1. What least number must be subtracted from 7250 to get a perfect square? Also, find the square root of this perfect square

2. What is the least number by which 12348must be divided to obtain a perfect square?

3. Find the cost of erecting a fence around a square field whose area is 9 hectares if fencing costs Rs 3.50 per metre

4. Find the least number of six digits which is a perfect square. Find the square root of this number.

5. Divide

(1) x3 - 1 by x - 1 (2) 7 +15x -13x2 +5x3 by 4 - 3x + x2

6. . x + y + z = 0, prove that x 3+ y3 + z3 = 3xyz. Or, Factorize : (a8 – b8)

7. If ( x2 + 1/X2) = 83 . Find X3- 1/X3

Or, , Factorize (i) 25a² – 4b² + 28bc – 49c² (ii) 5y² – 20y – 8z + 2yz

8. A motor boat covers a certain distance downstream in a river in 5 hours. It covers the same distance upstream in 6 hours. The speed of water is 2 km/hr . Find the speed of the boat in still water.

9. Three prizes are to be distributed in a quiz contest. The value of the second prize is five sixths the value of the first prize and the value of the third prize is four – fifths that of the second prize. If the total value of three prizes is Rs. 150, find the value of each prize.

10. (a) Each side of a triangle is increased by 10 cm. If the ratio of the perimeters of the new triangle and the given triangle is 5 : 4, find the perimeter of the given triangle

(b) . The difference between two positive integers is 36. The quotient, when one integer is divided by the other is 4. Find the two integers.

11. Factorize

(1) x4 – (y + z)4 (2) y2 –7y +12 (3) 6xy – 4y + 6 – 9x (4) a4 – 2a²b² + b4 (5) (x² – 2xy + y²) – z²

(1) x4 – (y + z)4 (2) y2 –7y +12 (3) 6xy – 4y + 6 – 9x (4) a4 – 2a²b² + b4 (5) (x² – 2xy + y²) – z²

**8th Factorization**

1. Factorization

(1) (a + b ) (1 – c ) – (b + c ) ( 1 – c ) (2) 1 6 a2 + 40 a b + 25 b2 (3) 4x2/9 - 2/3 x y + y2 /4

(4) 5x2yz - 5 x3y (5) 18 q2 + 338 p2 - 1 5 6 p q (6) -108 x2 - 363 y2 + 369 x y

2. Factorize

(1) 16- 4x2 (2) 20 x3 – 45 b4x (3) 4a2 – 9 b2 – c2 - 6bc

4) 25 ( x + 2y )2 - 36 (2x-5y)2 (5) a2 + 2 a b + b2 – c2 -2cd –d2

3. Factorize using a2+b2+c2 +2ab+2bc+2ca

(1) x2 + y2 + 25 z2 – 2 x y – 10 y z + 10 z x (2) 9x2 + 4y2 + 49z2 – 12 x y + 28 y z – 42 z x

(3) 4x6 + 9y6 + 16 x 6 + 12 x3 y3 + 16 x3 z3 + 24 y3z3 (4) a8 + 256 b8 + 96 a4b4-16a3b2 – 256a2b6

4. Factorize (x + a) (x + b) = x2 + (a + b) x +a b

(1) x2+7x+ 10 (2) x2+x-20 (3) x2-4x-21 (4) 15x2 + 13x + 2 (5) -6x2 - 13x+5

5. Factorize

(1) 125 a3 + 150 a2b + 60 ab3 + 8ab3 (2) x6 – 12 x4 b4 c + 6a2b5c2 + b6c3 (3) 81a3 + 24b3

(4) 64a3b2 – 125 b5 (5) 16 a3 – 54 b3 (6) 8X + 1

(7) a3 - 27b3 (8) 729a6 - 1 (9) 8m3 + 64 (10) 1000 – 343 a9

6. Find the following products:

(1) (9m + 2m )( 81m2 -18mn + 4n2) (2) (5 - 2x ) (25 +10x + 4x2) (3) (3 + 5/x ) ( 9 – 15/x + 25/x2)

7. Find the value of 27x2 + 64y2 + 36xy(3x + 4y) , when x = 5 and y = -3.

8. Using the identity (x + a) (x + b) = x2 + (a + b)x + a b, evaluate 98 x 97

9. x + y + z = 0, prove that x 3+ y3 + z3 = 3xyz.

