Subba Rao started work in 1995 at an annual salary of Rs. 5000 and received an increment of Rs.200 each year. In which year did his income reach Rs.7000

 

cbse ncert 10th mathematics

 chapter 5 arithmetic progressions, exercise 5.2

 

19. Subba Rao started work in 1995 at an annual salary of Rs. 5000 and received an increment of Rs.200 each year. In which year did his income reach Rs.7000?

 

first term a= 5000

d =200

tn=7000

n= ?

using the formula for the nth term of an arithmetic progression AP

tn= a+ (n-1)d

7000 = 5000 +(n-1)(200)

7000-5000 =(n-1)(200)

 

2000 =(n-1)(200) 

 

(n-1) = 2000/200

n-1 =10

 

n=10+1

n=11 

18. The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP.

given

t4 +t8 =24

t6+t10 =44

 

using the formula for the nth term of an arithmetic progression AP

tn= a+ (n-1)d

replace n with 4,8,6,10

t4=a+(4-1)d =a+3d

t8=a+7d

t6=a+5d

t10=a+9d

t4 +t8 =24 gives

[a+3d] +[a+7d] =24

2a+10d =24-------------------------(1)

t6+t10 =44 gives

[a+5d]+[a+9d] =44

2a+14d=44 ---------------------------------(2)

2a+10d =24-------------------------(1)

2a+14d=44 ---------------------------------(2)

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

      (-4)d =(-20)

d =(-20)/(-4)

d=5

substitute in (1)

2a +10(5)=24

2a = 24-50

2a =(-26)

a=(-26)/2

a=(-13)

first three terms of the AP are a, (a+d), (a+2d)

= (-13),[(-13)+5],[(-13+2(5)] =(-13),(-8),(-3)

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

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

11.If the sum of the first n terms of an AP is 4n –(n^2) , what is the first term (that is S1 )? What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms.

solution

10.  Show that a1 , a2 , . . ., an , . . . form an AP where a n is defined as below :
 an = 3 + 4n
(ii) an = 9 – 5n
Also find the sum of the first 15 terms in each case.

solution

 

9. If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.

  solution

 8. Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.

solution

 7. Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.

 solution

6. The first and the last terms of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?

solution

5. The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.

solution 

 

4. How many terms of the AP : 9, 17, 25, . . . must be taken to give a sum of 636?

solution 

 

 3.

given a = 5, d = 3, an = 50, find n and Sn

solution

 (ii) given a = 7, a13 = 35, find d and  S13

solution

(iii) given a(12) = 37, d = 3, find a and S(12 )

solution

(iv) given a3 = 15, S(10) = 125, find d and a(10)

 solution 

(v) given d = 5, S9 = 75, find a and a9 .

solution 

(vi) given a = 2, d = 8, Sn = 90, find n and an .

solution 

 vii) given a = 8, an = 62, Sn = 210, find n and d

solution

 (vii) given an = 4, d = 2, Sn = –14, find n and a.

solution 

 ix) given a = 3, n = 8, S = 192, find d.

solution

(x) given L= 28, S = 144, and there are total 9 terms. Find a.

solution

find the sums given below :

7 + [10 +(1/2) ] +14 + ...+84

solution

(ii) 34 + 32 + 30 + . . . + 10

 solution

(iii) –5 + (–8) + (–11) + . . . + (–230)

solution  

 Find the sum of the following APs:
 2, 7, 12, . . ., to 10 terms.

solution   

 (ii) –37, –33, –29, . . ., to 12 terms.

solution 

(iii) 0.6, 1.7, 2.8, . . ., to 100 terms

solution  

 (iv) (1/15) +(1/12) +(1/10) + .... 11terms

solution

 

Aruna saved Rs. 5 in the first week of a year and then increased her weekly savings by Rs. 1.75. If in the nth week, her weekly savings become Rs.20.75, find n.

solution

exercise 5.2

19. Subba Rao started work in 1995 at an annual salary of Rs. 5000 and received an increment of Rs.200 each year. In which year did his income reach Rs.7000?

