Algebra - Study Mode

[#366] If $$3{a^2} = {b^2}
e 0{ ext{,}}$$ xa0 then the value of $$frac{{{{left( {a + b}
ight)}^3} - {{left( {a - b}
ight)}^3}}}{{{{left( {a + b}
ight)}^2} + {{left( {a - b}
ight)}^2}}}$$ xa0xa0 is?
Correct Answer

(A) $$frac{{3b}}{2}$$

Explanation

Solution: $$eqalign{
& 3{a^2} = {b^2}{ ext{ }}left( {{ ext{Given}}}
ight) cr
& { ext{ }}frac{{{{left( {a + b}
ight)}^3} - {{left( {a - b}
ight)}^3}}}{{{{left( {a + b}
ight)}^2} + {{left( {a - b}
ight)}^2}}} cr} $$ $$ = frac{{{a^3} + {b^3} + 3ableft( {a + b}
ight), - ,left( {{a^3} - {b^3} - 3ableft( {a - b}
ight)}
ight){ ext{ }}}}{{{a^2} + {b^2} + 2ab + { ext{ }}{a^2} + {b^2} - 2ab}}$$ $$eqalign{
& = frac{{2{b^3}{ ext{ + 6}}{{ ext{a}}^2}{ ext{b }}}}{{2{a^2} + 2{b^2}{ ext{ }}}} cr
& = frac{{{b^3}{ ext{ + 3}}{{ ext{a}}^2}{ ext{b }}}}{{{a^2} + {b^2}{ ext{ }}}} cr
& = frac{{{b^3} + {b^3}{ ext{ }}}}{{frac{{{b^2}}}{3} + {b^2}{ ext{ }}}} cr
& = frac{{2{b^3}}}{{{b^2}left( {frac{1}{3} + 1}
ight)}} cr
& = frac{{2b}}{{frac{4}{3}}} cr
& = frac{{3b}}{2} cr} $$

[#367] The value of $$frac{{4{x^3} - x}}{{left( {2x + 1}
ight)left( {6x - 3}
ight)}}$$ xa0xa0 when x = 9999 is?
Correct Answer

(C) 3333

Explanation

Solution: $$eqalign{
& x = 9999{ ext{ }}left( {{ ext{ Given}}}
ight) cr
& frac{{4{x^3} - x}}{{left( {2x + 1}
ight)left( {6x - 3}
ight)}} cr
& = frac{{xleft( {4{x^2} - 1}
ight)}}{{3left( {2x + 1}
ight)left( {2x - 1}
ight)}} cr
& = frac{{xleft( {4{x^2} - 1}
ight)}}{{3left( {4{x^2} - 1}
ight)}} cr
& = frac{x}{3} cr
& herefore frac{{9999}}{3} = 3333 cr} $$

[#368] If $$x + frac{1}{x} = 99{ ext{,}}$$ xa0 find the value of xa0 $$frac{{100x}}{{2{x^2} + 2 + 102x}}$$ xa0xa0 is?
Correct Answer

(C) $$frac{1}{3}$$

Explanation

Solution: $$eqalign{
& x + frac{1}{x} = 99 cr
& herefore {x^2} + 1 = 99x cr
& Rightarrow 2left( {{x^2} + 1}
ight) = 2 imes 99x cr
& Rightarrow 2{x^2} + 2 = 198x cr
& = frac{{100x}}{{2{x^2} + 2 + 102x}} cr
& = { ext{ }}frac{{100x}}{{198x + 102x}} cr
& = frac{{100x}}{{300x}} cr
& = frac{1}{3} cr} $$

[#369] If $$frac{{4x - 3}}{x}$$ xa0 + $$frac{{4y - 3}}{y}$$ xa0 + $$frac{{4z - 3}}{z} = 0{ ext{,}}$$ xa0 then the value of $$frac{1}{x} + frac{1}{y} + frac{1}{z}$$ xa0 is?
Correct Answer

(C) 4

Explanation

Solution: $$eqalign{
& frac{{4x - 3}}{x} + frac{{4y - 3}}{y} + frac{{4z - 3}}{z} = 0 cr
& Rightarrow frac{{4x}}{x} - frac{3}{x} + frac{{4y}}{y} - frac{3}{y} + frac{{4z}}{z} - frac{3}{z} = 0 cr
& Rightarrow 4 - frac{3}{x} + 4 - frac{3}{y} + 4 - frac{3}{z} = 0 cr
& Rightarrow 12 - 3left( {frac{1}{x} + frac{1}{y} + frac{1}{z}}
ight) = 0 cr
& Rightarrow - 3left( {frac{1}{x} + frac{1}{y} + frac{1}{z}}
ight) = - 12 cr
& Rightarrow frac{1}{x} + frac{1}{y} + frac{1}{z} = 4 cr} $$

[#370] If $$frac{{xy}}{{x + y}} = a,$$ xa0 $$frac{{xz}}{{x + z}} = b$$ xa0 and $$frac{{yz}}{{y + z}} = c{ ext{,}}$$ xa0 where a, b, c are all non - zero numbers, then x equals to?
Correct Answer

(C) $$frac{{2abc}}{{bc + ac - ab}}$$

Explanation

Solution: $$eqalign{
& frac{{xy}}{{x + y}} = a,,frac{{xz}}{{x + z}} = b,,frac{{yz}}{{y + z}} = c{ ext{ }} cr
& { ext{Now,}} cr
& frac{{x + y}}{{xy}} = frac{1}{a} cr
& frac{{x + z}}{{xz}} = frac{1}{b} cr
& frac{{y + z}}{{yz}} = frac{1}{c} cr
& Rightarrow frac{1}{y} + frac{1}{x} = frac{1}{a},frac{1}{z} + frac{1}{x} = frac{1}{b},frac{1}{x} + frac{1}{y} = frac{1}{c} cr
& { ext{Now we have to find the value of }}x cr
& herefore frac{1}{a} + frac{1}{b} - frac{1}{c} = frac{1}{y} + frac{1}{x} + frac{1}{z} + frac{1}{x} - frac{1}{y} - frac{1}{z} cr
& herefore frac{1}{a} + frac{1}{b} - frac{1}{c} = frac{2}{x} cr
& Rightarrow frac{{bc + ac - ab}}{{abc}} = frac{2}{x} cr
& Rightarrow x = frac{{2abc}}{{bc + ac - ab}} cr} $$