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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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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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