IDENTITIES - Q&A
EXERCISE 12(A)
1. Multiple Choice Type:
Choose the correct answer from the options given below.
(i) (x + 2y)2 + (x - 2y)2 is equal to:
(a) 2x2 + 8y2 + 8xy
(b) x2 + 4y2
(c) 2x2 + 8y2
(d) 2x2 - 8y2
Solution:
Using the identity (a + b)2 + (a - b)2 = 2(a2 + b2)
Here a = x and b = 2y
= 2[x2 + (2y)2]
= 2(x2 + 4y2)
= 2x2 + 8y2
Answer: (c)
(ii) (a + b)(a - b) + (b - c)(b + c) + (c + a)(c - a) is equal to:
(a) 2a2 + 2b2 + 2c2
(b) a2 + b2 + c2 - 2ab - 2bc - 2ca
(c) 0
(d) none of these
Solution:
Using the identity (x + y)(x - y) = x2 - y2
Expression = (a2 - b2) + (b2 - c2) + (c2 - a2)
= a2 - b2 + b2 - c2 + c2 - a2
= 0
Answer: (c)
(iii) (3x - 4y)2 - (3x + 4y)2 is equal to:
(a) 18x2 + 32y2
(b) 18x2 - 32y2
(c) -48xy
(d) 48xy
Solution:
Using the identity (a - b)2 - (a + b)2 = -4ab
Here a = 3x and b = 4y
= -4(3x)(4y)
= -48xy
Answer: (c)
(iv) The value of (0.8)2 - 0.32 + (0.2)2 is equal to:
(a) 1
(b) 3.6
(c) 0.36
(d) 0.036
Solution:
Expression = (0.8)2 - 2(0.8)(0.2) + (0.2)2 [Since 2 × 0.8 × 0.2 = 0.32]
This is in the form a2 - 2ab + b2 = (a - b)2
= (0.8 - 0.2)2
= (0.6)2
= 0.36
Answer: (c)
(v) The value of (a - b - c)(a - b + c) is:
(a) a2 + b2 + c2 + 2ab
(b) a2 + b2 - c2 - 2ab
(c) a2 + b2 + c2 - 2ab
(d) a2 + b2 - c2 + 2ab
Solution:
Let x = (a - b)
Expression = (x - c)(x + c) = x2 - c2
Substitute x = a - b
= (a - b)2 - c2
= (a2 - 2ab + b2) - c2
= a2 + b2 - c2 - 2ab
Answer: (b)
2. Use direct method to evaluate the following products:
(i) (a - 8)(a + 2)
Solution:
Using (x + a)(x + b) = x2 + (a + b)x + ab
= a2 + (-8 + 2)a + (-8)(2)
= a2 - 6a - 16
(ii) (b - 3)(b - 5)
Solution:
Using (x - a)(x - b) = x2 - (a + b)x + ab
= b2 - (3 + 5)b + (3)(5)
= b2 - 8b + 15
(iii) (3x - 2y)(2x + y)
Solution:
= 3x(2x + y) - 2y(2x + y)
= 6x2 + 3xy - 4xy - 2y2
= 6x2 - xy - 2y2
(iv) (5a + 16)(3a - 7)
Solution:
= 5a(3a - 7) + 16(3a - 7)
= 15a2 - 35a + 48a - 112
= 15a2 + 13a - 112
(v) (8 - b)(3 + b)
Solution:
= 8(3 + b) - b(3 + b)
= 24 + 8b - 3b - b2
= 24 + 5b - b2
3. Evaluate :
(i) (2a + 3)(2a - 3)
Solution:
Using (x + y)(x - y) = x2 - y2
= (2a)2 - (3)2
= 4a2 - 9
(ii) (xy + 4)(xy - 4)
Solution:
= (xy)2 - (4)2
= x2y2 - 16
(iii) (ab + x2)(ab - x2)
Solution:
= (ab)2 - (x2)2
= a2b2 - x4
(iv) (3x2 + 5y2)(3x2 - 5y2)
Solution:
= (3x2)2 - (5y2)2
= 9x4 - 25y4
(v) (z - 2/3)(z + 2/3)
Solution:
= (z)2 - (2/3)2
= z2 - 4/9
(vi) (3/5 a + 1/2)(3/5 a - 1/2)
Solution:
= (3/5 a)2 - (1/2)2
= 9/25 a2 - 1/4
(vii) (0.5 - 2a)(0.5 + 2a)
Solution:
= (0.5)2 - (2a)2
= 0.25 - 4a2
(viii) (a/2 - b/3)(a/2 + b/3)
Solution:
= (a/2)2 - (b/3)2
= a2/4 - b2/9
