Algebra - Study Mode
[#201] If a + b + c = 15 and a 2 + b 2 + c 2 = 83 then the value of a 3 + b 3 + c 3 - 3abc = ?
Correct Answer
(B) 180
Explanation
Solution: $$eqalign{
& a + b + c = 15{ ext{ }} cr
& {a^2} + {b^2} + {c^2} = 83{ ext{ }}left( {{ ext{Given}}}
ight) cr
& herefore a + b + c = 15 cr
& left( {{ ext{Squaring both sides}}}
ight){ ext{ }} cr
& Rightarrow {left( {a + b + c}
ight)^2} = {left( {15}
ight)^2}{ ext{ }} cr
& Rightarrow {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ca = 225 cr
& Rightarrow 83 + 2left( {ab + bc + ca}
ight) = 225 cr
& Rightarrow 2left( {ab + bc + ca}
ight) = 142 cr
& Rightarrow ab + bc + ca = 71 cr} $$ $$ herefore {a^3} + {b^3} + {c^3} - 3abc = $$ xa0 xa0 $$left( {a + b + c}
ight)$$xa0 $$left( {{a^2} + {b^2} + {c^2} - ab - bc - ca}
ight)$$ $$eqalign{
& Rightarrow {a^3} + {b^3} + {c^3} - 3abc = 15left( {83 - 71}
ight) cr
& Rightarrow {a^3} + {b^3} + {c^3} - 3abc = 15 imes 12 cr
& Rightarrow {a^3} + {b^3} + {c^3} - 3abc = 180 cr} $$
[#202] If $$x + frac{1}{{x + 1}} = 1,$$ xa0xa0 then $${left( {x + 1}
ight)^5}$$ xa0 + $$frac{1}{{{{left( {x + 1}
ight)}^5}}}$$ xa0 equals?
Correct Answer
(B) 2
Explanation
Solution: $$eqalign{
& x + frac{1}{{x + 1}} = 1 cr
& { ext{Adding both 1 sides}} cr
& Rightarrow x + 1 + frac{1}{{x + 1}} = 1 + 1 cr
& Rightarrow left( {x + 1}
ight) + frac{1}{{left( {x + 1}
ight)}} = 2 cr
& { ext{Put }}x + 1 = 1 cr
& { ext{And }}frac{1}{{x + 1}} = 1 cr
& herefore {left( {x + 1}
ight)^5}{ ext{ + }}frac{1}{{{{left( {x + 1}
ight)}^5}}} cr
& = 1 + 1 cr
& = 2 cr} $$
[#203] If a + b + c = 0, then a 3 + b 3 + c 3 is equal to?
Correct Answer
(D) 3abc
Explanation
Solution: $$eqalign{
& { ext{If }}a + b + c = 0 cr
& { ext{then,}} cr
& {a^3} + {b^3} + {c^3} - 3abc = 0 cr
& Leftrightarrow {a^3} + {b^3} + {c^3} = 3abc cr} $$
[#204] If x = y = 333 and z = 334, then the value of x 3 + y 3 + z 3 - 3xyz is?
Correct Answer
(C) 1000
Explanation
Solution: x = y = 333, xa0 xa0 z = 334 ⇒ x 3 + y 3 + z 3 - 3xyz = $$frac{1}{2}$$ (x + y + z) [(x - y) 2 + (y - z) 2 + (z - x) 2 ] ⇒ x 3 + y 3 + z 3 - 3xyz = $$frac{1}{2}$$ (333 + 333 + 334) (333 - 333) 2 + (333 - 334) 2 + (334 - 333) 2 ⇒ x 3 + y 3 + z 3 - 3xyz = $$frac{1}{2}$$ (1000) (0 + 1 + 1) ⇒ x 3 + y 3 + z 3 - 3xyz = 1000
[#205] If $$a = frac{{{b^2}}}{{b - a}}{ ext{,}}$$ xa0 then the value of a 3 + b 3 is?
Correct Answer
(B) 0
Explanation
Solution: $$eqalign{
& a = frac{{{b^2}}}{{b - a}} cr
& Rightarrow aleft( {b - a}
ight) = {b^2} cr
& Rightarrow ab - {a^2} = {b^2} cr
& Rightarrow {a^2} + {b^2} - ab = 0 cr
& Rightarrow {a^3} + {b^3} = left( {a + b}
ight)left( {{a^2} + {b^2} - ab}
ight) cr
& Rightarrow {a^3} + {b^3} = 0 cr} $$