EXPANSIONS - Questions & Answers

EXERCISE 4 (A)

1. Multiple Choice Type : Choose the correct answer from the options given below.
(a) For x = 9 and y = 4, the value of x² + 2xy + y² - 3 is :
(i) 172   (ii) 100   (iii) 166   (iv) 169
Answer:
Given expression: x² + 2xy + y² - 3
Using identity: x² + 2xy + y² = (x + y)²
Expression becomes: (x + y)² - 3
Substitute x = 9, y = 4: (9 + 4)² - 3
= (13)² - 3
= 169 - 3 = 166
Correct option: (iii)

(b) For x = 5 and y = 3, the value of x² + y² - 2xy + 7 is :
(i) 11   (ii) 169   (iii) 71   (iv) 1
Answer:
Given expression: x² + y² - 2xy + 7
Using identity: x² - 2xy + y² = (x - y)²
Expression becomes: (x - y)² + 7
Substitute x = 5, y = 3: (5 - 3)² + 7
= (2)² + 7
= 4 + 7 = 11
Correct option: (i)

(c) If x + 1/x = 2, the value of x² + 1/x² + 5 is :
(i) -1   (ii) 2   (iii) 9   (iv) 7
Answer:
Given: x + 1/x = 2
Squaring both sides: (x + 1/x)² = 2²
x² + 1/x² + 2(x)(1/x) = 4
x² + 1/x² + 2 = 4
x² + 1/x² = 2
Now evaluate x² + 1/x² + 5:
= 2 + 5 = 7
Correct option: (iv)

(d) If x - 1/x = 8, the value of x² + 1/x² - 8 is :
(i) 56   (ii) 58   (iii) 70   (iv) -4
Answer:
Given: x - 1/x = 8
Squaring both sides: (x - 1/x)² = 8²
x² + 1/x² - 2 = 64
x² + 1/x² = 66
Now evaluate x² + 1/x² - 8:
= 66 - 8 = 58
Correct option: (ii)

(e) If x² + 1/x² = 9, the value of x⁴ + 1/x⁴ + 5 is :
(i) 78   (ii) 86   (iii) 84   (iv) 81
Answer:
Given: x² + 1/x² = 9
Squaring both sides: (x² + 1/x²)² = 9²
x⁴ + 1/x⁴ + 2 = 81
x⁴ + 1/x⁴ = 79
Now evaluate x⁴ + 1/x⁴ + 5:
= 79 + 5 = 84
Correct option: (iii)

(f) If x² - 3x + 1 = 0, the value of x² + 1/x² + 1 is :
(i) 8   (ii) 10   (iii) 5   (iv) 10/9
Answer:
Given: x² - 3x + 1 = 0
Divide by x: x - 3 + 1/x = 0
x + 1/x = 3
Squaring both sides: x² + 1/x² + 2 = 9
x² + 1/x² = 7
Now evaluate x² + 1/x² + 1:
= 7 + 1 = 8
Correct option: (i)

2. Evaluate :
(i) (7/8 x + 4/5 y)²
Answer:
Using identity (a + b)² = a² + 2ab + b²
= (7/8 x)² + 2(7/8 x)(4/5 y) + (4/5 y)²
= 49/64 x² + 56/40 xy + 16/25 y²
= 49/64 x² + 7/5 xy + 16/25 y²

(ii) (2x/7 - 7y/4)²
Answer:
Using identity (a - b)² = a² - 2ab + b²
= (2x/7)² - 2(2x/7)(7y/4) + (7y/4)²
= 4x²/49 - 28xy/28 + 49y²/16
= 4x²/49 - xy + 49y²/16

3. Evaluate :
(a/2b + 2b/a)² - (a/2b - 2b/a)² - 4
Answer:
Using identity (x + y)² - (x - y)² = 4xy
Let x = a/2b and y = 2b/a
Expression becomes: 4(a/2b)(2b/a) - 4
= 4(1) - 4
= 4 - 4 = 0

4. If x + y = 7/2 and xy = 5/2; find :
(i) x - y
Answer:
Using identity: (x - y)² = (x + y)² - 4xy
(x - y)² = (7/2)² - 4(5/2)
(x - y)² = 49/4 - 10
(x - y)² = 49/4 - 40/4 = 9/4
x - y = ±3/2

(ii) x² - y²
Answer:
Using identity: x² - y² = (x + y)(x - y)
= (7/2)(±3/2)
= ±21/4

5. If a - b = 0.9 and ab = 0.36; find :
(i) a + b
Answer:
Using identity: (a + b)² = (a - b)² + 4ab
(a + b)² = (0.9)² + 4(0.36)
(a + b)² = 0.81 + 1.44
(a + b)² = 2.25
a + b = ±1.5

(ii) a² - b²
Answer:
Using identity: a² - b² = (a - b)(a + b)
= 0.9 × (±1.5)
= ±1.35

6. If a - b = 4 and a + b = 6; find :
(i) a² + b²
Answer:
Using identity: (a + b)² + (a - b)² = 2(a² + b²)
(6)² + (4)² = 2(a² + b²)
36 + 16 = 2(a² + b²)
52 = 2(a² + b²)
a² + b² = 26

(ii) ab
Answer:
Using identity: (a + b)² - (a - b)² = 4ab
(6)² - (4)² = 4ab
36 - 16 = 4ab
20 = 4ab
ab = 5

7. If a + 1/a = 6 and a ≠ 0; find :
(i) a - 1/a
Answer:
Using identity: (a - 1/a)² = (a + 1/a)² - 4
(a - 1/a)² = (6)² - 4
(a - 1/a)² = 36 - 4 = 32
a - 1/a = ±√32 = ±4√2

