RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1

RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1

These Solutions are part of RD Sharma Class 10 Solutions. Here we have given RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1

Other Exercises

Question 1.
Find the circumference and area of a circle of radius 4.2 cm.
Solution:
Radius of a circle = 4.2 cm
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 1

Question 2.
Find the circumference of a circle whose area is 301.84 cm2.
Solution:
Area of a circle = 301.84 cm2
Let r be the radius, then πr2 = 301.84
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 2

Question 3.
Find the area of a circle whose circum­ference is 44 cm.
Solution:
Circumference of a circle = 44 cm
Let r be the radius,
then 2πr = circumference
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 3

Question 4.
The circumference of a circle exceeds the diameter by 16.8 cm. Find the circum­ference of the circle. (C.B.S.E. 1996)
Solution:
Let r be the radius of the circle
∴  Circumference = 2r + 16.8 cm
⇒  2πr = 2r + 16.8
⇒  2πr – 2r = 16.8
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 4

Question 5.
A horse is tied to a pole with 28 m long string. Find the area where the horse can graze. (Take π = 22/7)
Solution:
Radius of the circle (r) = Length of the rope = 28 m .
Area of the place where the horse can graze
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 5

Question 6.
A steel wire when bent in the form of a square encloses an area of 121 cm2. If the same wire is bent in the form of a circle, find the area of the circle.  (C.B.S.E. 1997)
Solution:
Area of square formed by a wire =121 cm2
∴ Side of square (a) = \(\sqrt { Area } \)  = \(\sqrt { 121 } \)  = 11 cm Perimeter of the square = 4 x side = 4 x 11 = 44 cm
∴Circumference of the circle formed by the wire = 44cm
Let r be the radius
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 6

Question 7.
The circumference of two circles are in the ratio 2 : 3. Find the ratio of their areas.
Solution:
Let R and r be the radii of two circles and their ratio between them circumference = 2 : 3
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 7

Question 8.
The sum of radii of two circles is 140 cm and the difference of their circum­ferences is 88 cm. Find the diameters of the circles.
Solution:
Let R and r be the radii of two circles Then R + r = 140 cm  …….(i)
and difference of their circumferences
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 8
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 9

Question 9.
Find the radius of a circle whose circumference is equal to the sum of the circumferences of two circles of radii 15 cm and 18 cm. [NCERT Exemplar]
Solution:
Let the radius of a circle be r.
Circumference of a circle = 2πr
Let the radii of two circles are r1 and r2 whose
values are 15 cm and 18 cm respectively.
i.e., r1 = 15 cm and r2 = 18 cm
Now, by given condition,
Circumference of circle = Circumference of first circle + Circumference of second circle
⇒   2πr = 2π r1 + 2πr2 =
⇒  r = r1 + r2
⇒   r = 15 + 18
∴ r = 33 cm
Hence, required radius of a circle is 33 cm.

Question 10.
The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having its area equal to the sum of the areas of the two circles.
Solution:
Radius of first circle (r1) = 8 cm
and radius of second circle (r2) = 6 cm
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 10
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 11

Question 11.
The radii of two circles are 19 cm and 9 cm respectively. Find the radius and area of the circle which has its circumference equal to the sum of the circumferences of the two circles.
Solution:
Radius of the first circle (r1) = 19 cm
and radius of the second circle (r2) = 9 cm S
um of their circumferences = 2πr1 + 2πr2
= 2π (r+ r2) = 2π (19 + 9) cm
= 2π x 28 = 56π cm
Let R be the radius of the circle whose circumference is the sum of the circum­ferences of given two circles, then
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 12

Question 12.
The area of a circular playground is 22176 m2. Find the cost of fencing this ground at the rate of ₹50 per metre.  [NCERT Exempiar]
Solution:
Given, area of a circular playground  = 22176 m2
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 13
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 14

Question 13.
The side of a square is 10 cm. Find the area of circumscribed and inscribed circles.
Solution:
ABCD is a square whose each side is 10 cm
∴  AB = BC = CD = DA = 10 cm
AC and BD are its diagonals
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 15
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 16

Question 14.
If a square is inscribed in a circle, find the ratio of the areas of the circle and the square.
Solution:
Let r be the radius of the circle a be the side of the square
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 17
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 18

Question 15.
The area of a circle inscribed in an equilateral triangle is 154 cm2. Find the perimeter of the triangle. (Use π = 22/7 and \(\sqrt { 3 } \)  = 1.73)
Solution:
Area of the inscribed circle of ΔABC = 154 cm2
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 19
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 20

Question 16.
A field is in the form of a circle. A fence is to be erected around the field. The cost of fencing would be ₹2640 at the rate of ₹12 per metre. Then, the field is to be thoroughly ploughed at the cost of ₹0.50 per m2. What is the amount required to plough the field ? (Take π = 22/7)
Solution:
Cost of the fencing the circular field = ₹2640
Rate = ₹12 per metre 2640
∴ Circumference = \((\frac { 2640 }{ 12 } )\) = 220 m
Let r be the radius of the field, then = 2πr = 220
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 21

Question 17.
A park is in the form of a rectangle 120 m x 100 m. At the centre of the park there is a circular lawn. The area of park excluding lawn is 8700 m2. Find the radius of the circular lawn. (Use π = 22/7).
Solution:
Area of the park excluding lawn = 8700 m2
Length of rectangular park = 120 m
and width = 100 m
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 22
∴ Area of lawn = l x b
= 120 x 100 m2 = 12000 m2
Let r be the radius of the circular lawn, then area of lawn = πr2
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 23

Question 18.
A car travels 1 kilometre distance in which each wheel makes 450 complete revolutions. Find the radius of the its wheels.
Solution:
Distance covered by the car in 450 revolutions = 1 km = 1000 m
∴ Distance covered in 1 revolution = \((\frac { 1000 }{ 450 } )\)
= \((\frac { 20 }{ 9 } )\) m
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 24

Question 19.
The area of enclosed between the concentric circles is 770 cm2. If the radius of the outer circle is 21 cm, find the radius of the inner circle.
Solution:
Area of enclosed between two concentric circles = 770 cm2
Radius of the outer circle (R) = 21 cm
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 25

Question 20.
An archery target has three regions formed by the concentric circles as shown in the figure. If the diameters of the concentric circles are in the ratio 1:2:3, then find the ratio of the areas of three regions.[NCERT Exemplar]
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 26
Solution:
Let the diameters of concentric circles be k, 2k , 3k
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 27

Question 21.
The wheel of a motor cycle is of radius 35 cm. How many revolutions per minute must the wheel make so as to keep a speed of 66 km/hr? [NCERT Exemplar]
Solution:
Given, radius of wheel, r = 35 cm
Circumference of the wheel = 2πr
= 2 x \((\frac { 22 }{ 7 } )\) x 35 = 220 cm
But speed of the wheel = 66 kmh-1
= \((\frac { 66 x 1000 }{ 60 } )\) m/ mm
= 1100 x 100 cm min-1
= 110000 cm min-1
∴ Number of revolutions in 1 min
= \((\frac { 110000 }{ 220 } )\)= 500 revolution
Hence, required number of revolutions per minute is 500.

