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**Exercise 4(A)**

**1. Find the square of:**

**(i) 2a + b**

**(ii) 3a + 7b**

**(iii) 3a – 4b**

**(iv) 3a/2b â€“ 2b/3aÂ **

**Solution:**

Using the identities,

(a + b)^{2} = a^{2} + b^{2} + 2ab and

(a – b)^{2} = a^{2} + b^{2} – 2ab

(i) (2a + b)^{2} = (2a)^{2} + b^{2} + 2(2a)(b)

= 4a^{2} + b^{2} + 4ab

(ii) (3a + 7b)^{2} = (3a)^{2} + (7b)^{2} + 2(3a)(7b)

= 9a^{2} + 49b^{2} + 42ab

(iii) (3a â€“ 4b)^{2} = (3a)^{2} + (4b)^{2} – 2(3a)(4b)

= 9a^{2} + 16b^{2} – 24ab

(iv) (3a/2b â€“ 2b/3a)^{2} = (3a/2b)^{2} + (2b/3a)^{2} â€“ 2(3a/2b)(2b/3a)

= 9a^{2}/4b^{2} + 4b^{2}/9a^{2} â€“ 2

**2. Use identities to evaluate:**

**(i) (101) ^{2}**

**(ii) (502) ^{2}**

**(iii) (97) ^{2}**

**(iv) (998) ^{2}**

**Solution:**

Using the identities,

(a + b)^{2} = a^{2} + b^{2} + 2ab and

(a – b)^{2} = a^{2} + b^{2} – 2ab

(i) (101)^{2} = (100 + 1)^{2}

= 100^{2} + 1^{2} + 2Ã—100Ã—1

= 10000 + 1 + 200

= 10201

(ii) (502)^{2} = (500 + 2)^{2}

= 500^{2} + 2^{2} + 2Ã—500Ã—2

= 250000 + 4 + 2000

= 252004

**Â **(iii) (97)^{2} = (100 – 3)^{2}

= 100^{2} + 3^{2} – 2Ã—100Ã—3

= 10000 + 9 – 600

= 9409

(iv) (998)^{2} = (1000 – 2)^{2}

= 1000^{2} + 2^{2} – 2Ã—1000Ã—2

= 100000 + 4 – 4000

= 996004

**3. Evalute:**

**(i) (7x/8 + 4y/5) ^{2}**

**(ii) (2x/7 â€“ 7y/4) ^{2}**

**Solution: **

Using the identities,

(a + b)^{2} = a^{2} + b^{2} + 2ab and

(a – b)^{2} = a^{2} + b^{2} – 2ab

(i) We have, (7x/8 + 4y/5)^{2}

= (7x/8)^{2} + (4y/5)^{2} + 2 Ã— (7x/8) Ã— (4y/5)

= 49x^{2}/64 + 16y^{2}/25 + 7xy/5

(ii)Â We have, (2x/7 – 7y/4)^{2}

= (2x/7)^{2} + (7y/4)^{2} â€“ 2 Ã— (2x/7) Ã— (7y/4)

= 4x^{2}/49 + 49y^{2}/16 â€“ xy

**4. Evaluate:**

**(i)Â (a/2b + 2b/a) ^{2} â€“ (a/2b â€“ 2b/a)^{2} – 4**

**(ii) (4a + 3b) ^{2}Â – (4a – 3b)^{2}Â + 48ab**

**Solution:**

Using the identities,

(a + b)^{2} = a^{2} + b^{2} + 2ab and

(a – b)^{2} = a^{2} + b^{2} – 2ab

(i) Given expression,** _{}**

On expanding the term using the identity, we have

(a/2b + 2b/a)^{2} = (a/2b)^{2} + (2b/a)^{2} + 2 Ã— (a/2b) Ã— (2b/a)

= a^{2}/4b + 4b^{2}/a^{2} + 2

Next, expanding the second term using the identity, we have

(a/2b â€“ 2b/a)^{2}** **= (a/2b)^{2} + (2b/a)^{2} – 2 Ã— (a/2b) Ã— (2b/a)

= a^{2}/4b + 4b^{2}/a^{2} â€“ 2

Now, using these results in the given expression

(a/2b + 2b/a)^{2} – (a/2b â€“ 2b/a)^{2} – 4 = (a^{2}/4b + 4b^{2}/a^{2} + 2) â€“ (a^{2}/4b + 4b^{2}/a^{2} â€“ 2) â€“ 4

= a^{2}/4b + 4b^{2}/a^{2} + 2 â€“ a^{2}/4b â€“ 4b^{2}/a^{2} + 2 â€“ 4

= 0

**Â **

(ii) Given expression, (4a + 3b)^{2}Â – (4a – 3b)^{2}Â + 48ab

On expanding the term using the identity, we have

(4a + 3b)^{2} = (4a)^{2} + (3b)^{2} + 2 Ã— (4a) Ã— (3b)

= 16a^{2} + 9b^{2} + 24ab

Next, expanding the second term using the identity, we have

(4a – 3b)^{2} = (4a)^{2} + (3b)^{2} – 2 Ã— (4a) Ã— (3b)

= 16a^{2} + 9b^{2} â€“ 24ab

Now, using these results in the given expression

(4a + 3b)^{2} – (4a – 3b)^{2} + 48ab = (16a^{2} + 9b^{2} + 24ab) â€“ (16a^{2} + 9b^{2} – 24ab) + 48ab

= 16a^{2} + 9b^{2} + 24ab â€“ 16a^{2} – 9b^{2} + 24ab + 48ab

= 96ab

**5. If a + b = 7 and ab = 10; find a – b.**

**Solution:**

Using the identities,

(a + b)^{2} = a^{2} + b^{2} + 2ab and

(a – b)^{2} = a^{2} + b^{2} – 2ab

Rewriting the above equation as

(a – b)^{2} = a^{2} + b^{2} + 2ab â€“ 4ab

= (a + b)^{2} â€“ 4ab â€¦ (i)

We have,

a + b = 7 and ab = 10

So, using these in equation (i), we get

(a – b)^{2} = (7)^{2} â€“ (4 Ã— 10)

= 49 â€“ 40

= 9

Then,

(a – b) = âˆš9

= Â±3

**6. If a – b = 7 and ab = 18; find a + b.**

**Solution:**

Using the identities,

(a – b)^{2} = a^{2} + b^{2} – 2ab and

(a + b)^{2} = a^{2} + b^{2} + 2ab

Rewriting the above equation as

(a + b)^{2} = a^{2} + b^{2} – 2ab + 4ab

= (a – b)^{2} + 4ab â€¦ (i)

We have,

a – b = 7 and ab = 18

So, using these in equation (i), we get

(a + b)^{2} = (7)^{2} + (4 Ã— 18)

= 49 + 72

= 121

Then,

(a + b) = âˆš121

= Â±11

**7. If x + y =Â 7/2 and xy =Â 5/2; find:**

**(i) x – yÂ **

**(ii) x ^{2 }– y^{2}**

**Solution:**

Using the identities,

(x + y)^{2} = x^{2} + y^{2} + 2xy and

(x – y)^{2} = x^{2} + y^{2} â€“ 2xy

Rewriting the above equation as

(x – b)^{2} = x^{2} + y^{2} + 2xy â€“ 4xy

= (x + y)^{2} â€“ 4xy â€¦ (1)

We have,

x + y = 7/2 and xy = 5/2

So, using these in equation (1), we get

(x â€“ y)^{2} = (7/2)^{2} â€“ (4 Ã— 5/2)

= 49/4 â€“ 10

= 9/4

Then,

(x – y) = âˆš(9/4)

= Â± 3/2 â€¦ (2)

**Â **

(ii) We know that,

x^{2} â€“ y^{2} = (x + y) (x – y)

Substituting values in RHS using given and (2), we get

x^{2} â€“ y^{2} = (7/2) (Â± 3/2)

= Â± 21/4

**8. If a – b = 0.9 and ab = 0.36; find:**

**(i) a + b**

**(ii) a ^{2}Â – b^{2}**

**Solution:**

Using the identities,

(a – b)^{2} = a^{2} + b^{2} â€“ 2ab and

(a + b)^{2} = a^{2} + b^{2} + 2ab

Rewriting the above equation as

(a + b)^{2} = a^{2} + b^{2} â€“ 2ab + 4ab

= (a – b)^{2} + 4ab â€¦ (1)

We have,

a – b = 0.9 and ab = 0.36

So, using these in equation (1), we get

(a + b)^{2} = (0.9)^{2} + (4 Ã— 0.36)

= 0.81 + 1.44

= 2.25

Then,

(a + b) = âˆš2.25

= Â±1.5 â€¦ (2)

**Â **

(ii) We know that,

a^{2} â€“ b^{2} = (a + b) (a – b)

