Page No 13.10:
Question 10:
Find the approximate value of f (2.01), where f (x) = 4x2 + 5x + 2.
Answer:
Page No 13.10:
Question 11:
Find the approximate value of f (5.001), where f (x) = x3 − 7x2 + 15.
Answer:
Page No 13.10:
Question 12:
Find the approximate value of log10 1005, given that log10e = 0.4343.
Answer:
Page No 13.10:
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.
Answer:
Let x be the radius and y be the surface area of the sphere.
Page No 13.10:
Question 14:
Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1%.
Answer:
Let y be the surface area of the cube.
Page No 13.10:
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.
Answer:
Let x be the radius of the sphere and y be its volume.
Page No 13.10:
Question 16:
Find the approximate change in the value V of a cube of side x metres caused by increasing the side by 1%.
Answer:
Page No 13.12:
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%
Answer:
(a) 1%
Let l be the length if the pendulum and T be the period.
Page No 13.12:
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)
(c) 3a%
(d) none of these
Answer:
(a) 2a%
Let x be the side of the cube and y be its surface area.
Page No 13.12:
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%
Answer:
(b) 3k%
Let x be the radius of the sphere and y be its volume.
Then,
Page No 13.12:
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
Answer:
(c) 3%
Let x be the radius, which is equal to the height of the cylinder. Let y be its volume.
Page No 13.12:
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)
(d) 3k %
Answer:
(b) 2k%
Let x be the side of the triangle and y be its area.
Page No 13.13:
Question 6:
If loge 4 = 1.3868, then loge 4.01 =
(a) 1.3968
(b) 1.3898
(c) 1.3893
(d) none of these
Answer:
(c) 1.3893
Page No 13.13:
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
Answer:
(c) 80000 π mm3
Let x be the radius of the sphere and y be its volume.
Page No 13.13:
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
Answer:
(c) 3 λ %
Let the radius of the cone be x, the height be 2x and the volume be y.
Page No 13.13:
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)
(b)
(c)
(d) none of these
Answer:
(c) %
We have
Page No 13.13:
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
Answer:
(d) n:1
Page No 13.13:
Question 11:
The approximate value of (33)1/5 is
(a) 2.0125
(b) 2.1
(c) 2.01
(d) none of these
Answer:
(a) 2.0125
Consider the function y= f(x)=.
Page No 13.13:
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)
(b) 0.01
(c)
(d) none of these
Answer:
(a)
Let x be the radius of the circle and y be its circumference.
Page No 13.13:
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
Answer:
Let
x = 2 and
x + â
x = 1.99.
∴ â
x = 1.99 − 2 = −0.01
(Given)
Differentiating both sides with respect to
x, we get
Thus, the change in
y is 0.32.
Hence, the correct answer is option (a).
Page No 13.13:
Question 1:
If y = x3 + 5 and x changes from 3 to 2.99, then the approximate change is y is _________________.
Answer:
Let
x = 3 and
x + â
x = 2.99.
∴ â
x = 2.99 − 3 = −0.01
y =
x3 + 5 (Given)
Differentiating both sides with respect to
x, we get
∴
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___.
Page No 13.13:
Question 2:
The approximate change in the volume of a cube of side x metres caused by increasing the side by 2%, is ______________.
Answer:
Let â
x be the change in
side
x and â
V be the change in the volume of the cube.
It is given that,
.....(1)
Now,
Volume of the cube of side
x,
V =
x3
Differentiating both sides with respect to
x, we get
[Using (1)]
Thus, the approximate change in volume of the cube is 0.06
x3 m
3.
The approximate change in the volume of a cube of side
x metres caused by increasing the side by 2%, is
___0.06x3 m3___.
Page No 13.13:
Question 1:
For the function y = x2, if x = 10 and âx = 0.1. Find ây.
Answer:
Page No 13.13:
Question 2:
If y = logex, then find ây when x = 3 and âx = 0.03.
Answer:
We have
Page No 13.13:
Question 3:
If the relative error in measuring the radius of a circular plane is α, find the relative error in measuring its area.
Answer:
Let x be the radius and y be the area of the circular plane.
Page No 13.13:
Question 4:
If the percentage error in the radius of a sphere is α, find the percentage error in its volume.
Answer:
Let V be the volume of the sphere.
Page No 13.13:
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.
Answer:
Let x be the side and V be the volume of the cube.
Page No 13.9:
Question 1:
If y = sin x and x changes from π/2 to 22/14, what is the approximate change in y?
Answer:
Hence, there is no change in the value of
y.
Page No 13.9:
Question 2:
The radius of a sphere shrinks from 10 to 9.8 cm. Find approximately the decrease in its volume.
Answer:
Page No 13.9:
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.
Answer:
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 cm2.
Page No 13.9:
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.
Answer:
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.
Page No 13.9:
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.
Answer:
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.
Page No 13.9:
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.
Answer:
Page No 13.9:
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?
Answer:
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.
Page No 13.9:
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.
Answer:
Let x be the radius of the sphere and y be its volume.
Hence proved.
Page No 13.9:
Question 9:
Using differentials, find the approximate values of the following:(i) (ii) (iii) (iv) (v) (vi) (vii) (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) (xiii) (xiv) (xv) (xvi) (xvii) (xviii) [CBSE 2000]
(xix) [CBSE 2000]
(xx) [CBSE 2002C]
(xxi) [CBSE 2005]
(xxii) (xxiii) (xxiv) (xxv) (xxvi) [CBSE 2012](xxvii) [CBSE 2014]
(xxviii) [NCERT EXEMPLAR]
(xxix) [NCERT EXEMPLAR]
Answer:
(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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