapter 10 exercise FBQ material is easily available at the Career 360 website for free of cost. No kind of payment is charged to utilize this resource material.

5. How many questions are answered in the RD Sharma Solutions Class 12 Maths Chapter 10 FBQ material?

The solutions for all the 26 questions asked in the textbook can be found at the Class 12 RD Sharma chapter 10 exercise FBQ material.