Algebra - Study Mode

[#61] If $$left( {0.4x + frac{1}{x}}
ight) = 5,$$ xa0xa0 what is the value of $$left( {0.064{x^3} + frac{1}{{{x^3}}}}
ight)?$$
Correct Answer

(C) 119

Explanation

Solution: $$eqalign{
& left( {0.4x + frac{1}{x}}
ight) = 5 cr
& { ext{cube both side}} cr
& {left( {0.4x}
ight)^3} + frac{1}{{{x^3}}} + 3left( {0.4x + frac{1}{x}}
ight) imes left( {0.4x}
ight)left( {frac{1}{x}}
ight) = {left( 5
ight)^3} cr
& 0.064{x^3} + frac{1}{{{x^3}}} + 3left( 5
ight)left( {0.4}
ight) = 125 cr
& 0.064{x^3} + frac{1}{{{x^3}}} + 6 = 125 cr
& 0.064{x^3} + frac{1}{{{x^3}}} = 119 cr} $$

[#62] If x + y + z = 0, then what is the value of $$frac{{xy + yz + zx}}{{{x^2} + {y^2} + {z^2}}}?$$
Correct Answer

(D) $$ - frac{1}{2}$$

Explanation

Solution: Given, x + y + z = 0 By putting values x = 2, y = -1, z = -1 $$eqalign{
& { ext{Now, }}frac{{xy + yz + zx}}{{{x^2} + {y^2} + {z^2}}} cr
& = frac{{ - 2 + 1 - 2}}{{4 + 1 + 1}} cr
& = frac{{ - 1}}{2} cr} $$

[#63] If P = 7 + 4√3 and PQ = 1, then what is the value of $$frac{1}{{{P^2}}} + frac{1}{{{Q^2}}}?$$
Correct Answer

(B) 194

Explanation

Solution: $$eqalign{
& P = 7 + 4sqrt 3 cr
& PQ = 1,,Q = frac{1}{P} cr
& Q = frac{1}{{7 + 4sqrt 3 }} = 7 - 4sqrt 3 cr
& P + Q cr
& = 7 + 4sqrt 3 + 7 - 4sqrt 3 cr
& = 14 cr
& Rightarrow frac{1}{{{P^2}}} + frac{1}{{{Q^2}}} cr
& = frac{{{P^2} + {Q^2}}}{{{{left( {PQ}
ight)}^2}}} cr
& = frac{{{{left( {P + Q}
ight)}^2} - 2PQ}}{{{{left( {PQ}
ight)}^2}}} cr
& = {left( {14}
ight)^2} - 2 cr
& = 196 - 2 cr
& = 194 cr} $$

[#64] 55 3 + 17 3 - 72 3 + 201960 is equal to
Correct Answer

(B) 0

Explanation

Solution: a = 55 b = 17 c = -72 a + b + c = 55 + 17 - 72 = 0 ∴ a 3 + b 3 + c 3 - 3abc = 0 (a + b + c) = 0 Answer = 0

[#65] If (a - b) = 4 and ab = 2, then (a 3 - b 3 ) is equal to:
Correct Answer

(B) 88

Explanation

Solution: (a - b) = 4 ab = 2 (a 3 - b 3 ) = (a - b)[(a - b) 2 + 3ab] = 4[4 2 + 3 × 2] = 4[16 + 6] = 88