Substituting values in RHS using given and (2), we get

a^{2} â€“ b^{2} = (Â±1.5) (0.9)

= Â± 1.35

**9. If a – b = 4 and a + b = 6; find**

**(i) a ^{2}Â + b^{2}**

**(ii) ab**

**Solution:**

Given, a – b = 4 and a + b = 6

We know that,

(a – b)^{2} = a^{2} + b^{2} – 2ab and

(a + b)^{2} = a^{2} + b^{2} + 2ab

Now, rewriting the above equation as

(a + b)^{2} = a^{2} + b^{2} â€“ 2ab + 4ab

â‡’ (a + b)^{2} = (a – b)^{2} + 4ab

Substituting the values in the above equation, we get

(6)^{2} = (4)^{2} + 4ab

36 = 16 + 4ab

4ab = 36 â€“ 16

4ab = 20

ab = 20/4

(ii) Thus, ab = 5

Now, in the identity: (a + b)^{2} = a^{2} + b^{2} + 2ab

(a + b)^{2} = (a^{2} + b^{2}) + 2ab

Letâ€™s substitute the values of the known terms,

(6)^{2} = (a^{2} + b^{2}) + 2 Ã— (5)

36 = (a^{2} + b^{2}) + 10

a^{2} + b^{2} = 36 â€“ 10

(i) Thus, a^{2} + b^{2} = 26

**10. If a +Â 1/a = 6 andÂ aÂ â‰ 0Â find:**

**(i)Â a â€“ 1/a**

**(ii)Â a ^{2} â€“ 1/a^{2}**

**Solution:**

Using the identities,

(a – b)^{2} = a^{2} + b^{2} – 2ab and

(a + b)^{2} = a^{2} + b^{2} + 2ab

(i) Now,

(a + 1/a)^{2} = a^{2} + 1/a^{2} + (2 Ã— a Ã— 1/a)

= a^{2} + 1/a^{2} + 2 â€¦ (1)

Substituting the value of (a + 1/a) in the equation (1), we get

6^{2} = a^{2} + 1/a^{2} + 2

36 = a^{2} + 1/a^{2} + 2

a^{2} + 1/a^{2} = 36 â€“ 2 = 34 â€¦ (2)

Similarly,

(a â€“ 1/a)^{2} = a^{2} + 1/a^{2} â€“ (2 Ã— a Ã— 1/a)

= (a^{2} + 1/a^{2}) â€“ 2

= 34 â€“ 2 â€¦ [From (2)]

= 32

â‡’ (a â€“ 1/a)^{2} = 32

a â€“ 1/aÂ = Â±âˆš32

= Â±4âˆš2 â€¦ (3)

Thus, a â€“ 1/aÂ = Â±4âˆš2

(ii) We know that,

a^{2} â€“ 1/a^{2} = (a â€“ 1/a) (a + 1/a)

Using the given and (3) in the above equation,

a^{2} â€“ 1/a^{2} = (Â±4âˆš2) (6)

= Â±24âˆš2

Thus, a^{2} â€“ 1/a^{2}Â = Â±24âˆš2

**11. If a â€“ 1/a = 8 and aÂ â‰ 0,Â find:**

**(i) a + 1/a**

**(ii) a ^{2} â€“ 1/a^{2}**

**Solution:**

Using the identities,

(a – b)^{2} = a^{2} + b^{2} – 2ab and

(a + b)^{2} = a^{2} + b^{2} + 2ab

(i) Now,

(a – 1/a)^{2} = a^{2} + 1/a^{2} – (2 Ã— a Ã— 1/a)

= a^{2} + 1/a^{2} – 2 â€¦ (1)

Substituting the value of (a + 1/a) in the equation (1), we get

8^{2} = a^{2} + 1/a^{2} – 2

64 = a^{2} + 1/a^{2} – 2

a^{2} + 1/a^{2} = 64 + 2 = 66 â€¦ (2)

Similarly,

(a + 1/a)^{2} = a^{2} + 1/a^{2} + (2 Ã— a Ã— 1/a)

= (a^{2} + 1/a^{2}) + 2

= 66 + 2 â€¦ [From (2)]

= 68

â‡’ (a + 1/a)^{2} = 68

a + 1/aÂ = âˆš68

= Â±2âˆš17 â€¦ (3)

Thus, a + 1/aÂ = Â±2âˆš17

(ii) We know that,

a^{2} â€“ 1/a^{2} = (a â€“ 1/a) (a + 1/a)

Using the given and (3) in the above equation,

a^{2} â€“ 1/a^{2} = (8) (Â±2âˆš17)

= Â±16âˆš17

Thus, a^{2} â€“ 1/a^{2}Â = Â±16âˆš17

**Â **

**12. If a ^{2}Â – 3a + 1 = 0, and a â‰ 0;Â find:**

**(i)Â a + 1/a**

**(ii)Â a ^{2} + 1/a^{2}**

**Solution:**

(i) Given equation,

a^{2}Â – 3a + 1 = 0

a^{2}Â + 1 = 3a

(a^{2} + 1)/a = 3

â‡’ a + 1/a = 3 â€¦ (1)

(ii) We know that,

(a + b)^{2} = a^{2} + b^{2} + 2ab

Now,

(a + 1/a)^{2} = a^{2} + 1/a^{2} + 2(a)(1/a)

= a^{2} + 1/a^{2} + 2

Using (1) in the above equation, we get

(3)^{2} = a^{2} + 1/a^{2} + 2

9 = a^{2} + 1/a^{2} + 2

a^{2} + 1/a^{2} = 9 â€“ 2

Thus, a^{2} + 1/a^{2} = 7

**13. If a ^{2}Â – 5a – 1 = 0 and a â‰ 0; find:**

**(i)Â a â€“ 1/a**

**(ii)Â a + 1/a**

**(iii) a ^{2} â€“ 1/a^{2}**

**Solution:**

(i) Given, a^{2}Â – 5a – 1 = 0

Rewriting the equation, we get

a^{2}Â – 1 = 5a

(a^{2}Â – 1)/a = 5

Hence, a â€“ 1/a = 5 â€¦ (1)

(ii) We know that,

(a + 1/a)^{2} = a^{2} + 1/a^{2} + 2

Manipulating the above as,

(a + 1/a)^{2} = a^{2} + 1/a^{2} â€“ 2 + 4

(a + 1/a)^{2} = (a â€“ 1/a)^{2} + 4

Now, using (1) in the above

(a + 1/a)^{2} = (5)^{2} + 4

(a + 1/a)^{2} = 25 + 4 = 29

Hence, a + 1/a = Â±âˆš29 â€¦ (2)

(iii) We know that,

a^{2} â€“ 1/a^{2} = (a + 1/a) (a â€“ 1/a)

Now, using (1) and (2) in the above equation, we get

a^{2} â€“ 1/a^{2} = (5) x (Â±âˆš29)

Hence, a^{2} â€“ 1/a^{2} = Â±5âˆš29

**14. If 3a + 4b = 16 and ab = 4; find the value of 9a ^{2}Â + 16b^{2}.**

**Solution:**

Given, 3a + 4b = 16 and ab = 4

Required to find: value of 9a^{2}Â + 16b^{2}

We know that,

(a + b)^{2} = a^{2} + b^{2} + 2ab

Now, the square of (3a + 4b) will be

(3a + 4b)^{2} = (3a)^{2} + (4b)^{2} + 2 Ã— (3a) Ã— (4b)

= 9a^{2} + 4b^{2} + 24ab

And, given 3a + 4b = 16

So, by squaring on both the sides

(3a + 4b)^{2} = 16^{2}

9a^{2} + 4b^{2} + 24ab = 256

9a^{2} + 4b^{2} + 24(4) = 256 [Given ab = 4]

9a^{2} + 4b^{2} = 256 â€“ 96

â‡’ 9a^{2} + 4b^{2} = 160

**15. The number a is 2 more than the number b. If the sum of the squares of a and b is 34, then find the product of a and b.**

**Solution:**

Given, a is 2 more than b

â‡’ a = b + 2

And, sum of squares of a and b is 34

â‡’ a^{2Â }+ b^{2Â }= 34

Letâ€™s replace a = (b + 2) in the above equation and solve for b

Then,

(b + 2)^{2Â }+ b^{2Â }= 34

2b^{2Â }+ 4b â€“ 30 = 0

b^{2Â }+ 2b – 15 = 0

(b + 5) (b – 3) = 0

So,

b = -5 or 3

Now,

For b = -5, a =-5 + 2 = -3

For b = 3, a = 3 + 2 = 5

Thus, the product of a and b is 15 in both cases.

