Algebra - Study Mode
[#316] If $$frac{{{a^2} + {b^2} + {c^2} - 1024}}{{ab - bc - ca}} = - 2$$ xa0 xa0 and a + b = 5c, where c > 0, then the value of c is . . . . . . . .
Correct Answer
(A) 8
Explanation
Solution: a + b = 5c $$frac{{{a^2} + {b^2} + {c^2} - 1024}}{{ab - bc - ca}} = - 2$$ a 2 + b 2 + c 2 + 2ab - 2bc - 2ac = 1024 (a + b) 2 + c(c - 2b - 2a) = 1024 (5c) 2 + c[c - 2(b + a)] = 1024 25c 2 + c[c - 2 × 5c) = 1024 25c 2 - 9c 2 = 1024 16c 2 = 1024 c 2 = 64 c = 8
[#317] If x = 2 + √3, y = 2 - √3, z = 1 then what is the value of $$frac{x}{{yz}} + frac{y}{{xz}} + frac{z}{{xy}} + 2left[ {frac{1}{x} + frac{1}{y} + frac{1}{z}}
ight]?$$
Correct Answer
(A) 25
Explanation
Solution: $$eqalign{
& x = 2 + sqrt 3 cr
& y = 2 - sqrt 3 cr
& z = 1 cr
& frac{x}{{yz}} + frac{y}{{xz}} + frac{z}{{xy}} + 2left[ {frac{1}{x} + frac{1}{y} + frac{1}{z}}
ight] cr
& = frac{{{x^2} + {y^2} + {z^2}}}{{xyz}} + 2left[ {frac{1}{x} + frac{1}{y} + frac{1}{z}}
ight] cr
& = frac{{{{left( {2 + sqrt 3 }
ight)}^2} + {{left( {2 - sqrt 3 }
ight)}^2} + {{left( 1
ight)}^2}}}{{left( {2 + sqrt 3 }
ight)left( {2 - sqrt 3 }
ight)left( 1
ight)}} + 2left[ {frac{1}{{left( {2 + sqrt 3 }
ight)}} + frac{1}{{left( {2 - sqrt 3 }
ight)}} + 1}
ight] cr
& = frac{{4 + 3 + 4sqrt 3 + 4 + 3 - 4sqrt 3 + 1}}{1} + 2left[ {2 - sqrt 3 + 2 + sqrt 3 + 1}
ight] cr
& = 15 + 10 cr
& = 25 cr} $$
[#318] If $$frac{1}{{{x^2} + {a^2}}} = {x^2} - {a^2},$$ xa0 xa0then the value of x is:
Correct Answer
(D) $${left( {{a^4} + 1}
ight)^{frac{1}{4}}}$$
Explanation
Solution: $$eqalign{
& frac{1}{{{x^2} + {a^2}}} = {x^2} - {a^2} cr
& 1 = {x^4} - {a^4} cr
& {x^4} = 1 + {a^4} cr
& x = {left( {1 + {a^4}}
ight)^{frac{1}{4}}} cr} $$
[#319] If 9x 2 + y 2 = 37 and xy = 2, x, y > 0, then the value of (27x 3 + y 3 ) is:
Correct Answer
(B) 217
Explanation
Solution: xy = 2 2 × 1 1 × 2 at x = 2, y = 1 9x 2 + y 2 = 37 9 × 4 + 1 = 37 37 = 37 it satisfy 27x 3 + y 3 = 27 × 2 3 + 1 3 = 27 × 8 + 1 = 217
[#320] If $${x^2} + frac{1}{{{x^2}}} = 66{ ext{,}}$$ xa0xa0 then the value of $$frac{{{x^2} - 1 + 2x}}{x}$$ xa0 = ?
Correct Answer
(B) 10, -6
Explanation
Solution: $$eqalign{
& {x^2} + frac{1}{{{x^2}}} = 66 cr
& ext{Subtract 2 from both sides} cr
& Rightarrow {x^2} + frac{1}{{{x^2}}} - 2 = 66 - 2 cr
& Rightarrow {left( {x + frac{1}{x}}
ight)^2} = 64 cr
& Rightarrow {left( {x + frac{1}{x}}
ight)^2} = {left( 8
ight)^2} cr
& Rightarrow x - frac{1}{x} = pm 8 cr
& Rightarrow frac{{{x^2} - 1 + 2x}}{x} cr
& Rightarrow frac{{frac{{{x^2}}}{x} - frac{1}{x} + frac{{2x}}{x}}}{{frac{x}{x}}} cr
& Rightarrow frac{{left( {x - frac{1}{x}}
ight) + 2}}{1} cr
& { ext{When , }}x - frac{1}{x} = + 8 cr
& { ext{Then,}} cr
& left( {x - frac{1}{x}}
ight) + 2 = 8 + 2 = 10 cr
& { ext{When }}x - frac{1}{x} = - 8 cr
& Rightarrow x - frac{1}{x} = - 8 + 2 cr
& Rightarrow x - frac{1}{x} = - 6 cr
& herefore left( {10, - 6}
ight) cr} $$