The graph of the function $y = \left( 3 ^ { x } - 3 ^ { - x } \right) \cos x$ on the interval $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ is approximately: (see figures A, B, C, D)
For the ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$, let $A$ be the left vertex. Points $P$ and $Q$ are both on $C$ and symmetric about the $y$-axis. If the product of the slopes of $AP$ and $AQ$ is $\frac { 1 } { 4 }$, then the eccentricity of $C$ is: A. $\frac { \sqrt { 3 } } { 2 }$ B. $\frac { \sqrt { 2 } } { 2 }$ C. $\frac { 1 } { 2 }$ D. $\frac { 1 } { 3 }$
Let vectors $\boldsymbol { a }$ and $\boldsymbol { b }$ have an angle whose cosine is $\frac { 1 } { 3 }$, and $| \boldsymbol { a } | = 1$, $| \boldsymbol { b } | = 3$. Then $( 2 \boldsymbol { a } + \boldsymbol { b } ) \cdot \boldsymbol { b } =$ $\_\_\_\_$
If the asymptotes of the hyperbola $y ^ { 2 } - \frac { x ^ { 2 } } { m ^ { 2 } } = 1 ( m > 0 )$ are tangent to the circle $x ^ { 2 } + y ^ { 2 } - 4 y + 3 = 0$, then $m =$ $\_\_\_\_$
Let $S _ { n }$ denote the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$. Given that $\frac { 2 S _ { n } } { n } + n = 2 a _ { n } + 1$. (1) Prove that $\left\{ a _ { n } \right\}$ is an arithmetic sequence; (2) If $a _ { 4 }$, $a _ { 7 }$, $a _ { 9 }$ form a geometric sequence, find the minimum value of $S _ { n }$.
In the pyramid $P - ABCD$, $PD \perp$ base $ABCD$, $CD \parallel AB$, $AD = DC = CB = 1$, $AB = 2$. (1) Prove that $BD \perp PA$; (2) Find the sine of the angle between $PD$ and plane $PAB$.
Given the function $f ( x ) = \frac { e ^ { x } } { x } - \ln x + x - a$. (1) If $f ( x ) \geq 0$, find the range of values for $a$; (2) Prove that if $f ( x )$ has two zeros $x _ { 1 }$ and $x _ { 2 }$, then $x _ { 1 } x _ { 2 } < 1$.