Question 22.
A circular pond is 17.5 m in diameter. It is surrounded by a 2 m wide path. Find the cost of constructing the path at the rate of ₹25 per m2. [NCERT Exemplar]
Solution:
Given that, a circular pond is surrounded by a wide path.
The diameter of circular pond = 17.5 m
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 28

Question 23.
A circular park is surrounded by a rod 21 m wide. If the radius of the park is 105 m, find the area of the road. [NCERT Exemplar]
Solution:
Given that, a circular park is surrounded by a road.
Width of the road = 21 m
Radius of the park (r1) = 105 m
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 29
.’. Radius of whole circular portion (park + road),
re = 105 + 21 = 126 m
Now, area of road = Area of whole circular portion – Area of circular park
= πr2 – πr2             [∵ area of circle = πr2]
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 30

Question 24.
A square of diagonal 8 cm is inscribed in a circle. Find the area of the region lying outside the circle and inside the square.  [NCERT Exemplar]
Solution:
Let the side of a square be a and the radius of circle be r.
Given that, length of diagonal of square = 8 cm
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 31

Question 25.
A path of 4 m width runs round a semi­circular grassy plot whose circumference is 81 \((\frac { 5 }{ 7 } )\)m. Find:
(i) the area of the path
(ii) the cost of gravelling the path at the rate of ₹1.50 per square metre
(iii) the cost of turfing the plot at the rate of 45 paise per m2.
Solution:
Width of path around the semicircular grassy plot = 4 m
Circumference of the plot = 81 \((\frac { 5 }{ 7 } )\)m
= \((\frac { 572 }{ 7 } )\) m
Let r be the radius of the plot, then
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 32
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 33

Question 26.
Find the area enclosed between two concentric circles of radii 3.5 cm and 7 cm. A third concentric circle is drawn outside the 7 cm circle, such that the area enclosed between it and the 7 cm circle is same as that between the two inner circles. Find the radius of the third circle correct to one decimal place.
Solution:
Radius of first circle (r1) = 3.5 cm
Radius of second circle (r2) = 7 cm
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 34
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 35

Question 27.
A path of width 3.5 m runs around a semi­circular grassy plot whose perimeter is 72 m. Find the area of the path. (Use π = 22/7)                   [CBSE 2015]
Solution:
Perimeter of semicircle grassy plot = 72 m
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 36
Let r be the radius of the plot
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 37
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 37.1

Question 28.
A circular pond is of diameter 17.5 m. It is surrounded by a 2 m wide path. Find the cost of constructing the path at the rate of ₹25 per square metre (Use π = 3.14)               [CBSE 2014]
Solution:
Diameter of circular pond (d) = 17.5 m
Radius (r) =\((\frac { 1725 }{ 2 } )\) = 8.75 m
Width of path = 2m
∴  Radius of outer cirlce (R) = 8.75 + 2 = 10.75 m
Area of path = (R2 – r2
= [(10.75)2 – (8.75)2](3.14)
= 3.14(10.75 + 8.75) (10.75 – 8.75)
= 3.14 x 19.5 x 2 = 122.46 m2
Cost of 1 m2 for constructing the path ₹25 m2
∴  Total cost = ₹ 122.46 x 25 = ₹3061.50

Question 29.
The outer circumference of a circular race-track is 528 m. The track is every­where 14 m wide. Calculate the cost of levelling the track at the rate of 50 paise per square metre (Use π= 22/7).
Solution:
Let R and r be the radii of the outer and inner of track.
Outer circumference of the race track = 528 m
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 38
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 39

Question 30.
A road which is 7 m wide surrounds a circular park whose circumference is 352 m. Find the area of the road.
Solution:
Width of the road = 7 m
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 40
Circumference of the park = 352 m
Let r be the radius, then 2πr = 352
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 41
RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 42

Question 31.
Prove that the area of a circular path of uniform width surrounding a circular region of radius r is πh(2r + h).
Solution:
Radius of inner circle = r
Width of path = h
∴ Outer radius (R) = (r + h)
∴ Area of path = πR2 – πr2
= π {(r + h)2 – r2}
= π {r2 + h2 + 2rh – r2}
= π {2rh + h2}
= πh (2r + h) Hence proved.

Hope given RD Sharma Class 10 Solutions Chapter 13 Areas Related to Circles Ex 13.1 are helpful to complete your math homework.

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RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS

RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS

These Solutions are part of RD Sharma Class 10 Solutions. Here we have given RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS

Other Exercises

Mark the correct alternative in each of the following :
Question 1.
If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is
(a) 87
(b) 88
(c) 89
(d) 90
Solution:
(c) 7th term (a7) = a + 6d = 34
13th term (a13) = a + 12d = 64
Subtracting, 6d = 30 => d = 5
and a + 12 x 5 = 64 => a + 60 = 64 => a = 64 – 60 = 4
18th term (a18) = a + 17d = 4 + 17 x 5 = 4 + 85 = 89

Question 2.
If the sum of p terms of an A.P. is q and the sum of q terms is p, then the sum of (p + q) terms will be
(a) 0
(b) p – q
(c) p + q
(d) – (p + q)
Solution:
(d)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 1

Question 3.
If the sum of n terms of an A.P. be 3n2 + n and its common difference is 6, then its first term is
(a) 2
(b) 3
(c) 1
(d) 4
Solution:
(d)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 2

Question 4.
The first and last terms of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be
(a) 5
(b) 6
(c) 7
(d) 8
Solution:
(b) First term of an A.P. (a) = 1
Last term (l) = 11
and sum of its terms = 36
Let n be the number of terms and d be the common difference, then
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 3

Question 5.
If the sum of n terms of an A.P. is 3n2 + 5n then which of its terms is 164 ?
(a) 26th
(b) 27th
(c) 28th
(d) none of these
Solution:
(b)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 4

Question 6.
If the sum of it terms of an A.P. is 2n2 + 5n, then its nth term is
(a) 4n – 3
(b) 3n – 4
(c) 4n + 3
(d) 3n + 4
Solution:
(c)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 5

Question 7.
If the sum of three consecutive terms of an increasing A.P. is 51 and the product of the first and third of these terms is 273, then the third term is :
(a) 13
(b) 9
(c) 21
(d) 17
Solution:
(c)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 6

Question 8.
If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are
(a) 5, 10, 15, 20
(b) 4, 10, 16, 22
(c) 3, 7, 11, 15
(d) None of these
Solution:
(a)
4 numbers are in A.P.
Let the numbers be
a – 3d, a – d, a + d, a + 3d
Where a is the first term and 2d is the common difference
Now their sum = 50
a – 3d + a – d + a + d + a + 3d = 50
and greatest number is 4 times the least number
a + 3d = 4 (a – 3d)
a + 3d = 4a – 12d
4a – a = 3d + 12d
=> 3a = 15d
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 7

Question 9.
Let S denotes the sum of n terms of an A.P. whose first term is a. If the common difference d is given by d = Sn – k Sn-1 + Sn-2 then k =
(a) 1
(b) 2
(c) 3
(d) None of these
Solution:
(b)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 8

Question 10.
The first and last term of an A.P. are a and l respectively. If S is the sum of all the terms of the A.P. and the common difference is given by
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 9
(a) S
(b) 2S
(c) 3S
(d) None of these
Solution:
(b)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 10

Question 11.
If the sum of first n even natural number is equal to k times the sum of first n odd natural numbers, then k =
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 11
Solution:
(d)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 12

Question 12.
If the first, second and last term of an A.P. are a, b and 2a respectively, its sum is
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 13
Solution:
(c)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 14

Question 13.
If S1 is the sum of an arithmetic progression of ‘n’ odd number of terms and S2 is the sum of the terms of the
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 15
Solution:
(a)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 16
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 17

Question 14.
If in an A.P., Sn = n2p and Sm = m2p, where S denotes the sum of r terms of the A.P., then Sp is equal to
(a) \(\frac { 1 }{ 2 }\) p3
(b) mnp
(c) p3
(d) (m + n) p2
Solution:
(c)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 18

Question 15.
If Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn , then S3n : Sn is equal to
(a) 4
(b) 6
(c) 8
(d) 10
Solution:
(b)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 19

Question 16.
In an AP, Sp = q, Sq = p and S denotes the sum of first r terms. Then, Sp+q is equal to
(a) 0
(b) – (p + q)
(c) p + q
(d) pq
Solution:
(c) In an A.P. Sp = q, Sq = p
Sp+q = Sum of (p + q) terms = Sum of p term + Sum of q terms = q + p

Question 17.
If Sn denotes the sum of the first r terms of an A.P. Then, S3n : (S2n – Sn) is
(a) n
(b) 3n
(b) 3
(d) None of these
Solution:
(c)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 20

Question 18.
If the first term of an A.P. is 2 and common difference is 4, then the sum of its 40 term is
(a) 3200
(b) 1600
(c) 200
(d) 2800
Solution:
(a)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 21