(ii) a² - 1/a²
Answer:
Using identity: a² - 1/a² = (a + 1/a)(a - 1/a)
= 6 × (±4√2)
= ±24√2

8. If a - 1/a = 8 and a ≠ 0; find :
(i) a + 1/a
Answer:
Using identity: (a + 1/a)² = (a - 1/a)² + 4
(a + 1/a)² = (8)² + 4
(a + 1/a)² = 64 + 4 = 68
a + 1/a = ±√68 = ±2√17

(ii) a² - 1/a²
Answer:
Using identity: a² - 1/a² = (a - 1/a)(a + 1/a)
= 8 × (±2√17)
= ±16√17

9. If a² - 3a + 1 = 0 and a ≠ 0; find :
(i) a + 1/a
Answer:
Given: a² - 3a + 1 = 0
Dividing by a: a - 3 + 1/a = 0
a + 1/a = 3

(ii) a² + 1/a²
Answer:
Squaring both sides of (a + 1/a) = 3
a² + 1/a² + 2 = 9
a² + 1/a² = 7

10. If a² - 5a - 1 = 0 and a ≠ 0; find :
(i) a - 1/a
Answer:
Given: a² - 5a - 1 = 0
Dividing by a: a - 5 - 1/a = 0
a - 1/a = 5

(ii) a + 1/a
Answer:
Using identity: (a + 1/a)² = (a - 1/a)² + 4
(a + 1/a)² = (5)² + 4
(a + 1/a)² = 25 + 4 = 29
a + 1/a = ±√29

(iii) a² - 1/a²
Answer:
Using identity: a² - 1/a² = (a - 1/a)(a + 1/a)
= 5 × (±√29)
= ±5√29

11. If 3x + 4y = 16 and xy = 4; find the value of 9x² + 16y².
Answer:
Given: 3x + 4y = 16
Squaring both sides: (3x + 4y)² = (16)²
9x² + 16y² + 2(3x)(4y) = 256
9x² + 16y² + 24xy = 256
Substitute xy = 4:
9x² + 16y² + 24(4) = 256
9x² + 16y² + 96 = 256
9x² + 16y² = 160

12. The difference between two positive numbers is 5 and the sum of their squares is 73. Find the product of these numbers.
Answer:
Let the numbers be x and y.
Given: x - y = 5 and x² + y² = 73
Using identity: (x - y)² = x² + y² - 2xy
(5)² = 73 - 2xy
25 = 73 - 2xy
2xy = 73 - 25
2xy = 48
xy = 24
The product of the numbers is 24.

EXERCISE 4 (B)

1. Multiple Choice Type : Choose the correct answer from the options given below.
(a) If x² - 2x + 1 = 0; the value of x⁴ + 1/x⁴ is equal to :
(i) 9   (ii) 7   (iii) 3   (iv) 2
Answer:
Given: x² - 2x + 1 = 0
(x - 1)² = 0 ⇒ x = 1
Value of x⁴ + 1/x⁴ = (1)⁴ + 1/(1)⁴ = 1 + 1 = 2
Correct option: (iv)

(b) If a = 5, b = -2 and c = -3, a³ + b³ + c³ is equal to :
(i) -90   (ii) 90   (iii) 60   (iv) -60
Answer:
Notice that a + b + c = 5 + (-2) + (-3) = 0
When a + b + c = 0, a³ + b³ + c³ = 3abc
= 3(5)(-2)(-3) = 3(30) = 90
Correct option: (ii)

(c) If x² + 1/x² = 2, the value of x² - 1/x² is :
(i) 0   (ii) 2   (iii) ±2   (iv) 8
Answer:
Using identity: (x² - 1/x²)² = (x² + 1/x²)² - 4(x²)(1/x²)
(x² - 1/x²)² = (2)² - 4 = 4 - 4 = 0
x² - 1/x² = 0
Correct option: (i)

(d) If x + 3y + 2z = 0, the value of x³ + 27y³ + 8z³ is :
(i) 9xyz   (ii) 6xyz   (iii) 18xyz   (iv) xyz
Answer:
Given: (x) + (3y) + (2z) = 0
Using conditional identity: A³ + B³ + C³ = 3ABC
x³ + (3y)³ + (2z)³ = 3(x)(3y)(2z)
x³ + 27y³ + 8z³ = 18xyz
Correct option: (iii)

2. Find the cube of :
(i) 3a - 2b
Answer:
(3a - 2b)³ = (3a)³ - (2b)³ - 3(3a)(2b)(3a - 2b)
= 27a³ - 8b³ - 18ab(3a - 2b)
= 27a³ - 8b³ - 54a²b + 36ab²

(ii) 5a + 3b
Answer:
(5a + 3b)³ = (5a)³ + (3b)³ + 3(5a)(3b)(5a + 3b)
= 125a³ + 27b³ + 45ab(5a + 3b)
= 125a³ + 27b³ + 225a²b + 135ab²

(iii) 2a + 1/2a (a ≠ 0)
Answer:
(2a + 1/2a)³ = (2a)³ + (1/2a)³ + 3(2a)(1/2a)(2a + 1/2a)
= 8a³ + 1/8a³ + 3(2a + 1/2a)
= 8a³ + 1/8a³ + 6a + 3/2a

(iv) 3a - 1/a (a ≠ 0)
Answer:
(3a - 1/a)³ = (3a)³ - (1/a)³ - 3(3a)(1/a)(3a - 1/a)
= 27a³ - 1/a³ - 9(3a - 1/a)
= 27a³ - 1/a³ - 27a + 9/a

3. If a² + 1/a² = 47 and a ≠ 0; find :
(i) a + 1/a
Answer:
(a + 1/a)² = a² + 1/a² + 2
(a + 1/a)² = 47 + 2 = 49
a + 1/a = ±7