4. Evaluate :
(i) (a + b)(a - b)(a2 + b2)
Solution:
First solve (a + b)(a - b) = a2 - b2
Now multiply by (a2 + b2)
= (a2 - b2)(a2 + b2)
= (a2)2 - (b2)2
= a4 - b4
(ii) (3 - 2x)(3 + 2x)(9 + 4x2)
Solution:
(3 - 2x)(3 + 2x) = 32 - (2x)2 = 9 - 4x2
Now multiply by (9 + 4x2)
= (9 - 4x2)(9 + 4x2)
= (9)2 - (4x2)2
= 81 - 16x4
(iii) (3x - 4y)(3x + 4y)(9x2 + 16y2)
Solution:
(3x - 4y)(3x + 4y) = (3x)2 - (4y)2 = 9x2 - 16y2
Now multiply by (9x2 + 16y2)
= (9x2 - 16y2)(9x2 + 16y2)
= (9x2)2 - (16y2)2
= 81x4 - 256y4
5. Use the formula: (a + b)(a - b) = a2 - b2 to evaluate :
(i) 21 × 19
Solution:
= (20 + 1)(20 - 1)
= 202 - 12
= 400 - 1 = 399
(ii) 33 × 27
Solution:
= (30 + 3)(30 - 3)
= 302 - 32
= 900 - 9 = 891
(iii) 103 × 97
Solution:
= (100 + 3)(100 - 3)
= 1002 - 32
= 10000 - 9 = 9991
(iv) 9.8 × 10.2
Solution:
= (10 - 0.2)(10 + 0.2)
= 102 - (0.2)2
= 100 - 0.04 = 99.96
(v) 7.7 × 8.3
Solution:
= (8 - 0.3)(8 + 0.3)
= 82 - (0.3)2
= 64 - 0.09 = 63.91
6. Evaluate
(i) (6 - xy)(6 + xy)
Solution:
= 62 - (xy)2
= 36 - x2y2
(ii) (7x + 2/3 y)(7x - 2/3 y)
Solution:
= (7x)2 - (2/3 y)2
= 49x2 - 4/9 y2
(iii) (a/2b + 2b/a)(a/2b - 2b/a)
Solution:
= (a/2b)2 - (2b/a)2
= a2/4b2 - 4b2/a2
(iv) (3x - 1/2y)(3x + 1/2y)
Solution:
= (3x)2 - (1/2y)2
= 9x2 - 1/4y2
(v) (2a + 3)(2a - 3)(4a2 + 9)
Solution:
= [(2a)2 - 32](4a2 + 9)
= (4a2 - 9)(4a2 + 9)
= (4a2)2 - (9)2
= 16a4 - 81
(vi) (a + bc)(a - bc)(a2 + b2c2)
Solution:
= (a2 - b2c2)(a2 + b2c2)
= (a2)2 - (b2c2)2
= a4 - b4c4
7. Expand:
(i) (a + 1/2a)2
Solution:
Using (x + y)2 = x2 + y2 + 2xy
= a2 + (1/2a)2 + 2(a)(1/2a)
= a2 + 1/4a2 + 1
(ii) (2a - 1/a)2
Solution:
Using (x - y)2 = x2 + y2 - 2xy
= (2a)2 + (1/a)2 - 2(2a)(1/a)
= 4a2 + 1/a2 - 4
(iii) (3x + 1/3x)2
Solution:
= (3x)2 + (1/3x)2 + 2(3x)(1/3x)
= 9x2 + 1/9x2 + 2
(iv) (a - b + c)2
Solution:
Using (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
Here x=a, y=-b, z=c
= a2 + (-b)2 + c2 + 2(a)(-b) + 2(-b)(c) + 2(c)(a)
= a2 + b2 + c2 - 2ab - 2bc + 2ca
(v) (a + b - c)2
Solution:
= a2 + b2 + (-c)2 + 2(a)(b) + 2(b)(-c) + 2(-c)(a)
= a2 + b2 + c2 + 2ab - 2bc - 2ca
8. Find the square of:
(i) a + 1/5a
Solution:
(a + 1/5a)2 = a2 + (1/5a)2 + 2(a)(1/5a)
= a2 + 1/25a2 + 2/5
(ii) 2a - 1/a
Solution:
(2a - 1/a)2 = (2a)2 + (1/a)2 - 2(2a)(1/a)
= 4a2 + 1/a2 - 4
(iii) x - 2y + 1
Solution:
(x - 2y + 1)2 = x2 + (-2y)2 + 12 + 2(x)(-2y) + 2(-2y)(1) + 2(1)(x)
= x2 + 4y2 + 1 - 4xy - 4y + 2x
(iv) 3a - 2b - 5c
Solution:
(3a - 2b - 5c)2 = (3a)2 + (-2b)2 + (-5c)2 + 2(3a)(-2b) + 2(-2b)(-5c) + 2(-5c)(3a)
= 9a2 + 4b2 + 25c2 - 12ab + 20bc - 30ca
(v) 2x + 1/x + 1
Solution:
