Rd Sharma XII Vol 1 2020 Solutions for Class 12 Commerce Maths Chapter 13 Differentials, Errors And Approximations are provided here with simple step-by-step explanations. These solutions for Differentials, Errors And Approximations are extremely popular among Class 12 Commerce students for Maths Differentials, Errors And Approximations Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma XII Vol 1 2020 Book of Class 12 Commerce Maths Chapter 13 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma XII Vol 1 2020 Solutions. All Rd Sharma XII Vol 1 2020 Solutions for class Class 12 Commerce Maths are prepared by experts and are 100% accurate.

#### Question 10:

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

#### Question 11:

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

#### Question 12:

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

#### Question 13:

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.

Let x be the radius and y be the surface area of the sphere.

#### Question 14:

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

Let y be the surface area of the cube.

#### Question 15:

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.

Let x be the radius of the sphere and y be its volume.

#### Question 16:

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

#### Question 1:

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%

(a) 1%
Let l be the length if the pendulum and T be the period.

#### Question 2:

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

(a) 2a%

Let x be the side of the cube and y be its surface area.

#### Question 3:

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%

(b) 3k%
Let x be the radius of the sphere and y be its volume.
Then,

#### Question 4:

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

(c) 3$\alpha$$\alpha$%
Let x be the radius, which is equal to the height of the cylinder. Let y be its volume.

#### Question 5:

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 %

(b) 2k%

Let x be the side of the triangle and y be its area.

#### Question 6:

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

(c) 1.3893

#### Question 7:

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

(c) 80000 π mm3

Let x be the radius of the sphere and y be its volume.

#### Question 8:

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

(c) 3 λ %

Let the radius of the cone be x, the height be 2x and the volume be y.
$\frac{∆x}{x}=\lambda %\phantom{\rule{0ex}{0ex}}⇒y=\frac{1}{3}\pi {x}^{2}×2x=\frac{2}{3}\pi {x}^{3}\phantom{\rule{0ex}{0ex}}⇒\frac{dy}{dx}=2\pi {x}^{2}\phantom{\rule{0ex}{0ex}}⇒\frac{∆y}{y}=\frac{2\pi {x}^{2}}{y}dx=\frac{3}{x}×\lambda x\phantom{\rule{0ex}{0ex}}⇒\frac{∆y}{y}=3\lambda %$

#### Question 9:

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

(c) $\frac{1}{8}$ %

We have

#### Question 10:

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

(d) n:1

#### Question 11:

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

(a) 2.0125
Consider the function y= f(x)=${x}^{\frac{1}{5}}$.

#### Question 12:

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

(a) $\frac{1}{14}$
Let x be the radius of the circle and y be its circumference.

#### Question 13:

If y = x4 - 10 and if x changes from 2 to 1.99, the change in y is
(a) 0.32            (b) 0.032             (c) 5.68             (d) 5.968

Let x = 2 and x + ∆x = 1.99.

∴ ∆x = 1.99 − 2 = −0.01

$y={x}^{4}-10$     (Given)

Differentiating both sides with respect to x, we get

$\frac{dy}{dx}=4{x}^{3}$

$⇒{\left(\frac{dy}{dx}\right)}_{x=2}=4×{\left(2\right)}^{3}=4×8=32$

$\therefore ∆y=\left(\frac{dy}{dx}\right)∆x$

$⇒∆y=32×\left(-0.01\right)=-0.32$

Thus, the change in y is 0.32.

Hence, the correct answer is option (a).

#### Question 1:

If y = x3 + 5 and x changes from 3 to 2.99, then the approximate change is y is _________________.

Let x = 3 and x + ∆x = 2.99.

∴ ∆x = 2.99 − 3 = −0.01

y = x3 + 5        (Given)

Differentiating both sides with respect to x, we get

$\frac{dy}{dx}=3{x}^{2}$

$⇒{\left(\frac{dy}{dx}\right)}_{x=3}=3×{\left(3\right)}^{2}=27$

$∆y=\left(\frac{dy}{dx}\right)∆x$

$⇒∆y=27×\left(-0.01\right)=-0.27$

Thus, the approximate change in y is −0.27.

If y = x3 + 5 and x changes from 3 to 2.99, then the approximate change is y is ___−0.27___.

#### Question 2:

The approximate change in the volume of a cube of side x metres caused by increasing the side by 2%, is ______________.

Let ∆x be the change in side x and ∆V be the change in the volume of the cube.

