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MATHSBOOKS - RS AGGARWAL MATHS

(HINGLISH)

OPERATIONS ON ALGEBRAICEXPRESSIONS

Exercise 6A

1. Add:

8ab, - 5ab, 3ab, - ab

Answer: 5ab

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2. Add:

7x, - 3x, 5x, - x, - 2x

Answer: 6x

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3. Add:

3a - 4b + 4c, 2a + 3b - 8c, a - 6b + c

Answer: 6a - 7b - 3c

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4. Add:

5x - 8y + 2z, 3z - 4y - 2x, 6y - z - x and 3x - 2z - 3y

Answer: 5x - 9y + 2z

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5. Add:

6ax - 2by + 3cz, 6by - 11ax - cz and 1 - cz - 2ax - 3by

Answer: -7ax + by + 12cz

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6. Add: 2x3 - 9x2 + 8, 3x2 - 6x - 5, 7x3 - 10x + 1 and

3 + 2x - 5x2 - 4x3

Answer: 5x2 - 11x2 - 14x + 7

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7. Add:

6p + 4q - r + 3, 3r - 5p - 6, 11q - 7p + 2r - 1 and 2q - 3r + 4

Answer: ]-6p + 17q

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8. Add:

4x2 - 7xy + 4y2 - 3, 5 + 6y2 - 8xy + x2 and

6 - 2xy + 2x2 - 5y2

Answer: 7x2 - 17xy + 5y2 + 8

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9. Subtract :

3a2b from -5a2b

Answer: -8a2b

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10. Subtract :

-8pq from 6pq

Answer: 14pq

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11. Subtract :

-2abc from - 8abc

Answer: -6abc

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12. Subtract :

16p from - 11p

Answer: 5p

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13. Subtract :

2a - 5b + 2c - 9 from 3a - 4b - c + 6

Answer: a + b - 3c + 15

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14. Subtract :

-6p + q + 3r + 8 from p - 2q - 5r - 8

Answer: 7p - 3q - 8r - 16

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15. Subtract : x3 + 3x2 - 5x + 4 from 3x3 - x2 + 2x - 4

Answer: 2x3 - 4x2 + 7x - 8

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16. Subtract :

5y4 - 3y3 + 2y2 + y - 1 from 4y4 - 2y3 - 6y2 - y + 5

Answer: -y4 + y3 - 8y2 - 2y + 6

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17. Subtract :

4p2 + 5q2 - 6r2 + 7 from 3p2 - 4q2 - 5r2 - 6

Answer: -p2 - 9q2 + r2 - 13

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18. what must be subtracted from 3a2 - 6ab - 3b2 - 1

to get 4a2 - 7ab - 4b2 + 1?

Answer: -a2 + ab + b2 - 2

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19. The two adjacent sides of rectangle are

5x2 - 3y2 and x2 + 2xy. Find the perimeter

Answer: 12x2 - 6y2 + 4xy

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Exercise 6B

20. The perimeter of a triangle is 6p2 - 4p + 9 and

two of its sides are p2 - 2p + 1 and 3p2 - 5p + 3. Find

the third side of the triangle.

Answer: 2p2 + 3p + 5

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1. Find each of the following products :

(5x + 7) × (3x + 4)

Answer: 15x2 + 41x + 28

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2. Find each of the following products :

(4x + 9) × (x - 6)

Answer: 4x2 - 15x - 54

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3. Find each of the following products :

(2x + 5) × (4x - 3)

Answer: 8x2 + 14x - 15

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4. Find each of the following products :

(3y - 8) × (5y - 1)

Answer: 15y2 - 43y + 8

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5. Find each of the following products :

(7x + 2y) × (x + 4y)

Answer: 7x2 + 8y2 + 30xy

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6. Find each of the following products :

(9x + 5y) × (4x + 3y)

Answer: 36x2 + 15y2 + 47xy

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7. Find each of the following products :

(3m - 4n) × (2m - 3n)

