The Great Pyramid of Egypt is one of the ancient wonders of the world. Its shape can be viewed as a regular square pyramid. The area of a square with side length equal to the height of the pyramid equals the area of one lateral triangular face of the pyramid. The ratio of the height of a lateral triangular face to the side length of the base square is
A. $\frac { \sqrt { 5 } - 1 } { 4 }$
B. $\frac { \sqrt { 5 } - 1 } { 2 }$
C. $\frac { \sqrt { 5 } + 1 } { 4 }$
D. $\frac { \sqrt { 5 } + 1 } { 2 }$

The Great Pyramid of Khufu in Egypt is one of the ancient wonders of the world. Its shape can be viewed as a regular square pyramid. The area of a square with side length equal to the height of the pyramid equals the area of one lateral triangular face of the pyramid. Then the ratio of the height of the lateral triangle to the base of the square is
A. $\frac { \sqrt { 5 } - 1 } { 4 }$
B. $\frac { \sqrt { 5 } - 1 } { 2 }$
C. $\frac { \sqrt { 5 } + 1 } { 4 }$
D. $\frac { \sqrt { 5 } + 1 } { 2 }$

For $0 < c < b < a$, let $( a + b - 2 c ) x ^ { 2 } + ( b + c - 2 a ) x + ( c + a - 2 b ) = 0$ and $\alpha \neq 1$ be one of its root. Then, among the two statements (I) If $\alpha \in ( -1, 0)$, then $b$ cannot be the geometric mean of $a$ and $c$. (II) If $\alpha \in (0, 1)$, then $b$ may be the geometric mean of $a$ and $c$.
(1) Both (I) and (II) are true
(2) Neither (I) nor (II) is true
(3) Only (II) is true
(4) Only (I) is true

A sales station sells three types of mobile phones: A, B, and C. The profit per unit is 100 yuan for type A, 400 yuan for type B, and 240 yuan for type C. Last year, $A, B, C$ units of types A, B, C were sold respectively, with an average profit of 260 yuan per unit. It is also known that the average profit for selling types A and B together ($A + B$ units) is 280 yuan per unit. The ratio of the quantities of the three types of mobile phones sold last year is $A : B : C =$ (13－1)：(13－2)：(13－3) (expressed as a ratio of integers in lowest terms)

For distinct positive real numbers $x$ and $y$,
$$\left( \frac { x } { y } - \frac { y } { x } \right) \cdot \frac { x y } { 4 } = ( x - y ) ^ { 2 }$$
Given that, what is the ratio $\frac { x } { y }$?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 4 } { 3 }$
E) $\frac { 5 } { 3 }$

For positive real numbers $a, x$ and $y$
$$\begin{aligned} & -2x^{2} + y^{2} = 2a \\ & 3x^{2} - 2y^{2} = -6a \end{aligned}$$
what is the ratio $\dfrac{y}{x}$?
A) 1 B) $\sqrt{2}$ C) $\sqrt{3}$ D) 2 E) 3