Algebra - Study Mode
[#156] If $$x - frac{1}{x} = 1,$$ xa0 then what is the value of $$frac{1}{x}left( {frac{1}{{x - 1}} - frac{1}{{x + 1}} + frac{1}{{{x^2} + 1}} - frac{1}{{{x^2} - 1}}}
ight)?$$
Correct Answer
(C) $$ pm frac{2}{{sqrt {5x} }}$$
Explanation
Solution: $$eqalign{
& x - frac{1}{x} = 1 cr
& Rightarrow {x^2} + frac{1}{{{x^2}}} - 2 = 1 cr
& Rightarrow {left( {x + frac{1}{x}}
ight)^2} - 2 = 3 cr
& Rightarrow x + frac{1}{x} = pm sqrt 5 cr
& frac{1}{x}left( {frac{1}{{x - 1}} - frac{1}{{x + 1}} + frac{1}{{{x^2} + 1}} - frac{1}{{{x^2} - 1}}}
ight) cr
& Rightarrow frac{1}{x}left( {frac{2}{{{x^2} - 1}} - frac{2}{{left( {{x^2} + 1}
ight)left( {{x^2} - 1}
ight)}}}
ight) cr
& Rightarrow frac{1}{x}left( {frac{{2{x^2} + 2 - 2}}{{left( {{x^2} + 1}
ight)left( {{x^2} - 1}
ight)}}}
ight) cr
& x08ecause ,{x^2} - 1 = x,,{x^2} + 1 = sqrt {5x} cr
& Rightarrow frac{1}{x}left( {frac{{2{x^2}}}{{x imes sqrt {5x} }}}
ight) cr
& Rightarrow pm frac{2}{{sqrt {5x} }} cr} $$
[#157] If 2x 2 + 5x + 1 = 0, then one of the value of $$x - frac{1}{{2x}}$$ xa0is:
Correct Answer
(A) $$frac{{sqrt {17} }}{2}$$
Explanation
Solution: $$eqalign{
& 2{x^2} + 5x + 1 = 0 cr
& 2{x^2} + 1 = - 5x cr
& 2x + frac{1}{x} = - 5 cr
& { ext{divide '2' both sides}} cr
& x + frac{1}{{2x}} = - frac{5}{2} cr
& a - b = sqrt {{{left( {a + b}
ight)}^2} - 4ab} cr
& x - frac{1}{{2x}} = sqrt {{{left( { - frac{5}{2}}
ight)}^2} - 4 imes x imes frac{1}{{2x}}} cr
& = sqrt {frac{{25}}{2} - 2} cr
& = frac{{sqrt {17} }}{2} cr} $$
[#158] If x 4 - 6x 2 - 1 = 0, then the value of $${x^6} - 5{x^2} + frac{5}{{{x^2}}} - frac{1}{{{x^6}}} + 5$$ xa0 xa0 is:
Correct Answer
(B) 209
Explanation
Solution: $$eqalign{
& {x^4} - 6{x^2} - 1 = 0, cr
& {x^2}left( {{x^2} - 6 - frac{1}{{{x^2}}}}
ight) = 0 cr
& {x^2} - frac{1}{{{x^2}}} = 6 cr
& {x^6} - frac{1}{{{x^6}}} = 216 + 18 = 234 cr
& {x^6} - frac{1}{{{x^6}}} - 5left( {{x^2} - frac{1}{{{x^2}}}}
ight) + 5 cr
& = 234 - 30 + 5 cr
& = 209 cr} $$
[#159] If a + b + c = -11, then what is the value of (a + 4) 3 + (b + 5) 3 + (c + 2) 3 - 3(a + 4)(b + 5)(c + 2)?
Correct Answer
(C) 0
Explanation
Solution: Given, a + b + c = -11 By putting value i.e. a = -4 b = -5 c = -2 Therefore, (a + 4) 3 + (b + 5) 3 + (c + 2) 3 - 3(a + 4)(b + 5)(c + 2) = 0 + 0 + 0 - 0 = 0
[#160] If $$x - frac{1}{x} = 5,$$ xa0 x ≠ 0, then what is the value of $$frac{{{x^6} + 3{x^3} - 1}}{{{x^6} - 8{x^3} - 1}}?$$
Correct Answer
(B) $$frac{{13}}{{12}}$$
Explanation
Solution: $$eqalign{
& { ext{Given, }}x - frac{1}{x} = 5 cr
& Rightarrow {x^3} - frac{1}{{{x^3}}} = {5^3} + 3 imes 5 = 140 cr
& frac{{{x^6} + 3{x^3} - 1}}{{{x^6} - 8{x^3} - 1}} cr
& = frac{{{x^3} + 3 - frac{1}{{{x^3}}}}}{{{x^3} - 8 - frac{1}{{{x^3}}}}} cr
& = frac{{{x^3} - frac{1}{{{x^3}}} + 3}}{{{x^3} - frac{1}{{{x^3}}} - 8}} cr
& = frac{{140 + 3}}{{140 - 8}} cr
& = frac{{143}}{{132}} cr
& = frac{{13}}{{12}} cr} $$