**Question 11. If 3x-2y=11 and xy = 12, find the value of 27x**^{3}-8y^{3}

^{3}-8y

^{3}

**Solution:**

Given, 3x-2y=11 and xy=12

we know that (a-b)

^{3}= a^{3}-b^{3}-3ab(a-b)(3x-2y)

^{3}= 11^{3}27x

^{3}-8y^{3}-3(12)(11)=133127x

^{3}-8y^{3}=1331+237627x

^{3}-8y^{3}=3707Hence, the value of 27x

^{3}-8y^{3}= 3707

**Question 12. If (x**^{4}+1/x^{4})=119, Find the value of (x^{3}-1/x^{3})

^{4}+1/x

^{4})=119, Find the value of (x

^{3}-1/x

^{3})

**Solution:**

Given, (x

^{4}+1/x^{4}) =119 —— 1we know that (x+y)

^{2}= x^{2}+y^{2}+2xysubstitute the given value in eq-1

(x

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

^{4}+1/x^{4}+2= 119+2

= 121

(x

^{2}+1/x^{2})^{2}= ±11Now, find (x-1/x)

we know that (x-y)

^{2}=x^{2}+y^{2}-2xy(x-1/x)

^{2}= x^{2}+1/x^{2}-2*x*1/x= x

^{2}+1/x^{2}-2= 11 -2

= 9

(x-1/x) = ±3

we need to find x

^{3}-1/x^{3}we know that , a

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

^{3}-(1/x)^{3}=(x-1/x)(x^{2}+(1/x)^{2}-x*1/x)Here, x

^{2}+1/x^{2}= 11 and x-1/x=3x

^{3}-1/x^{3}= 3(11+1)= 3(12)

= 36

Hence, the value of x

^{3}-1/x^{3}=36

**Question 13. Evaluate each of the following:**

**(a) (103) ^{3}**

**(b) (98) ^{3}**

**(c) (9.9) ^{3}**

**(d) (10.4) ^{3}**

**(e) (598) ^{3}**

**(f) (99) ^{3}**

**Solution:**

Given:

(a)(103)^{3}we know that (a+b)

^{3}= a^{3}+b^{3}+3ab(a+b)(103)

^{3}can be written as (100+3)^{3}Here, a=100 and b=3

(103)

^{3}= (100+3)^{3}=100

^{3}+3^{3}+3(100)(3)(100+3)=1000000+27+900(103)

=1092727

The value of (103)

^{3}=1092727

(b)(98)^{3}we know that (a-b)

^{3}= a^{3}-b^{3}-3ab(a-b)(98)

^{3}= (100-2)^{3}= 100

^{3}-2^{3}-3(100)(2)(100-2)= 1000000-8-600(98)

= 941192

The value of (98)

^{3}= 941192

(c) (9.9)^{3}we know that (a-b)

^{3}= a^{3}-b^{3}-3ab(a-b)(10-0.1)

^{3}= (10)^{3}-(0.1)^{3}-3(10)(0.1)(10-0.1)= 1000 – 0.001-3(9.9)

= 970.299

The value of (9.9)

^{3}=970.299

(d)(10.4)^{3}we know that (a+b)

^{3}= a^{3}+b^{3}+3ab(a+b)(10+0.4)

^{3}=(10)^{3}+(0.4)^{3}+3(10)(0.4)(10+0.4)= 1000+0.064+12(10.4)

= 1124.864

The value of (10.4)

^{3}=1124.864

(e) (598)^{3}we know that (a-b)

^{3}= a^{3}-b^{3}-3ab(a-b)(600-2)

^{3}= (600)^{3}-2^{3}-3(600)(2)(600-2)= 216000000 – 8 -(3600*598)

= 216000000 -8 – 2152800

= 213847192

The value of (598)

^{3}= 213847192

(f) (99)^{3}we know that (a-b)

^{3}= a^{3}-b^{3}-3ab(a-b)(100-1)

^{3}= (100)^{3}-1^{3}-3(100)(1)(100-1)= 1000000 – 1 -300*99

= 1000000 – 1 -29700

= 970299

The value of (99)

^{3}= 970299

**Question 14. Evaluate each of the following**

**(a) 111 ^{3} – 89^{3}**

**(b) 46 ^{3} +34^{3}**

**(c) 104 ^{3}+96^{3}**

**(d) 93 ^{3} – 107^{3}**

**Solution:**

Given:

(a) 111^{3}– 89^{3}The above equation can be written as (100+11)

^{3}– (100-11)^{3}we know that , (a+b)

^{3}-(a-b)^{3}= 2[b^{3}+3a^{2}b]Here, a=100 b=11

(100+11)

^{3}– (100-11)^{3}= 2[11^{3}+3(100)^{2}(11)]= 2[1331 + 330000]

= 2[331331]

= 662662

The value of 111

^{3}-89^{3}= 662662

(b) 46^{3}+ 34^{3}The above equation can be written as (40+6)

^{3}– (40-6)^{3}we know that , (a+b)

^{3}+(a-b)^{3}= 2[a^{3}+3ab^{2}]Here, a = 40 and b=6

(40+6)

^{3}– (40-6)^{3}= 2[(40)^{3}+3(6)^{2}(40)]= 2[64000+3*36*40]

=2[68320]

= 136640

The value of 46

^{3}+34^{3}=136640

(c) 104^{3}+96^{3}The above equation can be written as (100+4)

^{3}+ (100-4)^{3}we know that, (a+b)

^{3}+(a-b)^{3}= 2[a^{3}+3ab^{2}]here, a = 100 , b= 4

(100+4)

^{3}+(100-4)^{3}= 2[(100)^{3}+3(100)(4)^{2}]= 2[1000000 + 300*16]

= 2[1004800]