Answer: 6x2 + 12n2 - 17mn

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8. Find each of the following products :

x2 - a2 × (x - a)

Answer: x3 - ax2 - a2x + a3

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9. Find each of the following products :

x2 - y2 × (x + 2y) ( )

Answer: x3 + 2x2y - xy2 - 2y3

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10. Find each of the following products :

3p2 + q2 × 2p2 - 3q2

Answer: 6p4 - 7p2q2 - 3q4

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( ) ( )

11. Find each of the following products :

2x2 - 5y2 × x2 + 3y2

Answer: 2x4 - 15y4 + x2y2

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( ) ( )

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12. Find each of the following products :

x3 - y3 × x2 + y2

Answer: x5 + x3y2 - x2y3 - y5

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( ) ( )

13. Find each of the following products :

x4 + y4 × x2 - y2

Answer: x6 - x4y2 + x2y4 - y6

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( ) ( )

14. x4 +1

x4 × x +1x

Answer: x5 + x3 +1

x3+

1

x5

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( ) ( )

15. Find each of the following products :

x2 - 3x + 7 × (2x + 3)

Answer: 2x3 - 3x2 + 5x + 21

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16. Find each of the following products :

3x2 + 5x - 9 × (3x - 5)

Answer: 9x3 - 52x + 45

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17. Find each of the following products :

x2 - xy + y2 × (x + y)

Answer: x3 + y3

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18. Find each of the following products :

x2 + xy + y2 × (x - y)

Answer: x3 - y3

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( )( )

19. Find each of the following products :

x3 - 2x2 + 5 × (4x - 1)

Answer: 4x4 - 9x3 + 2x2 + 20x - 5

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20. Find each of the following products :

9x2 - x + 15 × x2 - 3

Answer: 9x2 - x3 + 12x2 + 3x - 45

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( ) ( )

21. Find each of the following products :

x2 - 5x + 8 × x2 + 2

Answer: x4 - 5x3 + 10x2 - 10x + 16

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( ) ( )

22. Find each of the following products :

x3 - 5x2 + 3x + 1 × x2 - 3

Answer: x5 - 5x4 + 16x2 - 9x - 3

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( ) ( )

23. Find each of the following products :

(3x + 2y - 4) × (x - y + 2)

Answer: 3x2 - xy + 2x - 2y2 + 8y - 8

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24. Find each of the following products :

x2 - 5x + 8 × x2 + 2x - 3( ) ( )

Answer: x4 - 3x3 - 5x2 + 31x - 24

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25. Find each of the following products :

2x2 + 3x - 7 × 3x2 - 5x + 4

Answer: 6x4 - x3 - 28x2 + 47x - 28

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( ) ( )

26. Find each of the following products :

9x2 - x + 15 × x2 - x - 1

Answer: 9x4 - 10x3 + 7x2 - 14x - 15

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Exercise 6C

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1. Divide:

(i) 24x2y3 by 3xy (ii) 36xyz2 by -9xz

(iii) -72x2y2z by - 12 xyz (iv) -56mnp2 by 7 mnp

Answer: (i) 8xy2 (ii) -4yz (iii) 6xy (iv) -8p

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2. Divide:

(i) 5m2 - 30m2 + 45m by 5m (ii) 8x2y2 - 6xy + 10x2y3

by 2xy

(iii) 9x2y - 6xy + 12xy2 by - 3xy (iv) 12x4 + 8x3 - 6x2

by -2x2

Answer: (i) m2 - 6m + 9 (ii) 4xy - 3y + 5xy2 (iii)

-3x + 2 - 4y (iv) -6x2 - 4x + 3

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( )

3. Write the quotient and remainder when we divide

:

x2 - 4x + 4 by (x - 2)

Answer: Quotient = (x - 2), remainder = 0

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( )

4. Write the quotient and remainder when we divide

:

x2 - 4 by (x + 2)

Answer: Quotient = (x - 2), remainder = 0

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5. Write the quotient and remainder when we divide