solution  

 

18. The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP.

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

five more than three times a number is 35. which is that number

 word problems involving a single unknown

five more than three times a number is 35. which is that number

 

let the number be x

 

three times a number means 3x

 

five more than three times a number gives ( 3x +5)

 

so  

( 3x +5) = 35

3x =35-5

3x =30

x=30/3

x=10

*

Two less than half a number is 23. Which is that number

let the number be x

 

half the number means(x/2)

 

 Two less than half the number gives [ (x/2) -2 ]

so

[ (x/2) -2 ] =23

(x/2) = 23+2

(x/2) =25

x =25*2

x=50

 

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

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

11.If the sum of the first n terms of an AP is 4n –(n^2) , what is the first term (that is S1 )? What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms.

solution

10.  Show that a1 , a2 , . . ., an , . . . form an AP where a n is defined as below :
 an = 3 + 4n
(ii) an = 9 – 5n
Also find the sum of the first 15 terms in each case.

solution

 

9. If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.

  solution

 8. Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.

solution

 7. Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.

 solution

6. The first and the last terms of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?

solution

5. The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.

solution 

 

4. How many terms of the AP : 9, 17, 25, . . . must be taken to give a sum of 636?

solution 

 

 3.

given a = 5, d = 3, an = 50, find n and Sn

solution

 (ii) given a = 7, a13 = 35, find d and  S13

solution

(iii) given a(12) = 37, d = 3, find a and S(12 )

solution

(iv) given a3 = 15, S(10) = 125, find d and a(10)

 solution 

(v) given d = 5, S9 = 75, find a and a9 .

solution 

(vi) given a = 2, d = 8, Sn = 90, find n and an .

solution 

 vii) given a = 8, an = 62, Sn = 210, find n and d

solution

 (vii) given an = 4, d = 2, Sn = –14, find n and a.

solution 

 ix) given a = 3, n = 8, S = 192, find d.

solution

(x) given L= 28, S = 144, and there are total 9 terms. Find a.

solution

find the sums given below :

7 + [10 +(1/2) ] +14 + ...+84

solution

(ii) 34 + 32 + 30 + . . . + 10

 solution

(iii) –5 + (–8) + (–11) + . . . + (–230)

solution  

 Find the sum of the following APs:
 2, 7, 12, . . ., to 10 terms.

solution   

 (ii) –37, –33, –29, . . ., to 12 terms.

solution 

(iii) 0.6, 1.7, 2.8, . . ., to 100 terms

solution  

 (iv) (1/15) +(1/12) +(1/10) + .... 11terms

solution

 

Aruna saved Rs. 5 in the first week of a year and then increased her weekly savings by Rs. 1.75. If in the nth week, her weekly savings become Rs.20.75, find n.

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

Find the sum of the following AP (iv) (1/15) +(1/12) +(1/10) + .... 11terms

 

cbse ncert 10th mathematics

 chapter 5 arithmetic progressions, exercise 5.3

 

Find the sum of the following AP

(iv) (1/15) +(1/12) +(1/10) + .... 11terms

first term a = (1/15)

difference d = t2 - t1 = (1/12) - (1/15)=(1/60)

 

n = 11terms

 

using the formula for sum of n terms of an arithmetic progression ( AP )

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

 

Sn = (11/2)*[2(1/15) + (11-1)(1/60) ]

Sn = (11/2)*[ (2/15) +(10/60)]

Sn =(11/2)*[3/10]

Sn= [33/20]

Aruna saved Rs. 5 in the first week of a year and then increased her weekly savings by Rs. 1.75. If in the nth week, her weekly savings become Rs.20.75, find n.

first term a = 5

d= 1.75

nth term tn=20.75

n=?

using the formula for the nth term of an AP

tn =a+(n-1)d

20.75 = 5 +(n-1)(1.75)

20.75 - 5 =(n-1)(1.75)

15.75 =(n-1)(1.75)

 

(n-1)  = 15.75 / 1.75

(n-1) =9

n=9+1

n=10

 