**Â **

**16. The difference between two positive numbers is 5 and the sum of their squares is 73. Find the product of these numbers.**

**Solution:**

Letâ€™s assume the two positive numbers asÂ a andÂ b

Given, the difference between them is 5 and the sum of their squares is 73

So, we have

a – b = 5 â€¦ (i) and

a^{2Â }+ b^{2Â }= 73 â€¦ (ii)

On squaring (i) on both sides, we get

(a – b)^{2Â }= 5^{2}

(a^{2Â }+ b^{2})^{Â }â€“ 2ab = 25

73^{Â }– 2ab = 25 â€¦ [Using (ii), given]

So,

2ab = 73 – 25 = 48

abÂ = 24Â

Therefore, the product of numbers is 24.

**Exercise 4(B)**

**1. Find the cube of:**

**(i) 3a – 2b**

**(ii) 5a + 3b**

**(iii)Â 2a + 1/2a **

**(iv)Â 3a â€“ 1/a (a** **â‰ 0)Â **

**Solution:**

Using the identities,

(a + b)^{3} = a^{3} + 3ab (a + b) + b^{3} and

(a – b)^{3 }= a^{3} – 3ab (a – b) + b^{3}

(i) (3a â€“ 2b)^{3} = (3a)^{3} â€“ 3 Ã— 3a Ã— 2b (3a â€“ 2b) – (2b)^{3}

= 27a^{3} â€“ 18ab (3a â€“ 2b) â€“ 8b^{3}

= 27a^{3} â€“ 54a^{2}b + 36ab^{2} â€“ 8b^{3}

(ii)Â (5a + 3b)^{3} = (5a)^{3} + 3 Ã— 5a Ã— 3b (5a + 3b) + (3b)^{3}

= 125a^{3} + 45ab (5a + 3b) + 27b^{3}

= 125a^{3} + 225a^{2}b + 135ab^{2} + 27b^{3}

(iii) (2a + 1/2a)^{3} = (2a)^{3} + 3 Ã— 2a Ã— 1/2a (2a + 1/2a) + (1/2a)^{3}

= 8a^{3} + 3 (2a + 1/2a) + 1/8a^{3}

= 8a^{3} + 6a + 3/2a + 1/8a^{3}

(iv) (3a â€“ 1/a)^{3} = (3a)^{3} â€“ 3 Ã— 3a Ã— 1/a (3a â€“ 1/a) – (1/a)^{3}

= 27a^{3} â€“ 9 (3a â€“ 1/a) â€“ 1/a^{3}

= 27a^{3} â€“ 27a + 9a â€“ 1/a^{3}

**2. If a ^{2}Â +Â 1/a^{2}_{ }= 47 andÂ aÂ â‰ 0Â find:**

**(i)Â a + 1/a**

**(ii)Â a ^{3} + 1/a^{3}**

**Solution:**

(i) Given, a^{2}Â +Â 1/a^{2}_{ }= 47

We know that,

(a + 1/a)^{2} = a^{2}Â +Â 1/a^{2} + 2 x a x 1/a

= (a^{2}Â +Â 1/a^{2}) + 2

= 47 + 2

= 49

So,

a + 1/a = âˆš49

= Â±7 â€¦ (1)

(ii) Using the identity

(a + b)^{3} = a^{3} + 3ab (a + b) + b^{3}

Now,

(a + 1/a)^{3} = a^{3} + 1/a^{3} + 3(a + 1/a)

a^{3} + 1/a^{3} = (a + 1/a)^{3} – 3(a + 1/a)

= (Â±7)^{3} â€“ 3(Â±7) â€¦ [From (1)]

= Â±343 – Â±21

Hence, a^{3} + 1/a^{3} = Â±322

**3. If a ^{2}Â +Â 1/a^{2} = 18; aÂ â‰ 0Â find:**

**(i) a â€“ 1/aÂ **

**(ii) a ^{3} â€“ 1/a^{3}Â **

**Solution:**

(i) Given, a^{2}Â +Â 1/a^{2} = 18

Using the identity (a + b)^{2} = a^{2} + b^{2} + 2ab

Now,

(a – 1/a)^{2} = a^{2}Â +Â 1/a^{2} – 2(a)(1/a)

= (a^{2}Â +Â 1/a^{2}) – 2

= 18 – 2

= 16

Hence,

a â€“ 1/a = âˆš16

= Â±4 â€¦ (1)

(ii) Using the identity,

(a – b)^{3 }= a^{3} – 3ab (a – b) + b^{3}

Now,

(a â€“ 1/a)^{3} = a^{3} – 3a(1/a) (a â€“ 1/a) + (1/a)^{3}

= a^{3} – 3 (a â€“ 1/a) + 1/a^{3}

a^{3} + 1/a^{3} = (a â€“ 1/a)^{3} + 3 (a â€“ 1/a)

= (Â±4)^{3} + 3(Â±4)

= Â±64 Â± 12

Hence,

a^{3} + 1/a^{3} = Â±76

**4. If a +Â 1/a = p and a â‰ 0; then show that:**

**a ^{3} + 1/a^{3} = p (p^{2} – 3)**

**Solution:**

Given, a +Â 1/a = p â€¦ (1)

Now, cubing on both sides

(a + 1/a)^{3} = p^{3}

a^{3} + 1/a^{3} + 3(a + 1/a) = p^{3}

a^{3} + 1/a^{3} = p^{3} – 3(a + 1/a)

= p^{3} â€“ 3(p) [From (1)]

= p (p^{2} – 3)

– Hence proved

**5. If a + 2b = 5; then show that:**

**a ^{3}Â + 8b^{3}Â + 30ab = 125.**

**Solution:**

Given, a + 2b = 5

Letâ€™s cube it on both sides,

(a + 2b)^{3} = 5^{3}

a^{3} + 3(a)(2b)(a + 2b) + (2b)^{3} = 125

a^{3} + 6ab(a + 2b) + 8b^{3} = 125

a^{3} + 8b^{3} = 125 – 6ab (a + 2b)

= 125 – 6ab (5) â€¦ [Given]

= 125 â€“ 30ab

So,

a^{3} + 8b^{3} + 30ab = 125

– Hence showed

**6. If (a + 1/a) ^{2} = 3 and a â‰ 0, then show: a^{3}Â + 1/a^{3}Â = 0.**

**Solution:**

Given, (a + 1/a)^{2} = 3

â‡’ a + 1/a = Â±âˆš3 â€¦ (1)

We know the identity,

(a + 1/a)^{3} = a^{3} + 1/a^{3} + 3(a + 1/a)

a^{3} + 1/a^{3} = (a + 1/a)^{3} – 3(a + 1/a)

= (Â±âˆš3)^{3} â€“ 3(Â±âˆš3)

= Â±3âˆš3 â€“ (Â±3âˆš3)

= 0

Thus, a^{3} + 1/a^{3} = 0

Â

**7. If a + 2b + c = 0; then show that:**

**a ^{3}Â + 8b^{3}Â + c^{3}Â = 6abc**

**Solution:**

We have, a + 2b + c = 0

a + 2b = -c

Now, on cubing it on both sides we get

(a + 2b)^{3} = (-c)^{3}

a^{3} + (2b)^{3} + 3(a)(2b)(a + 2b) = -c^{3}

a^{3} + 8b^{3} + 6ab (a + 2b) = -c^{3}

a^{3} + 8b^{3} + 6ab (-c) = -c^{3}

a^{3} + 8b^{3} – 6abc = -c^{3}

Hence,

a^{3} + 8b^{3} + c^{3 }= 6abc

**8. Use property to evaluate:**

**(i) 13 ^{3}Â + (-8)^{3}Â + (-5)^{3}**

**(ii) 7 ^{3}Â + 3^{3}Â + (-10)^{3}Â **

**(iii) 9 ^{3}Â – 5^{3}Â – 4^{3}**

**(iv) 38 ^{3}Â + (-26)^{3}Â + (-12)^{3}**

**Solution:**

The property is if a + b + c = 0 then

a^{3Â }+ b^{3Â }+ c^{3Â }= 3abc

Now,

(i) a = 13, b = -8 and c = -5

â‡’ 13^{3}Â + (-8)^{3}Â + (-5)^{3Â }= 3(13) (-8) (-5) â€¦ [Since, 13 + (-8) + (-5) = 0]

= 1560

Â

(ii) a = 7, b = 3, c = -10

â‡’ 7^{3}Â + 3^{3}Â + (-10)^{3Â }= 3(7) (3) (-10) â€¦ [Since, 7 + 3 + (-10) = 0]

= -630

(iii)a = 9, b = -5, c = -4

â‡’ 9^{3}Â – 5^{3}Â – 4^{3Â }= 9^{3}Â + (-5)^{3}Â + (-4)^{3Â } â€¦ [Since, 9 + (-5) + (-4) = 0]

