Find the sum of the first 15 multiples of 8.

 

 cbse ncert 10th mathematics

 chapter 5 arithmetic progressions, exercise 5.3

 

 13. Find the sum of the first 15 multiples of 8.

 the first 15 multiples of 8 are terms in an arithmetic progression (AP)

a= t1 =1*8 =8

t2 = 2*8 =16 .....

last term L =15*8=120

d = t2-t1 =8

using the formula for the sum of n terms of an AP with n=15

Sn = (n/2) [a +L]

S(15) = (15/2) [8 +120]

=(15/2)*128

=15*64

=960 

 

12. Find the sum of the first 40 positive integers divisible by 6.

the first 40 positive multiples of 8 are terms in an arithmetic progression (AP)

with a = t1 =6

t2 = 12

....

last term L=t(40)=40*6 =240

n=40

 

using the formula for the sum of n terms of an AP with n=40

Sn = (n/2) [a +L]

S(40) = (40/2) [6+240]

=20*246=4920

=================================================

ncert cbse 10th mathematics

chapter 5  arithmetic progressions 

exercise 5.4 optional exercise

Which term of the AP : 121, 117, 113, . . ., is its first negative term? 

solution

2. The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.

solution

 3. A ladder has rungs 25 cm apart. The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and last rungs are [ 2 and(1/2) ]m apart, what is the length of the wood required for the rungs?

solution

 4. The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x.

solution

5. A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of (1/4) m and a tread of (1/2)m.   Calculate the total volume of concrete required to build the terrace.

 solution

 

chapter 5 arithmetic progressions, exercise 5.3

 exercise 5.3

 20. In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato,and the other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line.A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?

 solution

 

19.

 200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on . In how many rows are the 200 logs placed and how many logs are in the top row?

solution 

 

 18. A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, . . .  What is the total length of such a spiral made up of thirteen consecutive semicircles.

solution 

 

17. In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of Class I will plant 1 tree, a section of Class II will plant 2 trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?

solution 

16. A sum of Rs.700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs.20 less than its preceding prize, find the value of each of the prizes. 

solution

15. A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: Rs. 200 for the first day, Rs. 250 for the second day, Rs.300 for the third day, etc., the penalty for each succeeding day being Rs.50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days? 

solution

 

14. Find the sum of the odd numbers between 0 and 50.

 solution

  13. Find the sum of the first 15 multiples of 8.

solution

 12. Find the sum of the first 40 positive integers divisible by 6.

solution

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: Rs. 200 for the first day, Rs. 250 for the second day, Rs.300 for the third day, etc., the penalty for each succeeding day being Rs.50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days?

 cbse ncert 10th mathematics

 chapter 5 arithmetic progressions, exercise 5.3

15. A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: Rs. 200 for the first day, Rs. 250 for the second day, Rs.300 for the third day, etc., the penalty for each succeeding day being Rs.50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days?

Using the concept of Arithmetic progressions (AP)

t1 =200, t2 = 250 , t3 =300 ...

with first term    a = 200

d =t2 - t1 = 250-200 = 50

n =30 

using the formula for sum of n terms of an AP

Sn = (n/2)*[ 2a + (n-1)d ]

total penalty = 

S(30) = (30/2) *[ 2(200) + (30-1)(50) ]

=(15)*[400 +29*50]

=15*[400+1450]

=15*1850

=Rs. 27,750 /- 

14. Find the sum of the odd numbers between 0 and 50.

selecting the odd numbers between 0 and 50 and adding 

1 + 3 +5 + 7 +11 +. . . +49

Using the concept of Arithmetic progressions (AP)

t1 =1, t2 = 3 , t3 =5 ...  last term L=49

with first term    a = 1

d =t2 - t1 = 3-1=2

n =?

use the concept of tn = a + (n-1)d

or n = [(L-a)/d] +1

n=[(49-1)/2] +1

n=[48/2] +1

n =24+1

n=25

using the formula for sum of n terms of an AP

Sn = (n/2) [a +L]

S(25) = (25/2) [ 1 +49]

= (25/2)(50)

=25 *25 

=625 

=================================================

ncert cbse 10th mathematics

chapter 5  arithmetic progressions 

exercise 5.4 optional exercise

Which term of the AP : 121, 117, 113, . . ., is its first negative term? 

solution

2. The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.

solution

 3. A ladder has rungs 25 cm apart. The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and last rungs are [ 2 and(1/2) ]m apart, what is the length of the wood required for the rungs?

solution

 4. The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x.

solution

5. A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of (1/4) m and a tread of (1/2)m.   Calculate the total volume of concrete required to build the terrace.

 solution

 

chapter 5 arithmetic progressions, exercise 5.3

 exercise 5.3

 20. In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato,and the other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line.A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?

 solution

 

19.

 200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on . In how many rows are the 200 logs placed and how many logs are in the top row?

solution 

 

 18. A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, . . .  What is the total length of such a spiral made up of thirteen consecutive semicircles.

solution 

 

17. In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of Class I will plant 1 tree, a section of Class II will plant 2 trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?

solution 

16. A sum of Rs.700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs.20 less than its preceding prize, find the value of each of the prizes. 

solution

15. A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: Rs. 200 for the first day, Rs. 250 for the second day, Rs.300 for the third day, etc., the penalty for each succeeding day being Rs.50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days? 

solution

 

14. Find the sum of the odd numbers between 0 and 50.

 solution

 

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of Class I will plant 1 tree, a section of Class II will plant 2 trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?

 cbse ncert 10th mathematics

 chapter 5 arithmetic progressions, exercise 5.3

 

 17. In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of Class I will plant 1 tree, a section of Class II will plant 2 trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?