It is given that, $\frac{∆x}{x}×100=2$         .....(1)

Now,

Volume of the cube of side x, V = x3

$V={x}^{3}$

Differentiating both sides with respect to x, we get

$\frac{dV}{dx}=3{x}^{2}$

$\therefore ∆V=\left(\frac{dV}{dx}\right)∆x$

$⇒∆V=3{x}^{2}∆x$

$⇒∆V=3{x}^{2}×\frac{2x}{100}$         [Using (1)]

$⇒∆V=\frac{6{x}^{3}}{100}$

$⇒∆V=0.06{x}^{3}$

Thus, the approximate change in volume of the cube is 0.06x3 m3.

The approximate change in the volume of a cube of side x metres caused by increasing the side by 2%, is ___0.06x3 m3___.

#### Question 1:

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

#### Question 2:

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

We have

#### Question 3:

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

Let x be the radius and y be the area of the circular plane.

#### Question 4:

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

Let V be the volume of the sphere.

#### Question 5:

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.

Let x be the side and V be the volume of the cube.

#### Question 1:

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

Hence, there is no change in the value of y.

#### Question 2:

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

#### Question 3:

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.

Let at any time, x be the radius and y be the area of the plate.

Hence, the approximate change in the area of the plate is 2k$\mathrm{\pi }$ cm2 .

#### Question 4:

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.

Let x be the edge of the cube and y be the surface area.

Hence, the percentage error in calculating the surface area is 2.

#### Question 5:

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.

Let x be the radius and y be the volume of the sphere.

Hence, the percentage error in the calculation of the volume of the sphere is 0.3.

#### Question 6:

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.

#### Question 7:

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?

Let h be the height, y be the surface area, V be the volume, l be the slant height and r be the radius of the cone.

#### Question 8:

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.

Let x be the radius of the sphere and y be its volume.

Hence proved.

#### Question 9:

Using differentials, find the approximate values of the following:

(i) $\sqrt{25.02}$

(ii) ${\left(0.009\right)}^{\frac{1}{3}}$

(iii) ${\left(0.007\right)}^{\frac{1}{3}}$

(iv) $\sqrt{401}$

(v) ${\left(15\right)}^{\frac{1}{4}}$

(vi) ${\left(255\right)}^{\frac{1}{4}}$

(vii) $\frac{1}{\left(2.002{\right)}^{2}}$

(viii) loge 4.04, it being given that log104 = 0.6021 and log10e = 0.4343

(ix) loge 10.02, it being given that loge10 = 2.3026

(x) log10 10.1, it being given that log10e = 0.4343

(xi) cos 61°, it being given that sin60° = 0.86603 and 1° = 0.01745 radian

(xii) $\frac{1}{\sqrt{25.1}}$

(xiii) $\mathrm{sin}\left(\frac{22}{14}\right)$

(xiv) $\mathrm{cos}\left(\frac{11\mathrm{\pi }}{36}\right)$

(xv) ${\left(80\right)}^{\frac{1}{4}}$

(xvi) ${\left(29\right)}^{\frac{1}{3}}$

(xvii) ${\left(66\right)}^{\frac{1}{3}}$

(xviii) $\sqrt{26}$                  [CBSE 2000]

(xix) $\sqrt{37}$                    [CBSE 2000]

(xx) $\sqrt{0.48}$                  [CBSE 2002C]

(xxi) ${\left(82\right)}^{\frac{1}{4}}$                  [CBSE 2005]

(xxii) ${\left(\frac{17}{81}\right)}^{\frac{1}{4}}$

(xxiii) ${\left(33\right)}^{\frac{1}{5}}$

(xxiv) $\sqrt{36.6}$

(xxv) ${25}^{\frac{1}{3}}$

(xxvi) $\sqrt{49.5}$                 [CBSE 2012]

(xxvii) ${\left(3.968\right)}^{\frac{3}{2}}$            [CBSE 2014]

(xxviii) ${\left(1.999\right)}^{5}$             [NCERT EXEMPLAR]

(xxix) $\sqrt{0.082}$               [NCERT EXEMPLAR]

(i)

(ii)

(iii)

(iv).

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

(xii)

(xiii)

(xiv)

(xv)

(xvi)

(xvii)

(xviii)

(xix)

(xx)

(xxi)

(xxii)

(xxiii)

(xxiv)

(xv)

(xxvi)

(xxvii)

(xxviii)

(xxix)

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