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

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

11.If the sum of the first n terms of an AP is 4n –(n^2) , what is the first term (that is S1 )? What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms.

solution

10.  Show that a1 , a2 , . . ., an , . . . form an AP where a n is defined as below :
 an = 3 + 4n
(ii) an = 9 – 5n
Also find the sum of the first 15 terms in each case.

solution

 

9. If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.

  solution

 8. Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.

solution

 7. Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.

 solution

6. The first and the last terms of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?

solution

5. The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.

solution 

 

4. How many terms of the AP : 9, 17, 25, . . . must be taken to give a sum of 636?

solution 

 

 3.

given a = 5, d = 3, an = 50, find n and Sn

solution

 (ii) given a = 7, a13 = 35, find d and  S13

solution

(iii) given a(12) = 37, d = 3, find a and S(12 )

solution

(iv) given a3 = 15, S(10) = 125, find d and a(10)

 solution 

(v) given d = 5, S9 = 75, find a and a9 .

solution 

(vi) given a = 2, d = 8, Sn = 90, find n and an .

solution 

 vii) given a = 8, an = 62, Sn = 210, find n and d

solution

 (vii) given an = 4, d = 2, Sn = –14, find n and a.

solution 

 ix) given a = 3, n = 8, S = 192, find d.

solution

(x) given L= 28, S = 144, and there are total 9 terms. Find a.

solution

find the sums given below :

7 + [10 +(1/2) ] +14 + ...+84

solution

(ii) 34 + 32 + 30 + . . . + 10

 solution

(iii) –5 + (–8) + (–11) + . . . + (–230)

solution  

 Find the sum of the following APs:
 2, 7, 12, . . ., to 10 terms.

solution   

 (ii) –37, –33, –29, . . ., to 12 terms.

solution 

(iii) 0.6, 1.7, 2.8, . . ., to 100 terms

solution  

 (iv) (1/15) +(1/12) +(1/10) + .... 11terms

solution

 

Aruna saved Rs. 5 in the first week of a year and then increased her weekly savings by Rs. 1.75. If in the nth week, her weekly savings become Rs.20.75, find n.

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

quadratic formula

 Quadratic formula

when solving ax^2 +bx +c = 0

x = { (-b) + sqrt[(b^2) - 4ac] } / {2a}

or

x = { (-b) - sqrt[(b^2) - 4ac] } / {2a}

 

 

 ----------------------------------------------------------------------------

If the equations (x^2) -ax+b=0 and (x^2)-ex+f =0 have a root in common and the second equation has equal roots show that ae =2(b+f)

solution 

The area of a right angle triangle is 63 sq.cm. The length of the base of the triangle is 5 cm. more than the altitute of the triangle. Find the length of the altitute of the triangle.

 solution

 

solve (x^2)- 2x - 8 = 0 by graphing

solution

 

solve 2x² -4x -7 = 0 by finishing the square

 

solution

 

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

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

11.If the sum of the first n terms of an AP is 4n –(n^2) , what is the first term (that is S1 )? What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms.

solution

10.  Show that a1 , a2 , . . ., an , . . . form an AP where a n is defined as below :
 an = 3 + 4n
(ii) an = 9 – 5n
Also find the sum of the first 15 terms in each case.

solution

 

9. If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.

  solution

 8. Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.

solution

 7. Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.

 solution

6. The first and the last terms of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?

solution

5. The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.

solution 

 

4. How many terms of the AP : 9, 17, 25, . . . must be taken to give a sum of 636?

solution 

 

 3.

given a = 5, d = 3, an = 50, find n and Sn

solution

 (ii) given a = 7, a13 = 35, find d and  S13

solution

(iii) given a(12) = 37, d = 3, find a and S(12 )

solution

(iv) given a3 = 15, S(10) = 125, find d and a(10)

 solution 

(v) given d = 5, S9 = 75, find a and a9 .

solution 

(vi) given a = 2, d = 8, Sn = 90, find n and an .

