Algebra - Study Mode
[#276] If $${y^4} + frac{1}{{{y^4}}} = 223$$ xa0 and y > 1, then find the value of $${y^2} + frac{1}{{{y^2}}}?$$
Correct Answer
(A) 15
Explanation
Solution: $$eqalign{
& {y^4} + frac{1}{{{y^4}}} = 223 cr
& Rightarrow {y^2} + frac{1}{{{y^2}}} = sqrt {223 + 2} cr
& Rightarrow {y^2} + frac{1}{{{y^2}}} = sqrt {225} cr
& Rightarrow {y^2} + frac{1}{{{y^2}}} = 15 cr} $$
[#277] If x - y + z = 0, then find the value of $$frac{{{y^2}}}{{2xz}} - frac{{{x^2}}}{{2yz}} - frac{{{z^2}}}{{2xy}}?$$
Correct Answer
(A) $$frac{3}{2}$$
Explanation
Solution: $$eqalign{
& frac{{{y^2}}}{{2xz}} - frac{{{x^2}}}{{2yz}} - frac{{{z^2}}}{{2xy}} cr
& frac{{{y^3} - {x^3} - {z^3}}}{{2xyz}},......,left( 1
ight) cr
& x - y + z = 0 cr
& x + z = y cr
& { ext{Cubing both side}} cr
& {left( {x + z}
ight)^3} = {y^3} cr
& {x^3} + {z^3} + 3left( {x + z}
ight)left( x
ight)left( z
ight) = {y^3} cr
& {x^3} + {z^3} + 3left( y
ight)left( x
ight)left( z
ight) = {y^3} cr
& 3xyz = {y^3} - {x^3} - {z^3} cr
& { ext{Put in equation }}left( 1
ight) cr
& frac{{3xyz}}{{2xyz}} = x08oxed{frac{3}{2}} cr} $$
[#278] If x 2 - 5x + 1 = 0, then the value of $$left( {{x^4} + frac{1}{{{x^2}}}}
ight) div left( {{x^2} + 1}
ight)$$ xa0 xa0 is:
Correct Answer
(B) 22
Explanation
Solution: $$eqalign{
& {x^2} - 5x + 1 = 0 cr
& left( {{x^4} + frac{1}{{{x^2}}}}
ight) div left( {{x^2} + 1}
ight) = ? cr
& {x^2} - 5x + 1 = 0 cr
& Rightarrow x + frac{1}{x} = 5 cr
& Rightarrow {x^3} + frac{1}{{{x^3}}} = 110 cr
& frac{{xleft( {{x^3} + frac{1}{{{x^3}}}}
ight)}}{{xleft( {x + frac{1}{x}}
ight)}} = frac{{110}}{5} = 22 cr} $$
[#279] If a + b + c = 6, a 3 + b 3 + c 3 - 3abc = 342, then what is the value of ab + bc + ca?
Correct Answer
(C) -7
Explanation
Solution: Let c = 0 a + b = 6 a 3 + b 3 = 342 ab = ? (a 3 + b 3 ) = (a + b)[(a + b) 2 - 3ab] 342 = 6[6 2 - 3ab] 57 = 36 - 3ab 3ab = -21 ab = -7
[#280] If 1 < x < 2, then the value of $$sqrt {{{left( {x - 1}
ight)}^2}} { ext{ + }}sqrt {{{left( {3 - x}
ight)}^2}} { ext{ is?}}$$
Correct Answer
(B) 2
Explanation
Solution: $$eqalign{
& 1 < x < 2 cr
& sqrt {{{left( {x - 1}
ight)}^2}} { ext{ + }}sqrt {{{left( {3 - x}
ight)}^2}} cr
& left( {{ ext{Square root cancel with square}}}
ight) cr
& herefore x - 1 + 3 - x cr
& = 2 cr} $$