Algebra - Study Mode
[#441] If a 2 + b 2 = 5ab, then the value of $$left( {frac{{{a^2}}}{{{b^2}}}{ ext{ + }}frac{{{b^2}}}{{{a^2}}}}
ight)$$ xa0 is?
Correct Answer
(C) 23
Explanation
Solution: $$eqalign{
& {a^2} + {b^2} = 5ab cr
& Rightarrow frac{{{a^2}}}{{ab}} + frac{{{b^2}}}{{ab}} = 5 cr
& Rightarrow frac{a}{b}{ ext{ + }}frac{b}{a}{ ext{ = 5}} cr
& { ext{Squaring the both sides}} cr
& Rightarrow {left( {frac{a}{b}}
ight)^2}{ ext{ + }}{left( {frac{b}{a}}
ight)^2} + 2 imes frac{a}{b} imes frac{b}{a} = 25 cr
& Rightarrow frac{{{a^2}}}{{{b^2}}}{ ext{ + }}frac{{{b^2}}}{{{a^2}}} = 25 - 2 cr
& Rightarrow frac{{{a^2}}}{{{b^2}}}{ ext{ + }}frac{{{b^2}}}{{{a^2}}} = 23 cr} $$
[#442] If xy + yz + zx = 0, then $$left( {frac{1}{{{x^2} - yz}} + frac{1}{{{y^2} - zx}} + frac{1}{{{z^2} - xy}}}
ight)$$ xa0 xa0 xa0 $$left( {x,y,z
e 0}
ight) = ?$$
Correct Answer
(D) 0
Explanation
Solution: $$eqalign{
& xy + yz + zx = 0 cr
& herefore xy + zx = - yz cr
& Rightarrow xy + yz = - zx cr
& Rightarrow yz + zx = - xy cr
& herefore frac{1}{{{x^2} - yz}} + frac{1}{{{y^2} - zx}} + frac{1}{{{z^2} - xy}} cr} $$ Putting values of -yz, -zx, -xy from above $$ Rightarrow frac{1}{{{x^2} + left( {xy + zx}
ight)}} + frac{1}{{{y^2} + left( {xy + yz}
ight)}}$$ xa0 xa0 xa0 $$ + frac{1}{{{z^2} + left( {yz + zx}
ight)}}$$ $$ Rightarrow frac{1}{{xleft( {x + y + z}
ight)}} + frac{1}{{yleft( {x + y + z}
ight)}}$$ xa0 xa0 xa0 $$ + frac{1}{{zleft( {x + y + z}
ight)}}$$ $$eqalign{
& Rightarrow frac{1}{{left( {x + y + z}
ight)}}left( {frac{1}{x} + frac{1}{y} + frac{1}{z}}
ight) cr
& Rightarrow frac{1}{{left( {x + y + z}
ight)}}left( {frac{{zy + xz + xy}}{{xyz}}}
ight) cr
& Rightarrow frac{1}{{x + y + z}} imes 0 cr
& Rightarrow 0 cr} $$
[#443] If a + b + c = 9 (where a, b, c are real numbers) then the minimum value of a 2 + b 2 + c 2 is?
Correct Answer
(C) 27
Explanation
Solution: $$eqalign{
& a + b + c = 9 cr
& { ext{For minimum value}} cr
& a = b = c cr
& Rightarrow 3a = 9 cr
& Rightarrow a = frac{9}{3} cr
& Rightarrow a = 3 cr
& { ext{For minimum value}} cr
& a = b = c = 3 cr
& herefore {a^2} + {b^2} + {c^2} cr
& Rightarrow {3^2} + {3^2} + {3^2} cr
& Rightarrow 9 + 9 + 9 cr
& Rightarrow 27 cr} $$
[#444] If a 2 + b 2 + 4c 2 = 2(a + b - 2c) - 3 and a, b, c are real, then the value of (a 2 + b 2 + c 2 ) is?
Correct Answer
(D) $${ ext{2}}frac{1}{4}$$
Explanation
Solution: $$eqalign{
& {a^2} + {b^2} + 4{c^2} = 2left( {a + b - 2c}
ight) - 3 cr
& Rightarrow {a^2} + {b^2} + 4{c^2} - 2a - 2b + 4c + 3 = 0 cr
& Rightarrow {a^2} - 2a + 1 + {b^2} - 2b + 1 + 4{c^2} + 4c + 1 = 0 cr
& Rightarrow {left( {a - 1}
ight)^2} + {left( {b - 1}
ight)^2} + {left( {2c + 1}
ight)^2} = 0 cr
& cr
& herefore a - 1 = 0,,,,,,,,,,,,,,,,,,,a = 1 cr
& ,,,,,,,b - 1 = 0,,,,,,,,,,,,,,,,,,,b = 1 cr
& ,,,,2c + 1 = 0,,,,,,,,,,,,,,,,,,,c = frac{{ - 1}}{2} cr
& cr
& herefore {a^2} + {b^2} + {c^2} cr
& Rightarrow 1 + 1 + frac{1}{4} cr
& Rightarrow 2 + frac{1}{4} cr
& Rightarrow frac{9}{4} cr
& Rightarrow 2frac{1}{4} cr} $$
[#445] If $$frac{{x - {a^2}}}{{b + c}}$$ xa0 + $$frac{{x - {b^2}}}{{c + a}}$$ xa0 + $$frac{{x - {c^2}}}{{a + b}}$$ xa0 = 4(a + b + c), then x = ?
Correct Answer
(A) (a + b + c) 2
Explanation
Solution: $$frac{{x - {a^2}}}{{b + c}} + frac{{x - {b^2}}}{{c + a}} + frac{{x - {c^2}}}{{a + b}} = 4left( {a + b + c}
ight)$$ Note : In such type of question to save your valuable time assume values as per your need which make your calculation easier. Assume a = 1, b = 0, c = 1 $$eqalign{
& { ext{Make sure there will be no }}left( {frac{0}{0}}
ight){ ext{ form}} cr
& herefore frac{{x - 1}}{{1 + 0}} + frac{{x - 0}}{{1 + 1}} + frac{{x - 1}}{{1 + 0}} = 4 cr
& Rightarrow x - 1 + frac{x}{2} + x - 1 = 4 imes 2 cr
& Rightarrow x + frac{x}{2} + x = 8 + 2 cr
& Rightarrow frac{{5x}}{2} = 10 cr
& Rightarrow x = 4 cr
& { ext{Now put values in options take option}} cr
& left( ext{A}
ight),{left( {a + b + c}
ight)^2} = {left( {1 + 0 + 1}
ight)^2} = 4 cr} $$