 

since there are 3 sections for each class 

 

Total number of trees

= 3*[ 1 + 2 + . . . + 12]  

using the formula for sum of n terms of an AP inside the  [ ... ]

 with n=12  first term a =1 last term L = 12

Sn = (n/2) [a +L]

Total number of trees

= 3*[ 1 + 2 + . . . + 12]  

= 3* [ (12/2) [1 +12]  ]

=3*[(6)*13]

=234 trees

16. A sum of Rs.700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs.20 less than its preceding prize, find the value of each of the prizes.

 A sum of Rs.700 is to be used means Sn= 700

given that, there are  seven cash prizes so n =7

each prize is Rs.20 less than its preceding prize

means that d =(-20) where the minus sign is used for " less "

if the first term a stands for the first prize.

using the formula for sum of n terms of an AP

Sn = (n/2)*[ 2a + (n-1)d ]

 

700 = (7/2) * [ 2a + (7-1)(-20) ]

700 = (7/2) * [ 2a -120]

2*700  = 7 [2a - 120]

1400 = 14a -840

1400+840 =14a

2240 = 14a 

a = 2240/14

a = 160

 

first prize = a = Rs.160

second prize =a+d = 160 + (-20) = Rs.140

third prize = a+2d = 160 + 2*(-20) = 160-40=Rs.120

fourth prize = a+3d =  160 + 3*(-20) = Rs.100

fifth prize = a+4d =  160 + 4*(-20) = Rs.80

sixth prize = a+5d = 160 + 5*(-20) = Rs.60

seventh prize =a+6d =160 + 6*(-20) = Rs.40

=================================================

ncert cbse 10th mathematics

chapter 5  arithmetic progressions 

exercise 5.4 optional exercise

Which term of the AP : 121, 117, 113, . . ., is its first negative term? 

solution

2. The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.

solution

 3. A ladder has rungs 25 cm apart. The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and last rungs are [ 2 and(1/2) ]m apart, what is the length of the wood required for the rungs?

solution

 4. The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x.

solution

5. A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of (1/4) m and a tread of (1/2)m.   Calculate the total volume of concrete required to build the terrace.

 solution

 

chapter 5 arithmetic progressions, exercise 5.3

 exercise 5.3

 20. In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato,and the other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line.A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?

 solution

 

19.

 200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on . In how many rows are the 200 logs placed and how many logs are in the top row?

solution 

 

 18. A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, . . .  What is the total length of such a spiral made up of thirteen consecutive semicircles.

solution 

 

17. In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of Class I will plant 1 tree, a section of Class II will plant 2 trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?

solution 

16. A sum of Rs.700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs.20 less than its preceding prize, find the value of each of the prizes. 

solution

 

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on. In how many rows are the 200 logs placed and how many logs are in the top row?

cbse ncert 10th mathematics

 chapter 5 arithmetic progressions, exercise 5.3

 

 

19.

 200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on . In how many rows are the 200 logs placed and how many logs are in the top row?

 

using arithmetic progressions AP

 t1 = 20

t2 = 19

t3=18

a =20 

d=t2 - t1 = 19 - 20 = (-1)

total number of logs is 200

so S(n) = 200 , n = ?

Sn = (n/2)*[ 2a + (n-1)d ]

 

200 = (n/2)*[ 2(20) + (n-1)(-1) ]

200 =  (n/2)*[ 40 - n +1 ]

200= (n/2)*[ 41 - n  ]

200*2 = n[41-n]

400 = 41n -(n^2)

 

(n^2) - 41n +400 =0

 

(n-16) (n-25) =0

n = 16, n=25

if n = 16

tn = a+(n-1)d

t(16) = 20 + (16-1)(-1) = 20+(15)(-1) = 20-15 = 5 logs in the top row

if n = 25

tn = a+(n-1)d

t(25) = 20 + (25-1)(-1) = 20+(24)(-1) = 20-24= (-4) logs which is not possible.

 

So number of rows is 16, and number of logs in the top row is 5.

18. A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, . . .  What is the total length of such a spiral made up of thirteen consecutive semicircles.

 

length of arc of semi circle   = pi*r

Total length of spiral = pi*0.5 +pi*1.0+pi*1.5 + .... (13 terms)

=pi [0.5 + 1.0 +1.5 + ...  (13 terms) ]

using formula for sum of n terms of an AP

Sn = (n/2)*[ 2a + (n-1)d ]

with n =13, a=0.5, d=1.0-0.5 = 0.5

 

Total length of spiral =pi { (13/2) [ 2(0.5)+(13-1)(0.5) ] }

=pi  { (13/2) [1.0 +12*0.5 ] }

=pi { (13/2) [1.0 +6.0 ] }

=pi {(13/2)[7]}

=(22/7) {(13/2)[7]}

=11*13

=143 cm

 

=================================================

ncert cbse 10th mathematics

chapter 5  arithmetic progressions 

exercise 5.4 optional exercise

Which term of the AP : 121, 117, 113, . . ., is its first negative term? 

solution

2. The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.

solution

 3. A ladder has rungs 25 cm apart. The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and last rungs are [ 2 and(1/2) ]m apart, what is the length of the wood required for the rungs?

solution

 4. The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x.

solution

5. A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of (1/4) m and a tread of (1/2)m.   Calculate the total volume of concrete required to build the terrace.

 solution

 

chapter 5 arithmetic progressions, exercise 5.3

 exercise 5.3

 20. In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato,and the other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line.A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?

 solution

 

19.