^{ }= 3(9) (-5) (-4) = 540

Â

(iv) a = 38, b = -26, c = -12

â‡’ 38^{3}Â + (-26)^{3}Â + (-12)^{3Â }= 3(38) (-26) (-12) â€¦ [Since, 38 + (-26) + (-12) = 0]

= 35568

Â

**9. If a â‰ 0 and a â€“ 1/a = 3; find:**

**(i) a ^{2} + 1/a^{2}**

**(ii) a ^{3} â€“ 1/a^{3}**

**Solution:**

(i) We have, a – 1/a = 3

On squaring on both sides, we get

(a â€“ 1/a)^{2} = 3^{2}

a^{2} + 1/a^{2} â€“ 2 = 9

a^{2} + 1/a^{2} = 9 + 2

Hence,

a^{2} + 1/a^{2} = 11

Â

(ii) We have, a – 1/a = 3

On cubing on both sides, we get

(a â€“ 1/a)^{3} = 3^{3}

a^{2} – 1/a^{3} â€“ 3(a â€“ 1/a) = 27

a^{2} – 1/a^{3} = 27 + 3(a â€“ 1/a)

= 27 + 3(3)

= 27 + 9

Hence,

a^{3} – 1/a^{3} = 36

**10. IfÂ a â‰ 0Â and a â€“Â 1/a = 4; find:**

**(i)Â a ^{2} + 1/a^{2}**

**(ii)Â a ^{4} + 1/a^{4}**

**(iii) a ^{3} â€“ 1/a^{3}**

**Solution:**

(i) We have, a â€“Â 1/a = 4 â€¦ (a)

On squaring it on both sides, we get

(a â€“Â 1/a)^{2} = 4^{2}

a^{2} + 1/a^{2} â€“ 2(a)(1/a) = 16

a^{2} + 1/a^{2} â€“ 2 = 16

a^{2} + 1/a^{2 }= 16 + 2 = 18 â€¦ (1)

Hence, a^{2} + 1/a^{2} = 18

(ii) Now, we know that

a^{4} + 1/a^{4} = (a^{2} + 1/a^{2})^{2} â€“ 2

= 18^{2} â€“ 2 â€¦ [From (1)]

= 324 â€“ 2

Hence, a^{4} + 1/a^{4} = 322

(iii) On cubing (i) on both sides, we get

(a â€“Â 1/a)^{3} = 4^{3}

a^{3} â€“ 1/a^{3} â€“ 3(a â€“ 1/a) = 64

a^{3} â€“ 1/a^{3} = 64 + 3(a â€“ 1/a)

= 64 + 3(4) â€¦ [Given]

= 64 + 12

Hence, a^{3} â€“ 1/a^{3} = 76

**11. If x â‰ 0Â andÂ x +Â 1/x = 2; then show that:**

**x ^{2} + 1/x^{2} = x^{3} + 1/x^{3} = x^{4} + 1/x^{4} **

**Solution:**

We have, x +Â 1/x = 2

We know that,

(x +Â 1/x)^{2} = x^{2} +Â 1/x^{2} + 2

(2)^{2} = x^{2} +Â 1/x^{2} + 2

x^{2} +Â 1/x^{2} = 4 â€“ 2

= 2 â€¦ (i)

Next, calculating

(x + 1/x)^{3} = x^{3} + 1/x^{3} + 3(x + 1/x)

(2)^{3} = x^{3} + 1/x^{3} + 3(2)

x^{3} + 1/x^{3} = 2^{3} â€“ 3(2)

= 8 â€“ 6

= 2 â€¦ (ii)

Next, we know that

x^{4} + 1/x^{4} = (x^{2} + 1/x^{2}) â€“ 2

= 2^{2} – 2 â€¦ [From (i)]

= 4 â€“ 2

= 2 â€¦ (iii)

Therefore, from (i), (ii) and (iii) we have

x^{2} + 1/x^{2} = x^{3} + 1/x^{3} = x^{4} + 1/x^{4}

Â

**12. If 2x – 3y = 10 andÂ xyÂ = 16; find the value of 8x ^{3}Â – 27y^{3}.**

**Solution:**

Given,Â

2xÂ –Â 3yÂ = 10 â€¦ (i) andÂ

xyÂ = 16 â€¦ (ii)

Now, on cubing (i) on both sides

(2x â€“ 3y)^{3} = 10^{3}

(2x)^{3} â€“ 3(2x)(3y) (2x â€“ 3y) â€“ (3y)^{3} = 1000 []

8x^{3} â€“ 18(xy) (2x â€“ 3y) â€“ 27y^{3} = 1000

8x^{3} â€“ 18 Ã— 16 Ã— 10 â€“ 27y^{3} = 1000

8x^{3} â€“ 2880 â€“ 27y^{3} = 1000

8x^{3} – 27y^{3} = 1000 + 2880

8x^{3} – 27y^{3} = 3880

**13. Expand:**

**(i) (3x + 5y + 2z) (3x – 5y + 2z)**

**(ii) (3x – 5y – 2z) (3x – 5y + 2z)**

**Solution:**

(i) We have, (3xÂ +Â 5yÂ +Â 2z) (3xÂ –Â 5yÂ +Â 2z)

= {(3xÂ +Â 2z)Â +Â (5y)} {(3xÂ +Â 2z)Â –Â (5y)} â€¦ [By grouping]

= (3xÂ +Â 2z)^{2}Â –Â (5y)^{2} â€¦ [As (aÂ +Â b) (aÂ –Â b)Â =Â a^{2}Â –Â b^{2}]

= 9x^{2}Â +Â 4z^{2}Â +Â (2 Ã—Â 3xÂ Ã— 2z) –Â 25y^{2}

= 9x^{2}Â +Â 4z^{2}Â +Â 12xzÂ – 25y^{2}

= 9x^{2}Â +Â 4z^{2Â }–Â 25y^{2}Â +Â 12xz

Â

(ii) We have, (3xÂ –Â 5yÂ –Â 2z) (3xÂ –Â 5yÂ +Â 2z)

= {(3xÂ –Â 5y)Â –Â (2z)} {(3xÂ –Â 5y)Â +Â (2z)} â€¦ [By grouping]

= (3xÂ –Â 5y)^{2}Â –Â (2z)^{2 }â€¦ [As (aÂ +Â b) (aÂ –Â b)Â =Â a^{2}Â –Â b^{2}]

= 9x^{2}Â +Â 25y^{2}Â – 2 Ã—Â 3xÂ Ã—Â 5yÂ –Â 4z^{2}

= 9x^{2}Â +Â 25y^{2}–Â 30xyÂ –Â 4z^{2}

= 9x^{2}Â +25y^{2}Â –Â 4z^{2}Â –Â 30xy

Â

**14. The sum of two numbers is 9 and their product is 20. Find the sum of their**

**(i) SquaresÂ (ii) Cubes**

**Solution:**

Given, the sum of two numbers is 9 and their product is 20

Letâ€™s assume the numbers toÂ â€˜aâ€™ andÂ â€˜bâ€™

So, we have

a + b = 9 â€¦ (1) and

abÂ = 20 â€¦ (2)

Now,

On squaring (1) on both sides gives, we get

(a + b)^{2Â }= 9^{2}

a^{2Â }+ b^{2Â }+ 2ab = 81

a^{2Â }+ b^{2Â }+ 2(20) = 81 â€¦ [From (2)]

a^{2Â }+ b^{2Â }+ 40 = 81

a^{2Â }+ b^{2Â }= 81 â€“ 40 = 41

(i) Hence, the sum of their squares is 41

Next,

On cubing (1) on both sides, we get

(a + b)^{3Â }= 9^{3}

a^{3Â }+ b^{3Â }+ 3ab (a + b) = 729

a^{3Â }+ b^{3Â }+ 3 Ã— (20) Ã— (9) = 729 â€¦ [From (1) and (2)]

a^{3Â }+ b^{3Â }= 729 – 540 = 189

(ii) Hence, the sum of their cubes is 189.

Â

**15. 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**

**(ii) Difference of their cubes**

**(iii) Sum of their cubes.**

**Solution:**

Given x – y = 5 andÂ xyÂ = 24 (x>y)

(x + y)^{2Â }= (x – y)^{2Â }+ 4xy = 25 + 96 = 121

So, x + y = 11; sum of these numbers is 11.

Cubing on both sides gives

(x – y)^{3Â }= 5^{3}

x^{3Â }– y^{3Â }– 3xy(x – y) = 125

x^{3Â }– y^{3Â }– 72(5) = 125

x^{3Â }– y^{3}= 125 + 360 = 485

So, difference of their cubes is 485.

Cubing both sides, we get

(x + y)^{3Â }= 11^{3}

x^{3Â }+ y^{3Â }+ 3xy(x + y) = 1331

x^{3Â }+ y^{3Â }= 1331 – 72(11) = 1331 – 792 = 539

So, sum of their cubes is 539.