:

x2 + 12x + 35 by (x + 7)

Answer: Quotient = (x + 5), remainder = 0

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6. Write the quotient and remainder when we divide

:

15x2 + x - 6 by (3x + 2)

Answer: Quotient = (5x - 3), remainder = 0

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7. Write the quotient and remainder when we divide

:

14x2 - 53x + 45 by (7x - 9)

Answer: Quotient = (2x - 5), remainder = 0

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( )

8. Write the quotient and remainder when we divide

:

6x2 - 31x + 47 by (2x - 5)

Answer: Quotient = (3x - 8), remainder = 7

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9. Write the quotient and remainder when we divide

: 2x3 + x2 - 5x - 2 by (2x + 3)

Answer: Quotient = x2 - x - 1 , remainder = 1

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( )( )

10. Write the quotient and remainder when we

divide :

x3 + 1 by (x + 1)

Answer: Quotient = x2 - x + 1 , remainder = 0

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( )( )

11. Write the quotient and remainder when we

divide :

x4 - 2x3 + 2x2 + x + 4 by x2 + x + 1

Answer: Quotient = x2 - 3x + 4 , remainder = 0

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( ) ( )( )

12. Write the quotient and remainder when we

divide :

x3 - 6x2 + 11x - 6 by x2 - 5x + 6

Answer: Quotient = (x - 1), remainder = 0

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( ) ( )

13. Write the quotient and remainder when we

divide :

5x3 - 12x2 + 12x + 13 by x2 - 3x + 4

Answer: Quotient = (5x + 3), remainder = (x + 1)

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( ) ( )

14. Write the quotient and remainder when we

divide : 2x3 - 5x2 + 8x - 5 by 2x2 - 3x + 5

Answer: Quotient = (x - 1), remainder = 0

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( ) ( )

15. Write the quotient and remainder when we

divide :

8x4 + 10x3 - 5x2 - 4x + 1 by 2x2 + x - 1

Answer: Quotient = 4x2 + 3x - 2 , remainder

= (x - 1)

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( ) ( )( )

Exercise 6D

1. Find each of the following products:

(i) (x + 6)(x + 6) (ii) (4x + 5y)(4x + 5y) (iii)

(7a + 9b)(7a + 9b)

(iv) 23x +

45y

23x +

45y (v) x2 + 7 x2 + 7 (vi)

56a2 + 2

56a2 + 2

Answer: (i) x2 + 12x + 36

(ii) 16x2 + 25y2 + 40xy

(iii) 49a2 + 81b2 + 126ab

(iv) 49x2 +

1625

y2 +1615

xy

(v) x4 + 49 + 14x2

(vi) 2536a

4 + 4 +103 a2

( )( ) ( )( )

( )( )

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2. Find each of the following products:

(i) (x - 4)(x - 4) (ii) (2x - 3y)(2x - 3y) (iii)

34x -

56y

34x -

56y

(iv) x -3x

x -3x

(v) 13x2 - 9

13x2 - 9 (vi)

12y2 -

13y

12y2 -

13y

Answer: (i) x2 - 8x + 16

(ii) 4x4 + 9y2 - 12xy

(iii) 916

x2 +2536

y2 -54xy

(iv) x2 +9

x2- 6

( )( )( )( ) ( )( )

( )( )

(v) 19x4 + 81 - 6x2

(vi) 14y

4 +19y

2 -13y

3

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3. Expand:

(i) (8a + 3b)2 (ii) (7x + 2y)2 (iii) (5x + 11)2

(iv) a2 +

2a

2 (v)

3x4 +

2y9

2 (vi) (9x - 10)2

(vii) x2y - yz2 2 (viii)

xy -

yx

2 (ix) 3m -

45n

2

Answer: (i) 64a2 + 9b2 + 48ab

(ii) 49x2 + 4y2 + 28xy

(iii) 25x2 + 110x + 12

(iv) a2

4+

4

a2+ 2

( ) ( )( ) ( ) ( )

