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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RD Sharma Solutions Class 12 Mathematics Chapter 13 VSA

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

Confused between CGPA and Percentage?

RD Sharma Chapter wise Solutions

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