 200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on . In how many rows are the 200 logs placed and how many logs are in the top row?

solution 

 

 18. A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, . . .  What is the total length of such a spiral made up of thirteen consecutive semicircles.

solution 

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

Free guide to ncert cbse 10th mathematics chapter 3 pair of linear equations in two variables selected questions

 

Free guide to ncert cbse 10th mathematics chapter 3 

pair of linear equations in two variables selected questions

 

 The ages of two friends Ani and Biju differ by 3 years. Ani’s father  is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Ani’s father differ by 30 years. Find the ages of Ani and Biju

solution

 

2. One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “If you give me ten, I shall be six times as rich as you”. Tell me what is the amount of their (respective) capital? 

solution

3. A train covered a certain distance at a uniform speed. If the train would have been 10 km/h faster, it would have taken 2 hours less than the scheduled time. And, if the train were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. Find the distance covered by the train. 

solution  

 

4. The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class.

solution  

5. In a ∆ ABC, ∠ C = 3 ∠ B = 2 (∠ A + ∠ B). Find the three angles.

solution

 

Solve the following pair of linear equations:

 px + qy = p – q 

 qx – py = p + q

solution

 

(ii) ax + by = c
     bx + ay = 1 + c

solution  

 

(iii) 

(x/a) -(y/b) = 0

ax +by = (a^2)  + (b^2)

solution  

 

(iv)

(a – b)x + (a + b) y = (a^2) – 2ab – (b^2)

(a + b)(x + y) = (a^2) + (b^2 )

solution

 

(v)

152x – 378y = – 74

–378x + 152y = – 604

solution   

 ABCD is a cyclic quadrilateral  Find the angles of the cyclic quadrilateral,

if angles are A =(4y+20) , B =(3y-5) , C=(-4x), D=(-7x+5)

solution

 

exercise 3.6

2

Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current. 

solution

 

 (ii) 2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to
finish the work, and also that taken by 1 man alone.

solution

 

(iii) 

Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and the remaining by bus. If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.

 

solution

 

exercise 3.5

4

A part of monthly hostel charges is fixed and the remaining depends on the
number of days one has taken food in the mess. When a student A takes food for
20 days she has to pay Rs.1000 as hostel charges whereas a student B, who takes
food for 26 days, pays Rs.1180 as hostel charges. Find the fixed charges and the
cost of food per day.

solution

 

(ii) A fraction changes to (1/3) when 1 is subtracted from the numerator and it changes to (1/4) when 8 is added to its denominator. Find the fraction. 

 solution

(iii)Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

solution 

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds,they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

solution 

 (v) The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle

solution

3. Solve the following pair of linear equations by the substitution and cross-multiplication methods :
 

8x + 5y = 9
3x + 2y = 4

solution

 

2

(i) For which values of a and b does the following pair of linear equations have an
infinite number of solutions?
2x + 3y = 7
(a – b) x + (a + b) y = 3a + b – 2

solution 

 

(ii) For which value of k will the following pair of linear equations have no solution?
 

3x + y = 1

(2k – 1) x + (k – 1) y = 2k + 1

  solution

 

Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using cross multiplication method.

x – 3y – 3 = 0
3x – 9y – 2 = 0

solution 

(ii) 

2x + y = 5
3x + 2y = 8

solution 

 

 

(iii) 

3x – 5y = 20
6x – 10y = 40

solution 

 

(iv) 

x – 3y – 7 = 0
3x – 3y – 15 = 0

 solution

exercise 3.4

2. Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method :

(v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs.27 for a book kept for seven days, while Susy paid Rs. 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day. 

solution

(iv) Meena went to a bank to withdraw Rs.2000. She asked the cashier to give her
Rs. 50 and Rs.100 notes only. Meena got 25 notes in all. Find how many notes of
Rs.50 and Rs.100 she received.

solution

 

(iii) 

(iii)The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number 

solution

 (ii) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?

solution

 

If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes (1/2) if we only add 1 to the denominator. What is the fraction?

solution  

 

Solve the following pair of linear equations by the elimination method and the substitution method : 

 x + y = 5 and 2x – 3y = 4

solution  

(ii) 3x + 4y = 10 and 2x – 2y = 2 

 

solution

 (iii) 3x – 5y – 4 = 0 and 9x = 2y + 7

  solution

 

 

iv)

(x/2)+(2y/3)=(-1)

 x -(y/3)=3

solution

 

exercise 3.3

solve by method of substitution 

(ii) 

s-t =3

(s/3)+(t/2)=6

solution

(iii)

 3x – y = 3

9x – 3y = 9

solution  

(iv) 

0.2x + 0.3y = 1.3
0.4x + 0.5y = 2.3 

solution

(v)

 sqrt(2)x +sqrt(3)y =0

sqrt(3)x -sqrt(8)y =0

solution

 

(vi)

(3x/2)-(5y/3)=(-2)

(x/3)+(y/2)=(13/6)

solution

 

3. Form the pair of linear equations for the following problems and find their solution by substitution method 

 The difference between two numbers is 26 and one number is three times the other. Find them.

solution

(ii) The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.

solution 

(iii) The coach of a cricket team buys 7 bats and 6 balls for Rs. 3800. Later, she buys 3 bats and 5 balls for Rs.1750. Find the cost of each bat and each ball.

 solution 

 

(iv) The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is Rs. 105 and for a journey of 15 km, the charge paid is Rs.155. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25 km?

solution

v) A fraction becomes (9/11) if 2 is added to both the numerator and the denominator.  If 3 is added to both the numerator and the denominator it becomes (5/6). Find the fraction.

solution  

 

(vi) Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his son. What are their present ages?

solution

 

exercise 3.2

form the equations and solve by graphical method

  10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.

solution

 

(ii) 5 pencils and 7 pens together cost Rs.50, whereas 7 pencils and 5 pens together cost  Rs.46. Find the cost of one pencil and that of one pen

solution

 

2. By using the ratios a1/a2 , b1/b2, c1/c2, find out if the pair of lines are 

intersecting at a point, or are parallel or are coincident

5x-4y+8=0

7x+6y-9=0

solution

ii) 

9x + 3y + 12 = 0
18x + 6y + 24 = 0

solution

 

(iii) 

6x – 3y + 10 = 0
2x – y + 9 = 0

solution 

 

5.

 Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.

 solution

 

7.

 Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region and find the area of the region.

solution

 
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There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of (1/4) m and a tread of (1/2)m. Calculate the total volume of concrete required to build the terrace.

 

 cbse ncert 10th mathematics

 chapter 5 arithmetic progressions, exercise 5.4 optional exercise

 

5. A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of (1/4) m and a tread of (1/2)m.   Calculate the total volume of concrete required to build the terrace.

 

thinking of each step as a cuboid with length =50m, breadth =(1/2)m

and  height (1/4)m for the first step with increase of (1/4)m for each further step

volume for  first step

V1 = length * breadth * height =50*(1/2)*(1/4)

 

for second step 

 length =50m, breadth =(1/2)m, height =(1/4)+(1/4) =(1/2)m

volume for  second step

V2 = length * breadth * height =50*(1/2)*(1/2)

 

for third step 

 length =50m, breadth =(1/2)m, height =(1/4)+(1/4)+(1/4) =(3/4)m

volume for  second step

V2 = length * breadth * height =50*(1/2)*(3/4)

 

and so on

 

so total volume

= [50*(1/2)*(1/4)]+[50*(1/2)*(1/2)]+[50*(1/2)*(3/4)] +... (15 terms)

=50*(1/2) { (1/4) +(1/2) +(3/4) + ....(15 terms) }

=25 *{ (1/4) +(1/2) +(3/4) + ....(15 terms) } ,

using sum of n=15 terms of an AP with first term a=(1/4),  d =(1/4) for { ...}

Sn = (n/2)*[ 2a + (n-1)d ]

total volume = 25 *{ (15/2) *[2(1/4) +(15-1)(1/4) ] }

=25* { (15/2) [ (1/2)+ (14/4)  ] }

=25* {15/2 [(1/2) +(7/2)}

=25* {(15/2)*(4)}

=25*15*2=750 cubic metres

 

 cbse ncert 10th mathematics

 chapter 5 arithmetic progressions, exercise 5.3

 exercise 5.3

 20. In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato,and the other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line.A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?

the distance covered to bring the first potato to the bucket = 5+5=10m

 

After that ,the distance covered for bringing the second potato to the bucket

=5 +3 + 3 +5 =16m

 

After that ,the distance covered for bringing the third potato to the bucket

=5 +3 + 3 +3+ 3 +5 =22m 

and so on for the 10 potatoes

 

so total distance = 10 +16 +22 + ...  (10 terms )

 

using sum of n=10 terms of an AP with first term a=10 , d=t2-t1=16-10=6

Sn = (n/2)*[ 2a + (n-1)d ]

 

total distance = 10 +16 +22 + ...  (10 terms )

= (10/2)*[2(10)+(10-1)(6)]

=5*74 =370m

=================================================

ncert cbse 10th mathematics

chapter 5  arithmetic progressions 

exercise 5.4 optional exercise

Which term of the AP : 121, 117, 113, . . ., is its first negative term? 

solution

2. The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.

solution

 3. A ladder has rungs 25 cm apart. The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and last rungs are [ 2 and(1/2) ]m apart, what is the length of the wood required for the rungs?

solution

 4. The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x.

solution

5. A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of (1/4) m and a tread of (1/2)m.   Calculate the total volume of concrete required to build the terrace.

 solution

 

chapter 5 arithmetic progressions, exercise 5.3

 exercise 5.3

 20. In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato,and the other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line.A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?

 solution

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

A ladder has rungs 25 cm apart. The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and last rungs are [ 2 and(1/2) ]m apart, what is the length of the wood required for the rungs?

 cbse ncert 10th mathematics

 chapter 5 arithmetic progressions, exercise 5.4 optional exercise

 

 3. A ladder has rungs 25 cm apart. The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and last rungs are [ 2 and(1/2) ]m apart, what is the length of the wood required for the rungs?

top and last rungs are [ 2 and(1/2) ]m apart means

top and last rungs are [5/2]m apart

 

changing to cm.

top and last rungs are [5/2]*100 =250 cm apart

 

now consecutive rungs are 25cm apart 

so number of rungs =[250/25] +1 = 10+1 =11 rungs

The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top means 

 

counting upwards

 the length of the rungs form an AP  with 

n=11terms

a=45cm 

last term (11th term ) t(11) =25cm

 

To find the total length of wood we find the sum of lengths S(11)

S(n) = (n/2) [a + L ] , L refers to the last term , L = t(11) =25cm

 

S(11) = (11/2) [ 45 + 25 ]

=(11/2)[70] = 11*35

=385 cm

 

 4. The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x.

a = 1

The house numbers form an AP with

a=1 and 

kth term =t(k) =k for k =1 to 49 

using S(n) = (n/2) [a + L ] , where 

where L refers to the last term 

 

Sx = (x/2)[1+x]

 

S(49) = (49/2) [1+49] = (49/2)[50]=49*25=1225

 

S(x-1) = [(x-1)/2][1+(x-1)]= [(x-1)/2][x]

 

by given conditions

 S(x-1) = S(49) - S(x)

[(x-1)/2][x] = 1225 - (x/2)[1+x]

 

multiply each term with 2

(x-1)x = 2450 -x(1+x)

(x^2) - x = 2450 -x - (x^2) 

2(x^2)  -2450 = 0

 dividing by 2

(x^2)  - 1225 = 0

 

