The line $x + y + 2 = 0$ intersects the $x$-axis and $y$-axis at points $A$ and $B$ respectively. Point $P$ is on the circle $( x - 2 ) ^ { 2 } + y ^ { 2 } = 2$. The range of the area of $\triangle ABP$ is A. $[ 2,6 ]$ B. $[ 4,8 ]$ C. $[ \sqrt { 2 } , 3 \sqrt { 2 } ]$ D. $[ 2 \sqrt { 2 } , 3 \sqrt { 2 } ]$
Each member of a certain group uses mobile payment with probability $p$. The payment methods of each member are independent. Let $X$ be the number of people among 10 members of the group who use mobile payment. If $D(X) = 2.4$ and $P ( X = 4 ) < P ( X = 6 )$, then $p =$ A. 0.7 B. 0.6 C. 0.4 D. 0.3
Points $A, B, C, D$ are on the surface of a sphere with radius 4. $\triangle ABC$ is an equilateral triangle with area $9 \sqrt { 3 }$. The maximum volume of the tetrahedron $D$-$ABC$ is A. $12 \sqrt { 3 }$ B. $18 \sqrt { 3 }$ C. $24 \sqrt { 3 }$ D. $54 \sqrt { 3 }$
Let $F _ { 1 }, F _ { 2 }$ be the left and right foci of the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ $(a > 0, b > 0)$, and $O$ be the origin. A perpendicular is drawn from $F _ { 2 }$ to an asymptote of $C$, with foot of perpendicular at $P$. If $| PF_2 | = \sqrt { 6 } | OP |$, then the eccentricity of $C$ is A. $\sqrt { 5 }$ B. 2 C. $\sqrt { 3 }$ D. $\sqrt { 2 }$
Given point $M ( - 1, 1 )$ and parabola $C : y ^ { 2 } = 4 x$. A line through the focus of $C$ with slope $k$ intersects $C$ at points $A$ and $B$. If $\angle AMB = 90 ^ { \circ }$, then $k = $ $\_\_\_\_$.
In a geometric sequence $\{ a _ { n } \}$, $a _ { 1 } = 1$ and $a _ { 4 } = 4 a _ { 2 }$. (1) Find the general term formula for $\{ a_n \}$; (2) Let $S _ { n }$ be the sum of the first $n$ terms of $\{ a_n \}$. If $S _ { m } = 63$, find $m$.
To improve production efficiency, a factory conducted technological innovation activities and proposed two new production methods for completing a production task. To compare the efficiency of the two methods, 40 workers were selected and randomly divided into two groups of 20 each. The first group used the first production method, and the second group used the second production method. Based on the time (in minutes) taken by workers to complete the production task, a stem-and-leaf plot was drawn. (1) Based on the stem-and-leaf plot, which production method has higher efficiency? Explain your reasoning. (2) Find the median $m$ of the time taken by all 40 workers to complete the production task, and fill in the contingency table with the number of workers whose completion time exceeds $m$ and does not exceed $m$ for each production method.