Algebra - Study Mode
[#236] If $${x^2} + frac{1}{{{x^2}}} = 38,$$ xa0 then what is the value of $$left( {x - frac{1}{x}}
ight)?$$
Correct Answer
(C) 6
Explanation
Solution: $$eqalign{
& {x^2} + frac{1}{{{x^2}}} = 38 cr
& {x^2} + frac{1}{{{x^2}}} - 2 = 38 - 2 cr
& {left( {x - frac{1}{x}}
ight)^2} = 36 cr
& x - frac{1}{x} = 6 cr} $$
[#237] If 5√5x 3 + 2√2y 3 = (Ax + √2y)(Bx 2 + 2y 2 + Cxy), then the value of (A 2 + B 2 - C 2 ) is:
Correct Answer
(B) 20
Explanation
Solution: $$eqalign{
& 5sqrt 5 {x^3} + 2sqrt 2 {y^3} = left( {Ax + sqrt 2 y}
ight)left( {B{x^2} + 2{y^2} + Cxy}
ight) cr
& {left( {sqrt 5 x}
ight)^3} + {left( {sqrt 2 y}
ight)^3} = left( {sqrt 5 x + sqrt 3 y}
ight)left( {5{x^2} + 9{y^2} - sqrt {10} xy}
ight) cr
& left( {sqrt 5 x + sqrt 3 y}
ight)left( {5{x^2} + 9{y^2} - sqrt {10} xy}
ight) = left( {Ax + sqrt 2 y}
ight)left( {B{x^2} + 2{y^2} + Cxy}
ight) cr
& { ext{Comparison both side:}} cr
& A = sqrt 5 cr
& B = 5 cr
& C = - sqrt {10} cr
& {A^2} + {B^2} + {C^2} = 5 + 25 - 10 = 20 cr} $$
[#238] If A = 1 + 2 P and B = 1 + 2 -P , then what is the value of B?
Correct Answer
(C) $$frac{A}{{A - 1}}$$
Explanation
Solution: $$eqalign{
& A = 1 + {2^P} cr
& B = 1 + {2^{ - P}} cr
& { ext{Put }}P = 1 cr
& A = 3,,B = frac{3}{2} cr
& { ext{Option C is correct}} cr
& cr
& {x08f{Alternate:}} cr
& A = 1 + {2^P},......,left( { ext{i}}
ight) cr
& B = 1 + {2^{ - P}} cr
& = 1 + frac{1}{{{2^P}}} cr
& = frac{{{2^P} + 1}}{{{2^P}}} cr
& = frac{A}{{A - 1}},,,,,left( {{ ext{from equation }}left( { ext{i}}
ight)}
ight) cr} $$
[#239] If $$x + frac{1}{x} = 8,$$ xa0 then find the value of $$frac{{5x}}{{{x^2} + 1 - 6x}}.$$
Correct Answer
(A) 2.5
Explanation
Solution: $$eqalign{
& x + frac{1}{x} = 8 cr
& {x^2} + 1 = 8x cr
& frac{{5x}}{{{x^2} + 1 - 6x}} cr
& = frac{{5x}}{{8x - 6x}} cr
& = frac{{5x}}{{2x}} cr
& = frac{5}{2} cr
& = 2.5 cr} $$
[#240] If $$x + frac{1}{x} = 2,$$ xa0 x ≠ 0, then the value of $${x^2}{ ext{ + }}frac{1}{{{x^3}}}$$ xa0 is equal to?
Correct Answer
(B) 2
Explanation
Solution: $$eqalign{
& { ext{ }}x + frac{1}{x} = 2{ ext{, }},,,x
e 0 cr
& { ext{Put }}x = 1 cr
& 1 + 1 = 2 cr
& herefore {x^2}{ ext{ + }}frac{1}{{{x^2}}} cr
& = 1 + 1 cr
& = 2 cr} $$