solution 

 vii) given a = 8, an = 62, Sn = 210, find n and d

solution

 (vii) given an = 4, d = 2, Sn = –14, find n and a.

solution 

 ix) given a = 3, n = 8, S = 192, find d.

solution

(x) given L= 28, S = 144, and there are total 9 terms. Find a.

solution

find the sums given below :

7 + [10 +(1/2) ] +14 + ...+84

solution

(ii) 34 + 32 + 30 + . . . + 10

 solution

(iii) –5 + (–8) + (–11) + . . . + (–230)

solution  

 Find the sum of the following APs:
 2, 7, 12, . . ., to 10 terms.

solution   

 (ii) –37, –33, –29, . . ., to 12 terms.

solution 

(iii) 0.6, 1.7, 2.8, . . ., to 100 terms

solution  

 (iv) (1/15) +(1/12) +(1/10) + .... 11terms

solution

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solve using elimination method

 solve using elimination method

 

10x +5y =110

5x+8y   =88 

first decide which unknow you are going to eliminate

If we decide to eliminate x

make the coefficient of x same

so multiply the second equation with 2

then subtract to eliminate x and solve for y

resubstitute in any one of the equations for x.

10x +5y =110

[5x+8y   =88 ]*2

 

 10x +5y   =110

10x + 16y =176 

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

      (-11)y =(-66)

          y = (-66)/(-11)

         y =6

resubstitute in 5x+8y   =88

5x +48 =88

5x=88-48

5x=40

x=8 

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

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

 

 
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Find the sum of the following APs: (ii) –37, –33, –29, . . ., to 12 terms.

 

 

cbse ncert 10th mathematics

 chapter 5 arithmetic progressions, exercise 5.3

 

Find the sum of the following APs:

 (ii) –37, –33, –29, . . ., to 12 terms.

 

first term a =(-37)

d =t2 - t1 = (-33) -  (-37)=-33+37 = 4

n = 12 terms

using the formula for sum of n terms of an arithmetic progression ( AP )

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

Sn = [12/2] [2(-37) +(12-1)(4)]

=6*[(-74)+44]=(-180)

(iii) 0.6, 1.7, 2.8, . . ., to 100 terms

first term a =0.6

 d =t2 - t1 =1.7-0.6 =1.1

n=100 terms

using the formula for sum of n terms of an arithmetic progression ( AP )

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

Sn=(100/2)*[2(0.6)+(100-1)(1.1)]

Sn=50*[1.2+ 108.9] =50*110.1 =5505

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

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

11.If the sum of the first n terms of an AP is 4n –(n^2) , what is the first term (that is S1 )? What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms.

solution

10.  Show that a1 , a2 , . . ., an , . . . form an AP where a n is defined as below :
 an = 3 + 4n
(ii) an = 9 – 5n
Also find the sum of the first 15 terms in each case.

solution

 

9. If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.

  solution

 8. Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.

solution

 7. Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.

 solution

6. The first and the last terms of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?

solution

5. The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.

solution 

 

4. How many terms of the AP : 9, 17, 25, . . . must be taken to give a sum of 636?

solution 

 

 3.

given a = 5, d = 3, an = 50, find n and Sn

solution

 (ii) given a = 7, a13 = 35, find d and  S13

solution

(iii) given a(12) = 37, d = 3, find a and S(12 )

solution

(iv) given a3 = 15, S(10) = 125, find d and a(10)

 solution 

(v) given d = 5, S9 = 75, find a and a9 .

solution 

(vi) given a = 2, d = 8, Sn = 90, find n and an .

solution 

 vii) given a = 8, an = 62, Sn = 210, find n and d

solution

 (vii) given an = 4, d = 2, Sn = –14, find n and a.

solution 

 ix) given a = 3, n = 8, S = 192, find d.

solution

(x) given L= 28, S = 144, and there are total 9 terms. Find a.

solution

find the sums given below :