(2x + 1/x + 1)2 = (2x)2 + (1/x)2 + 12 + 2(2x)(1/x) + 2(1/x)(1) + 2(1)(2x)
= 4x2 + 1/x2 + 1 + 4 + 2/x + 4x
= 4x2 + 1/x2 + 4x + 2/x + 5
(vi) 5 - x + 2/x
Solution:
(5 - x + 2/x)2 = 52 + (-x)2 + (2/x)2 + 2(5)(-x) + 2(-x)(2/x) + 2(2/x)(5)
= 25 + x2 + 4/x2 - 10x - 4 + 20/x
= x2 + 4/x2 - 10x + 20/x + 21
(vii) 2x - 3y + z
Solution:
(2x - 3y + z)2 = (2x)2 + (-3y)2 + z2 + 2(2x)(-3y) + 2(-3y)(z) + 2(z)(2x)
= 4x2 + 9y2 + z2 - 12xy - 6yz + 4zx
(viii) x + 1/x - 1
Solution:
(x + 1/x - 1)2 = x2 + (1/x)2 + (-1)2 + 2(x)(1/x) + 2(1/x)(-1) + 2(-1)(x)
= x2 + 1/x2 + 1 + 2 - 2/x - 2x
= x2 + 1/x2 - 2x - 2/x + 3
9. Evaluate using expansion of (a + b)2 or (a - b)2:
(i) (208)2
Solution:
= (200 + 8)2
= 2002 + 82 + 2(200)(8)
= 40000 + 64 + 3200
= 43264
(ii) (92)2
Solution:
= (90 + 2)2
= 902 + 22 + 2(90)(2)
= 8100 + 4 + 360
= 8464
(iii) (9.4)2
Solution:
= (9 + 0.4)2
= 92 + (0.4)2 + 2(9)(0.4)
= 81 + 0.16 + 7.2
= 88.36
(iv) (20.7)2
Solution:
= (20 + 0.7)2
= 202 + (0.7)2 + 2(20)(0.7)
= 400 + 0.49 + 28
= 428.49
10. Expand:
(i) (2a + b)3
Solution:
Using (x + y)3 = x3 + y3 + 3xy(x + y)
= (2a)3 + b3 + 3(2a)(b)(2a + b)
= 8a3 + b3 + 6ab(2a + b)
= 8a3 + b3 + 12a2b + 6ab2
(ii) (a - 2b)3
Solution:
Using (x - y)3 = x3 - y3 - 3xy(x - y)
= a3 - (2b)3 - 3(a)(2b)(a - 2b)
= a3 - 8b3 - 6ab(a - 2b)
= a3 - 8b3 - 6a2b + 12ab2
(iii) (3x - 2y)3
Solution:
= (3x)3 - (2y)3 - 3(3x)(2y)(3x - 2y)
= 27x3 - 8y3 - 18xy(3x - 2y)
= 27x3 - 8y3 - 54x2y + 36xy2
(iv) (x + 5y)3
Solution:
= x3 + (5y)3 + 3(x)(5y)(x + 5y)
= x3 + 125y3 + 15xy(x + 5y)
= x3 + 125y3 + 15x2y + 75xy2
(v) (a + 1/a)3
Solution:
= a3 + (1/a)3 + 3(a)(1/a)(a + 1/a)
= a3 + 1/a3 + 3(a + 1/a)
(vi) (2a - 1/2a)3
Solution:
= (2a)3 - (1/2a)3 - 3(2a)(1/2a)(2a - 1/2a)
= 8a3 - 1/8a3 - 3(2a - 1/2a)
11. Find the cube of:
(i) a + 2
Solution:
= (a + 2)3
= a3 + 23 + 3(a)(2)(a + 2)
= a3 + 8 + 6a(a + 2)
= a3 + 6a2 + 12a + 8
(ii) 2a - 1
Solution:
= (2a - 1)3
= (2a)3 - 13 - 3(2a)(1)(2a - 1)
= 8a3 - 1 - 6a(2a - 1)
= 8a3 - 12a2 + 6a - 1
(iii) 2a + 3b
Solution:
= (2a + 3b)3
= (2a)3 + (3b)3 + 3(2a)(3b)(2a + 3b)
= 8a3 + 27b3 + 18ab(2a + 3b)
= 8a3 + 27b3 + 36a2b + 54ab2
(iv) 3b - 2a
Solution:
= (3b - 2a)3
= (3b)3 - (2a)3 - 3(3b)(2a)(3b - 2a)
= 27b3 - 8a3 - 18ab(3b - 2a)
= 27b3 - 8a3 - 54b2a + 36a2b
(v) 2x + 1/x
Solution:
= (2x + 1/x)3
= (2x)3 + (1/x)3 + 3(2x)(1/x)(2x + 1/x)
= 8x3 + 1/x3 + 6(2x + 1/x)
(vi) x - 1/2
Solution:
= (x - 1/2)3
= x3 - (1/2)3 - 3(x)(1/2)(x - 1/2)
= x3 - 1/8 - 3/2 x(x - 1/2)
= x3 - 1/8 - 3/2 x2 + 3/4 x
EXERCISE 12(B)
1. Multiple Choice Type:
Choose the correct answer from the options given below :