Â

**16. If 4x ^{2Â }+ y^{2Â }= a andÂ xyÂ = b, find the value of 2x + y.**

**Solution:**

Given, xyÂ =Â bÂ â€¦ (i) and 4x^{2Â }+ y^{2Â }= aÂ â€¦ (ii)

Now,

(2x + y)^{2}Â = (2x)^{2}Â + 4xy + y^{2}

= (4x^{2}Â + y^{2})Â + 4xy

= a + 4bÂ â€¦ [Using (i) and (ii)]

Hence,

2x + y = Â±âˆš(a + 4b)

**Exercise 4(C)**

**1. Expand:**

**(i) (x + 8) (x + 10)**

**(ii) (x + 8) (x – 10)**

**(iii) (x – 8) (x + 10)**

**(iv) (x – 8) (x – 10)Â **

**Solution:**

Using the identity, (x + a) (x + b) = x^{2} + (a + b) x + ab

(i) We have, (x + 8) (x + 10)

= x^{2} + (8 + 10) x + 8 Ã— 10

= x^{2} + 18x + 80

(ii) We have, (x + 8) (x – 10)

= x^{2} + (8 – 10) x + 8 Ã— (-10)

= x^{2} â€“ 2x â€“ 80

(iii) (ii) We have, (x – 8) (x + 10)

= x^{2} + (-8 + 10) x + (-8) Ã— 10

= x^{2} + 2x â€“ 80

(iv) We have, (x – 8) (x – 10)

= x^{2} + (-8 – 10) x + (-8) Ã— (-10)

= x^{2} â€“ 18x + 80

**2. Expand:**

**(i) (2x â€“ 1/x) (3x + 2/x)**

**(ii) (3a + 2/b) (2a â€“ 3/b)**

**Solution:**

(i) We have, (2x â€“ 1/x) (3x + 2/x)

= (2x)(3x) + (2x)(2/x) â€“ (1/x)(3x) â€“ (1/x)(2/x)

= 6x^{2} + 4 â€“ 3 â€“ 2/x^{2}

= 6x^{2} + 1 â€“ 2/x^{2}

(ii) We have, (3a + 2/b) (2a â€“ 3/b)

= (3a)(2a) – (3a)(3/b) + (2/b)(2a) â€“ (2/b)(3/b)

= 6a^{2} â€“ 9a/b + 4a/b â€“ 6/b^{2}

= 6a^{2} â€“ 5a/b â€“ 6/b^{2}

**3. Expand:**

**(i) (x + y – z) ^{2}**

**(ii) (x – 2y + 2) ^{2}**

**(iii) (5a â€“ 3b + c) ^{2}**

**(iv) (5x – 3y – 2) ^{2}**

**(v) (x â€“ 1/x + 5) ^{2}**

**Solution:**

(i) (x + y – z)^{2} = x^{2} + y^{2} + z^{2} + 2(x)(y) â€“ 2(y)(z) â€“ 2(z)(x)

= x^{2} + y^{2} + z^{2} + 2xy â€“ 2yz â€“ 2zx

(ii) (x – 2y + 2)^{2} = x^{2} + (-2y)^{2} + 2^{2} + 2(x)(-2y) + 2(-2y)(2) + 2(2)(x)

= x^{2} + 4y^{2} + 4 – 4xy â€“ 8y + 4x

(iii) (5a â€“ 3b + c)^{2}** **= (5a)^{2} + (-3b)^{2} + c^{2} + 2(5a)(-3b) + 2(-3b)(c) + 2(c)(5a)

= 25a^{2} + 9b^{2} + c^{2} â€“ 30ab â€“ 6bc + 10ac

(iv) (5x – 3y – 2)^{2 }= (5x)^{2} + (-3y)^{2} + (-2)^{2} + 2(5x)(-3y) + 2(-3y)(-2) + 2(-2)(5x)

= 25x^{2} + 9y^{2} + 4 â€“ 30xy + 12y â€“ 20x

(v) (x â€“ 1/x + 5)^{2} = (x)^{2} + (-1/x)^{2} + (5)^{2} + 2(x)(-1/x) + 2(-1/x)(5) + 2(5)(x)

= x^{2} + 1/x^{2} + 25 â€“ 2 â€“ 10/x + 10x

= x^{2} + 1/x^{2} + 23 â€“ 10/x + 10x

Â

**4. If a + b + c = 12 and a ^{2}Â + b^{2}Â + c^{2}Â = 50; find ab + bc + ca.**

**Solution:**

Given, a + b + c = 12 and a^{2}Â + b^{2}Â + c^{2}Â = 50

We know that,

(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2(ab + bc + ca)

12^{2} = 50 + 2(ab + bc + ca)

144 = 50 + 2(ab + bc + ca)

ab + bc + ca = (144 â€“ 50)/ 2

= 94/2

Thus,

ab + bc + ca = 47

**5. If a ^{2}Â + b^{2}Â + c^{2}Â = 35 and ab + bc + ca = 23; find aÂ + bÂ + c.**

**Solution:**

Given, a^{2}Â + b^{2}Â + c^{2}Â = 35 and ab + bc + ca = 23

We know that,

(a + b + c)^{2} = (a^{2} + b^{2} + c^{2}) + 2(ab + bc + ca)

(a + b + c)^{2} = 35 + 2(23)

(a + b + c)^{2} = 35 + 46

(a + b + c)^{2} = 81

(a + b + c) = Â±âˆš81

Thus,

a + b + c = Â±9

**6. If a + b + c = p and ab + bc + ca = q; find a ^{2}Â + b^{2}Â + c^{2}.**

**Solution:**

Given, a + b + c = p and ab + bc + ca = q

We know that,

(a + b + c)^{2} = (a^{2} + b^{2} + c^{2}) + 2(ab + bc + ca)

(p)^{2} = (a^{2} + b^{2} + c^{2}) + 2(q)

â‡’ a^{2} + b^{2} + c^{2} = p^{2} â€“ 2q

**7. If a ^{2}Â + b^{2}Â + c^{2}Â = 50 andÂ abÂ +Â bcÂ + ca = 47,Â find a + b + c.**

**Solution:**

Given, a^{2}Â + b^{2}Â + c^{2}Â = 50 andÂ abÂ +Â bcÂ + ca = 47

We know that,

(a + b + c)^{2} = (a^{2} + b^{2} + c^{2}) + 2(ab + bc + ca)

(a + b + c)^{2} = 50 + 2(47)

(a + b + c)^{2} = 50 + 94

= 144

â‡’ (a + b + c) = âˆš144

Thus,

a + b + c = Â±12

**8. If x + y – z = 4 and x ^{2}Â + y^{2}Â + z^{2}Â = 30, then find the value ofÂ xyÂ –Â yzÂ –Â zx.**

**Solution:**

Given, x + y – z = 4 and x^{2}Â + y^{2}Â + z^{2}Â = 30

We know that,

(x + y – z)^{2} = x^{2} + y^{2} + z^{2} + 2(xy – yz – zx)

4^{2} = 30 + 2(xy – yz – zx)

16 â€“ 30 = 2(ab + bc + ca)

xyÂ –Â yzÂ –Â zx = -14/ 2

Thus,

xyÂ –Â yzÂ –Â zx = -7

**Exercise 4(D)**

**1. If x + 2y + 3z = 0 and x ^{3}Â + 4y^{3}Â + 9z^{3}Â = 18xyz; evaluate:**

**Solution:**

Given, x^{3}Â +Â 4y^{3}Â +Â 9z^{3}Â =Â 18xyz andÂ xÂ +Â 2yÂ +Â 3z = 0

So,

xÂ +Â 2yÂ = –Â 3z, 2yÂ +Â 3z =Â -xÂ and 3zÂ +Â xÂ =Â -2y

Now,

**2. If a +Â 1/a = m andÂ aÂ â‰ 0;Â find in terms of ‘m’; the value of:**

**(i)Â a â€“ 1/a**

**(ii)Â a ^{2} â€“ 1/a^{2}**

**Solution:**

(i) Given, a +Â 1/a = m

On squaring on both sides, we get

(a +Â 1/a)^{2} = m^{2}

a^{2} + 1/a^{2} + 2 = m^{2}

a^{2} + 1/a^{2} = m^{2} â€“ 2 â€¦ (1)

Now, consider the expansion

(a â€“ 1/a)^{2} = a^{2} + 1/a^{2} â€“ 2

= m^{2} â€“ 2 â€“ 2 â€¦ [From (1)]

= m^{2} â€“ 4

So,

(a â€“ 1/a) = Â±âˆš(m^{2} â€“ 4) â€¦ (2)

(ii) We know that,

a^{2} â€“ 1/a^{2} = (a â€“ 1/a) (a + 1/a)

= m [Â±âˆš(m^{2} â€“ 4)]