10. Factorize

(1) m4 – 256 (2) y2 –7y +12 (3) 6xy – 4y + 6 – 9x

(4) x4 – (y + z)4 (5) a4 – 2a²b² + b4 (6) (l + m) ² – 4lm

(7) (x² – 2xy + y²) – z² (8) 25a² – 4b² + 28bc – 49c² (9) 5y² – 20y – 8z + 2yz

(10) a8 – b8

(1) (a + b ) (1 – c ) – (b + c ) ( 1 – c ) (2) 1 6 a2 + 40 a b + 25 b2 (3) 4x2/9 - 2/3 x y + y2 /4

(4) 5x2yz - 5 x3y (5) 18 q2 + 338 p2 - 1 5 6 p q (6) -108 x2 - 363 y2 + 369 x y

2. Factorize

(1) 16- 4x2 (2) 20 x3 – 45 b4x (3) 4a2 – 9 b2 – c2 - 6bc

4) 25 ( x + 2y )2 - 36 (2x-5y)2 (5) a2 + 2 a b + b2 – c2 -2cd –d2

3. Factorize using a2+b2+c2 +2ab+2bc+2ca

(1) x2 + y2 + 25 z2 – 2 x y – 10 y z + 10 z x (2) 9x2 + 4y2 + 49z2 – 12 x y + 28 y z – 42 z x

(3) 4x6 + 9y6 + 16 x 6 + 12 x3 y3 + 16 x3 z3 + 24 y3z3 (4) a8 + 256 b8 + 96 a4b4-16a3b2 – 256a2b6

4. Factorize (x + a) (x + b) = x2 + (a + b) x +a b

(1) x2+7x+ 10 (2) x2+x-20 (3) x2-4x-21 (4) 15x2 + 13x + 2 (5) -6x2 - 13x+5

5. Factorize

(1) 125 a3 + 150 a2b + 60 ab3 + 8ab3 (2) x6 – 12 x4 b4 c + 6a2b5c2 + b6c3 (3) 81a3 + 24b3

(4) 64a3b2 – 125 b5 (5) 16 a3 – 54 b3 (6) 8X + 1

(7) a3 - 27b3 (8) 729a6 - 1 (9) 8m3 + 64 (10) 1000 – 343 a9

6. Find the following products:

(1) (9m + 2m )( 81m2 -18mn + 4n2) (2) (5 - 2x ) (25 +10x + 4x2) (3) (3 + 5/x ) ( 9 – 15/x + 25/x2)

7. Find the value of 27x2 + 64y2 + 36xy(3x + 4y) , when x = 5 and y = -3.

8. Using the identity (x + a) (x + b) = x2 + (a + b)x + a b, evaluate 98 x 97

9. x + y + z = 0, prove that x 3+ y3 + z3 = 3xyz.

10. Factorize

(1) m4 – 256 (2) y2 –7y +12 (3) 6xy – 4y + 6 – 9x

(4) x4 – (y + z)4 (5) a4 – 2a²b² + b4 (6) (l + m) ² – 4lm

(7) (x² – 2xy + y²) – z² (8) 25a² – 4b² + 28bc – 49c² (9) 5y² – 20y – 8z + 2yz

(10) a8 – b8

8th Linear Equations In One and two Variable

1. The perimeter of a rectangular swimming pool is 154 metres. Its length is 2m more than twice its breadth. What are the length and breadth of the pool.

2. Sum of two numbers is 95. If one exceeds the other by 15 find the numbers.

3. Two numbers are in the ration 5:3. If they differ by 18, find these numbers

4. Three consecutive integers add up to 51. What are these integers?

5. The sum of three consecutive multiples of 8 is 888. Find the multiple.

6. Three consecutive integers are as such when they are taken in increasing order and multiplied by 2, 3, and 4 respectively, they add up to 74. Find these numbers.

