Algebra - Study Mode
[#476] If $${left[ {a + frac{1}{a}}
ight]^2} - 2left[ {a - frac{1}{a}}
ight] = 12,$$ xa0 xa0 xa0then which of the following is a value of 'a'?
Correct Answer
(D) None of these
Explanation
Solution: $$eqalign{
& {left[ {a + frac{1}{a}}
ight]^2} - 2left[ {a - frac{1}{a}}
ight] = 12 cr
& {left[ {a - frac{1}{a}}
ight]^2} + 4 - 2left[ {a - frac{1}{a}}
ight] = 12 cr
& {left[ {a - frac{1}{a}}
ight]^2} - 2left[ {a - frac{1}{a}}
ight] - 8 = 0 cr
& { ext{Let }}a - frac{1}{a} = x cr
& herefore ,{x^2} - 2x - 8 = 0 cr
& {x^2} - 4x + 2x - 8 = 0 cr
& left( {x - 4}
ight)left( {x + 2}
ight) = 0 cr
& x = 4,,x = - 2 cr
& a - frac{1}{a} = - 2,,a - frac{1}{a} = 4 cr
& herefore ,a + frac{1}{a} = sqrt {{4^2} + 4} cr
& a + frac{1}{a} = 2sqrt 5 cr
& a - frac{1}{a} = 4 cr
& herefore ,2a = 2sqrt 5 + 4 cr
& a = 2 + sqrt 5 cr} $$
[#477] If $$sqrt x - frac{1}{{sqrt x }} = sqrt 5 ,$$ xa0 xa0then $${x^2} + frac{1}{{{x^2}}}$$ xa0is equal to:
Correct Answer
(C) 47
Explanation
Solution: $$eqalign{
& sqrt x - frac{1}{{sqrt x }} = sqrt 5 cr
& { ext{Square both side}} cr
& x + frac{1}{x} - 2.x.frac{1}{x} = 5 cr
& x + frac{1}{x} = 5 + 2 cr
& { ext{Square both side}} cr
& {x^2} + frac{1}{{{x^2}}} + 2.x.frac{1}{x} = 49 cr
& {x^2} + frac{1}{{{x^2}}} = 47 cr} $$
[#478] The value of $$frac{{5.35 imes 5.35 imes 5.35 + 3.65 imes 3.65 imes 3.65}}{{53.5 imes 53.5 + 36.5 imes 36.5 - 53.5 imes 36.5}}{ ext{is:}}$$
Correct Answer
(C) 0.09
Explanation
Solution: $$eqalign{
& frac{{5.35 imes 5.35 imes 5.35 + 3.65 imes 3.65 imes 3.65}}{{53.5 imes 53.5 + 36.5 imes 36.5 - 53.5 imes 36.5}} cr
& = frac{{{{5.35}^3} + {{3.65}^3}}}{{{{53.5}^2} + {{36.5}^2} - 53.5 imes 36.5}} cr
& = frac{{left( {5.35 + 3.65}
ight)left[ {{{left( {5.35}
ight)}^2} + {{left( {3.65}
ight)}^2} - 5.35 imes 3.65}
ight]}}{{{{10}^2}left[ {{{left( {5.35}
ight)}^2} + {{left( {3.65}
ight)}^2} - 5.35 imes 3.65}
ight]}} cr
& = frac{9}{{{{10}^2}}} cr
& = 0.09 cr} $$
[#479] If x 2 - 3x - 1 = 0, then the value of (x 2 + 8x - 1)(x 3 + x -1 ) -1 is:
Correct Answer
(C) 1
Explanation
Solution: $$eqalign{
& {x^2} - 3x - 1 = 0 cr
& xleft( {x - 3 - frac{1}{x}}
ight) = 0 cr
& x - frac{1}{x} = 3 cr
& {x^2} + frac{1}{{{x^2}}} = 11 cr
& frac{{left( {{x^2} + 8x - 1}
ight)}}{{{x^3} + frac{1}{x}}} cr
& = frac{{xleft( {frac{{x - 1}}{{x + 8}}}
ight)}}{{xleft( {frac{{{x^2} + 1}}{{{x^2}}}}
ight)}} cr
& = frac{{3 + 8}}{{11}} cr
& = 1 cr} $$
[#480] If a + b + c = 3 and none of a, b and c is equal to 1, then what is the value of $$frac{1}{{left( {1 - a}
ight)left( {1 - b}
ight)}} + frac{1}{{left( {1 - b}
ight)left( {1 - c}
ight)}} + frac{1}{{left( {1 - c}
ight)left( {1 - a}
ight)}}?$$
Correct Answer
(A) 0
Explanation
Solution: $$eqalign{
& a + b + c = 3 cr
& { ext{Put }}a = 4,,b = - 1,,c = 0 cr
& frac{1}{{left( {1 - a}
ight)left( {1 - b}
ight)}} + frac{1}{{left( {1 - b}
ight)left( {1 - c}
ight)}} + frac{1}{{left( {1 - c}
ight)left( {1 - a}
ight)}} cr
& = frac{1}{{ - 3 imes 2}} + frac{1}{{2 imes 1}} - frac{1}{3} cr
& = frac{1}{2} - frac{1}{3} - frac{1}{6} cr
& = frac{0}{6} cr
& = 0 cr} $$