= Â±mâˆš(m^{2} â€“ 4)

Â

**3. In the expansion of (2x ^{2}Â – 8) (x – 4)^{2}; find the value of**

**(i) coefficient of x ^{3}**

**(ii) coefficient of x ^{2}**

**(iii) constant term**

**Solution:**

We have, (2x^{2}Â – 8) (x – 4)^{2}

= (2x^{2}Â – 8) (x^{2 }â€“ 2 Ã— 4 Ã— x + 4^{2})

= (2x^{2}Â – 8) (x^{2 }â€“ 8x + 16)

= 2x^{2}(x^{2 }â€“ 8x + 16) â€“ 8(x^{2 }â€“ 8x + 16)

= 4x^{4} â€“ 16x^{3} + 32x^{2} â€“ 8x^{2} + 64x â€“ 128

= 4x^{4} â€“ 16x^{3} + 24x^{2} + 64x â€“ 128

Now,

(i) coefficient of x^{3} = -16

(ii) coefficient of x^{2}Â = 24

(iii) constant term = -128

**4. If x > 0 and x ^{2} + 1/9x^{2} = 25/36. Find: x^{3} + 1/27x^{3}**

**Solution:**

Given, x^{2} + 1/9x^{2} = 25/36 â€¦ (1)

Now, consider the expansion

(x + 1/3x)^{2} = x^{2} + (1/3x)^{2} + (2 Ã— x Ã— 1/3x)

= (x^{2} + 1/9x^{2}) + 2/3

= 25/36 + 2/3 â€¦ [From (1)]

= 49/36

So,

(x + 1/3x) = Â±âˆš(49/36)

= Â±7/6 â€¦ (2)

Now, consider the expansion

(x + 1/3x)^{3} = x^{3} + (1/3x)^{3} + 3(x + 1/3x)

(7/6)^{3} = x^{3} + (1/3x)^{3} + 3(7/6) â€¦â€¦ [From (2)]

343/216 = x^{3} + 1/27x^{3} + 21/6

x^{3} + 1/27x^{3} = 343/216 â€“ 21/6

= (343 – 252)/216

= 91/216

Thus, x^{3} + 1/27x^{3} = 91/216

**5. If 2(x ^{2}Â + 1) = 5x, find:**

**(i) x â€“ 1/xÂ **

**(ii) x ^{3} â€“ 1/x^{3}**

**Solution:**

(i) Given, 2(x^{2}Â + 1) = 5x

x^{2}Â + 1 = 5x/2

On dividing by x on both sides, we have

(x^{2} + 1)/x = 5/2

â‡’ (x + 1/x) = 5/2 â€¦ (1)

Â

Now, consider the expansion of (x + 1/x)^{2}

(x + 1/x)^{2} = x^{2} + 1/x^{2} + 2

(5/2)^{2} = x^{2} + 1/x^{2} + 2 â€¦ [From (1)]

x^{2} + 1/x^{2} = 25/4 â€“ 2

= (25 – 8)/4

= 17/4 â€¦ (2)

Now,

(x â€“ 1/x)^{2} = x^{2} + 1/x^{2} â€“ 2

= 17/4 â€“ 2 â€¦ [From (2)]

= (17 – 8)/4

= 9/4

So,

x â€“ 1/x = âˆš9/4

Thus,

(i) x â€“ 1/x = Â±3/2 â€¦ (3)

Â

Next, we know that

(x^{3} â€“ 1/x^{3}) = (x â€“ 1/x)^{3} + 3(x â€“ 1/x)

= (Â±3/2)^{3} + 3(Â±3/2) â€¦ [From (3)]

= Â± 27/8 Â± 9/2

= Â± (27 + 36)/8

= Â± 63/8

(ii) Thus, x^{3} â€“ 1/x^{3} = Â±63/8

Â

**6. If a ^{2}Â + b^{2}Â = 34 andÂ abÂ = 12; find:**

**(i) 3(a + b) ^{2}Â + 5(a – b)^{2}**

**(ii) 7(a – b) ^{2}Â – 2(a + b)^{2}**

**Solution:**

We have, a^{2}Â +Â b^{2}Â = 34 and ab= 12

We know that,

(aÂ +Â b)^{2}Â =Â (a^{2}Â +Â b^{2})Â +Â 2ab

= 34Â +Â 2 x 12

= 34Â +Â 24

= 58Â Â

Also, we know that

(aÂ –Â b)^{2}Â =Â (a^{2}Â +Â b^{2})Â –Â 2abÂ

= 34 – 2Â xÂ 12

= 34- 24

= 10

(i)Â 3(aÂ +Â b)^{2Â }+Â 5(aÂ –Â b)^{2}

= 3 x 58Â +Â 5 x 10

= 174Â +Â 50

= 224

Â

(ii)Â 7(aÂ –Â b)^{2}Â –Â 2(aÂ +Â b)^{2}

= 7Â xÂ 10 – 2Â xÂ 58

= 70 – 116

= -46

Â

**7. If 3x â€“ 4/x = 4 andÂ xÂ â‰ 0; find: 27x ^{3} â€“ 64/x^{3}.**

**Solution:**

Given, 3x â€“ 4/x = 4

Now, letâ€™s consider the expansion of (3x â€“ 4/x)^{3}

(3x â€“ 4/x)^{3} = 27x^{3} â€“ 64/x^{3} â€“ 3 Ã— 3x Ã— 4/x(3x â€“ 4/x)

(4)^{3} = 27x^{3} â€“ 64/x^{3} â€“ 36(3x â€“ 4/x)

64 = 27x^{3} â€“ 64/x^{3} â€“ 36(4)

64 = 27x^{3} â€“ 64/x^{3} â€“ 144

27x^{3} â€“ 64/x^{3} = 144 + 64

Hence,

27x^{3} â€“ 64/x^{3} = 208

**8. If x ^{2}Â +Â 1/x^{2} = 7 andÂ xÂ â‰ 0;Â find the value of: 7x^{3} + 8x â€“ 7/x^{3}Â â€“ 8/x.**

**Solution:**

Given,Â x^{2}Â +Â 1/x^{2} = 7

On subtracting 2 from both sides, we get

x^{2}Â +Â 1/x^{2} â€“ 2 = 7 â€“ 2

(x â€“ 1/x)^{2} = 5

x â€“ 1/x = Â±âˆš5 â€¦ (1)

Now, consider

(x â€“ 1/x)^{3} = x^{3} â€“ 1/x^{3} â€“ 3(x â€“ 1/x)

(Â±âˆš5)^{3} = x^{3} â€“ 1/x^{3} â€“ 3(Â±âˆš5)

x^{3} â€“ 1/x^{3} = (Â±âˆš5)^{3} + 3(Â±âˆš5) â€¦ (2)

Taking,

7x^{3} + 8x â€“ 7/x^{3}Â â€“ 8/x

= 7x^{3} â€“ 7/x^{3}Â + 8x â€“ 8/x

= 7(x^{3} â€“ 1/x^{3}) + 8(x â€“ 1/x)

= 7[(Â±âˆš5)^{3} + 3(Â±âˆš5)] + 8(Â±âˆš5)

= Â±35âˆš5** **Â± 21âˆš5** **Â± 8âˆš5

= Â±64âˆš5

**9. If x = 1/(x – 5)Â and xÂ â‰ 5,Â findÂ x ^{2} â€“ 1/x^{2}.**

**Solution:**

Given, x = 1/(x – 5)

By cross multiplying, we have

x (xÂ – 5)Â =Â 1Â

x^{2}Â –Â 5xÂ =Â 1

x^{2}Â –Â 1Â =Â 5x

Dividing both sides byÂ x,

(x^{2} – 1)/x = 5

(x â€“ 1/x) = 5 â€¦ (1)

Now,

(x â€“ 1/x)^{2} = 5^{2}

x^{2} + 1/x^{2} â€“ 2 = 25

x^{2} + 1/x^{2} = 25 + 2

= 27 â€¦ (2)

Considering the expansion (x + 1/x)^{2}

(x + 1/x)^{2} = x^{2} + 1/x^{2} + 2

(x + 1/x)^{2} = 27 + 2 â€¦ [From (1)]

(x + 1/x)^{2} = 29

x + 1/x = Â±âˆš29 â€¦ (3)

We know that,

x^{2} â€“ 1/x^{2} = (x + 1/x) (x â€“ 1/x)

= (Â±âˆš29) (5) â€¦ [From (3)]

= Â±5âˆš29

**10. If xÂ = 1/(5 – x) and xÂ â‰ 5;Â find x ^{3} + 1/x^{3}.**

**Solution:**

Given, xÂ = 1/(5 â€“ x)