7. The number of boys and girls in a class is in 7:5 ratio. The number of boys is 8 more than that of girls. Fin their numbers.

8. The ages of Rahul and Haroon are in the ratio of 5:7. Four years from now sum of their ages will be 56 years. Find their present age.

9. Baichung’s father is 26 years younger than Baichung’s grandfather and 29 years older than Baichung. The sum of their ages is 135. Find their ages.

10. Fifteen years from now Ravi’s age will be 4 times his current age. What is his current age.

11. Lakshmi is a cashier in a bank. She has notes of denominations of Rs. 100, 50 and 10 respectively. The ratio of number of these notes is 2:3:5 respectively. The total cash with Lakshmi is 4,00,000. How many notes of each denomination does she have?

12. I have total Rs 300 in coins of denominations of Rs.1, Rs.2, and Rs. 5.The number of Rs. 2 coins is 3 times the number of Rs. 5 coins. The total number of coins is 160. How many coins of each denomination are with me.

13. The organizers in an essay competition decide that winner will get a prize of Rs. 100 and a participation who doesn’t win gets a prize of Rs. 25. The total prize money distributed is Rs. 3,000. Find the number of winners if the total number of participants is 63.

14. If in a rational number denominator is greater than numerator by 8. If you increase the numerator by 17 and decrease the denominator by 1, you get 3/2 as result. Find the number.

15. Amina thinks of a number and subtracts 5/2 from it. She multiplies the result by 8. The final result is 3 times her original number. Find the number

16. A positive number is 5 times another number. If 21 is added to both the numbers then one of the new numbers becomes twice of another new numbers. Find the original numbers.

17. Sum of the digits of a two digit number is 9. When we interchange the digits the new number is 27 greater than the earlier number. Find the number.

18. One of the digits of a two digit number is three times the other digit. If you interchange the digits and add the resulting number to original number you get 88 as final result. Find the numbers.

19. Sahoo’s mother’s present age is six times Sahoo’s present age. Five year from now Sahoo’s age will be one-third of his mother’s age. Find their current age.

20. There is a narrow rectangular plot. The length and breadth of the plot are in the ratio of 11:4. At the rate of Rs. 100 per metre it will cost village panchayat Rs.75000 to fence the plot. What are the dimensions of the plot.

21. Hasan buys two kinds of cloth materials for school uniform. Shirt material cost him Rs. 50 per metre and trousers material cost him Rs. 90 per metre. For every 2 metres of the trousers material he buys 3 metres of shirt material. He sells them at 12% and 10% profit respectively. His total sale is Rs. 36,660. How much trousers material did he buy? ( 200m)

22. Half of a herd of deer are grazing in the field and three fourths of the remaining are playing nearby. The rest 9 are drinking water from the pond. Find the total number of deer in the herd.

23. A grandfather is 10 times older than his granddaughter. He is also 54 years older than her. Find their age.

24. A man’s age is three times his son’s age. Ten years ago his age was five times his son’s age. Find their current age.

25. Hari and Harry’s age are in the ratio of 5:7. Four years later the ratio of their ages will be 3:4. Find their current age.

8th Algebraic Expression

1. If ( x + 1 / x ) = 4, Find the value of ( x2 + 1/x2 ) and (x4 + 1/x4 )

2. If ( x - 1/ x ) = 3 .Find the value of (x3 + 1/x3 )

3. Find the remainder obtained by dividing x3 + 3 x2 - 5x + 4 by x + 1

4. Evaluate using algebraic identities (54)2 ; (78)2; (999)2

5. If x - y = 7, x y = 9 Find the value of x2 + y2

6. If x + y = 12 , x y = 27 Find the value of x3 + y3

7. If a2 + b2 + c2 =20 , a + b + c = 6 find a b + b c + ca

8. If ( x2 + 1 / x2 ) = 83 . Find (x3 - 1 / x3 )

9. What must be subtracted from 4p2 - 2pq - 6q2 - r +5 to get - p2 + p q - 8q2 - 2r+5

10. Factorise a3 + b3 + c3 - 3 a b c

11. Devide

(1) x3 - 1 by x - 1

(2) 7 +15x -13x2 +5x3 by 4 - 3x + x2

12. Evaluate

(1) 1.5 3 - 0.93 - 0.63

(2) (a - b) 3 + (a + b) 3

(3) (x + 2y -3z)2 + (x - 2y +3z)2

13.If (x4 + 1 / x4 ) = 47 find the value of (x3 + 1 / x3 )

