Algebra - Study Mode
[#351] If $$x - frac{1}{x} = 5{ ext{,}}$$ xa0 then $${x^2}{ ext{ + }}frac{1}{{{x^2}}}$$ xa0 is?
Correct Answer
(C) 27
Explanation
Solution: $$eqalign{
& x - frac{1}{x} = 5 cr
& left[ {{ ext{Squaring both sides}}}
ight] cr
& Rightarrow {x^2}{ ext{ + }}frac{1}{{{x^2}}} - 2 = 25 cr
& Rightarrow {x^2}{ ext{ + }}frac{1}{{{x^2}}} = 27 cr} $$
[#352] If $$n = 7 + 4sqrt 3 { ext{,}}$$ xa0 then the value of $$left( {sqrt n + frac{1}{{sqrt n }}}
ight)$$ xa0 is:
Correct Answer
(B) 4
Explanation
Solution: $$eqalign{
& n = 7 + 4sqrt 3 cr
& Rightarrow n = 4 + 3 + 4sqrt 3 cr
& Rightarrow n = {left( 2
ight)^2} + {left( {sqrt 3 }
ight)^2} + 2 imes 2 imes sqrt 3 cr
& Rightarrow n = {left( {2 + sqrt 3 }
ight)^2} cr
& Rightarrow sqrt n = 2 + sqrt 3 cr
& Rightarrow frac{1}{{sqrt n }} = 2 - sqrt 3 cr
& herefore sqrt n + frac{1}{{sqrt n }} cr
& = 2 + sqrt 3 + 2 - sqrt 3 cr
& = 4 cr} $$
[#353] If $$x = sqrt 3 + sqrt 2 { ext{,}}$$ xa0xa0 then the value of $$left( {x + frac{1}{x}}
ight),{ ext{is?}}$$
Correct Answer
(B) $${ ext{2}}sqrt 3 $$
Explanation
Solution: $$eqalign{
& { ext{ }}x = sqrt 3 + sqrt 2 cr
& frac{1}{x} = frac{1}{{sqrt 3 + sqrt 2 }} imes frac{{sqrt 3 - sqrt 2 }}{{sqrt 3 - sqrt 2 }} cr
& frac{1}{x} = sqrt 3 - sqrt 2 cr
& herefore x + frac{1}{x} cr
& = sqrt 3 + sqrt 2 + sqrt 3 - sqrt 2 cr
& = 2sqrt 3 cr} $$
[#354] If p + q = 10 and pq = 5, then the numerical value of $$frac{p}{q}{ ext{ + }}frac{q}{p}$$ xa0 will be?
Correct Answer
(D) 18
Explanation
Solution: $$eqalign{
& p + q = 10,.........{ ext{(i)}} cr
& pq = 5 cr
& { ext{Squaring both sides of equation (i)}} cr
& {left( {p + q}
ight)^2} = {left( {10}
ight)^2} cr
& {p^2} + {q^2} + 2pq = 100 cr
& {p^2} + {q^2} + 2 imes 5 = 100 cr
& {p^2} + {q^2} = 90 cr
& { ext{Now,}} cr
& herefore frac{p}{q}{ ext{ + }}frac{q}{p} cr
& = frac{{{p^2} + {q^2}}}{{pq}} cr
& = frac{{90}}{5} cr
& = 18 cr} $$
[#355] If x = 3 + 2$$sqrt 2 $$ and xy = 1, then the value of $$frac{{{x^2} + 3xy + {y^2}}}{{{x^2} - 3xy + {y^2}}}$$ xa0 is?
Correct Answer
(D) $$frac{{37}}{{31}}$$
Explanation
Solution: $$eqalign{
& x = 3 + 2sqrt 2 { ext{ and }}xy = 1 cr
& {y^2} = frac{1}{{{x^2}}} cr
& y = frac{1}{x} = frac{1}{{3 + 2sqrt 2 }} = 3 - 2sqrt 2 cr
& herefore x + frac{1}{x} = 3 + 2sqrt 2 + 3 - 2sqrt 2 = 6 cr
& herefore {x^2} + frac{1}{{{x^2}}} = 36 - 2 = 34 cr
& frac{{{x^2} + 3xy + {y^2}}}{{{x^2} - 3xy + {y^2}}} cr
& = frac{{{x^2} + frac{1}{{{x^2}}} + 3}}{{{x^2} + frac{1}{{{x^2}}} - 3}} cr
& = frac{{34 + 3}}{{34 - 3}} cr
& = frac{{37}}{{31}} cr} $$