Algebra - Study Mode

[#266] If a 2 + b 2 + c 2 + 27 = 6(a + b + c), then what is the value of $$
oot 3 of {{a^3} + {b^3} - {c^3}} ?$$
Correct Answer

(A) 3

Explanation

Solution: $$eqalign{
& {a^2} + {b^2} + {c^2} + 27 = 6left( {a + b + c}
ight) cr
& {left( {a - 3}
ight)^2} + {left( {b - 3}
ight)^2} + {left( {c - 3}
ight)^2} = 0 cr
& a = 3,,b = 3,,c = 3 cr
&
oot 3 of {{a^3} + {b^3} - {c^3}} cr
& =
oot 3 of {27 + 27 - 27} cr
& =
oot 3 of {27} cr
& = 3 cr} $$

[#267] If x 2 - √7x + 1 = 0, then what is the value of $${x^5} + frac{1}{{{x^5}}}?$$
Correct Answer

(A) 19√7

Explanation

Solution: $$eqalign{
& {x^2} - sqrt 7 x + 1 = 0 cr
& {x^2} + 1 = sqrt 7 x cr
& x + frac{1}{x} = sqrt 7 cr
& {x^5} + frac{1}{{{x^5}}} = left( {{x^2} + frac{1}{{{x^2}}}}
ight)left( {{x^3} + frac{1}{{{x^3}}}}
ight) - left( {x + frac{1}{x}}
ight) cr
& = left( {{{left( {sqrt 7 }
ight)}^2} - 2}
ight)left( {{{left( {sqrt 7 }
ight)}^3} - 3 imes sqrt 7 }
ight) - sqrt 7 cr
& = 5 imes 4sqrt 7 - sqrt 7 cr
& = 19sqrt 7 cr} $$

[#268] If $$x + frac{1}{{16x}} = 3,$$ xa0 then the value of $$16{x^3} + frac{1}{{256{x^3}}}$$ xa0 is:
Correct Answer

(A) 423

Explanation

Solution: $$eqalign{
& x + frac{1}{{16x}} = 3 cr
& 2x + frac{1}{{8x}} = 6 cr
& { ext{Cube both side}} cr
& 8{x^3} + frac{1}{{512{x^3}}} + 3 imes 2 imes frac{1}{8} imes 6 = 216 cr
& 8{x^3} + frac{1}{{512{x^3}}} = 216 - frac{9}{2} cr
& { ext{Multiply by '2' both side}} cr
& 16{x^3} + frac{1}{{256{x^3}}} = 432 - 9 = 423 cr} $$

[#269] If $$x + frac{1}{x} = 5,$$ xa0 then what is the value of $${x^6} + frac{1}{{{x^6}}}?$$
Correct Answer

(C) 12098

Explanation

Solution: $$eqalign{
& x + frac{1}{x} = 5 cr
& { ext{Cubing both sides, we get}} cr
& {x^3} + frac{1}{{{x^3}}} + 3 imes x imes frac{1}{x}left( {x + frac{1}{x}}
ight) = 125 cr
& {x^3} + frac{1}{{{x^3}}} + 3 imes 5 = 125 cr
& {x^3} + frac{1}{{{x^3}}} = 110 cr
& { ext{Squaring both sides, we get}} cr
& {x^6} + frac{1}{{{x^6}}} + 2 = 12100 cr
& {x^6} + frac{1}{{{x^6}}} = 12098 cr} $$

[#270] If x + y + z = 0, then the value of (x 2 + y 2 + z 2 ) ÷ (z 2 - xy) is:
Correct Answer

(B) 2

Explanation

Solution: $$eqalign{
& x + y + z = 0 cr
& { ext{Let }}x = y = 1 cr
& z = - 2 cr
& frac{{{x^2} + {y^2} + {z^2}}}{{{z^2} - xy}} cr
& = frac{{1 + 1 + 4}}{{4 - 1}} cr
& = 2 cr} $$