NCERT full solutions here you can find full detailed solutions for all the questions with proper format. class 10 maths ncert solutions chapter 5 exercise 5.1.
You can see the solution for complete chapter here –
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Complete 10th Class Maths Solution
- NCERT Class 10th – Chapter 3- Exercise 3.6 Pair of Two Equations in Two Variables
- NCERT Class 10th – Chapter 4 – Exercise 4.1 Quadratic Equations
- NCERT Class 10th – Chapter 4 – Exercise 4.2 Quadratic Equations
- NCERT Class 10th – Chapter 4 – Exercise 4.3 Quadratic Equations
- NCERT Class 10th – Chapter 4 – Exercise 4.4 Quadratic Equations
EXERCISE :-5.1
QUE :-1. In which of the following situations, does the list of numbers involved make an arithmetic
progression, and why?
(i) The taxi fare after each km when the fare is ` 15 for the first km and ` 8 for each
additional km.
(ii) The amount of air present in a cylinder when a vacuum pump removes1/4of theair remaining in the cylinder at a time.
(iii) The cost of digging a well after every metre of digging, when it costs ` 150 for the
First metre and rises by ` 50 for each subsequent metre.
(iv) The amount of money in the account every year, when ` 10000 is deposited at
compound interest at 8 % per annum.
SOL:-
(i) taxi fare for 1st km =15 =a1
taxi fare for two km =15+8 =23=a2
taxi fare for three km =23+8=31=a3
taxi fare for four km = 31+8=39 =a4
a2 -a1=23-15=8
a3-a2 =31-23=8
a4 -a3 =39-31=8
diffrence between the succsessive term are same there for given situation is an arithmetic progression .
(ii) let air present in cylinder is a1=x
air left after using first time vaccume pump a2= x -(1/4)x =(3/4)x
air left after using second time vaccume pumpa3= 3/4x -1/4(3/4x) =(13/16)x
air left after using third time vaccume pump a4= (13/16)x -1/4((13/16)x)=(39/64)x
a2 -a1=(3/4)x -x =-(1/4)x
a3-a2 =(13/16)x -(3/4)x =-(1/16)x
a4 -a3 = (39/64)x-(13/16)x =-(13/64)x
diffrence between the succsessive term are not same there for given situation is not an arithmetic progression .
(iii) cost digging well for 1 m. a1 =150
cost digging well for 2 m. a2 =150+50 =200
cost digging well for 3 m. a3 =200+50 =250
cost digging well for 4 m. a4 = 250+50=300
a2 -a1=200 -150 =50
a3-a2 =250 -200 =50
a4 -a3 = 300-250 =50
diffrence between the succsessive term are same there for given situation is an arithmetic progression .
(iv) principle =10000, rate =8%
a1 =10000
amount after 1 year a2 =10000( 1+8/100)¹ =10800
amount after 2 year a3 =10800( 1+8/100)² = 12597
amount after 3 year a4 =12597( 1+8/100)³ =15869
a2 -a1=10800 -10000 =800
a3-a2 =12597 -10800 =1797
a4 -a3 = 15869-12597 =3272
diffrence between the succsessive term are not same there for given situation is not an arithmetic progression .
QUE:-2. Write first four terms of the AP, when the first term a and the common difference d are
given as follows:
(i) a = 10, d = 10 (ii) a = –2, d = 0
(iii) a = 4, d = – 3 (iv) a = – 1, d =1/2
(v) a = – 1.25, d = – 0.25
SOL:-
(i) a = 10, d = 10
first term a =a1 =10
second term a2 =a1 +d = 10+10=20
third term a3 =a2 +d = 20+10=30
fourth term a4 =a3 +d =30+10 =40
(ii) a = –2, d = 0
first term a =a1 =-2
second term a2 =a1 +d = -2+0=-2
third term a3 =a2 +d =-2+0=-2
fourth term a4 =a3 +d =-2+0 =-2
(iii) a = 4, d = – 3
first term a =a1 =4
second term a2 =a1 +d = 4+-3=1
third term a3 =a2 +d =1+-3=-2
fourth term a4 =a3 +d =-2+-3 =-5
(iv) a = – 1, d =1/2
first term a =a1 =-1
second term a2 =a1 +d = -1+1/2=-1/2
third term a3 =a2 +d =-1/2+1/2=0
fourth term a4 =a3 +d =0+1/2 =1/2
(v) a = – 1.25, d = – 0.25
first term a =a1 =-1.25
second term a2 =a1 +d = -1.25+-0.25=-1.50
third term a3 =a2 +d =-1.50+-0.25=-1.75
fourth term a4 =a3 +d =-1.75+-0.25=-2.0
QUE:-3. For the following APs, write the first term and the common difference:
(i) 3, 1, – 1, – 3, . . . (ii) – 5, – 1, 3, 7, . . .
(iii)1/3 , 5 /3 ,9 /3 ,13/3. . . (iv) 0.6, 1.7, 2.8, 3.9, . .
SOL:-
(i) 3, 1, – 1, – 3, . . .
common difference in arithmetic progression is a2-a1
a2-a1 = 1 -3 =-2
(ii) – 5, – 1, 3, 7, . . .
common difference in arithmetic progression is a2-a1
a2-a1 = -1 -(-5)=4
(iii)1/3 , 5 /3 ,9 /3 ,13/3. .
common difference in arithmetic progression is a2-a1
a2-a1 = 5/3 -(1/3)=4/3
(iv) 0.6, 1.7, 2.8, 3.9, . .
common difference in arithmetic progression is a2-a1
a2-a1 = 1.7 -0.6=1.1
QUE:-4. Which of the following are APs ? If they form an AP, find the common difference d and
write three more terms.