Question 19.
The number of terms of the A.P. 3, 7,11, 15, … to be taken so that the sum is 406 is
(a) 5
(b) 10
(c) 12
(d) 14
Solution:
(d)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 22

Question 20.
Sum of n terms of the series
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 23
Solution:
(c)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 24
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 25

Question 21.
The 9th term of an A.P. is 449 and 449th term is 9. The term which is equal to zero is
(a) 50th
(b) 502th
(c) 508th
(d) None of these
Solution:
(d)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 26

Question 22.
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 27
Solution:
(c)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 28
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 29

Question 23.
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 30
Solution:
(b) Sn is the sum of first n terms
Last term nth term = Sn – Sn-1

Question 24.
The common difference of an A.P., the sum of whose n terms is Sn, is
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 31
Solution:
(a)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 32

Question 25.
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 33
Solution:
(b)
In first A.P. let its first term be a1 and common difference d1
and in second A.P., first term be a2 and common difference d2, then
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 34

Question 26.
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 35
Solution:
(b)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 36
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 37

Question 27.
If the first term of an A.P. is a and nth term is b, then its common difference is
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 38
Solution:
(b)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 39

Question 28.
The sum of first n odd natural numbers is
(a) 2n – 1
(b) 2n + 1
(c) n2
(d) n2 – 1
Solution:
(c)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 40

Question 29.
Two A.P.’s have the same common difference. The first term of one of these is 8 and that of the other is 3. The difference between their 30th terms is
(a) 11
(b) 3
(c) 8
(d) 5
Solution:
(d) In two A.P.’s common-difference is same
Let A and a are two A.P. ’s
First term of A is 8 and first term of a is 3
A30 – a30 = 8 + (30 – 1) d – 3 – (30 – 1) d
= 5 + 29d – 29d = 5

Question 30.
If 18, a, b – 3 are in A.P., the a + b =
(a) 19
(b) 7
(c) 11
(d) 15
Solution:
(d) 18, a, b – 3 are in A.P., then a – 18 = -3 – b
=> a + b = -3 + 18 = 15

Question 31.
The sum of n terms of two A.P.’s are in the ratio 5n + 4 : 9n + 6. Then, the ratio of their 18th term is
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 41
Solution:
(a)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 42
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 43

Question 32.
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 44
Solution:
(b)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 45
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 46

Question 33.
The sum of n terms of an A.P. is 3n2 + 5n, then 164 is its
(a) 24th term
(b) 27th term
(c) 26th term
(d) 25th term
Solution:
(b)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 47

Question 34.
If the nth term of an A.P. is 2n + 1, then the sum of first n terms of the A.P. is
(a) n (n – 2)
(b) n (n + 2)
(c) n (n + 1)
(d) n (n – 1)
Solution:
(b)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 48

Question 35.
If 18th and 11th term of an A.P. are in the ratio 3 : 2, then its 21st and 5th terms are in the ratio
(a) 3 : 2
(b) 3 : 1
(c) 1 : 3
(d) 2 : 3
Solution:
(b)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 49

Question 36.
The sum of first 20 odd natural numbers is
(a) 100
(b) 210
(c) 400
(d) 420 [CBSE 2012]
Solution:
(c)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 50

Question 37.
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 51
Solution:
(a)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 52

Question 38.
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 53
Solution:
(c)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 54

Question 39.
The common difference of the A.P. \(\frac { 1 }{ 2b }\) ,
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 55
Solution:
(d)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 56

Question 40.
If k, 2k – 1 and 2k + 1 are three consecutive terms of an AP, the value of k is
(a) -2
(b) 3
(c) -3
(d) 6 [CBSE 2014]
Solution:
(b) (2k – 1) – k = (2k + 1) – (2k- 1)
2k – 1 – k = 2
=> k = 3

Question 41.
The next term of the A.P. , √7 , √28, √63, …………
(a) √70
(b) √84
(c) √97
(d) √112 [CBSE 2014]
Solution:
(d)
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS 57
= √(l6 x 7)= √112

Question 42.
The first three terms of an A.P. respectively are 3y – 1, 3y + 5 and 5y + 1. Then, y equals
(a) -3
(b) 4
(c) 5
(d) 2 [CBSE 2014]
Solution:
(c) 2 (3y + 5) = 3y – 1 + 5y + 1
(If a, b, c are in A.P., b – a = c – b=> 2b = a + c)
=> 6y + 10 = 8y
=> 10 = 2y
=> y = 5

Hope given RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions MCQS are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

CA Foundation Business Economics Study Material – Theory of Consumer Behaviour

CA Foundation Business Economics Study Material Chapter 2 Theory of Demand and Supply – Theory of Consumer Behaviour

Theory of Consumer Behaviour

NATURE OF HUMAN WANTS

All wish, desires, tastes and motives of human beings are called wants in Economics. Human wants show some well marked characteristics as follows-

  • Wants are unlimited
  • Particular want is satiable
  • Wants are complementary
  • Wants are competitive
  • Some wants are both complementary and competitive
  • Wants are alternative
  • Wants vary with time, place, and person
  • Wants vary in urgency and intensity
  • Wants recur
  • Wants are influenced by advertisement
  • Wants become habits and customs
  • Present wants appear to be more important than future wants

Classification of wants:

1. Necessaries:

  • Necessaries of existence – These are the things without which we cannot exist. E.g. minimum of food, clothing and shelter.
  • Necessaries of Efficiency – These are the things which are not necessary to enable us to live, but are necessary to make us efficient workers and to take up any productive activities.
  • Conventional Necessaries – These are the things which are needed either because of social custom or traditions and because the people around us expect us to so.

2. Comforts:
Those goods and services which make for a fuller life and happy life are called comforts. E.g. for a student book is a necessity, a table and a chair are necessaries of efficiency, but cushioned chair is comfort.

3. Luxuries:
Luxuries are those wants which are superfluous and expensive. They are something we could easily do without. E.g. jewellery, big house, luxurious car, dining in a five star hotel etc.

What is Utility?

  • The demand of a commodity depends on the utility of that commodity to a consumer.
  • The want satisfying capacity or power of a commodity is called utility. It is anticipated satisfaction by a consumer.
  • It is a subjective and relative term and varies from person to person, place to place and time
    to time.
  • Utility does not mean the same things as usefulness. E.g. Liquor, Cigarettes, etc. have utility as people are ready to buy them but they are harmful for the health.
  • Therefore, utility has no moral or ethical significance.
  • To study the consumer behaviour, the two important theories are –
    – Marginal utility analysis, given by Dr. Alfred Marshall, and
    – Indifference curve analysis given by Hicks and Allen.

Marginal Utility Analysis

  • The theory of Marginal utility Analysis of demand was given by Alfred Marshall, a British economist.
  • He explained how a consumer spends his money income on different goods and services in order to get maximum satisfaction ie. how a consumer reaches equilibrium.
  • Dr. Alfred Marshall assumes that the utility derived from the consumption of a commodity is measurable. Hence, this approach is called CARDINAL APPROACH.

Marginal utility Analysis is based on the following assumptions:-

  • The Cardinal Measurability of utility: According to this theory, utility is a cardinal concept, ie. it is possible to measure and quantify satisfaction derived from the consumption of various commodities. According to Marshall, money is the measuring rod of marginal utility. E.g. – If a person is ready to pay Rs. 10 for pastry and Rs. 6 for burger, we can say that price represents the utility which he is expecting from these commodities.
  • Constancy of the Marginal Utility of Money: The marginal utility of money remain constant during the time when the consumer is spending money on a good and as a result of which the amount of money is reducing. This is so because money is used as a measuring rod of utility. If the money which is a unit of measurement itself varies, it cannot give correct measurement of the marginal utility of a good.
  • Independent Utilities: According to this assumption, the amount of utility which a consumer gets from one commodity, does not depend upon the quantity of other commodities consumed. E.g. – If a person is consuming Rooh Hafza Sharbat, its utility is not affected by the availability of sugar or Rose Sharbat. It just depends upon the availability of Rooh Hafza Sharbat only. This assumption, in other words, totally ignores the presence of complementary and substitute goods.
  • Rationality: The consumer is assumed to be rational whose aim is to maximise his utility subject to the constraint imposed by his given income. He makes all calculations
    carefully and then purchases the commodities.