(x^2) = 1225

taking square root

x = 35 , because x cannot be negative

=================================================

ncert cbse 10th mathematics

chapter 5  arithmetic progressions 

exercise 5.4 optional exercise

Which term of the AP : 121, 117, 113, . . ., is its first negative term? 

solution

2. The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.

solution

 3. A ladder has rungs 25 cm apart. The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and last rungs are [ 2 and(1/2) ]m apart, what is the length of the wood required for the rungs?

solution

 4. The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x.

solution

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

Which term of the AP : 121, 117, 113, . . ., is its first negative term?

 cbse ncert 10th mathematics

 chapter 5 arithmetic progressions, exercise 5.4 optional exercise

 

Which term of the AP : 121, 117, 113, . . ., is its first negative term? 

first term a =121

d= t2 - t1 = 117-121 =(-4)

nth term tn = a + (n-1)d

tn =121 + (n-1)(-4) 

tn = 121-4n+4

tn = 125 -4n

for a negative term 

tn < 0

[ 125 -4n ] <0

 

125 <4n

n>125/4

n>31.25

 

n is a natural number

so the first negative term is the 32nd term 

2. The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.

nth term

tn =a +(n-1)d

 n=3

3rd term

t3 =a+(3-1)d=a+2d

n= 7

7th term

t7=a+(7-1)d =a+6d

 

sum of the third and the seventh terms of an AP is 6

means

t3+t7 =6

t3+t7=  [a+2d] +[a+6d]= 2a+8d

so

2a +8d= 6

dividing by 2

a+4d=3

or a = (3-4d)----------------------(1)

their product is 8 means

t3 *t7 =8

[a+2d] * [a+6d]= 8-----------------------(2)

use equation (1) in (2) to eliminate a

 [(3-4d)+2d]*[(3-4d)+6d]=8

[3-2d][3+2d]=8

9-4(d^2) =8

4(d^2) = 9-8

4(d^2) =1

(d^2) =1/4

d = (1/2) or ( -1/2)

use in (1)

if d = (1/2)

a = 3-4d = 3-4(1/2) = 3-2 = 1

 

if

if d = (-1/2)

a = 3-4d = 3-4(-1/2) = 3+2 = 5

 

 

sum of first sixteen terms of the AP

Sn =(n/2)[ 2a+(n-1)d ] 

if a=1 , d=(1/2)

n=16

 

S(16) = (16/2) [ 2*1 +(16-1)(1/2) ] 

=8*[ 2 +(15/2)]

=8*[19/2]

=4*19=76

if a=5 , d=(-1/2)

n=16

 

S(16) = (16/2) [ 2*5 +(16-1)(-1/2) ] 

=8*[ 10 +(-15/2)]

=8*[5/2]

=4*5=20

=================================================

ncert cbse 10th mathematics

chapter 5  arithmetic progressions 

exercise 5.4 optional exercise

Which term of the AP : 121, 117, 113, . . ., is its first negative term? 

solution

2. The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.

solution

 

 

ncert cbse 10th mathematics chapter 3 optional exercise 3.7 

 

 The ages of two friends Ani and Biju differ by 3 years. Ani’s father  is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Ani’s father differ by 30 years. Find the ages of Ani and Biju

solution

 

2. One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “If you give me ten, I shall be six times as rich as you”. Tell me what is the amount of their (respective) capital? 

solution

3. A train covered a certain distance at a uniform speed. If the train would have been 10 km/h faster, it would have taken 2 hours less than the scheduled time. And, if the train were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. Find the distance covered by the train. 

solution  

 

4. The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class.

solution  

5. In a ∆ ABC, ∠ C = 3 ∠ B = 2 (∠ A + ∠ B). Find the three angles.

solution

 

Solve the following pair of linear equations:

 px + qy = p – q 

 qx – py = p + q

solution

 

(ii) ax + by = c
     bx + ay = 1 + c

solution  

 

(iii) 

(x/a) -(y/b) = 0

ax +by = (a^2)  + (b^2)

solution  

 

(iv)

(a – b)x + (a + b) y = (a^2) – 2ab – (b^2)

(a + b)(x + y) = (a^2) + (b^2 )

solution

 

(v)

152x – 378y = – 74

–378x + 152y = – 604

solution   

 ABCD is a cyclic quadrilateral  Find the angles of the cyclic quadrilateral,

if angles are A =(4y+20) , B =(3y-5) , C=(-4x), D=(-7x+5)

solution

 

exercise 3.6

2

Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current. 

solution

 

 (ii) 2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to
finish the work, and also that taken by 1 man alone.

solution

 

(iii) 

Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and the remaining by bus. If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.

 

solution

 

exercise 3.5

4

A part of monthly hostel charges is fixed and the remaining depends on the
number of days one has taken food in the mess. When a student A takes food for
20 days she has to pay Rs.1000 as hostel charges whereas a student B, who takes
food for 26 days, pays Rs.1180 as hostel charges. Find the fixed charges and the
cost of food per day.

solution

 

(ii) A fraction changes to (1/3) when 1 is subtracted from the numerator and it changes to (1/4) when 8 is added to its denominator. Find the fraction. 

 solution

(iii)Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

solution 

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds,they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

solution 

 (v) The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle

solution

3. Solve the following pair of linear equations by the substitution and cross-multiplication methods :
 

8x + 5y = 9
3x + 2y = 4

solution

 

2

(i) For which values of a and b does the following pair of linear equations have an
infinite number of solutions?
2x + 3y = 7
(a – b) x + (a + b) y = 3a + b – 2

solution 

 

(ii) For which value of k will the following pair of linear equations have no solution?
 

