Algebra - Study Mode
[#296] If $$frac{{{ ext{ }}{x^2} + 1}}{{{x^2}}} = 2{ ext{,}}$$ xa0 then the value of $$frac{{x - 1}}{x}$$ xa0is?
Correct Answer
(B) 0
Explanation
Solution: $$eqalign{
& {x^2} + frac{{{ ext{ }}1}}{{{x^2}}} = 2 cr
& Rightarrow {left( {x - frac{1}{x}}
ight)^2} + 2.x.frac{1}{x} = 2 cr
& Rightarrow x - frac{1}{x} = 2 - 2 cr
& Rightarrow x - frac{1}{x} = 0 cr} $$
[#297] If pq(p + q) = 1, then the value of $$frac{1}{{{p^3}{q^3}}}$$ xa0- $${p^3}$$ - $${q^3}{ ext{,}}$$ xa0is equal to?
Correct Answer
(C) 3
Explanation
Solution: $$eqalign{
& pqleft( {p + q}
ight) = 1 cr
& Rightarrow p + q = frac{1}{{pq}} cr
& ,,,,,,, ext{Cubing both side} cr
& Rightarrow {p^3} + {q^3} +3 pqleft( {p + q}
ight) = frac{1}{{{p^3}{q^3}}} cr
& ,,,,,,, ext{Puting the value of } pq = frac{1}{left({p + q}
ight)} cr
& Rightarrow frac{1}{{{p^3}{q^3}}} - {p^3} - {q^3} = left( {frac{{3}}{{p + q}}}
ight)left( {p + q}
ight) cr
& Rightarrow frac{1}{{{p^3}{q^3}}} - {p^3} - {q^3} = 3 cr} $$
[#298] If p 3 - q 3 = (p - q){(p + q) 2 - xpq}, then the value of x is?
Correct Answer
(A) 1
Explanation
Solution: $${p^3} - {q^3} = left( {p - q}
ight)left{ {{p^2} + {q^2} + pq}
ight}$$ $$ Rightarrow left( {p - q}
ight)left{ {{{left( {p + q}
ight)}^2} - xpq}
ight} = $$ xa0 xa0 xa0 $$left( {p - q}
ight)$$ $$left( {{p^2} + {q^2} + pq}
ight)$$ $$eqalign{
& Rightarrow {p^2} + {q^2} + 2pq - xpq = {p^2} + {q^2} + pq cr
& Rightarrow 2pq - pq = xpq cr
& Rightarrow pq = xpq cr
& Rightarrow x = 1 cr} $$
[#299] If $$x = {left( {0.25}
ight)^{frac{1}{2}}},$$ xa0 $$y = {left( {0.4}
ight)^2},$$ xa0 $$z = {left( {0.216}
ight)^{frac{1}{2}}}$$ xa0 then-
Correct Answer
(D) x > z > y
Explanation
Solution: $$eqalign{
& x = {left( {0.25}
ight)^{frac{1}{2}}} cr
& x =
oot 2 of {0.25} = 0.5{ ext{ }} cr
& y = {left( {0.4}
ight)^2} cr
& y = 0.16 cr
& z = {left( {0.216}
ight)^{frac{1}{2}}} cr
& z =
oot 2 of {0.216} = 0.464 cr
& herefore x > z > y cr} $$
[#300] a + b + c = 0, then the value of $$frac{{{a^2} + {b^2} + {c^2}}}{{ab + bc + ca}}$$ xa0 is?
Correct Answer
(B) -2
Explanation
Solution: $$eqalign{
& { ext{If }}a + b + c = 0 cr
& { ext{Put }}a = 1,{ ext{ }}b = 1{ ext{ and }}c = - 2 cr
& herefore frac{{{a^2} + {b^2} + {c^2}}}{{ab + bc + ca}} cr
& = frac{{1 + 1 + 4}}{{1 - 2 - 2}} cr
& = frac{6}{{ - 3}} cr
& = - 2 cr} $$