(ii) a³ + 1/a³
Answer:
Using identity: a³ + 1/a³ = (a + 1/a)³ - 3(a + 1/a)
= (±7)³ - 3(±7)
= ±343 ∓ 21
= ±322

4. If a² + 1/a² = 18 and a ≠ 0; find :
(i) a - 1/a
Answer:
(a - 1/a)² = a² + 1/a² - 2
(a - 1/a)² = 18 - 2 = 16
a - 1/a = ±4

(ii) a³ - 1/a³
Answer:
Using identity: a³ - 1/a³ = (a - 1/a)³ + 3(a - 1/a)
= (±4)³ + 3(±4)
= ±64 ± 12
= ±76

5. If a + 1/a = p and a ≠ 0; then show that : a³ + 1/a³ = p(p² - 3)
Answer:
Given: a + 1/a = p
Cubing both sides: (a + 1/a)³ = p³
a³ + 1/a³ + 3(a)(1/a)(a + 1/a) = p³
a³ + 1/a³ + 3(p) = p³
a³ + 1/a³ = p³ - 3p
a³ + 1/a³ = p(p² - 3)
Hence proved.

6. If a + 2b = 5; then show that : a³ + 8b³ + 30ab = 125.
Answer:
Given: a + 2b = 5
Cubing both sides: (a + 2b)³ = 5³
a³ + (2b)³ + 3(a)(2b)(a + 2b) = 125
a³ + 8b³ + 6ab(5) = 125
a³ + 8b³ + 30ab = 125
Hence proved.

7. If (a + 1/a)² = 3 and a ≠ 0; then show that : a³ + 1/a³ = 0.
Answer:
Given: (a + 1/a)² = 3 ⇒ a + 1/a = ±√3
Using identity: a³ + 1/a³ = (a + 1/a)³ - 3(a + 1/a)
= (±√3)³ - 3(±√3)
= ±3√3 ∓ 3√3 = 0
Hence proved.

8. If a + 2b + c = 0; then show that : a³ + 8b³ + c³ = 6abc.
Answer:
Given: a + 2b + c = 0
Let A = a, B = 2b, C = c. Thus A + B + C = 0
When sum is zero, A³ + B³ + C³ = 3ABC
a³ + (2b)³ + c³ = 3(a)(2b)(c)
a³ + 8b³ + c³ = 6abc
Hence proved.

9. Use property to evaluate :
(i) 9³ - 5³ - 4³
Answer:
Expression can be written as 9³ + (-5)³ + (-4)³
Let a = 9, b = -5, c = -4
a + b + c = 9 - 5 - 4 = 0
Since a + b + c = 0, a³ + b³ + c³ = 3abc
= 3(9)(-5)(-4) = 540

(ii) 38³ + (-26)³ + (-12)³
Answer:
Let a = 38, b = -26, c = -12
a + b + c = 38 - 26 - 12 = 0
Since a + b + c = 0, a³ + b³ + c³ = 3abc
= 3(38)(-26)(-12) = 35568

10. If a ≠ 0 and a - 1/a = 3; find :
(i) a² + 1/a²
Answer:
Given: a - 1/a = 3
Squaring both sides: (a - 1/a)² = 9
a² + 1/a² - 2 = 9
a² + 1/a² = 11

(ii) a³ - 1/a³
Answer:
Using identity: a³ - 1/a³ = (a - 1/a)³ + 3(a - 1/a)
= (3)³ + 3(3)
= 27 + 9 = 36

11. If a ≠ 0 and a - 1/a = 4; find :
(i) a² + 1/a²
Answer:
Given: a - 1/a = 4
Squaring both sides: (a - 1/a)² = 16
a² + 1/a² - 2 = 16
a² + 1/a² = 18

(ii) a⁴ + 1/a⁴
Answer:
Squaring (a² + 1/a² = 18):
(a² + 1/a²)² = 324
a⁴ + 1/a⁴ + 2 = 324
a⁴ + 1/a⁴ = 322

(iii) a³ - 1/a³
Answer:
Using identity: a³ - 1/a³ = (a - 1/a)³ + 3(a - 1/a)
= (4)³ + 3(4)
= 64 + 12 = 76

12. If x ≠ 0 and x + 1/x = 2; then show that : x² + 1/x² = x³ + 1/x³ = x⁴ + 1/x⁴
Answer:
Given: x + 1/x = 2
Multiplying by x: x² + 1 = 2x ⇒ x² - 2x + 1 = 0
(x - 1)² = 0 ⇒ x = 1
Substitute x = 1 into each expression:
x² + 1/x² = 1 + 1 = 2
x³ + 1/x³ = 1 + 1 = 2
x⁴ + 1/x⁴ = 1 + 1 = 2
Therefore, x² + 1/x² = x³ + 1/x³ = x⁴ + 1/x⁴ = 2. Hence proved.

13. If 2x - 3y = 10 and xy = 16; find the value of 8x³ - 27y³.
Answer:
Given: 2x - 3y = 10
Cubing both sides: (2x - 3y)³ = 1000
(2x)³ - (3y)³ - 3(2x)(3y)(2x - 3y) = 1000
8x³ - 27y³ - 18xy(2x - 3y) = 1000
Substitute xy = 16 and (2x - 3y) = 10:
8x³ - 27y³ - 18(16)(10) = 1000
8x³ - 27y³ - 2880 = 1000
8x³ - 27y³ = 3880

14. Expand :
(i) (3x + 5y + 2z)(3x - 5y + 2z)
Answer:
Rearrange terms: ((3x + 2z) + 5y)((3x + 2z) - 5y)
Using identity (A + B)(A - B) = A² - B²
= (3x + 2z)² - (5y)²
= 9x² + 12xz + 4z² - 25y²

