RD Sharma Solutions Class 12 Mathematics Chapter 13 VSA

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
$F(2+0.1)=F(2)+F^{\prime}(2) \Delta x$
$\begin{aligned} &=2^{4}-10+4(2)^{3}(0.1) \\\\ &=16-10+(32)(0.1) \\\\ &=16-10+3.2 \\\\ &=9.2 \end{aligned}$
$\mathrm{So}_{,} \; F(2.1) \text { is } 9.2$

The students who are preparing for their public exams lookout for reliable resources to study. Many such students of class12 depend on the RD Sharma class 12th exercise VSA book during their preparation. Here are a few reasons on why there is a huge demand for the RD Sharma books:

• 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.

• Homework and assignments can also be completed by referring to the RD Sharma Class 12 Solutions Differentials, Errors, and Approximations ex VSA resource material.

• 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.