The Law of Diminishing Marginal Utility

The Law of Diminishing Marginal Utility is based on two important facts, namely:
(a) Human wants are unlimited
(b) Each separate human want is limited. The amount of any commodity which a man can consume, in a given period of time is limited and hence each single want is satiable.

The law describes that, as the consumer has more and more of a commodity, the additional utility which he derives from an additional unit of commodity goes on falling. Marshall stated the law as follows

“The additional benefit which a person derives from a given increase in stock of a thing diminishes with every increase in the stock that he already has.” The law can be explained with the help of following table:

ca-foundation-business-economics-study-material-theory-of-consumer-behaviour-1

  • The above table shows that as the consumer goes on consuming rossgullas, the additional or marginal utility goes on diminishing.
  • The consumption of 3rd unit of rossgulla gives no additional utility and the 4th unit is giving negative utility.
  • The 4th unit instead of giving satisfaction causes dissatisfaction.
  • Total utility goes on increasing as long as MU is positive, but at diminishing rate.
  • When total utility is highest, marginal utility is zero. This is the point of full satisfaction.
  • When marginal utility becomes negative, total utility starts falling.
  • MU is the rate of change in TU or slope of TU curve.
  • MU can be positive, zero or negative.

We can show the information given in the table on a graph as follows:-
The figure shows that marginal utility curve goes on declining as the consumption increases. It even crosses the X-axis and suggest negative marginal utility. Total utility curve rises upto a point and then starts falling.ca-foundation-business-economics-study-material-theory-of-consumer-behaviour-2

The Law of Diminishing Marginal Utility helps us to understand how a consumer reaches equilibrium in ONE COMMODITY CASE.

  • A consumer tries to equalize marginal utility of a commodity with its price in order to maximize the satisfaction. A consumer thus compares the price with the marginal utility of a commodity.
  • He keep on purchasing a commodity till MU > P. In other words, so long as price is less, he buys more which is also the basis of the law of demand.
  • The consumer is at equilibrium where:
    Marginal Utility of the commodity = Price of the commodity
    ca-foundation-business-economics-study-material-theory-of-consumer-behaviour-3

In reality, a consumer spends his money income to buy different commodities. In case of many commodities, consumer equilibrium is explained with the Law of Equi-Marginal Utility.

  • The law states that a consumer will allocate his expenditure in a way that the utility gained from the last rupee spent on each commodity is equal or the marginal utility each commodity is proportional to its price.
  • The consumer is said to be equilibrium when the following condition is met-
    ca-foundation-business-economics-study-material-theory-of-consumer-behaviour-4

The Law of Diminishing Marginal Utility is based on the following assumptions:-

  1. Homogeneous Units:
    It is assumed that all the units of the commodity are homogeneous ie. identical in every respect like size, taste, colour, quality, blend, etc.
    E.g. – If a consumer is consuming CADBURY DAIRY MILK CHOCOLATE (40 gms.), then all bars of chocolate must be of Dairy Milk Chocolates and not of any other type.
  2. Continuous Consumption:
    There should not be any time gap or interval between the consumption of one unit and another unit.
  3. Rationality:
    The consumer is assumed to be rational.
  4. Cardinal Measurement:
    The utility is measurable and quantifiable.
  5. Constancy of Marginal Utility of Money:
    The marginal utility of money remain unchanged throughout when the consumer is spending on a commodity.
  6. The tastes of consumers should remain constant.

Exceptions to and limitations of the Law of Diminishing Marginal Utility:
In some cases a consumer gets increasing marginal utility with the increase in consumption.
Such cases are called as exception which are as follows-

  1. Hobbies and Rare Collections: The law does not hold good in case of hobbies and rare collections like reading, collection of stamps, coins, etc. Every additional unit gives more satisfaction ie. the marginal utility tends to increase.
  2. Abnormal Persons: The law does not apply to abnormal persons like misers, drunkards, musicians, drug addicts, etc. who want more and more of the commodity they are in love with.
  3. Indivisible Goods: The law cannot be applied in case of indivisible bulky goods like T. V. set, house, scooter, etc. No one purchases more than one unit of such goods at a time.

The limitations of the Law of Diminishing Marginal Utility are as follows –

  1. Cardinal Measurement Unrealistic: The law assumes cardinal measurement of utility. This is unrealistic, because, utility being a subjective or psychological phenomenon, cannot be measured numerically. The feeling experienced by a consumer cannot be quantified.
  2. Unrealistic Conditions: The law is based on unrealistic assumptions. It is not possible to meet all conditions like homogeneous goods, continuous consumption, rationality, etc. at the same time.
  3. Constant Marginal Utility of Money: The law assumes that marginal utility of money remains constant. Therefore, the utility of the commodity depends on its quantity alone. But, the marginal utility of money never remains constant.
  4. Inapplicable to Indivisible Goods: The assumptions of the Law of DMU cannot be made applicable to indivisible bulky goods like T. V. Sets, scooter, house, etc. because no one purchases more than one unit of such goods at a time.
  5. Single Commodity Model: The Law of DMU is a single commodity model. Marginal utility of each commodity is measured independently. But, a consumer may purchase more than one commodity. Also, utilities of goods such as complementary or substitutes are interdependent.

Consumer’s surplus

Consumers Surplus is one of the important concept in economic theory and in economic policy making. It was given by Dr. Alfred Marshall. Marshall’s concept of consumer surplus is based on the following assumptions:

  • Utility can be cardinally measured in monetary units.
  • Marginal utility of money remains constant.
  • Income, fashion and taste of consumer remains constant.
  • Independent marginal utility of each unit of the commodity. .
  • The law of diminishing marginal utility holds good.

EXPLANATION-

  • In our daily expenditure, we often find that the price we pay for a commodity is less than the satisfaction derived from its consumption.
  • Therefore, we are ready to pay much higher price for a commodity than we actually have to pay.
    E.g. Commodities like salt, newspaper, match box, etc. are very useful, but they are also very cheap.
  • From the purchase of such commodities we derive a good deal of extra satisfaction or surplus over and above the price that we pay for them. This is consumer’s surplus.
  • Marshall defined consumer surplus “as the excess of the price which a person would be willing to pay rather than go without the thing over that which he actually does pay”.
  • Thus, it is the difference between what a consumer is ready to pay and what he actually pays.
    Consumer Surplus = What a consumer is ready to pay – What he actually pays = Sum of Marginal Utilities – (Price X Units Purchased)
    = Total Utility – Total amount spent.

We can illustrate the concept of consumer’s surplus with the help of following table-

ca-foundation-business-economics-study-material-theory-of-consumer-behaviour-5

When the consumer buy first unit of commodity he is ready to pay Rs. 25 for it as he expects satisfaction worth Rs. 25 from it and thus gets a surplus worth Rs. 15. For second unit he is ready to pay only Rs. 20 for it as he expects lesser satisfaction from it and thus gets surplus worth Rs. 10 only. The consumer will go on buying the commodity till Marginal Utility = Price & consumer surplus is Zero i.e. upto 4th unit.
Here, Consumer Surplus = Total Utility – Total Amt. Spent = Rs. 70 – Rs. 40 = Rs. 30.
We can represent consumers surplus with the following diagram.

ca-foundation-business-economics-study-material-theory-of-consumer-behaviour-6
In the diagram MU is the marginal utility curve. OP (Rs. 10) is the market price. In equilibrium, consumer would buy OQ (4) units (at this MU = P). For OQ (4) units he is required to pay OQ (4 units) X OP (Rs. 10) = OQSP(Rs. 40). The consumer was ready to pay (by MU curve) OQ SA(Rs. 70). Thus, he derives surplus of satisfaction. OQSA(Rs. 70) – OQSP(Rs. 40) = PSA(Rs. 30)

The uses/importance of the consumer surplus concept are as follows:-

1. Distinction between Value-in-Useand Value-in-exchange: Consumer’s surplus draws a clear distinction between value-in-use and value-in-exchange. E.g. – SALT have great value -in-use but much less value-in-exchange. Being necessity and cheap thing, it yield a large consumer surplus. The consumer’s surplus depends on total utility, whereas price depends on marginal utility. The total utility of salt consumed is much greater but its marginal utility (and price) is low due to its excess supply.