(ii) (3x - 5y - 2z)(3x - 5y + 2z)
Answer:
Rearrange terms: ((3x - 5y) - 2z)((3x - 5y) + 2z)
Using identity (A - B)(A + B) = A² - B²
= (3x - 5y)² - (2z)²
= 9x² - 30xy + 25y² - 4z²

15. The sum of two whole numbers is 9 and their product is 20. Find the sum of their :
(i) squares
Answer:
Let numbers be a and b.
Given: a + b = 9 and ab = 20
Using identity: a² + b² = (a + b)² - 2ab
= (9)² - 2(20)
= 81 - 40 = 41

(ii) cubes
Answer:
Using identity: a³ + b³ = (a + b)³ - 3ab(a + b)
= (9)³ - 3(20)(9)
= 729 - 540 = 189

16. Two positive numbers x and y are such that x > y. If the difference of these numbers is 5 and their product is 24, find :
(i) sum of these numbers.
Answer:
Given: x - y = 5 and xy = 24
Using identity: (x + y)² = (x - y)² + 4xy
(x + y)² = (5)² + 4(24)
(x + y)² = 25 + 96 = 121
Since numbers are positive, x + y = 11

(ii) difference of their cubes.
Answer:
Using identity: x³ - y³ = (x - y)³ + 3xy(x - y)
= (5)³ + 3(24)(5)
= 125 + 360 = 485

(iii) sum of their cubes.
Answer:
Using identity: x³ + y³ = (x + y)³ - 3xy(x + y)
= (11)³ - 3(24)(11)
= 1331 - 792 = 539

17. If 4x² + y² = a and xy = b, find the value of 2x + y.
Answer:
Using identity: (2x + y)² = (2x)² + y² + 2(2x)(y)
(2x + y)² = 4x² + y² + 4xy
Substitute given values: (2x + y)² = a + 4b
2x + y = ±√(a + 4b)

EXERCISE 4 (C)

1. Multiple Choice Type : Choose the correct answer from the options given below.
(a) The expansion of (x - 3y)(x + 5y) is :
(i) x² + 2xy + 15y²   (ii) x² + 2xy - 15y²   (iii) x² - 2xy + 15y²   (iv) x² - 2xy - 15y²
Answer:
(x - 3y)(x + 5y) = x² + 5xy - 3xy - 15y²
= x² + 2xy - 15y²
Correct option: (ii)

(b) x² - (a+b)x + ab is the expansion of :
(i) (x - b)(x - a)   (ii) (x - b)(x + a)   (iii) (x + b)(x - a)   (iv) (x + a)(x + b)
Answer:
(x - a)(x - b) = x(x - b) - a(x - b) = x² - bx - ax + ab
= x² - (a+b)x + ab
Correct option: (i)

(c) If a + b - c = 4 and a² + b² + c² = 14, the value of ab - bc - ca is :
(i) 2   (ii) 1   (iii) 0.5   (iv) -0.5
Answer:
Squaring both sides of a + b - c = 4:
(a + b - c)² = 16
a² + b² + c² + 2(ab - bc - ca) = 16
Substitute 14: 14 + 2(ab - bc - ca) = 16
2(ab - bc - ca) = 2
ab - bc - ca = 1
Correct option: (ii)

(d) (a - 1/2a)² is equal to :
(i) a² + 1/4a² - 2   (ii) a² + 1/4a² + 2   (iii) a² + 1/4a² - 1   (iv) a² - 1/4a² - 2
Answer:
(a - 1/2a)² = a² - 2(a)(1/2a) + (1/2a)²
= a² - 1 + 1/4a²
= a² + 1/4a² - 1
Correct option: (iii)

2. Expand :
(i) (x + 8)(x + 10)
Answer:
Using (x + a)(x + b) = x² + (a + b)x + ab
= x² + (8 + 10)x + (8)(10)
= x² + 18x + 80

(ii) (x + 8)(x - 10)
Answer:
= x² + (8 - 10)x + (8)(-10)
= x² - 2x - 80

(iii) (x - 8)(x + 10)
Answer:
= x² + (-8 + 10)x + (-8)(10)
= x² + 2x - 80

(iv) (x - 8)(x - 10)
Answer:
= x² + (-8 - 10)x + (-8)(-10)
= x² - 18x + 80

3. Expand :
(i) (2x - 1/x)(3x + 2/x)
Answer:
Multiply directly:
= (2x)(3x) + (2x)(2/x) - (1/x)(3x) - (1/x)(2/x)
= 6x² + 4 - 3 - 2/x²
= 6x² + 1 - 2/x²

(ii) (3a + 2/b)(2a - 3/b)
Answer:
Multiply directly:
= (3a)(2a) - (3a)(3/b) + (2/b)(2a) - (2/b)(3/b)
= 6a² - 9a/b + 4a/b - 6/b²
= 6a² - 5a/b - 6/b²

4. Expand :
(i) (x + y - z)²
Answer:
= x² + y² + (-z)² + 2(x)(y) + 2(y)(-z) + 2(x)(-z)
= x² + y² + z² + 2xy - 2yz - 2zx

(ii) (x - 2y + 2)²
Answer:
= x² + (-2y)² + 2² + 2(x)(-2y) + 2(-2y)(2) + 2(x)(2)
= x² + 4y² + 4 - 4xy - 8y + 4x

(iii) (5a - 3b + c)²
Answer:
= (5a)² + (-3b)² + c² + 2(5a)(-3b) + 2(-3b)(c) + 2(5a)(c)
= 25a² + 9b² + c² - 30ab - 6bc + 10ac

(iv) (5x - 3y - 2)²
Answer:
= (5x)² + (-3y)² + (-2)² + 2(5x)(-3y) + 2(-3y)(-2) + 2(5x)(-2)
= 25x² + 9y² + 4 - 30xy + 12y - 20x

