Algebra - Study Mode
[#501] If $$frac{{2x - y}}{{x + 2y}} = frac{1}{2}{ ext{,}}$$ xa0 then value of $$frac{{3x - y}}{{3x + y}},{ ext{is?}}$$
Correct Answer
(B) $$frac{3}{5}$$
Explanation
Solution: $$eqalign{
& frac{{2x - y}}{{x + 2y}}, imes frac{1}{2}left( {{ ext{Cross multiply}}}
ight) cr
& Rightarrow 4x - 2y = x + 2y cr
& Rightarrow 3x = 4y cr
& Rightarrow x:y = 4:3 cr
& herefore frac{{3x - y}}{{3x + y}}{ ext{ }} cr
& { ext{ = }}frac{{3 imes 4 - 3}}{{3 imes 4 + 3}} cr
& { ext{ = }}frac{{12 - 3}}{{12 + 3}} cr
& { ext{ = }}frac{9}{{15}}{ ext{ }} cr
& { ext{ = }}frac{3}{5} cr} $$
[#502] If $$x + frac{1}{x} = 5{ ext{,}}$$ xa0 then $$frac{{2x}}{{3{x^2} - 5x + 3}}$$ xa0 is equal to?
Correct Answer
(B) $$frac{1}{5}$$
Explanation
Solution: $$eqalign{
& x + frac{1}{x} = 5 cr
& herefore frac{{2x}}{{3{x^2} - 5x + 3}},left( {{ ext{Divide by }}x}
ight) cr
& = frac{{frac{{2x}}{x}}}{{frac{{3{x^2}}}{x} - frac{{5x}}{x} + frac{3}{x}}} cr
& = frac{2}{{3x + frac{3}{x} - 5}} cr
& = frac{2}{{3left( {x + frac{1}{x}}
ight) - 5}} cr
& = frac{2}{{3 imes 5 - 5}} cr
& = frac{2}{{10}} cr
& = frac{1}{5} cr} $$
[#503] $${ ext{If }}sqrt {1 - frac{{{x^3}}}{{100}}} = frac{3}{5}{ ext{,}}$$ xa0 xa0 then x equals to?
Correct Answer
(B) 4
Explanation
Solution: $$eqalign{
& sqrt {1 - frac{{{x^3}}}{{100}}} = frac{3}{5} cr
& Rightarrow 1 - frac{{{x^3}}}{{100}} = {left( {frac{3}{5}}
ight)^2} cr
& Rightarrow 1 - frac{9}{{25}} = frac{{{x^3}}}{{100}} cr
& Rightarrow frac{{16}}{{25}} = frac{{{x^3}}}{{100}} cr
& Rightarrow {x^3} = frac{{16 imes 100}}{{25}} cr
& Rightarrow {x^3} = 16 imes 4 cr
& Rightarrow {x^3} = 64 cr
& Rightarrow x08oxed{x = 4} cr} $$
[#504] $${ ext{If }},sqrt {1 + frac{x}{9}} = frac{{13}}{3}{ ext{,}}$$ xa0xa0 then the value of x is?
Correct Answer
(B) 160
Explanation
Solution: $$eqalign{
& sqrt {1 + frac{x}{9}} = frac{{13}}{3} cr
& { ext{By option }} cr
& { ext{Put }}x = 160 cr
& sqrt {1 + frac{{160}}{9}} = frac{{13}}{3} cr
& Rightarrow sqrt {frac{{169}}{9}} = frac{{13}}{3} cr
& Rightarrow frac{{13}}{3} = frac{{13}}{3} cr
& cr
& {x08f{Alternate:}} cr
& { ext{Squaring both sides}} cr
& {left( {sqrt {1 + frac{{x}}{9}} }
ight)^2} = {left( {frac{{13}}{3}}
ight)^2} cr
& Rightarrow 1 + frac{x}{9} = frac{{169}}{9} cr
& Rightarrow frac{{9 + x}}{9} = frac{{169}}{9} cr
& Rightarrow 9 + x = 169 cr
& Rightarrow x08oxed{x = 160} cr} $$
[#505] $${ ext{If }}frac{{4sqrt 3 + 5sqrt 2 }}{{sqrt {48} + sqrt {18} }} = a + bsqrt 6 { ext{,}}$$ xa0 xa0 xa0 then the value of a and b are respectively?
Correct Answer
(D) $$frac{3}{5},frac{4}{{15}}$$
Explanation
Solution: $$eqalign{
& frac{{4sqrt 3 + 5sqrt 2 }}{{sqrt {48} + sqrt {18} }} = a + bsqrt 6 cr
& Rightarrow frac{{4sqrt 3 + 5sqrt 2 }}{{sqrt {16 imes 3} + sqrt {9 imes 2} }} = a + bsqrt 6 cr
& Rightarrow frac{{4sqrt 3 + 5sqrt 2 }}{{4sqrt 3 + 3sqrt 2 }} = a + bsqrt 6 cr
& Rightarrow frac{{4sqrt 3 + 5sqrt 2 }}{{4sqrt 3 + 3sqrt 2 }} imes frac{{4sqrt 3 - 3sqrt 2 }}{{4sqrt 3 - 3sqrt 2 }} = a + bsqrt 6 cr
& Rightarrow frac{{left( {4sqrt 3 + 5sqrt 2 }
ight)left( {4sqrt 3 - 3sqrt 2 }
ight)}}{{48 - 18}} = a + bsqrt 6 cr
& Rightarrow frac{{8sqrt 6 + 18}}{{30}} = a + bsqrt 6 cr
& Rightarrow frac{{8sqrt 6 }}{{30}} + frac{{18}}{{30}} = a + bsqrt 6 cr
& Rightarrow frac{4}{{15}}sqrt 6 + frac{3}{5} = a + bsqrt 6 cr
& Rightarrow frac{3}{5} + frac{4}{{15}}sqrt 6 = a + bsqrt 6 cr} $$ By comparing coefficients of rational and irrational parts. $$eqalign{
& Rightarrow a = frac{3}{5}{ ext{ , }}b = frac{4}{{15}} cr
& herefore left( {frac{3}{5},frac{4}{{15}}}
ight) cr} $$