Algebra - Study Mode
[#271] The value of $$frac{{0.325 imes 0.325 + 0.175 imes 0.175 + 25 imes 0.00455}}{{5 imes 0.0065 imes 3.25 - 7 imes 0.175 imes 0.025}} - frac{{0.5}}{{1.5}}{ ext{is:}}$$
Correct Answer
(B) 3
Explanation
Solution: $$eqalign{
& frac{{0.325 imes 0.325 + 0.175 imes 0.175 + 25 imes 0.00455}}{{5 imes 0.0065 imes 3.25 - 7 imes 0.175 imes 0.025}} - frac{{0.5}}{{1.5}} cr
& = frac{{{{left( {3.35 + 0.175}
ight)}^2}}}{{left( {0.325 - 0.175}
ight)left( {0.325 + 0.175}
ight)}} - frac{1}{3} cr
& = frac{{0.500}}{{0.150}} - frac{1}{3} cr
& = frac{{10}}{3} - frac{1}{3} cr
& = frac{9}{3} cr
& = 3 cr} $$
[#272] If $$sqrt x + frac{1}{{sqrt x }} = 2sqrt 2 ,$$ xa0 xa0then $${x^2} + frac{1}{{{x^2}}}$$ xa0is equal to:
Correct Answer
(A) 34
Explanation
Solution: $$eqalign{
& sqrt x + frac{1}{{sqrt x }} = 2sqrt 2 cr
& x + frac{1}{x} + 2 = 8 cr
& x + frac{1}{x} = 6 cr
& {x^2} + frac{1}{{{x^2}}} + 2 = 36 cr
& {x^2} + frac{1}{{{x^2}}} = 34 cr} $$
[#273] If x 4 - 83x 2 + 1 = 0, then a value of x 3 - x -3 can be:
Correct Answer
(B) 756
Explanation
Solution: $$eqalign{
& {x^4} - 83{x^2} + 1 = 0 cr
& {x^2} - 83 + frac{1}{{{x^2}}} = 0 cr
& {x^2} + frac{1}{{{x^2}}} = 83 cr
& {x^2} + frac{1}{{{x^2}}} - 2 = 83 - 2 cr
& {left( {x - frac{1}{x}}
ight)^2} = 81 cr
& x - frac{1}{x} = 9 cr
& {left( {x - frac{1}{x}}
ight)^3} = {9^3} cr
& {x^3} - frac{1}{{{x^3}}} - 3 imes 9 = 729 cr
& {x^3} - frac{1}{{{x^3}}} = 756 cr} $$
[#274] If 1 + 9r 2 + 81r 4 = 256 and 1 + 3r + 9r 2 = 32, then find the value of 1 - 3r + 9r 2 .
Correct Answer
(A) 8
Explanation
Solution: 1 + 9r 2 + 81r 4 = 256 1 + 3r + 9r 2 = 32 So, 1 - 3r + 9r 2 $$ = frac{{256}}{{32}} = 8$$
[#275] If $${x^2} + frac{1}{{{x^2}}} = frac{7}{4}$$ xa0 for x > 0 then what is the value of $${x^3} + frac{1}{{{x^3}}} = ?$$
Correct Answer
(C) $$frac{{3sqrt {15} }}{8}$$
Explanation
Solution: $$eqalign{
& { ext{Given, }}{x^2} + frac{1}{{{x^2}}} = frac{7}{4} cr
& { ext{Adding 2 to both sides}} cr
& {x^2} + frac{1}{{{x^2}}} + 2 = frac{7}{4} + 2 cr
& {left( {x + frac{1}{x}}
ight)^2} = frac{{15}}{4} cr
& x + frac{1}{x} = frac{{sqrt {15} }}{2} cr
& x08ecause ,{x^3} + frac{1}{{{x^3}}} = {left( {x + frac{1}{x}}
ight)^3} - 3left( {x + frac{1}{x}}
ight) cr
& herefore ,{x^3} + frac{1}{{{x^3}}} = frac{{15 imes sqrt {15} }}{8} - 3 imes frac{{sqrt {15} }}{2} cr
& = frac{{15sqrt {15} - 12sqrt {15} }}{8} cr
& = frac{{3sqrt {15} }}{8} cr} $$