RD Sharma Solutions Class 12 Mathematics Chapter 13 VSA

Edited By Satyajeet Kumar | Updated on Jan 27, 2022 10:01 AM IST

Students in class 12 will have to face tough competition in their board exams. Due to the minimal time, it becomes almost impossible for them to complete the syllabus and understand all the concepts clearly. Hence, students are recommended to use the RD Sharma class 12th exercise VSA solutions for their home practice. RD Sharma Solutions It will help them to improve their performance and enhance their math skills. Everyone knows that RD Sharma class 12 chapter 13 exercise VSA is the top choice of students who have scored high in their board exams.
The 13th chapter of the NCERT recommended Mathematics Textbook has Differentials, Errors, and Approximations which is a complex topic. The concepts that are covered in this section are Absolute error, Relative error, Percentage error. It will also talk about the Geometrical meaning of differentials and many more. The exercise VSA in the book and the RD Sharma class 12 chapter 13 exercise VSA contain six questions based on the entire chapter.

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RD Sharma Class 12 Solutions Chapter 13 VSA Differentials, Errors and Approximations - Other Exercise
Differentials, Errors and Approximations Excercise: VSA
RD Sharma Chapter wise Solutions

RD Sharma Class 12 Solutions Chapter 13 VSA Differentials, Errors and Approximations - Other Exercise

Differentials, Errors and Approximations Excercise: VSA

Differentials Errors and Approximations exercise very short answer question 1

Answer: 2
Hint: Here we use this below formula,
$\Delta y=F(x+\Delta x)-F(x)$
Given:
$\begin{aligned} &y=x^{2} \\ &x=10 \text { and } \Delta x=0.1 \end{aligned}$
Solution:
$\begin{aligned} &y=x^{2} \\\\ &\text { and } \Delta x=0.1 \end{aligned}$
Let’s take Differentiate of y
$\begin{aligned} &y=x^{2} \\\\ &\frac{d y}{d x}=2 x \end{aligned}$
$\begin{aligned} &\Rightarrow\left(\frac{d y}{d x}\right)=2 \times 10 \quad(x=10)(\text { given }) \\\\ &\Rightarrow \Delta y=d y=\frac{d y}{d x} \times \Delta x=20 \times 0.1=2 \quad(\mathrm{~d} \mathrm{x}=\Delta x=0.1)(\text { Given }) \end{aligned}$

Differentials Errors and Approximations exercise very short answer question 2

Answer: 0.01
Hint: Here we use basic formula,
$\Delta y=d y=\frac{d y}{d x} \times d x$
Given:
$\begin{aligned} &x=3, \Delta x=0.03 \\\\ &y=\log _{e} x \end{aligned}$
Solution:
For
$\begin{aligned} &x=3 \\\\ &y=\log _{e} 3 \end{aligned}$
Also, $\frac{d y}{d x}=\frac{1}{x}$
So, $\left(\frac{d y}{d x}\right)_{x=3}=\frac{1}{3} \Delta y=d y=\frac{d y}{d x} \times \Delta x \Delta y=\frac{1}{3} \times 0.03 \Delta y=0.01$
So, our answer is 0.01

Differentials Errors and Approximations exercise very short answer question 3

Answer:
$2 \alpha \text { ( } 2 \text { alpha) }$
Hint:
Here, we use the concept of approximation.
Given:
Relative error in measuring the radius of circular plan is $\alpha$ .
Solution:
Let $x$ be the radius and $y$ be the area of the circular plan.
We have,
$\begin{aligned} &\frac{\Delta x}{x}=\alpha \quad \text { and } y=x^{2} \\\\ &\frac{d y}{d x}=2 x \quad \quad \frac{\Delta y}{y}=\frac{2 x}{y} \Delta x=\frac{2 x}{x^{2}} \Delta x \end{aligned}$
$\frac{\Delta y}{y}=2 \alpha$ $\left[\because \frac{d\left(x^{n}\right)}{d x}=n x^{n-1}\right]$
Hence the relative error in the area of the circular plan is $2\alpha$ .

Differentials Errors and Approximations exercise very short answer question 4

Answer: $3\alpha$
Hint:

Here we use the basic concept of approximation.

Given:
The percentage error in radius of sphere is $\alpha$
Solution:
Let $V$ be the volume of the sphere
$V=\frac{4}{3} \pi r^{3}$
let radius $r =x$
We have,
$\frac{\Delta x}{x} \times 100=\alpha$
After differentiating,
$\begin{aligned} &\frac{d V}{d x}=4 \pi x^{2} \\\\ &\frac{d V}{V}=\frac{4 \pi x^{2}}{V} d x \end{aligned}$
$\begin{aligned} &\frac{\Delta V}{V}=\frac{4 \pi x^{2}}{\frac{4}{3} \pi x^{3}} \times \frac{x \alpha}{100} \\\\ &\frac{\Delta V}{V} \times 100=3 \alpha \end{aligned}$
Hence, the percentage error in volume of sphere is $3\alpha$ .

Differentials Errors and Approximations exercise very short answer question 5
Answer: $3a$

Hint:
Here we use basic formula of cube,
$V=x^{3}$
Given:
The percentage error in the edge of cube is $a$
Solution:
Let $x$ be the side and $V$ be the volume of the cube
$V=x^{3}$
We have,
$\frac{\Delta x}{x} \times 100=a$
$\frac{d V}{d x}=3 x^{2}$
$\frac{\Delta V}{V}=\frac{3 x^{2}}{V} \times \Delta x$
$\frac{\Delta V}{V}=\frac{3 x^{2}}{x^{3}} \times \frac{\Delta x}{100}$
$\frac{\Delta V}{V} \times 100=3 a$ $\left[\because \frac{d\left(x^{n}\right)}{d x}=n x^{n-1}\right]$
Hence, the percentage error in volume of cube is $3a$

Differentials Errors and Approximations exercise very short answer question 6

Answer: $F(2.1)=9.2$
Hint: Here we use this below formula,
$F(x+\Delta x)=F(x)+F^{\prime}(x) \Delta x$
Given:
$F(x)=x^{4}-10$
Solution:
We have,
$F(x)=x^{4}-10$
So, $F^{\prime}(x)=4 x^{3}$
and $x=2, \Delta x=0.1$
The let’s put the value in formula
$F(2+0.1)=F(2)+F^{\prime}(2) \Delta x$
$\begin{aligned} &=2^{4}-10+4(2)^{3}(0.1) \\\\ &=16-10+(32)(0.1) \\\\ &=16-10+3.2 \\\\ &=9.2 \end{aligned}$
$\mathrm{So}_{,} \; F(2.1) \text { is } 9.2$

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The answers for the book back questions are framed by certain experts who have an in-depth knowledge in the field of mathematics. They have provided various tricks and techniques in the RD Sharma Class 12th Exercise VSA book that can be used to solve the sums easily.
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The Class 12 RD Sharma Chapter 13 Exercise VSA solution material is available in the updated according to the current academic year’s textbook.
The Career 360 website gives the access for everyone to download the PDF format of the RD Sharma Class 12 Solutions Chapter 13 Ex VSA book.

RD Sharma Chapter wise Solutions

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