7 + [10 +(1/2) ] +14 + ...+84

solution

(ii) 34 + 32 + 30 + . . . + 10

 solution

(iii) –5 + (–8) + (–11) + . . . + (–230)

solution  

 Find the sum of the following APs:
 2, 7, 12, . . ., to 10 terms.

solution   

 (ii) –37, –33, –29, . . ., to 12 terms.

solution 

(iii) 0.6, 1.7, 2.8, . . ., to 100 terms

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 the equation of line having intercepts 3 and 2 on the x and y axes respectively

 Find the equation of line having intercepts  3 and 2 on the x and y axes respectively

 

using intercept form of straight line with

x-intercept, a = 3

y-intercept, b=2

(x/a) +(y/b)=1

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

mulitply by 6

2x + 3y=6

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

ncert cbse chapter 10 straight lines miscellaneous exercise

24.  A person standing at the junction (crossing) of two straight paths represented by the equations 2x – 3y + 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. Find equation of the path that he should follow.

solution

 

22. A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.

 

solution 

 21. Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 =0

solution

 

19. If the lines y = 3x +1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value of m.

solution

18.Find the image of the point (3, 8) with respect to the line x +3y = 7 assuming the line to be a plane mirror.

solution 

 

17. The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (– 4, 1). Find an equation of the legs (perpendicular sides) of the triangle 

 solution

 14. In what ratio, the line joining (–1, 1) and (5, 7) is divided by the 

line x + y = 4 ?

solution

12.Find the equation of the line passing through the point of intersection of the lines 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axes

solution

 

11. Find the equation of the lines through the point (3, 2) which make an angle of 45 degrees with the line x – 2y = 3.

solution

 

8. Find the area of the triangle formed by the lines y – x = 0, x + y = 0 

and x – k = 0

solution

9. Find the value of p so that the three lines 3x + y – 2 = 0, px + 2 y – 3 = 0 and
2x – y – 3 = 0 may intersect at one point.

 solution

6. Find the equation of the line parallel to y-axis and drawn through the point of
intersection of the lines x – 7y + 5 = 0 and 3x + y = 0.

solution

 

4. What are the points on the y-axis whose distance from the line
[x/3] + [y/4]=1 is 4 units.

solution

 

 3. Find the equations of the lines, which cut-off intercepts on the axes whose sum
and product are 1 and – 6, respectively.

solution

 2. Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line [sqrt(3)] x + y + 2 = 0.

solution 

 

 Find the values of k for which the line 

(k–3) x – (4 –( k^ 2) ) y + (k^2) –7k + 6 = 0 is

(a) Parallel to the x-axis,
(b) Parallel to the y-axis,
(c) Passing through the origin.

 solution  

 

ncert cbse chapter 10 exercise 10.3

  17. In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length of altitude from the vertex A.

solution

 

14. Find the coordinates of the foot of perpendicular from the point (–1, 3) to the
line 3x – 4y – 16 = 0.

solution

 

 13. Find the equation of the right bisector of the line segment joining the points

 (3, 4) and (–1, 2).

solution

 

10. The line through the points (h, 3) and (4, 1) intersects the line 

7 x − 9 y − 19 = 0 at right angle. Find the value of h. 

solution

 

 

8. Find equation of the line perpendicular to the line x – 7y + 5 = 0 and having
x intercept 3.

solution  

 

exercise 10.2

19. Point R (h, k) divides a line segment between the axes in the ratio 1: 2. Find
equation of the line.

solution

17.The owner of a milk store finds that, he can sell 980 litres of milk each week at Rs 14/litre and 1220 litres of milk each week at Rs 16/litre. Assuming a linear
relationship between selling price and demand, how many litres could he sell
weekly at Rs 17/litre?

solution

13. Find equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.  

solution 

 

12.Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3).

 solution

 

11.A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1: n. Find the equation of the line.   

 solution 

10. Find the equation of the line passing through (–3, 5) and perpendicular to the line through the points (2, 5) and (–3, 6). 

solution

 

9. The vertices of ∆ PQR are P (2, 1), Q (–2, 3) and R (4, 5). Find equation of the
median through the vertex R.

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