(i) If x - 1/x = 3, the value of x2 + 1/x2 is:
(a) 0
(b) 7
(c) 11
(d) 9
Solution:
(x - 1/x)2 = x2 + 1/x2 - 2
32 = x2 + 1/x2 - 2
9 = x2 + 1/x2 - 2
x2 + 1/x2 = 9 + 2 = 11
Answer: (c)
(ii) If a + b = 7 and ab = 10 the value of a2 + b2 is equal to:
(a) 29
(b) 49
(c) 39
(d) 69
Solution:
(a + b)2 = a2 + b2 + 2ab
72 = a2 + b2 + 2(10)
49 = a2 + b2 + 20
a2 + b2 = 49 - 20 = 29
Answer: (a)
(iii) The value of (95 × 95 - 5 × 5) / (95 - 5) is equal to:
(a) 100
(b) 90
(c) 95
(d) none of these
Solution:
(952 - 52) / (95 - 5)
Using a2 - b2 = (a + b)(a - b)
= [(95 + 5)(95 - 5)] / (95 - 5)
= 95 + 5 = 100
Answer: (a)
(iv) If a - b = 1 and a + b = 3, the value of ab is:
(a) 4
(b) 2
(c) -2
(d) 0
Solution:
Using 4ab = (a + b)2 - (a - b)2
4ab = 32 - 12
4ab = 9 - 1 = 8
ab = 2
Answer: (b)
(v) If x + 1/x = 2, the value of (x3 + 1/x3) - (x2 + 1/x2) is:
(a) 0
(b) 4
(c) 2
(d) 6
Solution:
If x + 1/x = 2, then x = 1 (since 1 + 1/1 = 2)
Substituting x = 1:
(13 + 1/13) - (12 + 1/12)
= (1 + 1) - (1 + 1)
= 2 - 2 = 0
Answer: (a)
2. If a + b = 5 and ab = 6, find a2 + b2
Solution:
(a + b)2 = a2 + b2 + 2ab
52 = a2 + b2 + 2(6)
25 = a2 + b2 + 12
a2 + b2 = 25 - 12 = 13
3. If a - b = 6 and ab = 16, find a2 + b2
Solution:
(a - b)2 = a2 + b2 - 2ab
62 = a2 + b2 - 2(16)
36 = a2 + b2 - 32
a2 + b2 = 36 + 32 = 68
4. If a2 + b2 = 29 and ab = 10, find:
(i) a + b
(ii) a - b
Solution:
(i) (a + b)2 = a2 + b2 + 2ab
= 29 + 2(10) = 29 + 20 = 49
a + b = ±√49 = ±7
(ii) (a - b)2 = a2 + b2 - 2ab
= 29 - 2(10) = 29 - 20 = 9
a - b = ±√9 = ±3
5. If a2 + b2 = 10 and ab = 3; find:
(i) a - b
(ii) a + b
Solution:
(i) (a - b)2 = a2 + b2 - 2ab
= 10 - 2(3) = 10 - 6 = 4
a - b = ±√4 = ±2
(ii) (a + b)2 = a2 + b2 + 2ab
= 10 + 2(3) = 10 + 6 = 16
a + b = ±√16 = ±4
6. If a + 1/a = 3, find a2 + 1/a2
Solution:
(a + 1/a)2 = a2 + 1/a2 + 2
32 = a2 + 1/a2 + 2
9 = a2 + 1/a2 + 2
a2 + 1/a2 = 7
7. If a - 1/a = 4, find: a2 + 1/a2
Solution:
(a - 1/a)2 = a2 + 1/a2 - 2
42 = a2 + 1/a2 - 2
16 = a2 + 1/a2 - 2
a2 + 1/a2 = 18
8. If a2 + 1/a2 = 23, find: a - 1/a
Solution:
(a - 1/a)2 = a2 + 1/a2 - 2
= 23 - 2 = 21
a - 1/a = ±√21
9. If a2 + 1/a2 = 11, find a - 1/a
Solution:
(a - 1/a)2 = a2 + 1/a2 - 2
= 11 - 2 = 9
a - 1/a = ±√9 = ±3
10. If a + b + c = 10 and a2 + b2 + c2 = 38, find: ab + bc + ca
Solution:
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
102 = 38 + 2(ab + bc + ca)
100 = 38 + 2(ab + bc + ca)
2(ab + bc + ca) = 100 - 38 = 62
ab + bc + ca = 31
11. Find a2 + b2 + c2 if a + b + c = 9 and ab + bc + ca = 24
Solution:
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
92 = a2 + b2 + c2 + 2(24)