14. Find the product of

(1) (x4 + 1/x4 ) and ( x + 1/x )

(2) (2x2 + 3x - 7)(3x2 -5x - 4)

15.Two adjacent side of a rectangle are 5x2-3y2 and x2 - 2xy Find its perimeter

16.The perimeter of a triangle is 6p2 - 4p + 9 and two of its adjacent side are

p2 - 2p + 1 and 3 p2 - 5p + 3. Find third side of triangle.

17. Find the least no. of 5 digits which is perfect square.

18. Find the greatest num. of 6 digits which is perfect square.

19. Evaluate

(1) (5-1 x 3-1 )-1 x 6-1 (2) ( 23x+1 +10 ) / 7 = 6

(3) [52x+1]/ 25 = 125 (4) (4/9)4 x (4/9) - 7 = (4/9) 2x – 1

Compound Interest:

1. You
invest Rs 5000 at 12% interest compounded annually. How much is in the account
after 2 years, assuming that you make no subsequent withdrawal or
deposit?

2. Find
the amount and the compound interest on Rs 4000 at 10% p.a. for 2½ years.

3. A
man invests Rs 5000 for three years at a certain rate of interest, compounded
annually. At the end of one year it amounts to Rs 5600. Calculate (i) the rate
of interest per annum, (ii) the interest accrued in the second year, (iii) the
amount at the end of the third year.

4. sum
of Rs 9600 is invested for 3 years at 10% per annum at compound interest. (i)
What is the sum due at the end of the first year? (ii) What is the sum due at
the end of the second year? (iii) Find the compound interest earned in two
years. (iv) Find the difference between the answers (ii) and (i) and find the
interest on this sum for one year. (v) Hence write down the compound interest
for the third year.

5. Find
the difference between the S.I. and C.I. on Rs 2500 for 2 years at 4% p.a.,
compound interest reckoned semi-annually.

6. Find
the amount and the compound interest on Rs 8000 in 2 years if the rate is 10%
for the first year and 12% for the second year.

7. A
man invests Rs 6500 for 3 years at 4·5% p.a. compound interest reckoned yearly.
Income tax at the rate of 20% is deducted at the end of each year. Find the
amount at the end of the third year.

8. Calculate
the compound interest for the second year on Rs 8000 invested for 3 years at
10% p.a.

9. Find
the sum which amounts to Rs 9261 at 10% p.a. compound interest for 18 months,
interest payable half-yearly.

10. On
what sum will the compound interest for 2 years at 5% p.a. be Rs 246?

11. On
what sum will the compound interest (reckoned yearly) for 3 years at 6¼% per
annum be Rs 408·50?

12. A
man invests Rs 1200 for two years at compound interest. After one year his
money amounts to Rs 1275. Find the rate of compound interest. Also find the
amount which the man will get after 2 years correct to the nearest paise.

13. At
what rate percent per annum compound interest will Rs 2000 amount to Rs 2315·25
in 3 years?

14. If
Rs 50000 amounts to Rs 73205 in 4 years, find the rate of compound interest
payable yearly.

15. In
what time will Rs 15625 amount to Rs 17576 at 4% per annum compound
interest?

16. In
what time will a sum of Rs 2500 produce Rs 309 at 6% per annum compound
interest?

17. In
what time will a sum of Rs 800 at 10% per annum compounded half-yearly produce
Rs 126·10?

18. The
simple interest on a sum of money for 2 years at 4% p.a. is Rs 450. Find the
compound interest on this sum of money at the same rate (i) for 1 year if the
interest is reckoned semi-annually. (ii) for 2 years if the interest is
reckoned annually.

19. At
what rate of compound interest will Rs 625 amount to Rs 729 after 2 years? Also
find the maturity value of Rs 625 after 2 years at the above rate of simple
interest.