(v) (x - 1/x + 5)²
Answer:
= x² + (-1/x)² + 5² + 2(x)(-1/x) + 2(-1/x)(5) + 2(x)(5)
= x² + 1/x² + 25 - 2 - 10/x + 10x
= x² + 1/x² + 23 + 10x - 10/x

5. If a + b + c = 12 and a² + b² + c² = 50; find ab + bc + ca.
Answer:
Using identity: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
(12)² = 50 + 2(ab + bc + ca)
144 = 50 + 2(ab + bc + ca)
94 = 2(ab + bc + ca)
ab + bc + ca = 47

6. If a² + b² + c² = 35 and ab + bc + ca = 23; find a + b + c.
Answer:
Using identity: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
(a + b + c)² = 35 + 2(23)
(a + b + c)² = 35 + 46 = 81
a + b + c = ±9

7. If a + b + c = p and ab + bc + ca = q; find a² + b² + c².
Answer:
Using identity: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
(p)² = a² + b² + c² + 2(q)
a² + b² + c² = p² - 2q

8. If a² + b² + c² = 50 and ab + bc + ca = 47, find a + b + c.
Answer:
Using identity: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
(a + b + c)² = 50 + 2(47)
(a + b + c)² = 50 + 94 = 144
a + b + c = ±12

9. If x + y - z = 4 and x² + y² + z² = 30, then find the value of xy - yz - zx.
Answer:
Using identity: (x + y - z)² = x² + y² + z² + 2(xy - yz - zx)
(4)² = 30 + 2(xy - yz - zx)
16 = 30 + 2(xy - yz - zx)
2(xy - yz - zx) = 16 - 30 = -14
xy - yz - zx = -7

10. The longest rod that can be placed in a rectangular box is 20 cm and the sum of its length, breadth and height is 30 cm. Find the total surface area of the box.
Answer:
Length of diagonal (longest rod) = √(l² + b² + h²) = 20
l² + b² + h² = 400
Sum of dimensions = l + b + h = 30
Total Surface Area (TSA) = 2(lb + bh + hl)
Using identity: (l + b + h)² = l² + b² + h² + 2(lb + bh + hl)
(30)² = 400 + TSA
900 = 400 + TSA
TSA = 500 cm²

EXERCISE 4 (D)

1. Multiple Choice Type : Choose the correct answer from the options given below.
(a) If a + 1/a = 2.5 and a - 1/a = 1.5, the value of (a + 1/a)² - (a - 1/a)² is :
(i) 4   (ii) 2   (iii) 1   (iv) 8.5
Answer:
Using formula (A)² - (B)²:
(2.5)² - (1.5)²
= 6.25 - 2.25 = 4
Correct option: (i)

(b) In (3x + 2)(x - 4), the coefficient of x is:
(i) -14   (ii) 8   (iii) -10   (iv) 3
Answer:
Expanding: (3x)(x) + (3x)(-4) + (2)(x) + (2)(-4)
= 3x² - 12x + 2x - 8
= 3x² - 10x - 8
Coefficient of x is -10.
Correct option: (iii)

(c) (x + y - z)(x - y + z) is equal to :
(i) x² - y² - z² - 2yz   (ii) x² - y² - z² + 2yz   (iii) x² - y² + z² - 2yz   (iv) x² - y² + z² + 2yz
Answer:
Rearrange grouping: (x + (y - z))(x - (y - z))
Using (A + B)(A - B) = A² - B²
= x² - (y - z)²
= x² - (y² + z² - 2yz)
= x² - y² - z² + 2yz
Correct option: (ii)

(d) If (3x - 4y)² = 9x² + axy + 16y²; the value of a is :
(i) -24   (ii) 24   (iii) -12   (iv) 12
Answer:
Expanding left side: (3x)² - 2(3x)(4y) + (4y)²
= 9x² - 24xy + 16y²
Comparing with 9x² + axy + 16y², we get a = -24.
Correct option: (i)

2. If x + 2y + 3z = 0 and x³ + 4y³ + 9z³ = 18xyz; evaluate : (x+2y)²/xy + (2y+3z)²/yz + (3z+x)²/zx
Answer:
From x + 2y + 3z = 0, we can write:
x + 2y = -3z
2y + 3z = -x
3z + x = -2y
Substitute these into the given expression:
(-3z)²/xy + (-x)²/yz + (-2y)²/zx
= 9z²/xy + x²/yz + 4y²/zx
Taking LCM as xyz:
= (9z³ + x³ + 4y³)/xyz
We are given x³ + 4y³ + 9z³ = 18xyz (Note: question likely implies (2y)^3 and (3z)^3 sum).
Substitute 18xyz in numerator:
= 18xyz / xyz = 18

3. If a + 1/a = m and a ≠ 0; find in terms of 'm'; the value of:
(i) a - 1/a
Answer:
Using identity: (a - 1/a)² = (a + 1/a)² - 4
(a - 1/a)² = m² - 4
a - 1/a = ±√(m² - 4)

(ii) a² - 1/a²
Answer:
Using identity: a² - 1/a² = (a + 1/a)(a - 1/a)
= m × (±√(m² - 4))
= ±m√(m² - 4)

4. In the expansion of (2x² - 8)(x - 4)²; find the value of :
(i) coefficient of x³
Answer:
First expand: (2x² - 8)(x² - 8x + 16)
= 2x⁴ - 16x³ + 32x² - 8x² + 64x - 128
= 2x⁴ - 16x³ + 24x² + 64x - 128
The coefficient of x³ is -16.

(ii) coefficient of x²
Answer:
From above expansion, the x² term is 24x².
The coefficient of x² is 24.

