The graph of the function $y = k \cdot 3 ^ { x } ( 0 < k < 1 )$ intersects the graphs of the two functions $y = 3 ^ { - x }$ and $y = - 4 \cdot 3 ^ { x } + 8$ at points $\mathrm { P }$ and $\mathrm { Q }$ respectively. When the ratio of the $x$-coordinates of points P and Q is $1 : 2$, find the value of $35 k$. [4 points]

The graph of the function $y = k \cdot 3 ^ { x }$ ($0 < k < 1$) meets the graphs of the two functions $y = 3 ^ { - x }$ and $y = - 4 \cdot 3 ^ { x } + 8$ at points P and Q, respectively. When the ratio of the $x$-coordinates of points P and Q is $1 : 2$, find the value of $35k$. [4 points]

When the graph of the function $y = 2 ^ { x } + 2$ is translated in the $x$-direction by $m$ units, and this graph is symmetric to the graph of the function $y = \log _ { 2 } 8 x$ translated in the $x$-direction by 2 units with respect to the line $y = x$, what is the value of the constant $m$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

The line $y = 2 x + k$ meets the graphs of the two functions $$y = \left( \frac { 2 } { 3 } \right) ^ { x + 3 } + 1 , \quad y = \left( \frac { 2 } { 3 } \right) ^ { x + 1 } + \frac { 8 } { 3 }$$ at points $\mathrm { P }$ and $\mathrm { Q }$ respectively. When $\overline { \mathrm { PQ } } = \sqrt { 5 }$, what is the value of the constant $k$? [4 points]
(1) $\frac { 31 } { 6 }$
(2) $\frac { 16 } { 3 }$
(3) $\frac { 11 } { 2 }$
(4) $\frac { 17 } { 3 }$
(5) $\frac { 35 } { 6 }$

Point A$(a, b)$ is on the curve $y = \log _ { 16 } ( 8 x + 2 )$ and point B is on the curve $y = 4 ^ { x - 1 } - \frac { 1 } { 2 }$, both in the first quadrant. The point obtained by reflecting A across the line $y = x$ lies on the line OB, and the midpoint of segment AB has coordinates $\left( \frac { 77 } { 8 } , \frac { 133 } { 8 } \right)$. When $a \times b = \frac { q } { p }$, find the value of $p + q$. (Here, O is the origin, and $p$ and $q$ are coprime natural numbers.) [4 points]

5. Determine the value of the real parameter $k$ so that the two curves $y = e ^ { x }$, $y = 6 - k e ^ { - x }$ are tangent to each other, finding the coordinates of the point of tangency.