(v) 9x2

16+4y2

81+13xy

(vi) 81x2 - 180x + 100

(vii) x4y2 + y2z4 - 2x2y2z2

(viii) x2

y2+

y2

x2- 2

(ix) 9m2 +1625

n2 -245mn

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4. Find each of the following products:

(i) (x + 3)(x - 3) (ii) (2x + 5)(2x - 5) (ii) (8 + x)(8 - x)

(iv) (7x + 11y)(7x - 11y) (v) 5x2 +34y2 5x2 -

34y2

(vi) 4x5

-5y3

4x5

+5y3

( )( )( )( )

(vii) x +1x

x -1x

(viii) 1x

+1y

1x

-1y

(ix)

2a +3b

2a -3b

Answer: (i) x2 - 9

(ii) 4x2 - 25

(iii) 64 - x2

(iv) 49x2 - 121y2

(v) 25x4 -916

y4

(vi) 16x2

25-25y2

9

(vii)x2 -1

x2

(viii) 1

x2-1

y2

(ix) 4a2 -9

b2

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( )( ) ( )( )( )( )

( )( )( )

5. Using the formula for squaring a binomial,

evaluate the following:

(i) (54)2 (ii) (82)2 (iii) (103)2 (iv) (704)2

Answer: (i) 2916

(ii) 6724

(iii) 10609

(iv) 495616

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6. Using the formula for squaring a binomial

evaluate the following:

(i) (69)2 (ii) (78)2 (iii) (197)2 (iv) (999)2

Answer: (i) 4761

(ii) 6084

(iii) 38809

(iv) 998001

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7. Find the value of:

(i) (82)2 - (18)2 (ii) (128)2 - (72)2 (iii) 197 × 203

(iv) 198 × 198 - 102 × 102

96 (v) (14.7 × 15.3) (vi)

(8.63)2 - (1.37)2

Answer: (i) 6400

(ii) 11200

(iii) 39991

(iv) 300

(v) 224.91

(vi) 72.6

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8. Find the value of the expression 9x2 + 24x + 16

, when x = 12

Answer: 1600

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( )

9. find the value of the expression

64x2 + 81y2 + 144xy when x = 11 and y =43

Answer: 10000

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10. Find the value of the expression

36x2 + 25y2 - 60xy , when x =23 and y =

15

Answer: 9

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( )

11. If x +1x

= 4; Find (i) x2 +1

x2 (ii) x4 +1

x4

Answer: (i) 14

(ii) 194

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12. If x -1x

= 5, find the value of x2 +1

x2

A. 34

B. 42

C. 27

D. 63

( ) ( )

Answer: C

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13. Find the continued product:

(i) (x + 1)(x - 1) x2 + 1

(ii) (x - 3)(x + 3) x2 + 9

(iii) (3x - 2y)(3x + 2y) 9x2 + 4y2

(iv) (2p + 3)(2p - 3) 4p2 + 9

Answer: (i) x4 - 1

(ii) x4 - 81

(iii) 81x4 - 16y4

(iv) 16p4 - 81

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( )( )

( )( )( )( )

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14. If x + y = 12 and xy = 14 find the values of

x2 + y2.

Answer: 116

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15. If x - y = 7 and xy = 9, find the value of x2 + y2

Answer: 67

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Exercise 6E (Tick the correct answer in each ofthe following:)

1. The sum of (6a + 4b - c + 3), (2b - 3c + 4), (11b -

7a + 2c - 1) and (2c - 5a - 6) is

A. (4a - 6b + 2)

B. (-3a + 14b - 3c + 2)

C. (-6a + 17b)

D. (-6a + 6b + c - 4)

Answer: C

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2. 3q + 7p2 - 2r3 + 4 - 4p2 - 2q + 7r3 - 3 = ?

A. p2 + 2q + 5r3 + 1

B. 11p2 + q + 5r3 + 1

C. - 3p2 - 5q + 9r3 - 7

D. 3p2 + 5q - 9r3 + 7

Answer: D

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( ) ( )

( )( )( )( )

3. (x + 5)(x - 3) = ?