3x + y = 1

(2k – 1) x + (k – 1) y = 2k + 1

  solution

 

Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using cross multiplication method.

x – 3y – 3 = 0
3x – 9y – 2 = 0

solution 

(ii) 

2x + y = 5
3x + 2y = 8

solution 

 

 

(iii) 

3x – 5y = 20
6x – 10y = 40

solution 

 

(iv) 

x – 3y – 7 = 0
3x – 3y – 15 = 0

 solution

exercise 3.4

2. Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method :

(v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs.27 for a book kept for seven days, while Susy paid Rs. 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day. 

solution

(iv) Meena went to a bank to withdraw Rs.2000. She asked the cashier to give her
Rs. 50 and Rs.100 notes only. Meena got 25 notes in all. Find how many notes of
Rs.50 and Rs.100 she received.

solution

 

(iii) 

(iii)The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number 

solution

 (ii) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?

solution

 

If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes (1/2) if we only add 1 to the denominator. What is the fraction?

solution  

 

Solve the following pair of linear equations by the elimination method and the substitution method : 

 x + y = 5 and 2x – 3y = 4

solution  

(ii) 3x + 4y = 10 and 2x – 2y = 2 

 

solution

 (iii) 3x – 5y – 4 = 0 and 9x = 2y + 7

  solution

 

 

iv)

(x/2)+(2y/3)=(-1)

 x -(y/3)=3

solution

 

exercise 3.3

solve by method of substitution 

(ii) 

s-t =3

(s/3)+(t/2)=6

solution

(iii)

 3x – y = 3

9x – 3y = 9

solution  

(iv) 

0.2x + 0.3y = 1.3
0.4x + 0.5y = 2.3 

solution

(v)

 sqrt(2)x +sqrt(3)y =0

sqrt(3)x -sqrt(8)y =0

solution

 

(vi)

(3x/2)-(5y/3)=(-2)

(x/3)+(y/2)=(13/6)

solution

 

3. Form the pair of linear equations for the following problems and find their solution by substitution method 

 The difference between two numbers is 26 and one number is three times the other. Find them.

solution

(ii) The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.

solution 

(iii) The coach of a cricket team buys 7 bats and 6 balls for Rs. 3800. Later, she buys 3 bats and 5 balls for Rs.1750. Find the cost of each bat and each ball.

 solution 

 

(iv) The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is Rs. 105 and for a journey of 15 km, the charge paid is Rs.155. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25 km?

solution

v) A fraction becomes (9/11) if 2 is added to both the numerator and the denominator.  If 3 is added to both the numerator and the denominator it becomes (5/6). Find the fraction.

solution  

 

(vi) Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his son. What are their present ages?

solution

 

exercise 3.2

form the equations and solve by graphical method

  10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.

solution

 

(ii) 5 pencils and 7 pens together cost Rs.50, whereas 7 pencils and 5 pens together cost  Rs.46. Find the cost of one pencil and that of one pen

solution

 

2. By using the ratios a1/a2 , b1/b2, c1/c2, find out if the pair of lines are 

intersecting at a point, or are parallel or are coincident

5x-4y+8=0

7x+6y-9=0

solution

ii) 

9x + 3y + 12 = 0
18x + 6y + 24 = 0

solution

 

(iii) 

6x – 3y + 10 = 0
2x – y + 9 = 0

solution 

 

5.

 Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.

 solution

 

7.

 Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region and find the area of the region.

solution

 
disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

 

Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.

 

7.

 Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region and find the area of the region.

 x – y + 1 = 0-------------------(1)

put y=0 ( x-axis) 

x-0+1=0 

x=(-1) 

point of intersection with x axis is (-1,0)

3x + 2y – 12 = 0-----------------(2)

put y=0 ( x-axis) 

3x +0-12=0

3x=12

x=12/3

x=4

 point of intersection with x axis is (4,0)

 

solving the two equations

[x  -  y = (-1) ]*3

3x+2y =12

------------------------eliminating x

3x - 3y = (-3)

3x +2y =12

--------------------subtracting

     -5y =( -15)

y=(15)/(-5)

y=3

substitute in x  -  y = (-1)

x-3=(-1) 

x= (-1) +3

x=2

point of intersection of the two lines is (2,3)

vertices of the triangle are (2,3),(-1,0),(4,0)

 

plot the vertices and the lines

so  using the graph

base=  5

height=3

area of triangle = (1/2)*base*height

=(1/2)*5*3=15/2

 

=================================================

ncert cbse 10th mathematics chapter 3 optional exercise 3.7 

 The ages of two friends Ani and Biju differ by 3 years. Ani’s father  is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Ani’s father differ by 30 years. Find the ages of Ani and Biju

solution

 

2. One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “If you give me ten, I shall be six times as rich as you”. Tell me what is the amount of their (respective) capital? 

solution

3. A train covered a certain distance at a uniform speed. If the train would have been 10 km/h faster, it would have taken 2 hours less than the scheduled time. And, if the train were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. Find the distance covered by the train. 

solution  

 

4. The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class.

solution  

5. In a ∆ ABC, ∠ C = 3 ∠ B = 2 (∠ A + ∠ B). Find the three angles.

solution

 

Solve the following pair of linear equations:

 px + qy = p – q 

 qx – py = p + q

solution

 

(ii) ax + by = c
     bx + ay = 1 + c

solution  

 

(iii) 

(x/a) -(y/b) = 0

ax +by = (a^2)  + (b^2)

solution  

 

(iv)

(a – b)x + (a + b) y = (a^2) – 2ab – (b^2)

(a + b)(x + y) = (a^2) + (b^2 )

solution

 

(v)

152x – 378y = – 74

–378x + 152y = – 604

solution   

 ABCD is a cyclic quadrilateral  Find the angles of the cyclic quadrilateral,

if angles are A =(4y+20) , B =(3y-5) , C=(-4x), D=(-7x+5)

solution

 

exercise 3.6

2

Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current. 

solution

 

 (ii) 2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to
finish the work, and also that taken by 1 man alone.

solution

 

(iii) 

Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and the remaining by bus. If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.