81 = a2 + b2 + c2 + 48
a2 + b2 + c2 = 81 - 48 = 33
12. Find a + b + c, if a2 + b2 + c2 = 83 and ab + bc + ca = 71.
Solution:
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
= 83 + 2(71)
= 83 + 142 = 225
a + b + c = ±√225 = ±15
13. If a + b = 6 and ab = 8, find: a3 + b3.
Solution:
a3 + b3 = (a + b)3 - 3ab(a + b)
= 63 - 3(8)(6)
= 216 - 144 = 72
14. If a - b = 3 and ab = 10, find: a3 - b3.
Solution:
a3 - b3 = (a - b)3 + 3ab(a - b)
= 33 + 3(10)(3)
= 27 + 90 = 117
15. Find : a3 + 1/a3 if a + 1/a = 5
Solution:
a3 + 1/a3 = (a + 1/a)3 - 3(a)(1/a)(a + 1/a)
= (a + 1/a)3 - 3(a + 1/a)
= 53 - 3(5)
= 125 - 15 = 110
16. Find: a3 - 1/a3 if a - 1/a = 4
Solution:
a3 - 1/a3 = (a - 1/a)3 + 3(a - 1/a)
= 43 + 3(4)
= 64 + 12 = 76
17. If 2x - 1/2x = 4, find:
(i) 4x2 + 1/4x2
(ii) 8x3 - 1/8x3
Solution:
(i) (2x - 1/2x)2 = (2x)2 + (1/2x)2 - 2(2x)(1/2x)
42 = 4x2 + 1/4x2 - 2
16 = 4x2 + 1/4x2 - 2
4x2 + 1/4x2 = 18
(ii) 8x3 - 1/8x3 = (2x)3 - (1/2x)3
= (2x - 1/2x)3 + 3(2x)(1/2x)(2x - 1/2x)
= 43 + 3(4)
= 64 + 12 = 76
18. If 3x + 1/3x = 3, find:
(i) 9x2 + 1/9x2
(ii) 27x3 + 1/27x3
Solution:
(i) (3x + 1/3x)2 = 9x2 + 1/9x2 + 2
32 = 9x2 + 1/9x2 + 2
9x2 + 1/9x2 = 9 - 2 = 7
(ii) 27x3 + 1/27x3 = (3x + 1/3x)3 - 3(3x + 1/3x)
= 33 - 3(3)
= 27 - 9 = 18
19. The sum of the squares of two numbers is 13 and their product is 6. Find:
(i) the sum of the two numbers.
(ii) the difference between them.
Solution:
Let numbers be a and b.
Given a2 + b2 = 13 and ab = 6
(i) (a + b)2 = a2 + b2 + 2ab
= 13 + 2(6) = 13 + 12 = 25
a + b = ±√25 = ±5
(ii) (a - b)2 = a2 + b2 - 2ab
= 13 - 2(6) = 13 - 12 = 1
a - b = ±√1 = ±1
Test yourself
1. Multiple Choice Type:
Choose the correct answer from the options given below.
(i) If a is positive and a2 + 1/a2 = 18; then the value of a - 1/a is:
(a) 4
(b) 16
(c) 20
(d) 2√5
Solution:
(a - 1/a)2 = a2 + 1/a2 - 2
= 18 - 2 = 16
a - 1/a = ±4
Since a is positive, the value is 4 (assuming a > 1/a).
