Algebra - Study Mode
[#71] If $${a^{frac{1}{3}}} = 11,$$ xa0 then the value of a 2 - 331a is?
Correct Answer
(B) 1331000
Explanation
Solution: $$eqalign{
& { ext{ }}{a^{frac{1}{3}}} = 11 cr
& Leftrightarrow a = {11^3} = 1331 cr
& {a^2} - 331a cr
& = aleft( {a - 331}
ight) cr
& = 1331left( {1331 - 331}
ight) cr
& = 1331left( {1000}
ight) cr
& = 1331000 cr} $$
[#72] If a 2 + b 2 = 2 and c 2 + d 2 = 1, then the value of (ad - bc) 2 + (ac - bd) 2 is?
Correct Answer
(D) 2
Explanation
Solution: $$eqalign{
& {a^2} + {b^2} = 2{ ext{ }} cr
& {c^2} + {d^2} = 1 cr
& { ext{Put values of }}a,{ ext{ }}b,{ ext{ }}c,{ ext{ }}d cr
& { ext{Take ,}}a = b = 1 cr
& c = 1 cr
& d = 0 cr
& Rightarrow {left( {ad - bc}
ight)^2}{ ext{ + }}{left( {ac - bd}
ight)^2} cr
& Rightarrow {left( {0 - 1}
ight)^2} + {left( {1 + 0}
ight)^2} cr
& Rightarrow {left( { - 1}
ight)^2} + {left( 1
ight)^2} cr
& Rightarrow 2 cr} $$
[#73] If $$x = frac{{4ab}}{{a + b}}{ ext{ }}a
e b,$$ xa0 xa0 the value of $$frac{{x + 2a}}{{x - 2a}}$$ xa0 + $$frac{{x + 2b}}{{x - 2b}}$$ xa0 is?
Correct Answer
(D) 2
Explanation
Solution: $$eqalign{
& { ext{ }}x = frac{{4ab}}{{a + b}}{ ext{ }} cr
& Rightarrow frac{x}{{2a}} = { ext{ }}frac{{2b}}{{a + b}}{ ext{ }} cr
& frac{{x + 2a}}{{x - 2a}} cr
& = frac{{2b + a + b}}{{2b - a - b}} cr
& = frac{{3b + a}}{{b - a}} cr} $$ ( By Componendo and Dividendo rule ) $$eqalign{
& Rightarrow { ext{again}}frac{x}{{2b}} = frac{{2a}}{{a + b}} cr
& frac{{x + 2b}}{{x - 2b}} cr
& = frac{{2a + a + b}}{{2a - a - b}} cr
& = frac{{3a + b}}{{a - b}} cr
& Rightarrow frac{{x + 2a}}{{x - 2a}}{ ext{ + }}frac{{x + 2b}}{{x - 2b}} cr
& Rightarrow frac{{3b + a}}{{b - a}} - frac{{3a + b}}{{a - b}} cr
& Rightarrow frac{{3b + a - 3a - b}}{{b - a}} cr
& Rightarrow frac{{2b - 2a}}{{b - a}} cr
& Rightarrow frac{{2left( {b - a}
ight)}}{{left( {b - a}
ight)}} cr
& Rightarrow 2 cr} $$
[#74] If $$m + frac{1}{{m - 2}} = 4{ ext{,}}$$ xa0xa0 find the value of $${left( {m - 2}
ight)^2}{ ext{ + }}frac{1}{{{{left( {m - 2}
ight)}^2}}}$$ xa0 xa0 is?
Correct Answer
(C) 2
Explanation
Solution: $$eqalign{
& m + frac{1}{{m - 2}} = 4 cr
& Rightarrow m - 2 + frac{1}{{m - 2}} = 2 cr
& left( {{ ext{Squaring the both sides}}}
ight) cr} $$ $$ Rightarrow {left( {m - 2}
ight)^2}{ ext{ + }}frac{1}{{{{left( {m - 2}
ight)}^2}}} + 2 imes $$ xa0 xa0xa0 $$left( {m - 2}
ight) imes $$ xa0$$frac{1}{{left( {m - 2}
ight)}}$$ xa0 $$ = 4$$ $$ Rightarrow {left( {m - 2}
ight)^2}{ ext{ + }}frac{1}{{{{left( {m - 2}
ight)}^2}}} = 2$$
[#75] If a 2 + b 2 + 2b + 4a + 5 = 0, then the value of $$frac{{a - b}}{{a + b}},{ ext{is?}}$$
Correct Answer
(C) $$frac{1}{3}$$
Explanation
Solution: $$eqalign{
& {a^2} + {b^2} + 2b + 4a + 5 = 0 cr
& Rightarrow {a^2} + {b^2} + 2b + 4a + 4 + 1 = 0 cr
& Rightarrow {a^2} + 4a + 4 + {b^2} + 2b + 1 = 0 cr
& Rightarrow {left( {a + 2}
ight)^2} + {left( {b + 1}
ight)^2} = 0 cr
& a + 2 = 0{ ext{ }} Rightarrow { ext{ a}} = - 2 cr
& b + 1 = 0,,,, Rightarrow ,,,,b = - 1 cr
& frac{{a - b}}{{a + b}} Rightarrow frac{{ - 2 + 1}}{{ - 2 - 1}} cr
& Rightarrow frac{{ - 1}}{{ - 3}} = frac{1}{3} cr} $$