 

solution

 

exercise 3.5

4

A part of monthly hostel charges is fixed and the remaining depends on the
number of days one has taken food in the mess. When a student A takes food for
20 days she has to pay Rs.1000 as hostel charges whereas a student B, who takes
food for 26 days, pays Rs.1180 as hostel charges. Find the fixed charges and the
cost of food per day.

solution

 

(ii) A fraction changes to (1/3) when 1 is subtracted from the numerator and it changes to (1/4) when 8 is added to its denominator. Find the fraction. 

 solution

(iii)Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

solution 

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds,they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

solution 

 (v) The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle

solution

3. Solve the following pair of linear equations by the substitution and cross-multiplication methods :
 

8x + 5y = 9
3x + 2y = 4

solution

 

2

(i) For which values of a and b does the following pair of linear equations have an
infinite number of solutions?
2x + 3y = 7
(a – b) x + (a + b) y = 3a + b – 2

solution 

 

(ii) For which value of k will the following pair of linear equations have no solution?
 

3x + y = 1

(2k – 1) x + (k – 1) y = 2k + 1

  solution

 

Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using cross multiplication method.

x – 3y – 3 = 0
3x – 9y – 2 = 0

solution 

(ii) 

2x + y = 5
3x + 2y = 8

solution 

 

 

(iii) 

3x – 5y = 20
6x – 10y = 40

solution 

 

(iv) 

x – 3y – 7 = 0
3x – 3y – 15 = 0

 solution

exercise 3.4

2. Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method :

(v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs.27 for a book kept for seven days, while Susy paid Rs. 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day. 

solution

(iv) Meena went to a bank to withdraw Rs.2000. She asked the cashier to give her
Rs. 50 and Rs.100 notes only. Meena got 25 notes in all. Find how many notes of
Rs.50 and Rs.100 she received.

solution

 

(iii) 

(iii)The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number 

solution

 (ii) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?

solution

 

If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes (1/2) if we only add 1 to the denominator. What is the fraction?

solution  

 

Solve the following pair of linear equations by the elimination method and the substitution method : 

 x + y = 5 and 2x – 3y = 4

solution  

(ii) 3x + 4y = 10 and 2x – 2y = 2 

 

solution

 (iii) 3x – 5y – 4 = 0 and 9x = 2y + 7

  solution

 

 

iv)

(x/2)+(2y/3)=(-1)

 x -(y/3)=3

solution

 

exercise 3.3

solve by method of substitution 

(ii) 

s-t =3

(s/3)+(t/2)=6

solution

(iii)

 3x – y = 3

9x – 3y = 9

solution  

(iv) 

0.2x + 0.3y = 1.3
0.4x + 0.5y = 2.3 

solution

(v)

 sqrt(2)x +sqrt(3)y =0

sqrt(3)x -sqrt(8)y =0

solution

 

(vi)

(3x/2)-(5y/3)=(-2)

(x/3)+(y/2)=(13/6)

solution

 

3. Form the pair of linear equations for the following problems and find their solution by substitution method 

 The difference between two numbers is 26 and one number is three times the other. Find them.

solution

(ii) The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.

solution 

(iii) The coach of a cricket team buys 7 bats and 6 balls for Rs. 3800. Later, she buys 3 bats and 5 balls for Rs.1750. Find the cost of each bat and each ball.

 solution 

 

(iv) The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is Rs. 105 and for a journey of 15 km, the charge paid is Rs.155. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25 km?

solution

v) A fraction becomes (9/11) if 2 is added to both the numerator and the denominator.  If 3 is added to both the numerator and the denominator it becomes (5/6). Find the fraction.

solution  

 

(vi) Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his son. What are their present ages?

solution

 

exercise 3.2

form the equations and solve by graphical method

  10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.

solution

 

(ii) 5 pencils and 7 pens together cost Rs.50, whereas 7 pencils and 5 pens together cost  Rs.46. Find the cost of one pencil and that of one pen

solution

 

2. By using the ratios a1/a2 , b1/b2, c1/c2, find out if the pair of lines are 

intersecting at a point, or are parallel or are coincident

5x-4y+8=0

7x+6y-9=0

solution

ii) 

9x + 3y + 12 = 0
18x + 6y + 24 = 0

solution

 

(iii) 

6x – 3y + 10 = 0
2x – y + 9 = 0

solution 

 

5.

 Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.

 solution

 

7.

 Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region and find the area of the region.

solution

 
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There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

 

find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: 9x + 3y + 12 = 0 18x + 6y + 24 = 0

 cbse 10th mathematics chapter 3 exercise 3.2 pair of linear equations in two variables

 

find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:

ii) 

9x + 3y + 12 = 0
18x + 6y + 24 = 0

a1=9 , b1=3, c1=12

a2=18,b2=6,c2=24

a1/a2 =9/18 = (1/2) 

b1/b2 = 3/6 =(1/2)

c1/c2 = 12/24 = (1/2)

therefore

a1/a2 =b1/b2 =c1/c2

the lines are coincident.