Answer: (a)
(ii) (x + y)(x - y)(x2 + y2)(x4 + y4) is equal to:
(a) x4 + y4
(b) x8 + y8
(c) x8 - y8
(d) 2x6y6
Solution:
(x + y)(x - y) = x2 - y2
Now, (x2 - y2)(x2 + y2) = x4 - y4
Now, (x4 - y4)(x4 + y4) = x8 - y8
Answer: (c)
(iii) The value of 102 × 98 is:
(a) 6999
(b) 6696
(c) 9696
(d) 9996
Solution:
= (100 + 2)(100 - 2)
= 1002 - 22
= 10000 - 4 = 9996
Answer: (d)
(iv) (x + 3)(x + 3) - (x - 2)(x - 2) is equal to:
(a) 10x + 5
(b) 10x - 5
(c) 5 - 10x
(d) none of these
Solution:
= (x + 3)2 - (x - 2)2
Using a2 - b2 = (a + b)(a - b)
a = x + 3, b = x - 2
= [(x + 3) + (x - 2)][(x + 3) - (x - 2)]
= (2x + 1)(x + 3 - x + 2)
= (2x + 1)(5)
= 10x + 5
Answer: (a)
(v) If 5a = 302 - 252, the value of a is:
(a) 5
(b) 11
(c) 55
(d) none of these
Solution:
5a = (30 + 25)(30 - 25)
5a = (55)(5)
a = 55
Answer: (c)
(vi) Statement 1: Cube of a binomial : (a - b)3 = a3 + 3a2b - 3ab2 - b3.
Statement 2: (a - b)2 - (a + b)2 = 4ab
Which of the following options is correct?
(a) Both the statements are true.
(b) Both the statements are false.
(c) Statement 1 is true, and statement 2 is false.
(d) Statement 1 is false, and statement 2 is true.
Solution:
Statement 1 is False. Correct is a3 - 3a2b + 3ab2 - b3.
Statement 2 is False. (a - b)2 - (a + b)2 = -4ab.
Answer: (b)
The following questions are Assertion-Reason based questions. Choose your answer based on the codes given below.
(1) Both A and R are correct, and R is the correct explanation for A.
(2) Both A and R are correct, and R is not the correct explanation for A.
(3) A is true, but R is false.
(4) A is false, but R is true.
(vii) Assertion (A): Using appropriate identity, we get 22.5 × 21.5 = 484.75
Reason (R): The product: (x + y)(x - y) = x2 - y2.
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Solution:
Check A: 22.5 × 21.5 = (22 + 0.5)(22 - 0.5) = 222 - 0.52 = 484 - 0.25 = 483.75. Assertion is False.
Check R: (x + y)(x - y) = x2 - y2. Reason is True.
Answer: (d)
(viii) Assertion (A): If we add 9 with 49x2 - 42x the resultant expression will be a perfect square expression.
Reason (R): The product of the sum and difference of the same two terms = Difference of their squares.
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Solution:
Check A: 49x2 - 42x + 9 = (7x)2 - 2(7x)(3) + 32 = (7x - 3)2. True.
Check R: (a + b)(a - b) = a2 - b2. True statement.
Explanation: R explains difference of squares, not perfect square trinomials. So not explanation.
Answer: (b)
(ix) Assertion (A): If the volume of a cube is a3 + b3 + 3ab(a + b), then the edge of the cube is (a + b).
Reason (R): (1stterm + 2ndterm)3 = (1stterm)3 + 3(1stterm)2(2ndterm) + 3(2ndterm)2(1stterm) + (2ndterm)3
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Solution:
Check A: a3 + b3 + 3ab(a + b) = (a + b)3. If volume = side3, then side = a + b. True.
Check R: This is the expansion formula for cube of sum. True.
Does R explain A? Yes.
Answer: (a)
(x) Assertion (A): 687 × 687 - 313 × 313 = 37400
Reason (R): The product of the sum and the difference of the same two terms = The square of their difference.
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Solution:
Check A: 6872 - 3132 = (687 - 313)(687 + 313) = 374 × 1000 = 374000. Value given is 37400. False.
Check R: (a + b)(a - b) = a2 - b2. Statement says "= Square of their difference", i.e., (a - b)2. This is False.
Since both are False, and no option covers this, this question likely contains errors in the text. Assuming the intended answer follows standard patterns where "Difference of squares" was meant for R and A was a typo:
Strictly mathematically: Both False.
Answer: Statement A is False, Statement R is False.
