Algebra - Study Mode

[#181] If x + y = 4, xy = 2, y + z = 5, yz = 3, z + x = 6 and zx = 4, then find the value of x 3 + y 3 + z 3 - 3xyz.
Correct Answer

(D) 153.75

Explanation

Solution: $$eqalign{
& x + y = 4,,y + z = 5,,z + x = 6 cr
& { ext{So, }}x + y + z = frac{{15}}{2} cr
& {left( {x - y}
ight)^2} = {left( {x + y}
ight)^2} - 4xy cr
& = {4^2} - 4 imes 2 cr
& = 8 cr
& {left( {y - z}
ight)^2} = {left( {y + z}
ight)^2} - 4yz cr
& = {5^2} - 4 imes 3 cr
& = 13 cr
& {left( {z - x}
ight)^2} = {left( {z + x}
ight)^2} - 4zx cr
& = {6^2} - 4 imes 4 cr
& = 20 cr
& {x^3} + {y^3} + {z^3} - 3xyz cr
& = frac{{left( {x + y + z}
ight)}}{2}left[ {{{left( {x - y}
ight)}^2} + {{left( {y - z}
ight)}^2} + {{left( {z - x}
ight)}^2}}
ight] cr
& = frac{1}{2} imes frac{{15}}{2}left[ {8 + 13 + 20}
ight] cr
& = frac{{15 imes 41}}{4} cr
& = 153.75 cr} $$

[#182] If x = 2 - p, then x 3 + 6xp + p 3 is equal to:
Correct Answer

(C) 8

Explanation

Solution: x = 2 - p x + p = 2 Cube both side (x + p) 3 = 2 3 x 3 + p 3 + 3xp(x + p) = 8 x 3 + p 3 + 2xp × 2 = 8 x 3 + 6xp + p 3 = 8

[#183] Simplify the following expression: $$frac{{{{left( {{a^2} - 4{b^2}}
ight)}^3} + 64{{left( {{b^2} - 4{c^2}}
ight)}^3} + {{left( {16{c^2} - {a^2}}
ight)}^3}}}{{{{left( {a - 2b}
ight)}^3} + {{left( {2b - 4c}
ight)}^3} + {{left( {4c - a}
ight)}^3}}}$$
Correct Answer

(B) 2(a + 2b) (b + 2c) (4c + a)

Explanation

Solution: $$eqalign{
& frac{{{{left( {{a^2} - 4{b^2}}
ight)}^3} + 64{{left( {{b^2} - 4{c^2}}
ight)}^3} + {{left( {16{c^2} - {a^2}}
ight)}^3}}}{{{{left( {a - 2b}
ight)}^3} + {{left( {2b - 4c}
ight)}^3} + {{left( {4c - a}
ight)}^3}}} cr
& { ext{Put }}a = b = c cr
& = frac{{{{left( { - 3{a^2}}
ight)}^3} + 64{{left( { - 3{a^2}}
ight)}^3} + {{left( {15{a^2}}
ight)}^3}}}{{{{left( { - a}
ight)}^3} + {{left( { - 2a}
ight)}^3} + {{left( {3a}
ight)}^3}}} cr
& = frac{{{a^6}left[ { - 27 - 27 imes 64 + {{left( {15}
ight)}^3}}
ight]}}{{{a^3}left[ { - 1 - 8 + 27}
ight]}} cr
& = frac{{3{a^3}left[ { - 9 - 576 + 1125}
ight]}}{{18}} cr
& = frac{{{a^3} imes 540}}{6} cr
& = 90{a^3} cr
& { ext{From option put }}a = b = c cr
& left( { ext{A}}
ight), - 45{a^3} cr
& left( { ext{B}}
ight),90{a^3} cr
& { ext{Hence option B is right answer}}{ ext{.}} cr} $$

[#184] x is a negative number such that k + k -1 = -2, then what is the value of $$frac{{{k^2} + 4k - 2}}{{{k^2} + k - 5}}?$$
Correct Answer

(B) 1

Explanation

Solution: $$eqalign{
& k + frac{1}{k} = 2 cr
& k = - 1 cr
& frac{{{k^2} + 4k - 2}}{{{k^2} + k - 5}} cr
& = frac{{{{left( { - 1}
ight)}^2} + 4left( { - 1}
ight) - 2}}{{{{left( { - 1}
ight)}^2} + left( { - 1}
ight) - 5}} cr
& = frac{{1 - 4 - 2}}{{1 - 1 - 5}} cr
& = frac{{ - 5}}{{ - 5}} cr
& = 1 cr} $$

[#185] If a + b + c = 2, $$frac{1}{a} + frac{1}{b} + frac{1}{c}$$ xa0 = 0, ac = $$frac{4}{b}$$ and a 3 + b 3 + c 3 = 28, find the value of a 2 + b 2 + c 2 .
Correct Answer

(D) 8

Explanation

Solution: $$eqalign{
& a + b + c = 2 cr
& ab + bc + ca = 0 cr
& abc = 4 cr
& {a^3} + {b^3} + {c^3} - 3abc = left( {a + b + c}
ight)left[ {left( {{a^2} + {b^2} + {c^2}}
ight) - left( {ab + bc + ca}
ight)}
ight] cr
& 28 - 3 imes 4 = 2left( {{a^2} + {b^2} + {c^2}}
ight) cr
& {a^2} + {b^2} + {c^2} = 8 cr} $$