(i) 2, 4, 8, 16, . . . (ii)2, 5/2 , 3, 7/2,………..
(iii) – 1.2, – 3.2, – 5.2, – 7.2, . . . (iv) – 10, – 6, – 2, 2, . . .
(v) 3, 3 +√2 , 3 + 2√2 , 3 + 3√2 , . . . (vi) 0.2, 0.22, 0.222, 0.2222, . . .
(vii) 0, – 4, – 8, –12, . . . (viii) –1/2 ,-1/2,-1/2,-1/2……………
(ix) 1, 3, 9, 27, . . . (x) a, 2a, 3a, 4a, . . .
(xi) a, a², a³, a4, . . . (xii)√2,√8,√18 ,√32, . . .
(xiii)√3,√6,√9 ,√12 , . . . (xiv) 1², 3², 5², 7², . . .
(xv) 1², 5², 7², 73, . . .
SOL:-
(i) 2, 4, 8, 16, . . .
a2 -a1=4 -2 =2
a3-a2 =8 -4=4
a4 -a3 = 16-8 =8
diffrence between the succsessive term are not same there for given progression is not an arithmetic progression .
(ii)2, 5/2 , 3, 7/2,………..
a2 -a1=5/2 -2 =1/2
a3-a2 =3 -5/2=1/2
a4 -a3 =7/2-3 =1/2
diffrence between the succsessive term are same there for given progression is an arithmetic progression .
and common difference is =1/2
(iii) – 1.2, – 3.2, – 5.2, – 7.2, . . .
a2 -a1=-3.2 -(-1.2) =-2
a3-a2 =-5.2 -(-3.2)=-2
a4 -a3 =-7.2-(-5.2)=-2
diffrence between the succsessive term are same there for given progression is an arithmetic progression .
and common difference is =-2
(iv) – 10, – 6, – 2, 2, . . .
a2 -a1=-6 -(-10) =4
a3-a2 =-2 -(-6)=4
a4 -a3 =2-(-2)=4
diffrence between the succsessive term are same there for given progression is an arithmetic progression .
and common difference is =4
(v) 3, 3 +√2 , 3 + 2√2 , 3 + 3√2 , . . .
a2 -a1=3 +√2 -3 =√2
a3-a2 =3 + 2√2 -(3 +√2)=√2
a4 -a3 =3 + 3√2-(3 + 2√2)=√2
diffrence between the succsessive term are same there for given progression is an arithmetic progression .
and common difference is =√2
(vi) 0.2, 0.22, 0.222, 0.2222, . . .
a2 -a1=0.22 -0.2 =0.02
a3-a2 =0.222 -0.22=0.002
a4 -a3 =0.2222-0.222=0.0002
diffrence between the succsessive term are not same there for given progression is not an arithmetic progression .
(vii) 0, – 4, – 8, –12, . . .
a2 -a1=-4 -0 =-4
a3-a2 =-8 -(-4)=-4
a4 -a3 =-12-(-8)-=-4
diffrence between the succsessive term are same there for given progression is an arithmetic progression .
and common difference is =-4
(viii) –1/2 ,-1/2,-1/2,-1/2……………
a2 -a1=-1/2 -(-1/2) =0
a3-a2 =-1/2 -(-1/2) =0
a4 -a3 =-1/2 -(-1/2) =0
diffrence between the succsessive term are same there for given progression is an arithmetic progression .
and common difference is =0
(ix) 1, 3, 9, 27, . . .
a2 -a1=3 -1 =2
a3-a2 =9 -3 =6
a4 -a3 =27 -9 =18
diffrence between the succsessive term are not same there for given progression is not an arithmetic progression .
(x) a, 2a, 3a, 4a, . . .
a2 -a1=2a-a=a
a3-a2 =3a-2a =a
a4 -a3 =4a -3a =a
diffrence between the succsessive term are same there for given progression is an arithmetic progression .
and common difference is =a
(xi) a, a², a³, a4, . . .
a2 -a1=a²-a
a3-a2 =a³-a²
a4 -a3 =4a -a³
diffrence between the succsessive term are not same there for given progression is not an arithmetic progression .
(xii)√2,√8,√18 ,√32, . . .
a2 -a1=√8-√2=2√2-√2=√2
a3-a2 =√18-√8=3√2-2√2=√2
a4 -a3 =√32-√18 =4√2-3√2=√2
diffrence between the succsessive term are same there for given progression is an arithmetic progression .
and common difference is =√2
(xiii)√3,√6,√9 ,√12 , . . .
a2 -a1=√6-√3
a3-a2 =√9-√6
a4 -a3 =√12-√9
diffrence between the succsessive term are not same there for given progression is not an arithmetic progression .
(xiv) 1², 3², 5², c, . . .
a2 -a1=3²-1²=9-1=8
a3-a2 =5²-3² =25-9=16
a4 -a3 =7²-5²= 49-25=24
diffrence between the succsessive term are not same there for given progression is not an arithmetic progression .
(xv) 1², 5², 7², 73, . . .
a2 -a1=5²-1²=25-1=24
a3-a2 =7²-5² =49-25=24
a4 -a3 =73-7²= 73-49=24
diffrence between the succsessive term are same there for given progression is an arithmetic progression .
and common difference is =24
You can see the solution for complete chapter here –
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