(iii) constant term.
Answer:
From above expansion, the term without x is -128.
The constant term is -128.

5. If x > 0 and x² + 1/9x² = 25/36, find : x³ + 1/27x³.
Answer:
Using (x + 1/3x)² = x² + 1/9x² + 2(x)(1/3x)
(x + 1/3x)² = 25/36 + 2/3
(x + 1/3x)² = 25/36 + 24/36 = 49/36
Since x > 0, x + 1/3x = 7/6
Now cube: (x + 1/3x)³ = x³ + 1/27x³ + 3(x)(1/3x)(x + 1/3x)
(7/6)³ = x³ + 1/27x³ + 1(7/6)
343/216 = x³ + 1/27x³ + 7/6
x³ + 1/27x³ = 343/216 - 252/216 = 91/216

6. If 2(x² + 1) = 5x, find :
(i) x - 1/x
Answer:
Given: 2x² + 2 = 5x
Divide by 2x: x + 1/x = 5/2
Using identity: (x - 1/x)² = (x + 1/x)² - 4
= (5/2)² - 4 = 25/4 - 16/4 = 9/4
x - 1/x = ±3/2

(ii) x³ - 1/x³
Answer:
Using identity: x³ - 1/x³ = (x - 1/x)³ + 3(x - 1/x)
= (±3/2)³ + 3(±3/2)
= ±27/8 ± 9/2 = ±27/8 ± 36/8 = ±63/8

7. If a² + b² = 34 and ab = 12; find :
(i) 3(a+b)² + 5(a-b)²
Answer:
First find (a+b)² and (a-b)²
(a+b)² = a² + b² + 2ab = 34 + 24 = 58
(a-b)² = a² + b² - 2ab = 34 - 24 = 10
Evaluate expression: 3(58) + 5(10)
= 174 + 50 = 224

(ii) 7(a-b)² - 2(a+b)²
Answer:
Using values from part (i):
= 7(10) - 2(58)
= 70 - 116 = -46

8. If 3x - 4/x = 4 and x ≠ 0; find : 27x³ - 64/x³.
Answer:
Cubing both sides: (3x - 4/x)³ = 4³
(3x)³ - (4/x)³ - 3(3x)(4/x)(3x - 4/x) = 64
27x³ - 64/x³ - 36(4) = 64
27x³ - 64/x³ - 144 = 64
27x³ - 64/x³ = 208

9. If x² + 1/x² = 7 and x ≠ 0; find the value of : 7x³ + 8x - 7/x³ - 8/x.
Answer:
Rearrange expression: 7(x³ - 1/x³) + 8(x - 1/x)
Given: x² + 1/x² = 7
(x - 1/x)² = 7 - 2 = 5 ⇒ x - 1/x = ±√5
x³ - 1/x³ = (x - 1/x)³ + 3(x - 1/x) = (±√5)³ + 3(±√5) = ±5√5 ± 3√5 = ±8√5
Value = 7(±8√5) + 8(±√5)
= ±56√5 ± 8√5 = ±64√5

10. If x = 1/(x-5) and x ≠ 5, find : x² - 1/x².
Answer:
Given: x = 1/(x - 5)
x(x - 5) = 1 ⇒ x² - 5x = 1 ⇒ x - 5 = 1/x ⇒ x - 1/x = 5
Now find x + 1/x: (x + 1/x)² = (x - 1/x)² + 4 = 25 + 4 = 29 ⇒ x + 1/x = ±√29
x² - 1/x² = (x - 1/x)(x + 1/x)
= 5(±√29) = ±5√29

11. If x = 1/(5-x) and x ≠ 5, find : x³ + 1/x³.
Answer:
Given: x = 1/(5 - x)
x(5 - x) = 1 ⇒ 5x - x² = 1 ⇒ x² - 5x + 1 = 0
Divide by x: x - 5 + 1/x = 0 ⇒ x + 1/x = 5
x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x)
= 5³ - 3(5) = 125 - 15 = 110

12. If 3a + 5b + 4c = 0, show that : 27a³ + 125b³ + 64c³ = 180abc.
Answer:
Let x = 3a, y = 5b, z = 4c.
Since x + y + z = 0, x³ + y³ + z³ = 3xyz
(3a)³ + (5b)³ + (4c)³ = 3(3a)(5b)(4c)
27a³ + 125b³ + 64c³ = 180abc.
Hence proved.

13. The sum of two whole numbers is 7 and the sum of their cubes is 133, find the sum of their squares.
Answer:
Let numbers be x and y.
Given: x + y = 7 and x³ + y³ = 133
x³ + y³ = (x + y)³ - 3xy(x + y)
133 = 7³ - 3xy(7)
133 = 343 - 21xy
21xy = 210 ⇒ xy = 10
Sum of squares: x² + y² = (x + y)² - 2xy
= (7)² - 2(10) = 49 - 20 = 29

14. In each of the following, find the value of 'a':
(i) 4x² + ax + 9 = (2x + 3)²
Answer:
Expand RHS: (2x + 3)² = 4x² + 12x + 9
Compare with LHS: 4x² + ax + 9 = 4x² + 12x + 9
Hence, a = 12

(ii) 4x² + ax + 9 = (2x - 3)²
Answer:
Expand RHS: (2x - 3)² = 4x² - 12x + 9
Compare with LHS: 4x² + ax + 9 = 4x² - 12x + 9
Hence, a = -12

(iii) 9x² + (7a - 5)x + 25 = (3x + 5)²
Answer:
Expand RHS: (3x + 5)² = 9x² + 30x + 25
Compare coefficients of x: 7a - 5 = 30
7a = 35 ⇒ a = 5