2. Comparing Advantages of Different Places: The concept of consumer’s surplus is useful when we compare the advantages of living in two different places. A place with greater amenities available at cheaper rates give large surplus of satisfaction to consumers than backward place or region. Consumer’s surplus thus indicates conjunctural advantages, Le. the advantages of environment arid opportunities.

3. To the Businessman and Monopolist: A businessman can raise prices of those goods in which there is a large consumer’s surplus. The seller will be able raise price especially if he is a monopolist and controls the supply of the commodity.

4. Useful to the government in determining taxes: The concept is very useful to Finance Minister in imposing taxes on various goods and fixing their rates. He will tax those goods in which the consumers enjoy large surplus. The consumers thus will have to pay more and their consumer’s surplus will fall. But at the same time it will raise the revenue of the government. The loss of consumers must be compared with the gains to the government. If the loss of consumers is greater than the gains to government, then, the tax is not proper and vice versa.

5. Measuring Benefits from International Trade: Through international trade, a country can import goods cheaply le. a country can get goods at lower price than they are prepared to pay for them. The imports, therefore give larger surplus of satisfaction to people. The larger this surplus, the more beneficial is the international trade.

6. Useful in cost-benefits analysis of projects: While undertaking any project the government
usually compare the cost of project and flow of benefits from project both in economic and in non-economic terms. E.g. For FLYOVER BRIDGE PROJECT the government will consider the consumers surplus Le. benefits in terms of time saving, fuel saving, etc. expected from flyover bridge project.

The CRITICISMS of the consumer’s surplus concept are as follows:-

  1. Imaginary: The concept of consumer’s surplus is quite imaginary idea. One has to imagine what you are prepared to pay and you proceed to deduct from that what you actually pay. It is all hypothetical and unreal.
  2. Cardinal measurement is not possible: Consumer’s surplus cannot be measured precisely because it is difficult to measure the total utilities and marginal utilities of the commodities consumed in quantitative terms.
  3. Ignores the interdependence between goods: The concept of consumer’s surplus does not consider the effect of availability and non-availability of substitutes and complementary goods on the consumption of a particular commodity. Actually consumer surplus derived from a commodity is affected by substitutes and complementary goods.
  4. Cannot be measured in terms of money: This is because the marginal utility of money changes as purchases are made and the consumer’s stock of money diminishes. But, Marshall assumed that-the marginal utility of money to be constant.
  5. Not applicable to Necessaries: It does not apply to the necessaries of life. In such cases the surplus is immeasurable e.g. – Food and Water. Consumer surplus is infinite because a consumer will stake whole of his income rather than go without them.
  6. Not applicable to prestige: e.g. – Diamonds jewellery, etc. fall in their prices lead to a fall in consumer’s surplus.

Indifference Curve Analysis

  • An indifference curve is a curve which represents combinations of two commodities that gives same level of satisfaction to the consumer.
  • As all the combinations give same level of satisfaction, the consumer becomes indifferent (Le. neutral) as to which combination he gets.
  • In other words, all the combinations lying on indifference curve are equally desirable and equally preferred by the consumer.

To Understand consider the following indifference schedule.

ca-foundation-business-economics-study-material-theory-of-consumer-behaviour-7

In the schedule I above, the consumer is indifferent whether he gets combination A, B, C or D. This is because all combinations give him same amount of satisfaction and therefore equally preferable to him. He gets as much satisfaction from 1 burger and 10 sandwiches as from 3 burgers and 3 sandwiches.

By plotting the above combinations on a graph, we can derive an indifference curve as shown in the following figure:

ca-foundation-business-economics-study-material-theory-of-consumer-behaviour-8

In the diagram, quantity of burger is measured on X-axis and quantity of sandwiches on Y-axis. The various combinations A, B, C, D are plotted and on joining them, we get a curve known as indifference curve. All combinations lying on the indifference curve give the same level of satisfaction to the consumer. Hence, the consumer is indifferent among them.

If the indifference schedule II is also plotted on the graph, we will get IC2. This will lie above the IC1 as all of IC2 combinations contain greater quantities of burgers and sandwiches. Similarly, we can draw IC3, IC4, etc… to make a complete indifference map as follows. Indifference map represents a full description of consumer’s tastes and preferences.

ca-foundation-business-economics-study-material-theory-of-consumer-behaviour-9

In the diagram, the various combinations E, F, G, H on IC2 give consumer same level of satisfaction and hence equally preferable to the consumer. The consumer is indifferent whether he gets combination E or F or any other combination.

The consumer however will prefer any combination lying on IC2 as it will give him more satisfaction than any combination lying on IC1.

This is because combinations lying on IC2 have larger quantity of burgers and sandwiches.

Thus, a higher indifference curve represents a higher level of satisfaction than lower indifference curve but HOW MUCH HIGHER cannot be indicated.

It is so because IC system is based on ORDINAL APPROACH according to which utility cannot be quantified but can only be compared.

Assumptions of indifference curve

  • Non-satiety: This assumption means that the consumer has not reached the point of full satisfaction in the consumption of any commodity. Therefore, a larger quantity of both commodities are preferred by the consumer. Larger the quantities of commodities, higher would be the total utility.
  • Rationality: Consumer is assumed to be rational. He aims at the maximisation of his total utility, given the market prices and money income. He is also assumed to have all relevant information like prices of goods, the markets where they are available, etc.
  • Consistency: The consumer is consistent in his choice Le. the preferences of consumers are consistent. If he prefers combination ‘A’ over combination ‘B’ in one period of time he will NOT prefer ‘B’ over ‘A’ in another period of time.
  • Transitivity: If combination A is preferred to B and ‘B’ is preferred to C, then, A is preferred to C. Symbolically, if A>B, and B>C, then A>C.
  • Ordinal Utility: It is assumed that consumer cannot measure precisely utility or satisfaction in absolute units Le. cardinally, but he can express utility ordinally. In other words, consumer is capable of comparing and ranking satisfaction derived from various goods and their combinations.
  • Diminishing Marginal Rate of Substitution: It means that as more and more units of ‘A’ are substituted for ‘B’ consumer will sacrifice lesser and lesser units of ‘B’ for each additional unit of ‘A’.

Marginal Rate of Substitution

  • The concept of marginal rate of substitution is the basis of Indifference Curves in the Theory of Consumer’s Behaviour.
  • IT MAY BE DEFINED AS THE RATE AT WHICH A CONSUMER WILL EXCHANGE SUCCESSIVE UNITS OF ONE GOOD (COMMODITY) FOR ANOTHER.

Consider the following schedule-

ca-foundation-business-economics-study-material-theory-of-consumer-behaviour-10

  • The above schedule shows the combinations of two goods ‘X’ and ‘Y’. Suppose the consumer wants more of ‘X’. To do so he must sacrifice some units of ‘Y’. in order to maintain same level of satisfaction.
  • Initially, the consumer sacrifices 4Y to get 1X, to obtain second unit of ‘X’ he sacrifices 2Y and so on.
  • This rate of sacrifice is technically called Marginal Rate of Substitution (MRS).
  • Thus, for any goods X and Y, the MRS is the loss of Y which can just be compensated by a gain of X. MRSxy goes on diminishing.

We can also measure MRS on an indifference curve. Consider the following diagram-

ca-foundation-business-economics-study-material-theory-of-consumer-behaviour-11

In the diagram, when the consumer moves from combination A to combination B, he gives up AC of Y and takes up CB of X and gets the same level of satisfaction.

ca-foundation-business-economics-study-material-theory-of-consumer-behaviour-12
MRSxy between two points is also the slope of the indifference curve between these two points. As the consumer moves from combination B to C, C to D, the MRSxy goes on diminishing.

