If ^@ \alpha ^@ and ^@ \beta ^@ are the zeros of polynomial ^@ x^2-x-2,^@ find a polynomial whose zeros are ^@ \dfrac{ \alpha^2 }{ \beta^2 } ^@ and ^@ \dfrac{ \beta^2 }{ \alpha^2 }. ^@
Solve the following system of linear equations by the method of cross-multiplication.
@^
\begin{aligned}
&\dfrac { x } { g } + \dfrac { y } { h } = g + h \\
&\dfrac { x } { g^2 } + \dfrac { y } { h^2 } = 9
\end{aligned}
@^
A.
^@ x = \dfrac { - 8h^3 + g^3 } { g^2 - h^2 } \text{ and } y = \dfrac { 8gh^2 - h^3 } { g - h }^@
B.
^@ x = \dfrac { - 8g^2h + g^3 } { g - h } \text{ and } y = \dfrac { 8gh^2 - h^3 } { g - h }^@
C.
^@ x = \dfrac { - 8g^3h + g^4 } { g^2 - h } \text{ and } y = \dfrac { 8g^2h^2 + h^3 } { g - h }^@
D.
^@ x = \dfrac { - 8g^2h + g^4 } { g + h } \text{ and } y = \dfrac { 8gh^2 + h^3 } { g - h }^@
The areas of two similar triangles are ^@ 49\space cm^2^@ and ^@16\space cm^2^@ respectively. If the altitude of the bigger triangle is ^@6.3 \space cm^@, find the corresponding altitude of the smaller triangle.
From the top of a tower ^@ h \space meter ^@ high, the angle of depression of two objects, which are in line to the foot of the tower is ^@ \alpha ^@ and ^@ \beta ^@ ^@( \beta > \alpha)^@. What is the distance between the two objects?
A circle is inscribed in a ^@ \triangle ABC ^@, touching ^@ BC, CA, ^@ and ^@ AB ^@ at points ^@ P, Q, ^@ and ^@ R ^@ respectively. If ^@ AB = 12 \space cm, AQ = 9 \space cm ^@ and ^@ CQ = 7 \space cm ^@ then find the length of ^@ BC ^@.
Riley is always curious in creating formulas, for solving mathematical problems, of his own by repeated trial and error method. One day he found a formula for finding the surface area of the hexagonal pyramid. What should be the correct formula for the surface area of the hexagonal pyramid?
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