15. If (x² + 1)/x = 3 1/3 and x > 1; find
(i) x - 1/x
Answer:
Given: x + 1/x = 10/3
Using identity: (x - 1/x)² = (x + 1/x)² - 4
(x - 1/x)² = (10/3)² - 4 = 100/9 - 36/9 = 64/9
Since x > 1, x - 1/x is positive, so x - 1/x = 8/3

(ii) x³ - 1/x³
Answer:
Using identity: x³ - 1/x³ = (x - 1/x)³ + 3(x - 1/x)
= (8/3)³ + 3(8/3)
= 512/27 + 8 = 512/27 + 216/27 = 728/27

16. The difference between two positive numbers is 4 and the difference between their cubes is 316. Find :
(i) their product.
Answer:
Let numbers be x and y. Given x - y = 4, x³ - y³ = 316
Using identity: x³ - y³ = (x - y)³ + 3xy(x - y)
316 = (4)³ + 3xy(4)
316 = 64 + 12xy
12xy = 252 ⇒ xy = 21
Their product is 21.

(ii) the sum of their squares.
Answer:
Using identity: x² + y² = (x - y)² + 2xy
= (4)² + 2(21)
= 16 + 42 = 58

TEST YOURSELF

1. Multiple Choice Type : Choose the correct answer from the options given below.
(a) If x² + 1/x² = 3, the value of x - 1/x is :
(i) 1   (ii) -1   (iii) ±1   (iv) 0
Answer:
(x - 1/x)² = x² + 1/x² - 2
(x - 1/x)² = 3 - 2 = 1
x - 1/x = ±1
Correct option: (iii)

(b) If a³ + b³ + c³ = 3abc, then a + b + c is equal to :
(i) 1   (ii) -1   (iii) ±1   (iv) 0
Answer:
The standard algebraic condition states if a + b + c = 0, then a³ + b³ + c³ = 3abc.
Therefore, a + b + c = 0.
Correct option: (iv)

(c) (196 × 196 × 196 + 204 × 204 × 204) / ((196)² + (204)² - 196 × 204) is equal to :
(i) 400   (ii) -8   (iii) 8   (iv) none of these
Answer:
The expression is of the form (a³ + b³) / (a² + b² - ab)
Using identity (a³ + b³) = (a + b)(a² - ab + b²)
The expression simplifies to (a + b)
= 196 + 204 = 400
Correct option: (i)

(d) If a = x + 1/x and b = x - 1/x, then a² - b² is :
(i) x² + 1/x²   (ii) x² - 1/x²   (iii) 4   (iv) 2(x² + 1/x²)
Answer:
a² - b² = (a + b)(a - b)
a + b = (x + 1/x) + (x - 1/x) = 2x
a - b = (x + 1/x) - (x - 1/x) = 2/x
(2x)(2/x) = 4
Correct option: (iii)

(e) Statement (1) : x > 0 and x² + 1/x² = 2, then x² - 1/x² = 0.
Statement (2) : (x + 1/x)² = x² + 1/x² + 2 = 2 + 2 = 4; (x - 1/x)² = x² + 1/x² - 2 = 2 - 2 = 0 and, x² - 1/x² = (x + 1/x)(x - 1/x).
(i) Both the statements are true.   (ii) Both the statements are false.   (iii) Statement 1 is true, and statement 2 is false.   (iv) Statement 1 is false, and statement 2 is true.
Answer:
Statement 2 shows the valid mathematical derivation identity.
By substituting the result into Statement 1, we see x² - 1/x² = (2)(0) = 0, which is true.
Both statements are mathematically sound and true.
Correct option: (i)

(f) Assertion (A) : x² - 5x - 1 = 0 ⇒ x - 1/x = 5 is true.
Reason (R) : x² - 5x - 1 = 0 ⇒ x - 1/x = 5. But x - 1/x = 5 is true when x ≠ 0
(i) (1)   (ii) (2)   (iii) (3)   (iv) (4)
Answer:
The division step by x requires that x ≠ 0. Since 0² - 5(0) - 1 ≠ 0, x is not 0.
Both Assertion and Reason are true and R provides the correct reason constraint.
Correct option: (iii)

(g) Assertion (A) : (ab+bc+ca-a²-b²-c²)/(3abc-a³-b³-c³) = 1/(a+b+c)
Reason (R) : (ab+bc+ca-a²-b²-c²)/(3abc-a³-b³-c³) = ... (identity simplification) = 1/(a+b+c)
(i) (1)   (ii) (2)   (iii) (3)   (iv) (4)
Answer:
The Reason demonstrates the step-by-step factorization identity for A.
Both A and R are true and R is the correct reason.
Correct option: (iii)

(h) Assertion (A) : If x > y, x + y = 6 and x - y = 2 then x² + y² = 40.
Reason (R) : (x + y)² + (x - y)² = 2(x² + y²)
(i) (1)   (ii) (2)   (iii) (3)   (iv) (4)
Answer:
Using Reason: 6² + 2² = 2(x² + y²) ⇒ 36 + 4 = 40 = 2(x² + y²) ⇒ x² + y² = 20.
Thus, Assertion states x² + y² = 40 which is false.
Assertion is false, Reason is true.
Correct option: (ii)

(i) Assertion (A) : 5³ - 3³ - 2³ = 3 × 5 × -3 × -2
Reason (R) : ∵ 5 - 3 - 2 = 0 ⇒ 5³ - 3³ - 2³ = 5³ + (-3)³ + (-2)³
(i) (1)   (ii) (2)   (iii) (3)   (iv) (4)
Answer:
Since a + b + c = 0, sum of cubes is 3abc. Reason establishes this conditional correctly.
Both A and R are true and R is correct reason.
Correct option: (iii)