The MRS goes on diminishing due to following reasons-

  1. The want for a particular commodity is satiable. So, as the consumer has more and more
    of that commodity, he is willing to take less and less of it. Thus, in the above example, when the consumer has more and more of ‘X’, his intensity of want for X’ diminishes but for ‘Y’ increases. Therefore, he does not want more of ‘X’ now and is not ready to H sacrifice more number of ‘Y’ for ‘X’.
  2. The second reason is that, the goods are not perfect substitutes of each other in the satisfaction of particular want. If they are perfect substitutes, the MRS would not fall and remain constant.

Properties of Indifference Curves

(Refer to Schedule I and above diagram)

Indifference curves always slope downwards from left to right:

  1. This means that an indifference curve has a negative slope.
  2. REASON – In order to maintain same level of satisfaction, as the quantity of burgers is increased in the combination, the quantity of sandwiches is reduced.
  3. Thus, this property follows from the definition of an IC and non-satiety assumption ie. more is preferred to less.
  4. Indifference curve cannot be horizontal straight line or vertical straight line or positively sloped.

Indifference Curves are convex to the origin:

  1. This means that IC is relatively steeper first in its left hand portion and tends to become relatively flatter in its right hand portion.
  2. REASON: Diminishing Marginal Rate of Substitution.
  3. The schedule and diagram shows that the consumer sacrifices less and less of sandwiches for every additional unit of burger.
  4. The convexity of an IC means that the two commodities can substitute each other but are not perfect substitute.
  5. If IC were concave to origin, it would mean increasing MRS. This is against the assumption of diminishing MRS.
    Similarly, IC cannot be straight lines as it would mean that MRS remains constant (for perfect substitute.)

Higher Indifference Curves Represents Higher Level of Satisfaction:

  1. In an indifference map, combinations lying on a higher IC gives higher level of satisfaction than the combinations lying on a lower IC. But how much higher cannot be indicated.
  2. REASON: This is because combinations on higher IC contains more quantity of either sandwiches or burger without having less of other as shown in the following diagram.
    ca-foundation-business-economics-study-material-theory-of-consumer-behaviour-13
  3. Combinations B and C on IC2 will be preferred by the consumer than the combination A on IC1.
  4. Combination B on IC2 contains more quantity of sandwiches without having less of burgers compared to combination A on IC1.
  5. Hence, all combinations on IC2 gives more satisfaction to consumer. Thus, higher IC represents higher satisfaction.

Indifference curves cannot intersect each other:

  1. It means that only one IC will pass through a point in the indifference map.
  2. In other words, ONE combination can lie only on one IC.
  3. Higher IC represents higher level of satisfaction and lower IC represents lower level of satisfaction. If they intersect each other, it would lead to illogical result.
  4. It can be proved with the help of following diagram –
    ca-foundation-business-economics-study-material-theory-of-consumer-behaviour-14

In the diagram two IC intersect each other at point A. On IC1; combinations A = B and on IC2, Combinations A = C. Therefore, by assumption of transitivity if, A = B and A = C. ∴ B = C. But C>B as it lie on higher IC giving higher satisfaction due to more quantity of sandwiches. So two IC cannot intersect.

Indifference curve will not touch either X-axis or Y-axis

ca-foundation-business-economics-study-material-theory-of-consumer-behaviour-15

  1. The indifference curve will not touch either X-axis or Y-axis because we have assumed that consumer is considering the different combinations of TWO commodities.
  2. If IC touches either of the axis, it would mean that consumer is interested in one commodity only.
  3. In the diagram IC touches X-axis at point B and Y-axis at point A.
  4. At point B the consumer is satisfied with OB quantity of X-commodity and zero quantity of A. This is against the definition of IC. Therefore, IC curve will not touch either axis.

Budget Line Or Price Line (Or Price Opportunity Line Or Expenditure Line Or Budget Constraint or Consumption Possibility Line)

  • A higher indifference curve shows a higher level of satisfaction than lower one.
  • Therefore, to maximize satisfaction consumer will try to reach the highest possible indifference curve.
  • He will try to buy more and more goods to get more and more satisfaction. But, what and how much a consumer can actually buy depends on –
    1. The money income of consumer,
    2. Prices of goods he wants to buy. They are the two objective factors which form the budgetary constraint of the consumer.

The budgetary position of the consumer can be graphically shown by BUDGET LINE. A budget line or price line shows maximum quantity of the different combinations of TWO GOODS that the consumer can purchase with his given money income and given market prices of goods.
Example:
The consumer’s money income is Rs. 100 to spend on X and Y.
Price of X is Rs. 5 per unit Price of Y is Rs. 2 per unit
Therefore, the consumer can get either 20 units of X and no Y.
OR
50 units of Y and no X.
OR
Combination of X and Y
Hence, 20 X and 50 Y form the two extreme limits of his expenditure. But the consumer can buy any ONE of the many combinations of X and Y within these limits. Graphically it can be shown as followsca-foundation-business-economics-study-material-theory-of-consumer-behaviour-16
This budget line corresponds to the following equation, called Budget Line Equation
Px. X + Py. Y = M
Where-
M = Total Money Income
Px = Price of commodity ‘X’
X = Quantity of X commodity
Py = Price of commodity
Y = Quantity of ‘Y’ commodity

It can be seen in the diagram that the consumer can buy a maximum of 20 units of X as denoted by points ‘L’ or buy a maximum of 50 units of Y as denoted by point ‘P’. On joining points P &L, we get a line PL called as budget line. It determines the limit or boundary of purchase.

The consumer can choose any combination of X and Y lying on budget line like combinations ‘a’ (8 X & 30 Y) or ‘b’ (12 X & 20 Y) or any other combination. However, the consumer cannot choose combination ‘Z’ as it is beyond his means i.e. budget. Any combination like ‘S’ lying within the budget line, shows under spending by consumer.

The slope of budget line is equal to the ratio of the prices of two goods Le. ratio of the prices of X to the price of Y. Thus, the slope of the budget line PL is Px/Py

Consumer’s equilibrium

  • The consumer is said to be in equilibrium when he maximizes his satisfaction (i.e. utility).
  • To explain the consumer’s equilibrium under ordinal approach, we have to make use of TWO TOOLS of indifference curve analysis namely-
    1. the consumer’s INDIFFERENCE MAP, and
    2. his PRICE/BUDGET LINE.

Assumptions:

  • The consumer has a fixed amount of money income to spend.
  • The consumer intends to buy TWO GOODS.
  • The Consumer is RATIONAL and tries to maximise his satisfaction.
  • The prices of two goods are GIVEN and are CONSTANT. Therefore, budget line has constant slope.
  • Goods are HOMOGENEOUS and DIVISIBLE.
  • The scale of preference of consumer Le. his taste & preferences remains unchanged. Scale of preference is expressed through indifference map.

The CONSUMER’S INDIFFERENCE MAP shows all indifference curves which rank the consumer’s preferences between various possible combinations of TWO commodities.

  • To maximises his satisfaction consumer would like to reach highest possible indifference curve.
  • The slope of IC at any one point shows the MARGINAL RATE OF SUBSTITUTION (which diminishes).
    ca-foundation-business-economics-study-material-theory-of-consumer-behaviour-17
  • To maximise satisfaction consumer will try to reach the highest possible IC and so will try to buy more and more of the two commodities.
  • But there are limits to which he can go on and on.
  • These limits are imposed (i) his money income, & (ii) prices of the commodities. These limits are described by PRICE/BUDGET LINE which shows the various combinations of two commodities the consumer can afford to buy.
  • All the combinations lying on the budget line are affordable by the consumer. Any, combination lying beyond budget line is unaffordable.
  • The slope of budget/price line shows the ratio of the prices of two commodities ie. Px/Py
  • Now we can show how a consumer reaches equilibrium ie., how he allocates his money expenditure between commodities X and Y and gets maximum satisfaction.