= 2009600

The value of 104

^{3}+ 96^{3}= 2009600

(d) 93^{3}– 107^{3}The above equation can be written as (100-7)

^{3}– (100+7)^{3}we know that, (a-b)

^{3}-(a+b)^{3}= -2[b^{3}+3a^{2}b]here, a = 100 , b= 7

(100-7)

^{3}– (100+7)^{3}= -2[7^{3}+3*(100)^{2}*7]= -2[210343]

= -420686

The value of 93

^{3}– 107^{3}= -420686

**Question 15. If x+1/x = 3, calculate x**^{2}+1/x^{2}, x^{3}+1/x^{3}, x^{4}+1/x^{4}

^{2}+1/x

^{2}, x

^{3}+1/x

^{3}, x

^{4}+1/x

^{4}

**Solution:**

Given, x+1/x=3

we know that (x+y)

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

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

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

^{2}+1/x^{2 }=7squaring on both the sid

(x

^{2}+1/x^{2})^{2}= 49x

^{4}+1/x^{4}+2(x^{2})(1/x^{2}) = 49x

^{4}+1/x^{4}= 49-2x

^{4}+1/x^{4}= 47again cubing on both the sides ,

(x+1/x)

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

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

^{3}+1/x^{3}=27-9x

^{3}+1/x^{3}= 18The value x

^{2}+1/x^{2}=7, x^{3}+1/x^{3}= 18, x^{4}+1/x^{4}= 47

**Question 16. If x**^{4}+1/x^{4}=194, calculate x^{2}+1/x^{2}, x^{3}+1/x^{3}, x+1/x

^{4}+1/x

^{4}=194, calculate x

^{2}+1/x

^{2}, x

^{3}+1/x

^{3}, x+1/x

**Solution:**

Given,

x

^{4}+1/x^{4}=194 —– 1add and substract (2*x

^{2}*1/x^{2}) on the left side in above given equationx

^{4}+1/x^{4}+2*x^{2}*1/x^{2}-2*x*1/x^{2}= 194x

^{4}+1/x^{4}+2*x^{2}*1/x^{2}-2 =194(x

^{2})^{2}+(1/x^{2})^{2}+ 2*x^{2}*1/x^{2}= 196(x

^{2}+1/x^{2})^{2}= 196(x

^{2}+1/x^{2}) = 14 ——— 2add and subtract (2*x*1/x) on the left side in above given equation

x

^{2}+1/x^{2}+2*x*1/x-2*x*1/x =14(x+1/x)

^{2}= 14 +2(x+1/x) = 4 ———— 3

Now cubing eq-3 on both sides.

(x+1/x)

^{3}= 4^{3}x

^{3}+1/x^{3}+3*x*1/x(x+1/x) = 64x

^{3}+1/x^{3}+3*4 = 64x

^{3}+1/x^{3}= 64 -12= 52

Hence, the values of (x

^{2}+1/x^{2}) = 14, (x^{3}+1/x^{3}) = 52, (x+1/x) = 4

**Question 17. Find the value of 27x**^{3}+8y^{3}, if

^{3}+8y

^{3}, if

**(a) 3x+2y=14 and xy = 8**

**(b) 3x+2y = 20 and xy=14/9**

**Solution:**

(a)Given, 3x+2y = 14 and xy = 8cubing on both the sides

(3x+2y)

^{3}= 14^{3}we know that, (a+b)

^{3}=a^{3}+b^{3}+3ab(a+b)27x

^{3}+8y^{3}+3(3x)(2y)(3x+2y) = 274427x

^{3}+8y^{3}+18xy(3x+2y) = 274427x

^{3}+8y^{3}+18*8*14 = 274427x

^{3}+8y^{3}= 2744 – 201627x

^{3}+8y^{3}= 728Hence, the value of 27x

^{3}+8y^{3}= 728

(b)Given, 3x+2y = 20 and xy=14/9cubing on both the sides

we know that, (a+b)

^{3}=a^{3}+b^{3}+3ab(a+b)27x

^{3}+8y^{3}+3(3x)(2y)(3x+2y) = 800027x

^{3}+8y^{3}+18xy(3x+2y) = 800027x

^{3}+8y^{3}+18*14/9*20 = 800027x

^{3}+8y^{3}= 8000 – 560= 7440

Hence, the value of 27x

^{3}+8y^{3}= 7440

**Question 18. Find the value of 64x**^{3}-125z^{3}, if 4x-5z=16 and xz=12

^{3}-125z

^{3}, if 4x-5z=16 and xz=12

**Solution:**

Given, 64x

^{3}– 125z^{3}Here, 4x -5z = 16 and xz = 12

cubing (4x-5z)

^{3}= 16^{3}we know that (a-b)

^{3}=a^{3}-b^{3}-3ab(a-b)(4x -5z)

^{3}= (4x)^{3}-(5z)^{3}-3(4x)(5z)(4x-5z)(16)

^{3}= 64x^{3}-125z^{3}-60(4x-5z)4096 = 64x

^{3}-125z^{3}-60(16)64x

^{3}-125z^{3}= 4096 + 960= 5056

Hence, the value of 64x

^{3}– 125z^{3}= 5056

**Question 19. If x-1/x =****, find the value of x**^{3}-1/x^{3}

^{3}-1/x

^{3}

**Solution:**

Given, x-1/x =

cubing both the sides,

we know that , (a-b)

^{3}= a^{3}-b^{3}-3ab(a-b)(x-1/x)

^{3}= x^{3}-1/x^{3}-3*x*1/x(x-1/x)(3+2\sqrt{2})^3 = x

^{3}-1/x^{3}-3(3)x

^{3}-1/x^{3 }= 108+Hence, the value of x

^{3}-1/x^{3}= 108+

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