Based on the bar chart of China's sulfur dioxide emissions (in units of 10,000 tons) from 2004 to 2013 shown below, which of the following conclusions is incorrect? (A) Year-on-year comparison shows that 2008 had the most significant reduction in sulfur dioxide emissions (B) China's treatment of sulfur dioxide emissions became evident in 2007 (C) Since 2006, China's annual sulfur dioxide emissions have shown a decreasing trend (D) Since 2006, China's annual sulfur dioxide emissions are positively correlated with the year
The circle passing through three points $A ( 1,3 ) , B ( 4,2 ) , C ( 1,7 )$ intersects the $y$-axis at points $\mathrm { M }$ and $\mathrm { N }$. Then $| M N | =$ (A) $2 \sqrt { 6 }$ (B) $8$ (C) $4 \sqrt { 6 }$ (D) $10$
As shown in the figure, rectangle $ABCD$ has sides $\mathrm { AB } = 2 , \mathrm { BC } = 1$, and $O$ is the midpoint of $AB$. Point $P$ moves along edges $\mathrm { BC } , \mathrm { CD }$, and $DA$, with $\angle \mathrm { BOP } = \mathrm { x }$. The sum of distances from moving point $P$ to points $A$ and $B$ is expressed as a function of $x$, denoted $f ( x )$. The graph of $f ( x )$ is approximately (A), (B), (C), or (D) [as shown in figures]
Points $A$ and $B$ are the left and right vertices of hyperbola $E$. Point $M$ is on $E$, and $\triangle A B M$ is an isosceles triangle with vertex angle $120 ^ { \circ }$. Then the eccentricity of $E$ is (A) $\sqrt { 5 }$ (B) $2$ (C) $\sqrt { 3 }$ (D) $\sqrt { 2 }$
Let $\mathrm { f } ^ { \prime } ( \mathrm { x } )$ be the derivative of the odd function $f ( x ) ( x \in \mathbf { R } )$. Given $\mathrm { f } ( - 1 ) = 0$, and when $\mathrm { x } > 0$, $x f ^ { \prime } ( x ) - f ( x ) < 0$. Then the range of $x$ for which $f ( x ) > 0$ holds is (A) $( - \infty , - 1 ) \cup ( 0,1 )$ (B) $( - 1,0 ) \cup ( 1 , + \infty )$ (C) $( - \infty , - 1 ) \cup ( - 1,0 )$ (D) $( 0,1 ) \cup ( 1 , + \infty )$
Let vectors $\mathrm { a }$ and $\mathrm { b }$ be non-parallel. If vector $\lambda a + b$ is parallel to $\boldsymbol { a } + 2 b$, then the real number $\lambda = $ $\_\_\_\_$.
If $\mathrm { x } , \mathrm { y }$ satisfy the constraint conditions $\left\{ \begin{array} { l } x - y + 1 \geqslant 0 , \\ x - 2 y \leqslant 0 , \\ x + 2 y - 2 \leqslant 0 , \end{array} \right.$ then the maximum value of $z = x + y$ is $\_\_\_\_$ .
Let $\mathrm { S } _ { \mathrm { n } }$ be the sum of the first $n$ terms of sequence $\left\{ \mathrm { a } _ { \mathrm { n } } \right\}$, and $a _ { 1 } = - 1 , a _ { \mathrm { n } + 1 } = S _ { n } S _ { n + 1 }$. Then $S _ { n } = $ $\_\_\_\_$ .
In $\triangle \mathrm { ABC }$, $D$ is a point on $BC$, $AD$ bisects $\angle \mathrm { BAC }$, and the area of $\triangle \mathrm { ABD }$ is 2 times the area of $\triangle \mathrm { ADC }$. (I) Find $\frac { \sin \angle B } { \sin \angle C }$ ; (II) If $A D = 1 , D C = \frac { \sqrt { 2 } } { 2 }$, find the lengths of $B D$ and $A C$.
A company conducted a survey of 20 users from regions A and B respectively to understand user satisfaction with its products. The satisfaction scores are as follows: Region A: 62, 73, 81, 92, 95, 85, 74, 64, 53, 76, 78, 86, 95, 66, 97, 78, 88, 82, 76, 89 Region B: 73, 83, 62, 51, 91, 46, 53, 73, 64, 82, 93, 48, 65, 81, 74, 56, 54, 76, 65, 79 (I) Complete the stem-and-leaf plot for user satisfaction scores in both regions based on the two sets of data, and compare the mean and dispersion of satisfaction scores between the two regions through the stem-and-leaf plot (no need to calculate exact values, just draw conclusions); (II) Based on user satisfaction scores, classify user satisfaction into three levels from low to high: Satisfaction Score: Below 70, 70 to 89, At least 90 Satisfaction Level: Dissatisfied, Satisfied, Very Satisfied Let event $C$: ``The satisfaction level of users in region A is higher than that of users in region B''. Assume the evaluation results from the two regions are independent. Based on the given data, using the frequency of event occurrence as the probability of the corresponding event, find the probability of event $C$.
As shown in the figure, in rectangular prism $\mathrm { ABCD } - \mathrm { A } _ { 1 } \mathrm { B } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$, we have $\mathrm { AB } = 16 , \mathrm { BC } = 10 , \mathrm { AA } _ { 1 } = 8$. Points $\mathrm { E }$ and $\mathrm { F }$ are on $\mathrm { A } _ { 1 } \mathrm { B } _ { 1 }$ and $\mathrm { D } _ { 1 } \mathrm { C } _ { 1 }$ respectively, with $\mathrm { A } _ { 1 } \mathrm { E } = \mathrm { D } _ { 1 } \mathrm { F }$. A plane $\alpha$ passes through points $E$ and $F$ and intersects the faces of the rectangular prism, with the intersection lines forming a square. (I) Draw this square in the figure (no need to explain the method or reasoning) (II) Find the sine of the angle between line