 

 

(iii) 

6x – 3y + 10 = 0
2x – y + 9 = 0

 

a1=6 , b1=(-3), c1=10

a2=2,b2=(-1),c2=9

 

a1/a2 =6/2 =3

b1/b2 =(-3) /(-1) =3

c1/c2 = 10/9

a1/a2 =b1/b2 =/=(not equal to) c1/c2

 

parallel lines 

=================================================

ncert cbse 10th mathematics chapter 3 optional exercise 3.7 

 The ages of two friends Ani and Biju differ by 3 years. Ani’s father  is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Ani’s father differ by 30 years. Find the ages of Ani and Biju

solution

 

2. One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “If you give me ten, I shall be six times as rich as you”. Tell me what is the amount of their (respective) capital? 

solution

3. A train covered a certain distance at a uniform speed. If the train would have been 10 km/h faster, it would have taken 2 hours less than the scheduled time. And, if the train were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. Find the distance covered by the train. 

solution  

 

4. The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class.

solution  

5. In a ∆ ABC, ∠ C = 3 ∠ B = 2 (∠ A + ∠ B). Find the three angles.

solution

 

Solve the following pair of linear equations:

 px + qy = p – q 

 qx – py = p + q

solution

 

(ii) ax + by = c
     bx + ay = 1 + c

solution  

 

(iii) 

(x/a) -(y/b) = 0

ax +by = (a^2)  + (b^2)

solution  

 

(iv)

(a – b)x + (a + b) y = (a^2) – 2ab – (b^2)

(a + b)(x + y) = (a^2) + (b^2 )

solution

 

(v)

152x – 378y = – 74

–378x + 152y = – 604

solution   

 ABCD is a cyclic quadrilateral  Find the angles of the cyclic quadrilateral,

if angles are A =(4y+20) , B =(3y-5) , C=(-4x), D=(-7x+5)

solution

 

exercise 3.6

2

Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current. 

solution

 

 (ii) 2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to
finish the work, and also that taken by 1 man alone.

solution

 

(iii) 

Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and the remaining by bus. If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.

 

solution

 

exercise 3.5

4

A part of monthly hostel charges is fixed and the remaining depends on the
number of days one has taken food in the mess. When a student A takes food for
20 days she has to pay Rs.1000 as hostel charges whereas a student B, who takes
food for 26 days, pays Rs.1180 as hostel charges. Find the fixed charges and the
cost of food per day.

solution

 

(ii) A fraction changes to (1/3) when 1 is subtracted from the numerator and it changes to (1/4) when 8 is added to its denominator. Find the fraction. 

 solution

(iii)Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

solution 

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds,they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

solution 

 (v) The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle

solution

3. Solve the following pair of linear equations by the substitution and cross-multiplication methods :
 

8x + 5y = 9
3x + 2y = 4

solution

 

2

(i) For which values of a and b does the following pair of linear equations have an
infinite number of solutions?
2x + 3y = 7
(a – b) x + (a + b) y = 3a + b – 2

solution 

 

(ii) For which value of k will the following pair of linear equations have no solution?
 

3x + y = 1

(2k – 1) x + (k – 1) y = 2k + 1

  solution

 

Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using cross multiplication method.

x – 3y – 3 = 0
3x – 9y – 2 = 0

solution 

(ii) 

2x + y = 5
3x + 2y = 8

solution 

 

 

(iii) 

3x – 5y = 20
6x – 10y = 40

solution 

 

(iv) 

x – 3y – 7 = 0
3x – 3y – 15 = 0

 solution

exercise 3.4

2. Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method :

(v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs.27 for a book kept for seven days, while Susy paid Rs. 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day. 

solution

(iv) Meena went to a bank to withdraw Rs.2000. She asked the cashier to give her
Rs. 50 and Rs.100 notes only. Meena got 25 notes in all. Find how many notes of
Rs.50 and Rs.100 she received.

solution

 

(iii) 

(iii)The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number 

solution

 (ii) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?

solution

 

If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes (1/2) if we only add 1 to the denominator. What is the fraction?

solution  

 

Solve the following pair of linear equations by the elimination method and the substitution method : 

 x + y = 5 and 2x – 3y = 4

solution  

(ii) 3x + 4y = 10 and 2x – 2y = 2 

 

solution

 (iii) 3x – 5y – 4 = 0 and 9x = 2y + 7

  solution

 

 

iv)

(x/2)+(2y/3)=(-1)

 x -(y/3)=3

solution

 

exercise 3.3

solve by method of substitution 

(ii) 

s-t =3

(s/3)+(t/2)=6

solution

(iii)

 3x – y = 3

9x – 3y = 9

solution  

(iv) 

0.2x + 0.3y = 1.3
0.4x + 0.5y = 2.3 

solution

(v)

 sqrt(2)x +sqrt(3)y =0

sqrt(3)x -sqrt(8)y =0

solution

 

(vi)

(3x/2)-(5y/3)=(-2)

(x/3)+(y/2)=(13/6)

solution

 

3. Form the pair of linear equations for the following problems and find their solution by substitution method 

 The difference between two numbers is 26 and one number is three times the other. Find them.

solution

(ii) The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.

solution 

(iii) The coach of a cricket team buys 7 bats and 6 balls for Rs. 3800. Later, she buys 3 bats and 5 balls for Rs.1750. Find the cost of each bat and each ball.

 solution 

 

(iv) The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is Rs. 105 and for a journey of 15 km, the charge paid is Rs.155. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25 km?

solution

v) A fraction becomes (9/11) if 2 is added to both the numerator and the denominator.  If 3 is added to both the numerator and the denominator it becomes (5/6). Find the fraction.

solution  

 

(vi) Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his son. What are their present ages?

solution

 

exercise 3.2

form the equations and solve by graphical method

  10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.

solution

 

(ii) 5 pencils and 7 pens together cost Rs.50, whereas 7 pencils and 5 pens together cost  Rs.46. Find the cost of one pencil and that of one pen

solution

 

2. By using the ratios a1/a2 , b1/b2, c1/c2, find out if the pair of lines are 

intersecting at a point, or are parallel or are coincident

5x-4y+8=0

7x+6y-9=0

solution

ii) 

9x + 3y + 12 = 0
18x + 6y + 24 = 0

solution

 

(iii) 

6x – 3y + 10 = 0
2x – y + 9 = 0

solution 

 

5.

 Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.

 solution

 

 
disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means