20. Ram
and Bhola each borrow equal sums for 3 years at 10% p.a. simple interest and
compound interest respectively. At the time of repayment, Bhola has to pay Rs
372 more than Ram. Find the sum borrowed and interest paid by each.

21. The
difference between the compound interest for a year payable half-yearly and the
simple interest on a certain sum of money lent out at 10% p.a. for a year is Rs
15. Find the sum of money lent out.

22. The
difference between compound interest and simple interest in 3 years at 10% p.a.
reckoned yearly is Rs 18·60. Find the sum and the compound interest.

23. The
amount at compound interest which is calculated yearly on a certain sum of
money is Rs 1250 in one year and Rs 1375 in two years. Calculate the rate of
interest.

24. A
certain sum of money amounts to Rs 10584 in two years and to Rs 11113·20 in
three years, interest being compounded annually. Find the interest rate percent
and the original sum.

25. The
compound interest and the simple interest on the same sum of money at the same
rate percent per annum are Rs 410 and Rs 400 respectively. Find the rate of
interest and the sum of money.

26. The
compound interest calculated yearly on a certain sum of money for the second
year is Rs 880 and for the third year it is Rs 968. Find the rate of interest
and the original money.

27. The
simple interest on a certain sum for 3 years is Rs 150 and the compound
interest on the same sum at the same rate for 2 years is Rs 110. Find the rate
of interest and the principal.

28. A
sum of money lent at C.I. on 1st April 96 amounts to Rs 2420 on 1st April 98
and to Rs 2662 on 1st April 99. Find the rate of interest and the sum.

29. The
simple interest in 3 years and the compound interest in 2 years on a certain
sum at the same rate are Rs 1200 and Rs 832 respectively. Find (i) the rate of
interest, (ii) the principal, (iii) the difference between C.I. and S.I. for
three years.

30. The
value of a machine depreciates every year at the rate of 10% of its value. The
machine was purchased for Rs 40000 when new and it was sold for Rs 29160. Find
the number of years that the machine was used.

31. A
man borrowed a sum of money and agrees to pay off by paying Rs 3150 at the end
of the first year and Rs 4410 at the end of the second year. If the rate of
compound interest is 5% p.a., find the sum borrowed.

32. A
man borrowed a certain sum of money and paid it back in 2 years in two equal
instalments. If the rate of compound interest was 4% p.a. and if he paid Rs 676
annually, what sum did he borrow?

33. A
sum of Rs 16400 is borrowed to be paid back in 2 years by two equal annual
instalments allowing 5% compound interest. Find the annual payment.

34. A
loan of Rs 4641 is to be paid back by 4 equal annual instalments. The interest
is compounded yearly at 10%. Find the value of each instalment.

35. A
man borrows Rs 6000 at 5% p.a. compound interest. If he repays Rs 1200 at the
end of each year, find the amount outstanding at the beginning of the third
year.

Answers

1. Rs 6272 2. Rs 5082; Rs 1082 3. (i) 12% (ii) Rs 672 (iii) Rs 6952·64

4. (i) Rs 10560 (ii) Rs 11616 (iii) Rs 2016 (iv) Rs 1056, Rs 105·60 (v) Rs 1161·60

5. Rs 6·08 6. Rs 9856; Rs 1856 7. Rs 7227·56

8. Rs 880 9. Rs 8000 10. Rs 2400 11. Rs 2048

12. 6¼%; Rs 1354·69 13. 5% 14. 10%

15. 3 years 16. 2 years 1 7. 1½ years

18. (i) Rs 227·25 (ii) Rs 459 19. 8%; Rs 725 20. Rs 12000; Rs 3600, Rs 3972

21. Rs 6000 22. Rs 600; Rs 196·60 23. 10% 24. 5% p.a., Rs 9600 25. 5% p.a., Rs 4000

26. 10%, Rs 8000 27. 20%; Rs 250 28. 10%, Rs 2000 29. (i) 8% (ii) Rs 5000 (iii) Rs 98·56

30. 3 years 31. Rs 7000 32. Rs 1275 33. Rs 8820 34. Rs 1464·10 35. Rs 4155

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