2. Simplify :
(i) (x + 6)(x + 4)(x - 2)
Answer:
First multiply (x + 6)(x + 4) = x² + 10x + 24
Now multiply by (x - 2): (x² + 10x + 24)(x - 2)
= x³ - 2x² + 10x² - 20x + 24x - 48
= x³ + 8x² + 4x - 48

(ii) (x - 6)(x - 4)(x + 2)
Answer:
First multiply (x - 6)(x - 4) = x² - 10x + 24
Now multiply by (x + 2): (x² - 10x + 24)(x + 2)
= x³ + 2x² - 10x² - 20x + 24x + 48
= x³ - 8x² + 4x + 48

(iii) (x - 6)(x - 4)(x - 2)
Answer:
First multiply (x - 6)(x - 4) = x² - 10x + 24
Now multiply by (x - 2): (x² - 10x + 24)(x - 2)
= x³ - 2x² - 10x² + 20x + 24x - 48
= x³ - 12x² + 44x - 48

(iv) (x + 6)(x - 4)(x - 2)
Answer:
First multiply (x + 6)(x - 4) = x² + 2x - 24
Now multiply by (x - 2): (x² + 2x - 24)(x - 2)
= x³ - 2x² + 2x² - 4x - 24x + 48
= x³ - 28x + 48

3. Simplify using following identity : (a ± b)(a² ∓ ab + b²) = a³ ± b³
(i) (2x + 3y)(4x² - 6xy + 9y²)
Answer:
Let a = 2x and b = 3y. The expression matches (a + b)(a² - ab + b²)
Result is a³ + b³
= (2x)³ + (3y)³
= 8x³ + 27y³

(ii) (a/3 - 3b)(a²/9 + ab + 9b²)
Answer:
Let x = a/3 and y = 3b. Matches (x - y)(x² + xy + y²)
Result is x³ - y³
= (a/3)³ - (3b)³
= a³/27 - 27b³

4. Using suitable identity, evaluate :
(i) (104)³
Answer:
(104)³ = (100 + 4)³
Using (a + b)³ = a³ + b³ + 3ab(a + b)
= (100)³ + (4)³ + 3(100)(4)(100 + 4)
= 1000000 + 64 + 1200(104)
= 1000000 + 64 + 124800
= 1124864

(ii) (97)³
Answer:
(97)³ = (100 - 3)³
Using (a - b)³ = a³ - b³ - 3ab(a - b)
= (100)³ - (3)³ - 3(100)(3)(100 - 3)
= 1000000 - 27 - 900(97)
= 1000000 - 27 - 87300
= 912673

5. Simplify : ((x² - y²)³ + (y² - z²)³ + (z² - x²)³) / ((x - y)³ + (y - z)³ + (z - x)³)
Answer:
Let A = x² - y², B = y² - z², C = z² - x². A + B + C = 0. Numerator = 3ABC.
Let P = x - y, Q = y - z, R = z - x. P + Q + R = 0. Denominator = 3PQR.
Expression = 3(x² - y²)(y² - z²)(z² - x²) / 3(x - y)(y - z)(z - x)
= ((x - y)(x + y)(y - z)(y + z)(z - x)(z + x)) / ((x - y)(y - z)(z - x))
Cancel common terms:
= (x + y)(y + z)(z + x)

6. Evaluate : (0.8 × 0.8 × 0.8 + 0.5 × 0.5 × 0.5) / (0.8 × 0.8 - 0.8 × 0.5 + 0.5 × 0.5)
Answer:
Let a = 0.8 and b = 0.5
Expression is (a³ + b³) / (a² - ab + b²)
This simplifies to (a + b)
= 0.8 + 0.5
= 1.3

7. If a - 2b + 3c = 0; state the value of a³ - 8b³ + 27c³.
Answer:
Let A = a, B = -2b, C = 3c. Given A + B + C = 0
Therefore, A³ + B³ + C³ = 3ABC
a³ + (-2b)³ + (3c)³ = 3(a)(-2b)(3c)
a³ - 8b³ + 27c³ = -18abc

8. If x + 5y = 10; find the value of x³ + 125y³ + 150xy - 1000.
Answer:
Rewrite equation: x + 5y - 10 = 0
Let A = x, B = 5y, C = -10. Since A + B + C = 0, A³ + B³ + C³ - 3ABC = 0
x³ + (5y)³ + (-10)³ - 3(x)(5y)(-10) = 0
x³ + 125y³ - 1000 + 150xy = 0
The value is 0.

9. If a + b = 11 and a² + b² = 65; find a³ + b³.
Answer:
First find ab: (a + b)² = a² + b² + 2ab
121 = 65 + 2ab
2ab = 56 ⇒ ab = 28
Using identity: a³ + b³ = (a + b)(a² - ab + b²)
= (11)(65 - 28)
= 11(37)
= 407

10. If x, y and z are three different numbers, then prove that : x² + y² + z² - xy - yz - zx is always positive.
Answer:
Multiply and divide expression by 2: 1/2 [2x² + 2y² + 2z² - 2xy - 2yz - 2zx]
Rearrange terms: 1/2 [(x² - 2xy + y²) + (y² - 2yz + z²) + (z² - 2zx + x²)]
= 1/2 [(x - y)² + (y - z)² + (z - x)²]
Since squares of real numbers are non-negative, and x, y, z are different, the sum of squares is > 0.
Hence, it is always positive. Proved.

11. Given x = √{65} and y = √{64}, find the value of (x - y) - 1 / (x² + xy + y²).
Answer:
Multiply and divide the second term by (x - y):
= (x - y) - (x - y) / ((x - y)(x² + xy + y²))
= (x - y) - (x - y) / (x³ - y³)
Substitute x³ = 65 and y³ = 64:
= (x - y) - (x - y) / (65 - 64)
= (x - y) - (x - y) / 1
= (x - y) - (x - y) = 0