By cross multiplying, we have

x (5Â – x)Â =Â 1

x^{2}Â –Â 5xÂ = -1

x^{2}Â +Â 1Â =Â 5x

Dividing both sides byÂ x,

(x^{2} + 1)/x = 5

x + 1/x = 5 â€¦ (1)

Now,

(x + 1/x)^{3} = x^{3} + 1/x^{3} + 3(x + 1/x)

x^{3} + 1/x^{3} = (x + 1/x)^{3} – 3(x + 1/x)

= 5^{3} â€“ 3(5)

= 125 â€“ 15

= 110

Thus, x^{3} + 1/x^{3} = 110

**11. If 3a + 5b + 4c = 0, **

**Show that: 27a ^{3}Â + 125b^{3}Â + 64c^{3}Â = 180abc**

**Solution:**

Given, 3aÂ +Â 5bÂ +Â 4c = 0

â‡’ 3aÂ +Â 5bÂ =Â -4c

On cubing on both sides, we have

(3aÂ +Â 5b)^{3}Â =Â (-4c)^{3}

(3a)^{3}Â +Â (5b)^{3}Â +Â 3Â xÂ 3aÂ xÂ 5b (3aÂ +Â 5b) = -64c^{3}

27a^{3}Â +Â 125b^{3}Â +Â 45abÂ (-4c)Â =Â -64c^{3}

27a^{3}Â +Â 125b^{3}Â –Â 180abc = -64c^{3}

27a^{3}Â +Â 125b^{3}Â +Â 64c^{3}Â =Â 180abcÂ

– Hence Proved.

**12. The sum of two numbers is 7 and the sum of their cubes is 133, find the sum of their square.**

**Solution:**

Letâ€™s assumeÂ a and bÂ to be the two numbers

So, aÂ +Â bÂ = 7 and a^{3}Â +Â b^{3}Â = 133

We know that,

(aÂ +Â b)^{3}Â =Â a^{3}Â +Â b^{3}Â +Â 3ab (aÂ +Â b)

(7)^{3}Â = 133Â +Â 3ab (7)

343Â =Â 133Â +Â 21ab

21abÂ =Â 343 – 133

= 210

â‡’ ab =Â 21

Now,Â

a^{2}Â +Â b^{2}Â =Â (aÂ +Â b)^{2}Â –Â 2ab

= 7^{2}Â – 2 x 10Â

=Â 49 – 20Â

=Â 29

**13. In each of the following, find the value of ‘a’:**

**(i) 4x ^{2}Â + ax + 9 = (2x + 3)^{2}**

**(ii) 4x ^{2}Â + ax + 9 = (2x – 3)^{2}**

**(iii) 9x ^{2}Â + (7a – 5)x + 25 = (3x + 5)^{2}**

**Solution:**

(i) 4x^{2}Â + ax + 9 = (2x + 3)^{2} = 4x^{2} + 12x + 9

On comparing coefficients of x terms, we get

ax = 12x

So,

a = 12

(ii) 4x^{2}Â + ax + 9 = (2x – 3)^{2} = 4x^{2} + 12x + 9

On comparing coefficients of x terms, we get

ax = -12x

So,

a = -12

(iii) 9x^{2}Â + (7a – 5)x + 25 = (3x + 5)^{2} = 9x^{2} + 30x + 25

On comparing coefficients of x terms, we get

(7a – 5)x = 30x

7a – 5 = 30

7a = 35

â‡’ a = 5

Â

**14. If (x ^{2} + 1)/x = 3 1/3 and x > 1; find**

**(i) x â€“ 1/xÂ **

**(ii) x ^{3} â€“ 1/x^{3}**

**Solution:**

Given,

(x^{2} + 1)/x = 3 1/3 = 10/3

x + 1/x = 10/3

On squaring on both sides, we get

(x + 1/x)^{2} = (10/3)^{2}

x^{2} + 1/x^{2} + 2 = 100/9

x^{2} + 1/x^{2} = 100/9 â€“ 2

= (100 – 18)/9

= 82/9

Now,

(x – 1/x)^{2} = x^{2} + 1/x^{2} â€“ 2

= 82/9 â€“ 2

= (82 – 18)/9

= 64/9

x â€“ 1/x = âˆš(64/9)

= Â±8/3

On cubing both sides, we get

(x â€“ 1/x)^{3} = (8/3)^{3}

x^{3} â€“ 1/x^{3} â€“ 3(x â€“ 1/x) = 512/27

x^{3} â€“ 1/x^{3} = 3(x â€“ 1/x) + 512/27

= 3(8/3) + 512/27

= 24/3 + 512/27

= (216 + 512)/27

= 728/27

Therefore, x^{3} â€“ 1/x^{3} = 728/27

**15. The difference between two positive numbers is 4 and the difference between their cubes is 316.**

**Find:**

**(i) Their product**

**(ii) The sum of their squares**

**Solution:**

Given, difference between two positive numbers is 4

And, the difference between their cubes is 316

Letâ€™s assume the positive numbers to be a and b

So,

a – b = 4

a^{3Â }– b^{3Â }= 316

On cubing both sides, we have

(a – b)^{3Â }= 64

a^{3Â }– b^{3Â }– 3ab(a – b) = 64

Also,

Given: a^{3Â }– b^{3Â }= 316

So,

316 – 64 = 3ab(4)

252 = 12ab

So,

abÂ = 21

Thus, the product of numbers is 21

Now,

On squaring both sides, we get

(a – b)^{2Â }= 16

a^{2Â }+ b^{2Â }– 2ab = 16

a^{2Â }+ b^{2Â }= 16 + 42 = 58

Thus, sum of their squares is 58.

**Exercise 4(E)**

**1. Simplify:**

**(i) (x + 6)(x + 4)(x – 2)**

**(ii) (x – 6)(x – 4)(x + 2)**

**(iii) (x – 6)(x – 4)(x – 2)**

**(iv) (x + 6)(x – 4)(x – 2)Â **

**Solution:**

Using identity:

(x + a)(x + b)(x + c) = x^{3}Â + (a + b + c)x^{2}Â + (abÂ +Â bcÂ + ca)x +Â abc

(i) We have, (x + 6)(x + 4)(x – 2)

= x^{3}Â + (6 + 4 – 2)x^{2}Â + [6 Ã— 4 + 4 Ã— (-2) + (-2) Ã— 6]x + 6 Ã— 4 Ã— (-2)

= x^{3}Â + 8x^{2}Â + (24 – 8 – 12)x – 48

= x^{3}Â + 8x^{2}Â + 4x – 48

(ii) We have, (x – 6)(x – 4)(x + 2)

= x^{3}Â + (-6 – 4 + 2)x^{2}Â + [-6 Ã— (-4) + (-4) Ã— 2 + 2 Ã— (-6)]x + (-6) Ã— (-4) Ã— 2

= x^{3}Â – 8x^{2}Â + (24 – 8 – 12)x + 48

= x^{3}Â – 8x^{2}Â + 4x + 48

(iii) We have, (x – 6)(x – 4)(x – 2)

= x^{3}Â + (-6 – 4 – 2)x^{2}Â + [-6 Ã— (-4) + (-4) Ã— (-2) + (-2) Ã— (-6)]x + (-6) Ã— (-4) Ã— (-2)

= x^{3}Â – 12x^{2}Â + (24 + 8 + 12)x – 48

= x^{3}Â – 12x^{2}Â + 44x – 48

Â

(iv) We have, (x + 6)(x – 4)(x – 2)

= x^{3}Â + (6 – 4 – 2)x^{2}Â + [6 Ã— (-4) + (-4) Ã— (-2) + (-2) Ã— 6]x + 6 Ã— (-4) Ã— (-2)

= x^{3}Â – 0x^{2}Â + (-24 + 8 – 12)x + 48

= x^{3}Â – 28x + 48Â

**2. Simply using following identity:**

**(a Â± b) (a ^{2} âˆ“ ab + b^{2}) = a^{3} Â± b^{3}**

**(i) (2x + 3y) (4x ^{2} â€“ 6xy + 9y^{2})**

**(ii) (3x â€“ 5/x) (9x ^{2} + 15 + 25/x^{2})**

**(iii) (a/3 â€“ 3b) (a ^{2} + ab + 9b^{2})**

**Solution:**

(i) We have, (2x + 3y) (4x^{2} â€“ 6xy + 9y^{2})

= (2x + 3y) [(2x)^{2} â€“ (2x)(3y) + (3y)^{2}]

= (2x)^{3} + (3y)^{3}

= 8x^{3} + 27y^{3}

(ii) We have, (3x â€“ 5/x) (9x^{2} + 15 + 25/x^{2})

= (3x â€“ 5/x) [(3x)^{2} + (3x)(5/x) + (5/x)^{2}]

= (3x)^{3} + (5/x)^{3}

= 27x^{3} + 125/x^{3}

(iii) We have, (a/3 â€“ 3b) (a^{2}/9 + ab + 9b^{2})