A. x2 + 5x - 15

B. x2 - 3x - 15

C. x2 + 2x + 15

D. x2 + 2x - 15

Answer: D

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4. (2x + 3)(3x - 1) = ?

A. 6x2 + 8x - 3

B. 6x2 + 7x - 3

C. 6x2 - 7x - 3

( )( )(

D. 6x2 - 7x + 3

Answer: B

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( )

5. (x + 4)(x + 4) = ?

A. x2 + 16

B. x2 + 4x + 16

C. x2 + 8x + 16

D. x2 + 16x

Answer: C

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( )( )( )( )

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6. (x - 6)(x - 6) = ?

A. x2 - 36

B. x2 + 36

C. x2 - 6x + 36

D. x2 - 12x + 36

Answer: D

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( )( )( )( )

7. (2x + 5)(2x - 5) = ?

A. 4x2 + 25

B. 4x2 - 25

C. 4x2 - 10x + 25

D. 4x2 + 10x - 25

Answer: B

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( )( )( )( )

8. 8a2b3

-2ab= ?

A. 4ab2

B. 4a2b

C. -4ab2

D. -4a2b

Answer: C

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9. 2x2 + 3x + 1 / (x + 1) = ?

A. (x + 1)

B. (2x + 1)

C. (x + 3)

D. (2x + 3)

( )

Answer: B

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10. x2 - 4x + 4 / (x - 2) = ?

A. (x - 2)

B. (x + 2)

C. (2 - x)

D. 2 + x + x2

Answer: A

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( )

( )

11. (a + 1)(a - 1) a2 + 1 = ?

A. a4 - 2a2 - 1

B. a4 - a2 - 1

C. a4 - 1

D. a4 + 1

Answer: C

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( )

( )( )( )( )

12. 1x

+1y

1x

-1y

= ? ( )( )

A. 1

x2 -1

y2

B. 1

x2 +1

y2

C. 1

x2 +1

y2 -1xy

D. 1

x2 -1

y2 +1xy

Answer: A

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( )( )( )( )

13. If x -1x

= 5 then x2 +1

x2 = ? ( ) ( )

A. 25

B. 27

C. 23

D. 251

25

Answer: C

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14. If x -1x

= 6, then x2 +1

x2 = ?

A. 36

B. 38

( ) ( )

C. 32

D. 361

36

Answer: B

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15. Simplify the following using the formula:

(a - b)(a + b) = a2 - b2 : (82)2 - (18)2 (ii) (467)2 - (33)2

A. 8218

B. 6418

C. 6400

D. 7204

Answer: C

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16. (197 × 203) = ?

A. 39991

B. 39999

C. 40009

D. 40001

Answer: A

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17. If (a + b) = 12 and ab = 14, then a2 + b2 = ?

A. 172

B. 116

C. 162

D. 126

Answer: B

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( )

18. If (a - b) = 7 and ab = 9, then a2 + b2 = ?

A. 67

( )

B. 31

C. 40

D. 58

Answer: A

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19. If x = 10, then the value of 4x2 + 20x + 25 = ?

A. 256

B. 425

C. 625

( )

Example

D. 575

Answer: C

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1. Add: 6a + 8b - 5c, 2b + c - 4a and a - 3b - 2c.

