4. Given that variables x and y satisfy the relationship $y = - 0.1 x + 1$, and variable y is positively correlated with z, the correct conclusion is A. x and y are positively correlated, x and z are negatively correlated B. x and y are positively correlated, x and z are positively correlated C. x and y are negatively correlated, x and z are negatively correlated D. x and y are negatively correlated, x and z are positively correlated
Q6
Composite & Inverse FunctionsDetermine Domain or Range of a Composite FunctionView
6. The domain of the function $f ( x ) = \sqrt { 4 - | x | } + \lg \frac { x ^ { 2 } - 5 x + 6 } { x - 3 }$ is A. $ ( 2,3 )$ B. $ ( 2,4 ]$ C. $ ( 2,3 ) \cup ( 3,4 ]$ D. $ ( - 1,3 ) \cup ( 3,6 ]$
Q7
Modulus functionAlgebraic identities and properties of modulusView
7. For $x \in R$, define the sign function $\operatorname { sgn } x = \left\{ \begin{array} { c } 1 , x > 0 \\ 0 , x = 0 \\ - 1 , x < 0 \end{array} \right.$, then A. $ \{ x | = x | \operatorname { sgn } x \mid \}$ B. $ \{ x | = \operatorname { sgn } | x \mid \}$ C. $ \{ x | = x | \operatorname { sgn } x \}$ D. $ \{ x \mid = x \operatorname { sgn } x \}$
8. Two numbers $\mathrm { x } , \mathrm { y }$ are randomly selected from the interval $[ 0,1 ]$. Let $p _ { 1 }$ be the probability of the event ``$x + y \leq \frac { 1 } { 2 }$'', and $P _ { 2 }$ be the probability of the event ``$x y \leq \frac { 1 } { 2 }$'', then A. $p _ { 1 } < p _ { 2 } < \frac { 1 } { 2 }$ B. $p _ { 2 } < \frac { 1 } { 2 } < p _ { 1 }$ C. $\frac { 1 } { 2 } < p _ { 2 } < p _ { 1 }$ D. $p _ { 1 } < \frac { 1 } { 2 } < p _ { 2 }$
Q9
Conic sectionsEccentricity or Asymptote ComputationView
9. The real semi-major axis length a and imaginary semi-minor axis length $\mathrm { b }$ of a hyperbola $C _ { 1 }$ with eccentricity $e _ { 1 }$ (where $a = b$) are both increased by $\mathrm { m } ( m > 0 )$ units of length to obtain a hyperbola $C _ { 2 }$ with eccentricity $e _ { 2 }$. Then A. For any $\mathrm { a } , \mathrm { b }$, $e _ { 1 } < e _ { 2 }$ B. When $a > b$, $e _ { 1 } < e _ { 2 }$ ; when $a < b$, $e _ { 1 } > e _ { 2 }$ C. For any a, b, $e _ { 1 } > e _ { 2 }$ D. When $a > b$, $e _ { 1 } > e _ { 2 }$ ; when $a < b$, $e _ { 1 } < e _ { 2 }$
11. Given that vectors $\overrightarrow { O A } \perp \overrightarrow { O B } , | \overrightarrow { O A } | = 3$, then $\overrightarrow { O A } \bullet \overrightarrow { O B } =$ $\_\_\_\_$.
Q13
Curve SketchingNumber of Solutions / Roots via Curve AnalysisView
13. The number of zeros of the function $f ( x ) = 2 \sin x \sin \left( x + \frac { \pi } { 2 } \right) - x ^ { 2 }$ is $\_\_\_\_$.
14. An e-commerce company conducted a statistical survey of 10,000 online shoppers' consumption in 2014. The consumption amount (in units of 10,000 yuan) falls within the interval [0.3, 0.9]. The frequency distribution histogram is shown in the figure below. (1) In the histogram, $\mathrm { a } =$ $\_\_\_\_$. (2) Among these shoppers, the number of shoppers with consumption amount in the interval [0.5, 0.9] is $\_\_\_\_$. [Figure]
Q15
Sine and Cosine RulesHeights and distances / angle of elevation problemView
15. As shown in the figure, a car is traveling due west on a horizontal road. At point A, the mountain peak D on the north side of the road is measured to be in the direction of $30 ^ { 0 }$ west of north. After traveling 600 m to reach point B, the mountain peak is measured to be in the direction of $75 ^ { 0 }$ west of north, with an angle of elevation of $30 ^ { 0 }$. Then the height of the mountain $\mathrm { CD } =$ $\_\_\_\_$ m. [Figure]
16. As shown in the figure, circle C is tangent to the x-axis at point $T ( 1,0 )$, and intersects the positive y-axis at two points $\mathrm { A } , \mathrm { B }$ (B is above A), with $| A B | = 2$. (1) The standard equation of circle C is $\_\_\_\_$. (2) The x-intercept of the tangent line to circle C at point B is $\_\_\_\_$. [Figure]
Q17
Stationary points and optimisationFind absolute extrema on a closed interval or domainView
17. Let a be a real number. The maximum value of the function $f ( x ) = \left| x ^ { 2 } - a x \right|$ on the interval $[ 0,1 ]$ is denoted by $g ( a )$. When $a =$ $\_\_\_\_$, $$\mathbf { y }$$ the value of $g ( a )$ is minimized. III. Solution Questions
18. A student uses the ``five-point method'' to draw the graph of the function $f ( \mathrm { x } ) = \mathrm { A } \sin ( \omega \mathrm { x } + \varphi ) \left( \omega > 0 , \varphi < \frac { \pi } { 2 } \right)$ during a certain period, creates a table and fills in partial data as follows:
(I) Please complete the above data, fill in the corresponding positions on the answer sheet, and directly write the analytical expression of the function $f ( \mathrm { x } )$; (II) Shift all points on the graph of $y = f ( \mathrm { x } )$ to the left by $\frac { \pi } { 6 }$ units to obtain the graph of $y = g ( \mathrm { x } )$. Find the center of symmetry of the graph of $y = g ( \mathrm { x } )$ that is closest to the origin O.
Q19
Arithmetic Sequences and SeriesArithmetic-Geometric Hybrid ProblemView
19. Let the common difference of an arithmetic sequence be d, the sum of the first n terms be, the common ratio of a geometric sequence be q. Given $= - = 2 , \mathrm { q } = \mathrm { d } , $ $= 100$. (I) Find the general term formulas of the sequences and. (II) When $\mathrm { d } > 1$, let $= c _ { n } = \frac { a _ { n } } { b _ { n } }$. Find the sum of the first n terms of the sequence.
Q20
Vectors 3D & LinesMulti-Part 3D Geometry ProblemView
20. In ``The Nine Chapters on the Mathematical Art,'' a quadrangular pyramid with a rectangular base and one lateral edge perpendicular to the base is called a ``yang ma'', and a tetrahedron with all four faces being right triangles is called a ``bie nao''. In the yang ma $\mathrm { P } - \mathrm { ABCD }$ shown in the figure, the lateral edge $\mathrm { PD } \perp$ base ABCD, and $\mathrm { PD } = \mathrm { CD }$. Point E is the midpoint of PC. Connect $\mathrm { DE } , \mathrm { BD } , \mathrm { BE }$. [Figure] Figure for Question 20 (I) Prove that $\mathrm { DE } \perp$ plane PBC. Determine whether the tetrahedron EBCD is a ``bie nao''. If yes, write out the right angle of each face (only conclusions are needed); if no, please explain the reason; (II) Let the volume of the yang ma $\mathr