= (a/3 â€“ 3b) [(a/3)^{2} + (a/3)(3b) + (3b)^{2}]

= (a/3)^{3} – (3b)^{3}

= a^{3}/27 â€“ 27b^{3}

Â

**3. Using suitable identity, evaluate**

**(i) (104) ^{3}**

**(ii) (97) ^{3}Â **

**Solution:**

Using identity: (a Â± b)^{3}Â = a^{3}Â Â± b^{3}Â Â± 3ab(a Â± b)

(i) (104)^{3}Â = (100 + 4)^{3}

= (100)^{3}Â + (4)^{3}Â + 3 Ã— 100 Ã— 4(100 + 4)

= 1000000 + 64 + 1200 Ã— 104

= 1000000 + 64 + 124800

= 1124864

Â

(ii) (97)^{3Â }= (100 – 3)^{3}

= (100)^{3}Â – (3)^{3}Â – 3 Ã— 100 Ã— 3(100 – 3)

= 1000000 – 27 – 900 Ã— 97

= 1000000 – 27 – 87300

= 912673

**4. Simply:**

**Solution:**

We know that,

If a + b + c = 0, then a^{3} + b^{3} + c^{3} = 3abc

Now, if

(x^{2} â€“ y^{2}) + (y^{2} â€“ z^{2}) + (z^{2} â€“ x^{2}) = 0

Then, we have

(x^{2} â€“ y^{2})^{3} + (y^{2} â€“ z^{2})^{3} + (z^{2} â€“ x^{2})^{3} = 3(x^{2} â€“ y^{2})(y^{2} â€“ z^{2})(z^{2} â€“ x^{2}) â€¦ (1)

Similarly, if

x â€“ y + y â€“ z + z â€“ x = 0

Then,

(x â€“ y)^{3} + (y â€“ z)^{3} + (z â€“ x)^{3} = 3(x â€“ y)(y â€“ z)(z â€“ x) â€¦ (2)

Now,

= (x + y)(y + z)(z + x)

Â

**5. Evaluate:**

**Solution:**

(i) We have,

= a + b

= 0.8 + 0.5

= 1.3

(ii) We have,

Â

**6. If a – 2b + 3c = 0; state the value of a ^{3Â }– 8b^{3}Â + 27c^{3}.**

**Solution:**

Given, a – 2b + 3c = 0

Then,

a^{3Â }– 8b^{3}Â + 27c^{3}Â = a^{3}Â + (-2b)^{3}Â + (3c)^{3} = 3(a)( -2b)(3c)

= -18abcÂ

**7. If x + 5y = 10; find the value of x ^{3}Â + 125y^{3}Â + 150xy – 1000.**

**Solution:**

Given, x + 5y = 10

On cubing both sides, we get

(x + 5y)^{3}Â = 10^{3}

x^{3}Â + (5y)^{3}Â + 3(x)(5y)(x + 5y) = 1000

x^{3}Â + (5y)^{3}Â + 3(x)(5y)(10) = 1000

x^{3}Â + (5y)^{3}Â + 150xy = 1000

Thus,

x^{3}Â + (5y)^{3}Â + 150xy – 1000 = 0Â

**8. If x = 3 + 2âˆš2, find:**

**(i) 1/x**

**(ii) x â€“ 1/x**

**(iii) (x â€“ 1/x) ^{3}**

**(iv) x ^{3} â€“ 1/x^{3}**

**Solution:**

We have, x = 3 + 2âˆš2

(i) 1/x = 1/(3 + 2âˆš2)

= (3 – 2âˆš2)/ [(3 + 2âˆš2) Ã— (3 – 2âˆš2)]

= (3 – 2âˆš2)/ [3^{2} â€“ (2âˆš2)^{2}]

= (3 – 2âˆš2)/ (9 – 8)

= 3 – 2âˆš2

(ii) x â€“ 1/x = (3 + 2âˆš2) â€“ (3 – 2âˆš2) â€¦ [From (i)]

= (3 + 2âˆš2 – 3 + 2âˆš2)

= 4âˆš2

(iii) (x â€“ 1/x)^{3} = (4âˆš2)^{3} â€¦ [From (ii)]

= (64 x 2âˆš2)

= 128âˆš2

(iv) (x^{3} â€“ 1/x^{3}) = (x â€“ 1/x)^{3} â€“ 3(x â€“ 1/x)

= 128âˆš2 â€“ 3(4âˆš2) â€¦ [From (iii) and (ii)]

= 128âˆš2 – 12âˆš2Â

**9. If a + b = 11 and a ^{2}Â + b^{2}Â = 65; find a^{3}Â + b^{3}.**

**Solution:**

Given, a + b = 11 and a^{2}Â + b^{2}Â = 65

Now, we know that

(a + b)^{2} = a^{2} + b^{2} + 2ab

(11)^{2} = 65 + 2ab

121 = 65 + 2ab

2ab = 121 â€“ 65

ab = (121 – 65)/2

= 56/2

= 28

Considering the expansion (a^{3} + b^{3})

(a^{3} + b^{3}) = (a + b) (a^{2 }+ b^{2} – ab)

= (11) (65 – 28)

= 11 Ã— 37

= 407

Thus, a^{3}Â + b^{3} = 407

**10. Prove that:**

**x ^{2}+ y^{2Â }+ z^{2Â }–Â xyÂ –Â yzÂ –Â zxÂ is always positive.**

**Solution:**

We have, x^{2Â }+ y^{2Â }+ z^{2Â }–Â xyÂ –Â yzÂ –Â zx

= 2(x^{2Â }+ y^{2Â }+ z^{2Â }–Â xyÂ –Â yzÂ –Â zx)

= 2x^{2Â }+ 2y^{2Â }+ 2z^{2Â }– 2xy – 2yz – 2zx

= x^{2Â }+ x^{2}Â + y^{2Â }+ y^{2}Â + z^{2}Â + z^{2Â }– 2xy – 2yz – 2zx

= (x^{2}Â + y^{2}Â – 2xy) + (z^{2}Â + x^{2}Â – 2zx) + (y^{2}Â + z^{2}Â – 2yz)

= (x – y)^{2}Â + (z – x)^{2}Â + (y – z)^{2}

As the square of any number is positive, the given equation is always positive.

**11. Find:**

**(i)Â (a + b)(a + b)**

**(ii)Â (a + b)(a + b)(a + b)**

**(iii)Â (a – b)(a – b)(a – b) by using the result of part (ii)**

**Solution:**

(i)Â We have, (a + b)(a + b)

= (a + b)^{2}

= a Ã— a + a Ã— b + b Ã— a + b Ã— b

= a^{2}Â +Â abÂ +Â abÂ + b^{2}

= a^{2}Â + b^{2}Â + 2ab

Â

(ii)Â We have, (a + b)(a + b)(a + b)

= (a Ã— a + a Ã— b + b Ã— a + b Ã— b)(a + b)

= (a^{2}Â +Â abÂ +Â abÂ + b^{2})(a + b)

= (a^{2}Â + b^{2}Â + 2ab)(a + b)

= a^{2}Â Ã— a + a^{2}Â Ã— b + b^{2}Â Ã— a + b^{2}Â Ã— b + 2ab Ã— a + 2ab Ã— b

= a^{3}Â + a^{2}Â b + ab^{2}Â + b^{3}Â + 2a^{2}b + 2ab^{2}

= a^{3}Â + b^{3}Â + 3a^{2}b + 3ab^{2}

^{Â }

(iii)Â We have, (a – b)(a – b)(a – b)

In result (ii), replacing b by -b, we get (a – b)(a – b)(a – b)

= a^{3}Â + (-b)^{3}Â + 3a^{2}(-b) + 3a(-b)^{2}

= a^{3}Â – b^{3}Â – 3a^{2}b + 3ab^{2}

## Selina Solutions for Class 9 Maths Chapter 4- Expansions

The Chapter 4, Expansions, contains 5 exercises and the Solutions given here contains the answers for all the questions present in these exercises. Let us have a look at some of the topics that are being discussed in this chapter.

4.1 Introduction

4.2 Identities

4.3 Expansions of (a+b)^{3}

4.4 Expansions of (x+a)(x+b)

4.5 Expansions of (a+b+c)^{2}

4.6 Using Expansions

4.7 Special Products

## Selina Solutions for Class 9 Maths Chapter 4- Expansions

In chapter 4 of Class 9, the students are taught about a process in which the contents of brackets are evaluated, the expansion. The chapter helps the students in recollecting the identities that were taught in earlier classes. An identity is an equation that is true for all values of its variables. Study the Chapter 4 of Selina textbook to understand more about Expansions. Learn the Selina Solutions for Class 9 effectively to come out with flying colours in the examinations.