2. Evaluate :
(i) (3x + 1/2)(2x + 1/3)
Solution:
= 3x(2x + 1/3) + 1/2(2x + 1/3)
= 6x2 + x + x + 1/6
= 6x2 + 2x + 1/6
(ii) (2a + 0.5)(7a - 0.3)
Solution:
= 2a(7a - 0.3) + 0.5(7a - 0.3)
= 14a2 - 0.6a + 3.5a - 0.15
= 14a2 + 2.9a - 0.15
(iii) (9 - y)(7 + y)
Solution:
= 9(7 + y) - y(7 + y)
= 63 + 9y - 7y - y2
= 63 + 2y - y2
(iv) (2 - z)(15 - z)
Solution:
= 2(15 - z) - z(15 - z)
= 30 - 2z - 15z + z2
= 30 - 17z + z2
(v) (a2 + 5)(a2 - 3)
Solution:
= a2(a2 - 3) + 5(a2 - 3)
= a4 - 3a2 + 5a2 - 15
= a4 + 2a2 - 15
(vi) (4 - ab)(8 + ab)
Solution:
= 4(8 + ab) - ab(8 + ab)
= 32 + 4ab - 8ab - a2b2
= 32 - 4ab - a2b2
(vii) (5xy - 7)(7xy + 9)
Solution:
= 5xy(7xy + 9) - 7(7xy + 9)
= 35x2y2 + 45xy - 49xy - 63
= 35x2y2 - 4xy - 63
(viii) (3a2 - 4b2)(8a2 - 3b2)
Solution:
= 3a2(8a2 - 3b2) - 4b2(8a2 - 3b2)
= 24a4 - 9a2b2 - 32a2b2 + 12b4
= 24a4 - 41a2b2 + 12b4
3. Find the square of:
(i) 3x + 2/y
Solution:
= (3x + 2/y)2
= (3x)2 + (2/y)2 + 2(3x)(2/y)
= 9x2 + 4/y2 + 12x/y
(ii) 5a/6b - 6b/5a
Solution:
= (5a/6b - 6b/5a)2
= (5a/6b)2 + (6b/5a)2 - 2(5a/6b)(6b/5a)
= 25a2/36b2 + 36b2/25a2 - 2
(iii) 2m2 - 2/3 n2
Solution:
= (2m2)2 + (2/3 n2)2 - 2(2m2)(2/3 n2)
= 4m4 + 4/9 n4 - 8/3 m2n2
(iv) 5x + 1/5x
Solution:
= (5x)2 + (1/5x)2 + 2(5x)(1/5x)
= 25x2 + 1/25x2 + 2
(v) 8x + 3/2 y
Solution:
= (8x)2 + (3/2 y)2 + 2(8x)(3/2 y)
= 64x2 + 9/4 y2 + 24xy
(vi) 607
Solution:
6072 = (600 + 7)2
= 6002 + 72 + 2(600)(7)
= 360000 + 49 + 8400
= 368449
(vii) 391
Solution:
3912 = (400 - 9)2
= 4002 + 92 - 2(400)(9)
= 160000 + 81 - 7200
= 152881
(viii) 9.7
Solution:
9.72 = (10 - 0.3)2
= 102 + 0.32 - 2(10)(0.3)
= 100 + 0.09 - 6
= 94.09
4. If a + 1/a = 2 find:
(i) a2 + 1/a2
(ii) a4 + 1/a4
Solution:
(i) (a + 1/a)2 = a2 + 1/a2 + 2
22 = a2 + 1/a2 + 2
4 = a2 + 1/a2 + 2
a2 + 1/a2 = 2
(ii) (a2 + 1/a2)2 = a4 + 1/a4 + 2
22 = a4 + 1/a4 + 2
4 = a4 + 1/a4 + 2
a4 + 1/a4 = 2
5. If m - 1/m = 5 find:
(i) m2 + 1/m2
(ii) m4 + 1/m4
(iii) m2 - 1/m2
Solution:
(i) (m - 1/m)2 = m2 + 1/m2 - 2
52 = m2 + 1/m2 - 2
m2 + 1/m2 = 25 + 2 = 27
(ii) (m2 + 1/m2)2 = m4 + 1/m4 + 2
272 = m4 + 1/m4 + 2
729 = m4 + 1/m4 + 2
m4 + 1/m4 = 727
(iii) m2 - 1/m2 = (m + 1/m)(m - 1/m)
We need m + 1/m. (m + 1/m)2 = m2 + 1/m2 + 2 = 27 + 2 = 29.
