Algebra - Study Mode

[#41] If $$x + frac{1}{{4x}} = frac{3}{2}{ ext{,}}$$ xa0 find the value of $${ ext{8}}{x^3}{ ext{ + }}frac{1}{{8{x^3}}} = ?$$
Correct Answer

(A) 18

Explanation

Solution: $$eqalign{
& x + frac{1}{{4x}} = frac{3}{2} cr
& { ext{Multiply by 2 both sides}} cr
& herefore 2x + frac{1}{{2x}} = 3 cr
& { ext{Take cube both sides}} cr
& Rightarrow {left( {2x + frac{1}{{2x}}}
ight)^3} = {left( 3
ight)^3} cr
& Rightarrow { ext{8}}{x^3}{ ext{ + }}frac{1}{{8{x^3}}} + 3.2x.frac{1}{{2x}}left( {2x{ ext{ + }}frac{1}{{2x}}}
ight) = 27 cr
& Rightarrow { ext{8}}{x^3}{ ext{ + }}frac{1}{{8{x^3}}} + 3left( 3
ight) = 27 cr
& Rightarrow { ext{8}}{x^3}{ ext{ + }}frac{1}{{8{x^3}}} = 18 cr} $$

[#42] If $$3x + frac{1}{{2x}} = 5{ ext{,}}$$ xa0 then the value of $${ ext{8}}{x^3}{ ext{ + }}frac{1}{{27{x^3}}},{ ext{is?}}$$
Correct Answer

(B) $${ ext{30}}frac{{10}}{{27}}$$

Explanation

Solution: $$eqalign{
& 3x + frac{1}{{2x}} = 5 cr
& Rightarrow { ext{Multiply both sides by }}frac{2}{3} cr
& herefore 3x imes frac{2}{3} + frac{1}{2}x imes frac{2}{3} = 5 imes frac{2}{3} cr
& Rightarrow 2x + frac{1}{{3x}} = frac{{10}}{3} cr
& herefore { ext{Taking cube on both sides}} cr
& Rightarrow { ext{8}}{x^3}{ ext{ + }}frac{1}{{27{x^3}}} + 3.2x.frac{1}{{3x}}left( {{ ext{ 2}}x{ ext{ + }}frac{1}{{3x}}}
ight) = {left( {frac{{10}}{3}}
ight)^3} cr
& Rightarrow { ext{8}}{x^3}{ ext{ + }}frac{1}{{27{x^3}}} + 2left( {frac{{10}}{3}}
ight) = left( {frac{{1000}}{{27}}}
ight) cr
& Rightarrow { ext{8}}{x^3}{ ext{ + }}frac{1}{{27{x^3}}} = frac{{1000}}{{27}} - frac{{20}}{3} cr
& Rightarrow { ext{8}}{x^3}{ ext{ + }}frac{1}{{27{x^3}}} = frac{{1000 - 180}}{{27}} cr
& Rightarrow { ext{8}}{x^3}{ ext{ + }}frac{1}{{27{x^3}}} = frac{{820}}{{27}} cr
& Rightarrow { ext{8}}{x^3}{ ext{ + }}frac{1}{{27{x^3}}} = 30frac{{10}}{{27}} cr} $$

[#43] If x + y = z, then the expression x 3 + y 3 - z 3 + 3xyz will be equal to?
Correct Answer

(A) 0

Explanation

Solution: $$eqalign{
& x + y = z cr
& x + y - z = 0 cr
& { ext{If }}a + b + c = 0 cr
& { ext{then }}{a^3} + {b^3} + {c^3} - 3abc = 0 cr
& Rightarrow {x^3} + {y^3} - {z^3} = - 3xyz cr
& herefore {x^3} + {y^3} - {z^3} + 3xyz = 0 cr
& Rightarrow 3xyz - 3xyz = 0 cr} $$

[#44] If a 3 - b 3 - c 3 - 3abc = 0, then -
Correct Answer

(D) a = b + c

Explanation

Solution: $$eqalign{
& {a^3} - {b^3} - {c^3} - 3abc = 0 cr
& herefore a - b - c = 0 cr
& Rightarrow a = b + c cr} $$

[#45] If a = 2.361, b = 3.263, and c = 5.624, then the value of a 3 + b 3 - c 3 + 3abc is?
Correct Answer

(C) 0

Explanation

Solution: $$eqalign{
& a = 2.361 cr
& b = 3.263 cr
& c = 5.624 cr
& a + b - c = 0 cr
& 2.361 + 3.263 - 5.624 = 0 cr
& 0 = 0 cr
& herefore {a^3} + {b^3} - {c^3} + 3abc cr
& Rightarrow 0 cr} $$