Algebra - Study Mode
[#301] If a + b + c = m and $$frac{1}{a}$$ + $$frac{1}{b}$$ + $$frac{1}{c}{ ext{,}}$$ then average of a 2 , b 2 , c 2 is?
Correct Answer
(B) $$frac{{{m^2}}}{3}$$
Explanation
Solution: $$eqalign{
& a + b + c = m cr
& Rightarrow frac{1}{a} + frac{1}{b} + frac{1}{c} = 0 cr
& Rightarrow frac{{ab + bc + ca}}{{abc}} = 0 cr
& Rightarrow ab + bc + ca = 0 cr
& {left( {a + b + c}
ight)^2} = { ext{ }}{a^2} + {b^2} + {c^2} + 2left( {ab + bc + ca}
ight) cr
& Rightarrow {m^2} = { ext{ }}{a^2} + {b^2} + {c^2} cr
& Rightarrow frac{{{m^2}}}{3} = { ext{ }}frac{{{a^2} + {b^2} + {c^2}}}{3} cr} $$
[#302] If $$left( {sqrt a + sqrt b }
ight)$$ xa0 = 15 and $$left( {sqrt a - sqrt b }
ight)$$ xa0 = 3, then the value of $$frac{{sqrt {ab} }}{4}$$xa0 is?
Correct Answer
(C) $$frac{{27}}{2}$$
Explanation
Solution: $$eqalign{
& left( {sqrt a + sqrt b }
ight) = 15{ ext{ }} cr
& { ext{Square both sides}} cr
& Rightarrow a + b + 2sqrt {ab} = 225 cr
& Rightarrow a + b = 225 - 2sqrt {ab} ,.....(i) cr
& { ext{ }}left( {sqrt a - sqrt b }
ight) = 3 cr
& { ext{Square both sides}} cr
& Rightarrow a + b - 2sqrt {ab} = 9 cr
& Rightarrow a + b = 9 + 2sqrt {ab} ,.....(ii) cr
& { ext{From (i) and (ii)}} cr
& Rightarrow 225 - 2sqrt {ab} = 9 + 2sqrt {ab} cr
& Rightarrow 216 = 4sqrt {ab} cr
& Rightarrow 54 = sqrt {ab} cr
& { ext{Divided by 4 on both sides}} cr
& Rightarrow frac{{sqrt {ab} }}{4} = frac{{54}}{4} cr
& Rightarrow frac{{sqrt {ab} }}{4} = frac{{27}}{2} cr} $$
[#303] If $$a + frac{1}{a} = 3,$$ xa0 then the value of $${a^3} + frac{1}{{{a^3}}}$$ xa0is?
Correct Answer
(D) 18
Explanation
Solution: $$eqalign{
& { ext{Given , }}a + frac{1}{a} = 3 cr
& { ext{Cube both sides}} cr
& {a^3} + frac{1}{{{a^3}}} + 3 imes a imes frac{1}{a}left( {a + frac{1}{a}}
ight) = {left( 3
ight)^3} cr
& Rightarrow {a^3} + frac{1}{{{a^3}}} + 3 imes 3 = 27 cr
& Rightarrow {a^3} + frac{1}{{{a^3}}} = 27 - 9 cr
& Rightarrow {a^3} + frac{1}{{{a^3}}} = 18 cr} $$
[#304] If $$x = sqrt {a
oot 3 of {absqrt {a
oot 3 of {ab} } } } .... propto $$ xa0 xa0 xa0then the value of x is?
Correct Answer
(D) $$
oot 5 of {{a^4}b} $$
Explanation
Solution: $$eqalign{
& x = sqrt {a
oot 3 of {absqrt {a
oot 3 of {ab} } } } ....... propto cr
& { ext{Square both sides}} cr
& {x^2} = a
oot 3 of {abx} { ext{ }}left( { herefore sqrt {a
oot 3 of {ab} } .... propto }
ight) cr
& { ext{Again cube both sides}} cr
& {x^6} = {a^3}abx cr
& {x^5} = {a^4}b cr
& x =
oot 5 of {{a^4}b} cr} $$
[#305] If $$frac{{m - 3{a^3}}}{{{b^3} + {c^3}}}$$ xa0$$+$$ $$frac{{m - 3{b^3}}}{{{c^3} + {a^3}}}$$ xa0$$+$$ $$frac{{m - 3{c^3}}}{{{a^3} + {b^3}}}$$ xa0 = 9, then the value of m is?
Correct Answer
(C) $${ ext{3}}{a^2} + 3{b^2} + 3{c^2}$$
Explanation
Solution: $$eqalign{
& { ext{Put }}a = b = c = 1 cr
& { ext{Then we have }} cr
& frac{{m - 3}}{2}{ ext{ + }}frac{{m - 3}}{2}{ ext{ + }}frac{{m - 3}}{2} = 9{ ext{ }} cr
& m = 9 cr} $$ Now, putting values of a, b, c in option, Only option (C) gives value of m = 9 So, option (C) is correct.