So m + 1/m = ±√29.
m2 - 1/m2 = ±√29 × 5 = ±5√29
6. If a2 + b2 = 41 and ab = 4 find:
(i) a - b
(ii) a + b
Solution:
(i) (a - b)2 = a2 + b2 - 2ab
= 41 - 2(4) = 41 - 8 = 33
a - b = ±√33
(ii) (a + b)2 = a2 + b2 + 2ab
= 41 + 2(4) = 41 + 8 = 49
a + b = ±√49 = ±7
7. If 2a + 1/2a = 8, find :
(i) 4a2 + 1/4a2
(ii) 16a4 + 1/16a4
Solution:
(i) (2a + 1/2a)2 = 4a2 + 1/4a2 + 2
82 = 4a2 + 1/4a2 + 2
64 - 2 = 4a2 + 1/4a2
4a2 + 1/4a2 = 62
(ii) (4a2 + 1/4a2)2 = 16a4 + 1/16a4 + 2
622 = 16a4 + 1/16a4 + 2
3844 - 2 = 16a4 + 1/16a4
16a4 + 1/16a4 = 3842
8. If 3x - 1/3x = 5, find :
(i) 9x2 + 1/9x2
(ii) 81x4 + 1/81x4
Solution:
(i) (3x - 1/3x)2 = 9x2 + 1/9x2 - 2
52 = 9x2 + 1/9x2 - 2
25 + 2 = 9x2 + 1/9x2
9x2 + 1/9x2 = 27
(ii) (9x2 + 1/9x2)2 = 81x4 + 1/81x4 + 2
272 = 81x4 + 1/81x4 + 2
729 - 2 = 81x4 + 1/81x4
81x4 + 1/81x4 = 727
9. Expand:
(i) (3x - 4y + 5z)2
Solution:
= (3x)2 + (-4y)2 + (5z)2 + 2(3x)(-4y) + 2(-4y)(5z) + 2(5z)(3x)
= 9x2 + 16y2 + 25z2 - 24xy - 40yz + 30zx
(ii) (2a - 5b - 4c)2
Solution:
= 4a2 + 25b2 + 16c2 - 20ab + 40bc - 16ca
(iii) (5x + 3y)3
Solution:
= (5x)3 + (3y)3 + 3(5x)(3y)(5x + 3y)
= 125x3 + 27y3 + 45xy(5x + 3y)
= 125x3 + 27y3 + 225x2y + 135xy2
(iv) (6a - 7b)3
Solution:
= (6a)3 - (7b)3 - 3(6a)(7b)(6a - 7b)
= 216a3 - 343b3 - 126ab(6a - 7b)
= 216a3 - 343b3 - 756a2b + 882ab2
10. If a + b + c = 9 and ab + bc + ca = 15, find: a2 + b2 + c2
Solution:
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
92 = a2 + b2 + c2 + 2(15)
81 = a2 + b2 + c2 + 30
a2 + b2 + c2 = 81 - 30 = 51
11. If a + b + c = 11 and a2 + b2 + c2 = 81 find: ab + bc + ca
Solution:
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
112 = 81 + 2(ab + bc + ca)
121 - 81 = 2(ab + bc + ca)
40 = 2(ab + bc + ca)
ab + bc + ca = 20
12. If 3x - 4y = 5 and xy = 3, find: 27x3 - 64y3.
Solution:
(3x - 4y)3 = (3x)3 - (4y)3 - 3(3x)(4y)(3x - 4y)
53 = 27x3 - 64y3 - 36xy(3x - 4y)
125 = 27x3 - 64y3 - 36(3)(5)
125 = 27x3 - 64y3 - 540
27x3 - 64y3 = 125 + 540 = 665
13. If a + b = 8 and ab = 15, find: a3 + b3.
Solution:
a3 + b3 = (a + b)3 - 3ab(a + b)
= 83 - 3(15)(8)
= 512 - 360 = 152
14. If 3x + 2y = 9 and xy = 3, find: 27x3 + 8y3
Solution:
(3x + 2y)3 = 27x3 + 8y3 + 3(3x)(2y)(3x + 2y)
93 = 27x3 + 8y3 + 18xy(3x + 2y)
729 = 27x3 + 8y3 + 18(3)(9)
729 = 27x3 + 8y3 + 486
27x3 + 8y3 = 729 - 486 = 243
15. If 5x - 4y = 7 and xy = 8, find: 125x3 - 64y3
Solution:
(5x - 4y)3 = 125x3 - 64y3 - 3(5x)(4y)(5x - 4y)
73 = 125x3 - 64y3 - 60xy(5x - 4y)
343 = 125x3 - 64y3 - 60(8)(7)
343 = 125x3 - 64y3 - 3360
125x3 - 64y3 = 343 + 3360 = 3703
16. The difference between two numbers is 5 and their product is 14. Find the difference between their cubes.
Solution:
Let numbers be a and b.
a - b = 5, ab = 14
a3 - b3 = (a - b)3 + 3ab(a - b)
= 53 + 3(14)(5)
= 125 + 210 = 335