Algebra - Study Mode
[#496] If $$a + frac{1}{a} + 1 = 0left( {a
e 0}
ight){ ext{,}}$$ xa0 xa0 then the value of $$left( {{a^4} - a}
ight),{ ext{is?}}$$
Correct Answer
(A) 0
Explanation
Solution: $$eqalign{
& a + frac{1}{a} + 1 = 0 cr
& a + frac{1}{a} = - 1 cr
& { ext{Squaring both sides}} cr
& Rightarrow {a^2} + frac{1}{{{a^2}}} + 2 = 1 cr
& Rightarrow {a^2} + frac{1}{{{a^2}}} = - 1 cr
& Rightarrow {a^2} + 1 = - frac{1}{{{a^2}}},.....(i) cr
& Rightarrow a + frac{1}{a} = - 1{ ext{ }}left( {{ ext{Given}}}
ight) cr
& herefore {a^2} + 1 = - a,.....(ii) cr
& Rightarrow - a = frac{{ - 1}}{{{a^2}}} cr
& { ext{For equation (i) and (ii)}} cr
& {{ ext{a}}^3} = 1 cr
& herefore {{ ext{a}}^3} - 1 = 0 cr
& Rightarrow {a^4} - a = aleft( {{a^3} - 1}
ight) cr
& Rightarrow {a^4} - a = a imes 0 cr
& Rightarrow {a^4} - a = 0 cr} $$
[#497] If p = 101, then the value of $$
oot 3 of {pleft( {{p^2} - 3p + 3}
ight) - 1} $$ xa0 xa0 is?
Correct Answer
(A) 100
Explanation
Solution: $$eqalign{
& p = 101 cr
&
oot 3 of {pleft( {{p^2} - 3p + 3}
ight) - 1} cr
& =
oot 3 of {{p^3} - 3{p^2} + 3p - 1} cr
& herefore left[ {{{left( {p - 1}
ight)}^3} = {p^3} - {{left( 1
ight)}^3} - 3pleft( {p - 1}
ight)}
ight] cr
& =
oot 3 of {{{left( {p - 1}
ight)}^3}} cr
& = p - 1 cr
& = 101 - 1 cr
& = 100{ ext{ }} cr} $$
[#498] If 50% of (p - q) = 30% of (p + q), then p : q is equal to?
Correct Answer
(B) 4 : 1
Explanation
Solution: $$eqalign{
& 50\% { ext{ of }}left( {p - q}
ight) = 30\% { ext{ of }}left( {p + q}
ight) cr
& frac{{p - q}}{2} = frac{3}{{10}}left( {p + q}
ight) cr
& left[ {50\% = frac{1}{2}}
ight] cr
& Rightarrow 5left( {p - q}
ight) = 3left( {p + q}
ight) cr
& Rightarrow 5p - 5q = 3p + 3q cr
& Rightarrow 2p = 8q cr
& Rightarrow 1p = 4q cr
& Rightarrow p:q = 4:1 cr} $$
[#499] If x : y = 2 : 1, then (5x 2 - 13xy + 6y 2 ) is equal to ?
Correct Answer
(C) 0
Explanation
Solution: $$eqalign{
& x:y = 2:1 cr
& { ext{then, }}5{x^2} - 13xy + 6{y^2} cr
& Rightarrow 5 imes 4 - 13 imes 2 imes 1 + 6 imes {1^2} cr
& Rightarrow 20 - 26 + 6 cr
& Rightarrow 0 cr} $$
[#500] If $$frac{a}{3} = frac{b}{2}{ ext{,}}$$ xa0 then the value of $$frac{{2a + 3b}}{{3a - 2b}},{ ext{is?}}$$
Correct Answer
(A) $$frac{{12}}{5}$$
Explanation
Solution: $$eqalign{
& frac{a}{3} = frac{b}{2} Rightarrow frac{a}{b} = frac{3}{2} cr
& herefore frac{{2a + 3b}}{{3a - 2b}} cr
& = frac{{2 imes 3 + 3 imes 2}}{{3 imes 3 - 2 imes 2}} cr
& = frac{{6 + 6}}{{9 - 4}} cr
& = frac{{12}}{5} cr} $$