The Circular Mound Altar at the Beijing Temple of Heaven is an ancient place for worshipping heaven, divided into three levels: upper, middle, and lower. At the center of the upper level is a circular stone slab (called the Heaven's Heart Stone), surrounded by 9 fan-shaped stone slabs forming the first ring, with each outer ring increasing by 9 slabs. On the next level, the first ring has 9 more slabs than the last ring of the upper level, and each outer ring also increases by 9 slabs. It is known that each level has the same number of rings, and the lower level has 729 more slabs than the middle level. The total number of fan-shaped stone slabs (excluding the Heaven's Heart Stone) in all three levels is A． 3699 slabs B． 3474 slabs C． 3402 slabs D． 3339 slabs
If a circle passing through point $(2,1)$ is tangent to both coordinate axes, then the distance from the center of the circle to the line $2 x - y - 3 = 0$ is A．$\frac { \sqrt { 5 } } { 5 }$ B．$\frac { 2 \sqrt { 5 } } { 5 }$ C．$\frac { 3 \sqrt { 5 } } { 5 }$ D．$\frac { 4 \sqrt { 5 } } { 5 }$
In the sequence $\left\{ a _ { n } \right\}$ , $a _ { 1 } = 2 , a _ { m + n } = a _ { m } a _ { n }$ . If $a _ { k + 1 } + a _ { k + 2 } + \cdots + a _ { k + 10 } = 2 ^ { 15 } - 2 ^ { 5 }$ , then $k =$ A． 2 B． 3 C． 4 D． 5
Let $O$ be the origin of coordinates. The line $x = a$ intersects the two asymptotes of the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ at points $D$ and $E$ respectively. If the area of $\triangle O D E$ is 8, then the minimum value of the focal distance of $C$ is A． 4 B． 8 C． 16 D． 32
Let the function $f ( x ) = \ln | 2 x + 1 | - \ln | 2 x - 1 |$ . Then $f ( x )$ A． is an even function and monotonically increasing on $\left( \frac { 1 } { 2 } , + \infty \right)$ B． is an odd function and monotonically decreasing on $\left( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$ C． is an even function and monotonically increasing on $\left( - \infty , - \frac { 1 } { 2 } \right)$ D． is an odd function and monotonically decreasing on $\left( - \infty , - \frac { 1 } { 2 } \right)$
If $2 ^ { x } - 2 ^ { y } < 3 ^ { - x } - 3 ^ { - y }$ , then A． $\ln ( y - x + 1 ) > 0$ B． $\ln ( y - x + 1 ) < 0$ C． $\ln | x - y | > 0$ D． $\ln | x - y | < 0$
Given unit vectors $\boldsymbol { a } , \boldsymbol { b }$ with an angle of $45 ^ { \circ }$ between them, and $k \boldsymbol { a } - \boldsymbol { b}$ is perpendicular to $\boldsymbol { a }$ , then $k = $ $\_\_\_\_$.
Four students participate in garbage classification publicity activities in 3 residential areas, with each student going to only 1 area and each area having at least 1 student assigned. The total number of different arrangement methods is $\_\_\_\_$.
In $\triangle A B C$ , $\sin ^ { 2 } A - \sin ^ { 2 } B - \sin ^ { 2 } C = \sin B \sin C$ . (1) Find $A$ ; (2) If $B C = 3$ , find the maximum value of the perimeter of $\triangle A B C$ .
After treatment, the ecosystem in a desert region has improved significantly, and the number of wild animals has increased. To investigate the population of a certain wild animal species in this region, the area is divided into 200 plots of similar size. A simple random sample of 20 plots is selected as sample areas. The sample data obtained is $\left( x _ { i } , y _ { i } \right) ( i = 1,2 , \cdots , 20 )$ , where $x _ { i }$ and $y _ { i }$ represent the plant coverage area (in hectares) and the number of this wild animal species in the $i$-th sample area, respectively. The following calculations are obtained: $\sum _ { i = 1 } ^ { 20 } x _ { i } = 60 , \sum _ { i = 1 } ^ { 20 } y _ { i } = 1200 , \sum _ { i = 1 } ^ { 20 } \left( x _ { i } - \bar { x } \right) ^ { 2 } = 80 , \sum _ { i = 1 } ^ { 20 } \left( y _ { i } - \bar { y } \right) ^ { 2 } = 9000 , \sum _ { i = 1 } ^ { 20 } \left( x _ { i } - \bar { x } \right) \left( y _ { i } - \bar { y } \right) = 800$ . (1) Find the estimated value of the population of this wild animal species in the region (the estimated value equals the average number of this wild animal species in the sample areas multiplied by the number of plots); (2) Find the correlation coefficient of the sample $\left( x _ { i } , y _ { i } \right) ( i = 1,2 , \cdots , 20 )$ (accurate to 0.01); (3) Based on current statistical data, there is great variation in plant coverage area among different plots. To improve the representativeness of the sample and obtain a more accurate estimate of the population of this wild animal species in the region, please suggest a more reasonable sampling method and explain your reasoning.
Given the function $f ( x ) = \left| x - a ^ { 2 } \right| + | x - 2 a + 1 |$ . (1) When $a = 2$, find the solution set of the inequality $f ( x ) \geqslant 4$; (2) If $f ( x ) \geqslant 4$ for all $x$, find the range of values of $a$.