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# RD Sharma solutions for Class 12 Mathematics chapter 14 - Differentials, Errors and Approximations

## Chapter 14: Differentials, Errors and Approximations

#### Chapter 14: Differentials, Errors and Approximations solutions [Pages 9 - 10]

Q 1 | Page 9

If y = sin x and x changes from π/2 to 22/14, what is the approximate change in y ?

Q 2 | Page 9

The radius of a sphere shrinks from 10 to 9.8 cm. Find approximately the decrease in its volume ?

Q 3 | Page 9

A circular metal plate expends under heating so that its radius increases by k%. Find the approximate increase in the area of the plate, if the radius of the plate before heating is 10 cm.

Q 4 | Page 9

Find the percentage error in calculating the surface area of a cubical box if an error of 1% is made in measuring the lengths of edges of the cube ?

Q 5 | Page 9

If there is an error of 0.1% in the measurement of the radius of a sphere, find approximately the percentage error in the calculation of the volume of the sphere ?

Q 6 | Page 9

The pressure p and the volume v of a gas are connected by the relation pv1.4 = const. Find the percentage error in p corresponding to a decrease of 1/2% in v ?

Q 7 | Page 9

The height of a cone increases by k%, its semi-vertical angle remaining the same. What is the approximate percentage increase (i) in total surface area, and (ii) in the volume, assuming that k is small ?

Q 8 | Page 9

Show that the relative error in computing the volume of a sphere, due to an error in measuring the radius, is approximately equal to three times the relative error in the radius ?

Q 9.01 | Page 9

1 Using differential, find the approximate value of the following:

$\sqrt{25 . 02}$

Q 9.02 | Page 9

Using differential, find the approximate value of the following:  $\left( 0 . 009 \right)^\frac{1}{3}$

Q 9.03 | Page 9

Using differential, find the approximate value of the following: $\left( 0 . 007 \right)^\frac{1}{3}$

Q 9.04 | Page 9

Using differential, find the approximate value of the $\sqrt{401}$ ?

Q 9.05 | Page 9

Using differential, find the approximate value of the $\left( 15 \right)^\frac{1}{4}$ ?

Q 9.06 | Page 9

Using differential, find the approximate value of the $\left( 255 \right)^\frac{1}{4}$ ?

Q 9.07 | Page 9

Using differential, find the approximate value of the $\frac{1}{(2 . 002 )^2}$ ?

Q 9.08 | Page 9

Using differential, find the approximate value of the loge 4.04, it being given that log104 = 0.6021 and log10e = 0.4343 ?

Q 9.09 | Page 9

Using differential, find the approximate value of the loge 10.02, it being given that loge10 = 2.3026 ?

Q 9.1 | Page 9

Using differential, find the approximate value of the  log10 10.1, it being given that log10e = 0.4343 ?

Q 9.11 | Page 9

Using differentials, find the approximate values of the cos 61°, it being given that sin60° = 0.86603 and 1° = 0.01745 radian ?

Q 9.12 | Page 9

Using differential, find the approximate value of the $\frac{1}{\sqrt{25 . 1}}$ ?

Q 9.13 | Page 9

Using differential, find the approximate value of the $\sin\left( \frac{22}{14} \right)$ ?

Q 9.14 | Page 9

Using differential, find the approximate value of the $\cos\left( \frac{11\pi}{36} \right)$ ?

Q 9.15 | Page 9

Using differential, find the approximate value of the $\left( 80 \right)^\frac{1}{4}$ ?

Q 9.16 | Page 9

Using differential, find the approximate value of the $\left( 29 \right)^\frac{1}{3}$ ?

Q 9.17 | Page 9

Using differential, find the approximate value of the $\left( 66 \right)^\frac{1}{3}$ ?

Q 9.18 | Page 9

Using differential, find the approximate value of the $\sqrt{26}$ ?

Q 9.19 | Page 9

Using differential, find the approximate value of the  $\sqrt{37}$ ?

Q 9.2 | Page 9

Using differential, find the approximate value of the  $\sqrt{0 . 48}$ ?

Q 9.21 | Page 9

Using differential, find the approximate value of the $\left( 82 \right)^\frac{1}{4}$ ?

Q 9.22 | Page 9

Using differential, find the approximate value of the $\left( \frac{17}{81} \right)^\frac{1}{4}$ ?

Q 9.23 | Page 9

Using differential, find the approximate value of the $\left( 33 \right)^\frac{1}{5}$ ?

Q 9.24 | Page 9

Using differential, find the approximate value of the $\sqrt{36 . 6}$ ?

Q 9.25 | Page 9

Using differential, find the approximate value of the ${25}^\frac{1}{3}$ ?

Q 9.26 | Page 9

Using differential, find the approximate value of the $\sqrt{49 . 5}$ ?

Q 9.27 | Page 9

Using differential, find the approximate value of the $\left( 3 . 968 \right)^\frac{3}{2}$ ?