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2. Add: 5x2 + 3x - 8, 4x + 7 - 2x2 and 6 - 5x + 4x2

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3. Add: 8x2 - 5xy - 3y2, 2xy - 6y2 + 3x2 and

y2 + xy - 6x2

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4. Subtract 4a + 5b - 3c from 6a - 3b + c

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5. Subtract 3x2 - 6x - 4 from 5 + x - 2x2

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6. Find the product of: (i) 6xy and -3x3y3 (ii)

7ab2, - 4a2b and - 5abc

Answer: (i) = - 18x [ 1 + 2 ]y [ 1 + 3 ] = - 18x3y4

(ii) = 140a [ 1 + 2+1 ]b [ 2 + 1+1 ]C = 140a4b4c

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7. Find each of the following products:

(i) 5a2b2 × 3a2 - 4ab + 6b2 (ii)

- 3x2y × 4x2y - 3xy2 + 4x - 5y

Answer: (i) = 15a4b2 = 20a3b3 + 30a2b4

(ii) = - 12x4y2 + 9x3y3 - 12x3y + 15x2y2

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( )( ) ( )

8. Multiply (3x + 5y) and (5x - 7y).

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9. Multiply 3x2 + y2 by 2x2 + 3y2

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( ) ( )

10. Multiply 5x2 - 6x + 9 by (2x - 3)

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( )

11. Multiply 2x2 - 5x + 4 by x2 + 7x - 8

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( ) ( )

12. Multiply 2x3 - 5x2 - x + 7 by 3 - 2x + 4x2

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( ) ( )

13. Divide:

(i) 8x2y2 by - 2xy (ii) -15x3yz3 by -5xyz2

Answer: (i) = - 4xy2

(ii)3x2z

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14. Divide:

(i) 6x5 + 18x4 - 3x2 by 3x2 (ii) 20x3y + 12x2y2 - 10xy

by 2xy

Answer: (i) = 2x3 + 6x2 - 1

(ii) = 10x2 + 6xy - 5

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15. Divide 2x2 + 3x + 1 by (x + 1)

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16. Divide 9x - 6x2 + x3 - 2 by (x - 2)

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17. Divide 29x - 6x2 - 28 by (3x - 4)

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( )

18. 5x3 - 4x2 + 3x + 18 by 3 - 2x + x2

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( ) ( )

19. Using division, show that (x - 1) is a factor of

x3 - 1

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( )

20. Find the quotient and remainder when

7 + 15x - 13x2 + 5x3 is divided by 4 - 3x + x2

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( ) ( )

21. Divide 10x4 + 17x3 - 62x2 + 30x - 3 by

2x2 + 7x - 1

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( )( )

22. Find each of the following products:

(i) (3x + 2y) (3x + 2y) (ii) 4x2 + 5 4x2 + 5

Answer: = 16x4 + 25 + 40x2

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( ) ( )

23. Expand : (i) (2x + 5y)2 (ii) 23a +

34b

2

Answer: =49a2 +

916

b2 + ab

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( )

24. Find each of the following products:

(i) (4x - 7y) (4x - 7y) (ii) 3x2 - 4y2 3x2 - 4y2

Answer: (i) ∴ (4x - 7y)(4x - 7y) = 16x2 + 49y2 - 56xy

(ii) ∴ 3x2 - 4y2 3x2 - 4y2 = 9x4 + 16y4 - 24x2y2

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( )( )

( ) ( )

25. Expand (i) (3x - 2y)2 (ii) 34p -

56q

2

Answer: (i) = 9x2 + 4y2 - 12xy

(ii) =916p

2 +2536q

2 -54pq

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( )

26. Find each of the following products:

(i) (4x + 5y) (4x - 5y) (ii) 3x2 + 2y2 3x2 - 2y2

Answer: (i) = 16x2 - 25y2

(ii) = 9x4 - 4y4 .

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( )( )

( )

27. Evaluate the following using identities:

(i) (105)2 (ii) (47)2 (iii) (8.3 × 7.7)

Answer: (i) = 1000 + 25 + 1000 = 11025

(ii) = 2500 + 9 - 300 = 2209

(iii) 64 - 0.09 = 63.91

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28. Find the value of the expression

25x2 + 9y2 + 30xy. When x = 8 and y = 10.

Answer: 4900

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29. Find the value of the expression

81x2 + 16y2 - 72xy when x =23

and y =34

Answer: 9

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( )

30. If x +1x

= 5 find the values of (i) x2 +1

x2 and (ii)

x4 +1

x4

Answer: 527

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