For showing this, we will have to superimpose the price line on the indifference map as follows-

ca-foundation-business-economics-study-material-theory-of-consumer-behaviour-18

  • In order to maximise his satisfaction, the consumer will try to reach highest IC ie. IC4.
  • But the budget constraint forces him to remain ON THE BUDGET LINE.
  • In the diagram, budget line PL shows all the combinations of X & Y that the consumer can buy. In diagram, we find combinations a, b, c, d, e lie on budget line PL and hence are affordable.
  • Points a,b,d and e lie on lower ICs and so are not the points of equilibrium as the consumer can get more satisfaction with the same amount of money.
  • Point ‘C’is the point of equilibrium as it lies on budget line and also on highest possible indifference curve IC3 giving maximum satisfaction.
  • At ‘point’ ‘C’, the budget line PL is TANGENT to indifference curve IC3.
  • At the point of tangency, Slope of indifference Curve = Slope of Budget Line
    ca-foundation-business-economics-study-material-theory-of-consumer-behaviour-19
  • Thus, the consumer is at equilibrium when
    ca-foundation-business-economics-study-material-theory-of-consumer-behaviour-20

 

RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions VSAQS

RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions VSAQS

These Solutions are part of RD Sharma Class 10 Solutions. Here we have given RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions VSAQS

Other Exercises

Answer each of the following questions either in one word or one sentence or as per requirement of the questions :
Question 1.
Define an arithmetic progression.
Solution:
A sequence a1, a2, a3, …, an is called an arithmetic progression of then exists a constant d
Such that a2 – a1 = d, a3 – a2 = d, ………… an – an-1 = d
and so on and d is called common difference

Question 2.
Write the common difference of an A.P. whose nth term is an = 3n + 7.
Solution:
an = 3n + 7
a1 = 3 x 1 + 7 = 3 + 7 = 10
a2 = 3 x 2 + 7 = 6 + 7 = 13
a3 = 3 x 3 + 7 = 9 + 7 = 16
d = a3 – a2 or a2 – a1 = 16 – 13 = 3 or 13 – 10 = 3

Question 3.
Which term of the sequence 114, 109, 104, … is the first negative term ?
Solution:
Sequence is 114, 109, 104, …..
Let an term be negative
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions VSAQS 1

Question 4.
Write the value of a30 – a10 for the A.P. 4, 9, 14, 19, …………
Solution:
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions VSAQS 2

Question 5.
Write 5th term from the end of the A.P. 3, 5, 7, 9,…, 201.
Solution:
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions VSAQS 3
= 3 + 190 = 193
5th term from the end = 193

Question 6.
Write the value of x for which 2x, x + 10 and 3x + 2 are in A.P.
Solution:
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions VSAQS 4

Question 7.
Write the nth term of an A.P. the sum of whose n terms is Sn.
Solution:
Sum of n terms = Sn
Let a be the first term and d be the common difference an =Sn – Sn-1

Question 8.
Write the sum of first n odd natural numbers.
Solution:
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions VSAQS 5

Question 9.
Write the sum of first n even natural numbers.
Solution:
First n even natural numbers are
2, 4, 6, 8, ……….
Here a = 2, d = 2
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions VSAQS 6

Question 10.
If the sum of n terms of an A.P. is Sn = 3n² + 5n. Write its common difference.
Solution:
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions VSAQS 7

Question 11.
Write the expression for the common difference of an A.P. Whose first term is a and nth term is b.
Solution:
First term of an A.P. = a
and an = a + (n – 1) d = b .
Subtracting, b – a = (n – 1) d
d = \(\frac { b – a }{ n – 1 }\)

Question 12.
The first term of an A.P. is p and its common difference is q. Find its 10th term. [CBSE 2008]
Solution:
First term of an A.P. (a) = p
and common difference (d) = q
a10 = a + (n – 1) d
= p + (10 – 1) q = p + 9q

Question 13.
For what value of p are 2p + 1, 13, 5p – 3 are three consecutive terms of an A.P.? [CBSE 2009]
Solution:
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions VSAQS 8

Question 14.
If \(\frac { 4 }{ 5 }\), a, 2 are three consecutive terms of an A.P., then find the value of a.
Solution:
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions VSAQS 9

Question 15.
If the sum of first p term of an A.P. is ap² + bp, find its common difference.
Solution:
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions VSAQS 10

Question 16.
Find the 9th term from the end of the A.P. 5, 9, 13, …, 185. [CBSE 2016]
Solution:
Here first term, a = 5
Common difference, d = 9 – 5 = 4
Last term, l = 185
nth term from the end = l – (n – 1) d
9th term from the end = 185 – (9 – 1) 4 = 185 – 8 x 4 = 185 – 32 = 153

Question 17.
For what value of k will the consecutive terms 2k + 1, 3k + 3 and 5k – 1 form on A.P.? [CBSE 2016]
Solution:
(3k + 3) – (2k + 1) = (5k – 1) – (3k + 3)
3k + 3 – 2k – 1 = 5k – 1 – 3k – 3
k + 2 = 2k – 4
2k – k = 2 + 4
k = 6

Question 18.
Write the nth term of the A.P.
\(\frac { 1 }{ m }\) , \(\frac { 1 + m }{ m }\) , \(\frac { 1 + 2m }{ m }\) , ……… [CBSE 2017]
Solution:
RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions VSAQS 11

Hope given RD Sharma Class 10 Solutions Chapter 5 Arithmetic Progressions VSAQS are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.

CA Foundation Business Economics Study Material – Demand Forecasting

CA Foundation Business Economics Study Material Chapter 2 Theory of Demand and Supply – Demand Forecasting

Demand Forecasting

Meaning:

  • Demand forecasting is an estimate of the future market demand for a product. The process of forecasting is based on reliable statistical data of past and present behaviour, trends, etc.
  • Demand forecasting cannot be hundred per cent correct. But, it gives a reliable estimates of the possible outcome with a reasonable accuracy.
  • Demand forecasting may be at international level or local level depending upon area of operation, cost, time, etc.

Usefulness:

Demand forecasting is an important function, of managers as it reduces uncertainty of environment in which DECISIONS are made. Further, it helps in PLANNING for future level of production. Its significance can be stated as follows:

  1. Production Planning: Demand forecasting is a pre-requisite for planning of production in a firm. Expansion of production capacity depends upon likely demand for its output. Otherwise, there may be overproduction or underproduction leading to losses.
  2. Sales Forecasting: Sales forecasting depends upon demand forecasting. Promotional efforts of the firm like advertisements, suitable pricing etc. should be based on demand forecasting.
  3. Control of Business: Demand forecast provide information for budgetary planning and cost control in functional area of finance and accounting.
  4. Inventory Control: Demand forecasting helps in exercising satisfactory control of business inventories like raw-materials, intermediate goods, semi-finished goods, spare parts, etc. Estimates of future requirement of inventories is to be done regularly and it can be known from demand forecasts.
  5. Capital Investments: Capital investments yield returns over many years in future. Decision about investment is to be taken by comparing rate of return on capital investment and current rate of interest. Demand forecasting helps in taking investment decisions.

Types of forecasts:

Macro-level forecasting deals with the general economic environment prevailing in the economy as measured by the Index of Industrial Production (IIP), national income and general level of employment, government expenditure, consumption level, consumers spending habits, etc.

  1. Industry-level forecasting refers to forecasting the demand of a good of a particular industry as a whole.
    E.g.- Demand for two-wheelers in India.
  2. Firm-level forecasting refers to forecasting the demand of a good of a particular firm.
    E.g.- Bajaj motor cycle.

Based on time period, demand forecasting may be-

  1. Short-term demand forecasting normally relates to a period not exceeding a year. It is also called as ‘operating forecast’. It is useful for estimating stock requirement, providing working capital, etc.
  2. Long-term demand forecasting may cover one to five years, depending on the nature of the firm. It provides information for taking decisions like expansion of plant capacity, man-power planning, long-term financial planning, etc.

Demand Distinctions

1. Producer’s goods and Consumer’s Goods

  • The goods which are used for the production of other goods are called producer’s goods. E.g. Machines
  • The goods which are used for final consumption are called consumer’s goods. E.g. readymade clothes, toothpaste, soap, house etc.