Q 9.28 | Page 9

Using differential, find the approximate value of the $\left( 1 . 999 \right)^5$ ?

Q 9.29 | Page 9

Using differential, find the approximate value of the  $\sqrt{0 . 082}$ ?

Q 10 | Page 10

Find the approximate value of f (2.01), where f (x) = 4x2 + 5x + 2 ?

Q 11 | Page 10

Find the approximate value of f (5.001), where f (x) = x3 − 7x2 + 15 ?

Q 12 | Page 10

Find the approximate value of log10 1005, given that log10 e = 0.4343 ?

Q 13 | Page 10

If the radius of a sphere is measured as 9 cm with an error of 0.03 m, find the approximate error in calculating its surface area ?

Q 14 | Page 10

Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1% ?

Q 15 | Page 10

If the radius of a sphere is measured as 7 m with an error of 0.02 m, find the approximate error in calculating its volume ?

Q 16 | Page 10

Find the approximate change in the value V of a cube of side x metres caused by increasing the side by 1% ?

#### Chapter 14: Differentials, Errors and Approximations solutions [Page 12]

Q 1 | Page 12

For the function y = x2, if x = 10 and ∆x = 0.1. Find ∆ y ?

Q 2 | Page 12

If y = loge x, then find ∆y when x = 3 and ∆x = 0.03 ?

Q 3 | Page 12

If the relative error in measuring the radius of a circular plane is α, find the relative error in measuring its area ?

Q 4 | Page 12

If the percentage error in the radius of a sphere is α, find the percentage error in its volume ?

Q 5 | Page 12

A piece of ice is in the form of a cube melts so that the percentage error in the edge of cube is a, then find the percentage error in its volume ?

#### Chapter 14: Differentials, Errors and Approximations solutions [Page 13]

Q 1 | Page 13

If there is an error of 2% in measuring the length of a simple pendulum, then percentage error in its period is
(a)1%
(b) 2%
(c) 3%
(d) 4%

Q 2 | Page 13

If there is an error of a% in measuring the edge of a cube, then percentage error in its surface is
(a) 2a%

(b)$\frac{a}{2} \%$

(c) 3a%

(d) none of these

Q 3 | Page 13

If an error of k% is made in measuring the radius of a sphere, then percentage error in its volume is
(a) k%
(b) 3k%
(c) 2k%
(d) k/3%

Q 4 | Page 13

The height of a cylinder is equal to the radius. If an error of α % is made in the height, then percentage error in its volume is
(a) α %
(b) 2α %
(c) 3α %
(d) none of these

Q 5 | Page 13

While measuring the side of an equilateral triangle an error of k % is made, the percentage error in its area is
(a) k %
(b) 2k %
(c) $\frac{k}{2}\%$

(d) 3k %

Q 6 | Page 13

If loge 4 = 1.3868, then loge 4.01 =
(a) 1.3968
(b) 1.3898
(c) 1.3893
(d) none of these

Q 7 | Page 13

A sphere of radius 100 mm shrinks to radius 98 mm, then the approximate decrease in its volume is
(a) 12000 π mm3
(b) 800 π mm3
(c) 80000 π mm3
(d) 120 π mm3

Q 8 | Page 13

If the ratio of base radius and height of a cone is 1 : 2 and percentage error in radius is λ %, then the error in its volume is
(a) λ %
(b) 2 λ %
(c) 3 λ %
(d) none of these

Q 9 | Page 13

The pressure P and volume V of a gas are connected by the relation PV1/4 = constant. The percentage increase in the pressure corresponding to a deminition of 1/2 % in the volume is

(a) $\frac{1}{2} \%$

(b) $\frac{1}{4} \%$

(c)  $\frac{1}{8} \%$

(d) none of these

Q 10 | Page 13

If y = xn, then the ratio of relative errors in y and x is
(a) 1 : 1
(b) 2 : 1
(c) 1 : n
(d) n : 1

Q 11 | Page 13

The approximate value of (33)1/5 is
(a) 2.0125
(b) 2.1
(c) 2.01
(d) none of these

Q 12 | Page 13

The circumference of a circle is measured as 28 cm with an error of 0.01 cm. The percentage error in the area is

(a) $\frac{1}{14}$

(b) 0.01

(c) $\frac{1}{7}$

(d) none of these

## RD Sharma solutions for Class 12 Mathematics chapter 14 - Differentials, Errors and Approximations

RD Sharma solutions for Class 12 Maths chapter 14 (Differentials, Errors and Approximations) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

Further, we at shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 12 Mathematics chapter 14 Differentials, Errors and Approximations are Maximum and Minimum Values of a Function in a Closed Interval, Maxima and Minima, Simple Problems on Applications of Derivatives, Graph of Maxima and Minima, Approximations, Tangents and Normals, Increasing and Decreasing Functions, Rate of Change of Bodies Or Quantities, Introduction to Applications of Derivatives.

Using RD Sharma Class 12 solutions Differentials, Errors and Approximations exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer RD Sharma Textbook Solutions to score more in exam.

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