2. Durable goods and Non-durable goods

  • Goods can be further divided into durable and non-durable goods.
  • Durable goods are those which can be consumed more than once and yield utility over a period of time.
    – Producer’s Durable Goods – E.g.- Building, Plant, Machinery etc.
    – Consumer’s Durable Goods – E.g.- Cars, TV, Refrigerators, etc.
  • Non-durable goods are those which cannot be consumed more than once. These will meet only the current demand.
    – Producer’s Non-durable Goods – E.g.- raw material, fuel, power, etc.
    – Consumer’s Non-durable Goods – E.g.- milk, bread, etc.

3. Derived demand and Autonomous demand

  • The demand for a commodity is said to be derived when its demand depends on the demand for some other commodity. In other words, it is the demand which has been derived from the demand for some other commodity called “parent product”.
    E.g. The demand for bricks, cement, steel, sand, etc. is derived demand because their demand depends on the demand for houses. Producer goods and complementary goods have derived demand.
  • When the demand of a commodity is independent of the demand for other commodity, then it is called autonomous demand.

4. Industry demand and Company demand

  • The total demand of a commodity of a particular industry is called industry demand.
    E.g. the total demand for shoes in the country.
  • The demand for the commodity of a particular company is called company demand.
    E.g. shoes produced by BATA.

5. Short-run demand and long-run demand

  • When the demand respond immediately to price changes, income changes etc. is referred as short-run demand.
  • When the demand still exist as a result of changes in pricing, sales promotion, quality improvement etc. after enough time is allowed to let the market adjust to the new situation is called long-run demand.

Factors affecting demands for non-durable goods
The factors which affects the demand of NON-DURABLE CONSUMER GOODS are as follows:-

  • Disposable Income: The income left with a person after paying direct taxes and other deductions is called as disposable income. Other things being equal, more the disposable income of the household, more is its demand for goods and vice versa.
  • Price: The demand for a commodity depends upon its price and the prices of its substitutes y and complements. The demand for a commodity is inversely related to its own price and the price of its complements. The demand for a commodity is positively related to its substitutes.
  • Demography: This involves the characteristics of the populations, human as well as non human which use the given product. E.g. – If forecast about the demand for toys is to be made, we will have to estimate the number and characteristics of children whose parents can afford toys.

Factors affecting the demand for consumer durable goods

  • Long time use or replacement: For how long a consumer can use a good depends on the factors like his status, prestige attached to good, his level of money income, etc. Replacement of a good depends upon the factors like the wear and tear rate, the rate of obsolescence, etc.
  • Special Facilities: Some goods need special facilities for their use. E.g. Roads for cars, electricity for T.V., refrigerators etc. The expansion of such facilities expands the demand for such goods.
  • Joint use of a good by household: As consumer durables are used by more than one person, the decision to purchase may be influenced by family characteristics like size of family, age and sex consumption.
  • Price and Credit facilities: Demand for consumer durables is very much influenced by their prices and credit facilities like hire purchase, low interest rates, etc. available to buy them. More the easy credit facilities higher is the demand for goods like two wheelers, cars TVs. etc.

Factors affecting the demand for producer goods

  • The demand for producer or capital goods is a derived demand. It is derived from the demand of consumer goods they produce.
  • The forecasting of a capital good depends upon the following factors:
    – Increase in the price of substitutable factor increases the demand for capital goods.
    – The existing stock of the capital goods.
    – Technology advancement leading to reduced cost of production.
    – Prevailing rates of interest etc.

Methods of Demand Forecasting

There is no easy method to predict the future with certainty. The firm has to apply a proper mix of methods of forecasting to predict the future demand for a product. The various methods of demand forecasting are as follows:

1. Survey of Buyer’s intentions: In this method, customers are asked what they are planning to buy for the forthcoming time period usually a year.

This method involve use of conducting direct interviews or mailing questionnaire asking customers about their intentions or plans to buy the product.

The survey may be conducted by any of the following methods:

  • Complete Enumeration where all potential customers of a product are interviewed about what they are planning or intending to buy in future. It is cumbersome, costly and time consuming method.
  • Sample Survey where only a few customers are selected and interviewed about their future plans. It is less cumbersome and less costly method.
  • End-use method or Input-output method where the bulk of good is made for industrial manufactures who usually have definite future plans.

This method is useful for short-term forecasts.
In this method burden of forecasting is put on the customers.

2. Collective opinion Method: The method is also known as sales force opinion method or grass roots approach.

  • Under this method, salesmen are asked to estimate expectations of sales in their territories. Salesmen are considered to be the nearest persons to the customers retailers and wholesalers and have good knowledge and information about the future demand trend.
  • The estimates of all the sales-force is collected are examined in the light of proposed changes in selling price, product design, expected competition, etc. and also factors like purchasing power, employment, population, etc.
  • This method is based on first hand knowledge of the salesmen. However, its main drawback is that it is subjective. Its accuracy depends on the intelligence, vision and his ability to foresee the influence of many unknown factors.

3. Expert Opinion Method (Delphi Method): Under this method of demand forecasting views of specialists/experts and consultants are sought to estimate the demand in future. These experts may be of the firm itself like the executives and sales managers or consultant firms who are professionally trained for forecasting demand.

  • The Delphi technique, developed by OLAF HEMLER at the Rand Corporation of the
    U.S.A. is used to get the opinion of a number of experts about future demand.
  • Experts are provided with information and opinion feedbacks of other experts at different rounds and are repeatedly questioned for their opinion and comments till consensus emerges.
  • It is a time saving method.

4. Statistical Method: Statistical method have proved to be very useful in demand forecasting. Statistical methods are superior, more scientific, reliable and free from subjectively. The important statistical methods of demand forecasting are:

  • Trend Projection Method: The method is also known as Classical Method. It is considered as a ‘naive’ approach to demand forecasting.
    • Under this, data on sales over a period of time is chronologically arranged to get a ‘time series’. The time series shows the past sales pattern. It is assumed that the past sales pattern will continue in the future also. The techniques of trend projection based on, time series data are Graphical Method and Fitting trend equation or Least Square Method.
  • Graphical Method: This is the simplest technique to determine the trend.
    • Under this method, all values of sales for different years are plotted and free hand curve is drawn passing through as many points as possible. The direction of the free hand curve shows the trend.
    • The main drawback of this method is that it may show trend but not measure it.
  • Fitting Trend Equation/Least Square Method: This method is based on the assumption that the past rate of change will continue in the future.
    • It is a mathematical procedure for fitting a time to a set of observed data points in such a way that the sum of the squared deviation between the calculated and observed values is minimized.
    • This method is popular because it is simple and inexpensive.
  • Regression Analysis: This is a very common method of forecasting demand.
  • Under this method, a quantitative relationship is established between quantity demanded (dependent variable) and the independent variables like income, price of good, price of related goods, etc. Based on this relationship, an estimate is made for future demand.
  • It can be expressed as follows-
    Y = a + b X
    Where
    X, Y are variables a, b are constants

5. Controlled Experiments: Under this method, an effort is made to vary certain determinants of demand like price, advertising, etc. and conduct the experiments assuming that the other factors remain constant.

  • The effect of demand determinants on sales can be assessed either by varying then in different markets or by varying over a period of time in the same market.
  • The responses of demand to such changes over a period of time are recorded and are used for estimating the future demand for the product.
  • This method is used less as it is expensive and time consuming.
  • This method is also called as market experiment method.

6. Barometric Method of forecasting: This method is based on the assumption that future can be predicted from certain events occurring in the present. We need not depend upon the past observations for demand forecasting.

There are economic ups and downs in an economy which indicate the turning points. There are many economic indicators like income, population, expenditure, investment, etc. which can be used to forecast demand. There are three types of economic indicators, viz.

  • Coincidental Indicators are those which move up and down simultaneously with aggregate economy. It measures the current economic activity. E.g.- rate of unemployment.
  • Leading Indicators reflect future change in the trend of aggregate economy.
  • Lagging Indicators reflect future changes in the trend of aggregate economic activities.