Given set $A = \left\{ x \mid x ^ { 2 } - 3 x - 4 < 0 \right\} , B = \{ - 4,1,3,5 \}$ , then $A \cap B =$ A. $\{ - 4,1 \}$ B. $\{ 1,5 \}$ C. $\{ 3,5 \}$ D. $\{ 1,3 \}$
The Great Pyramid of Egypt is one of the ancient wonders of the world. Its shape can be viewed as a regular square pyramid. The area of a square with side length equal to the height of the pyramid equals the area of one lateral triangular face of the pyramid. The ratio of the height of a lateral triangular face to the side length of the base square is A. $\frac { \sqrt { 5 } - 1 } { 4 }$ B. $\frac { \sqrt { 5 } - 1 } { 2 }$ C. $\frac { \sqrt { 5 } + 1 } { 4 }$ D. $\frac { \sqrt { 5 } + 1 } { 2 }$
Let $O$ be the center of square $A B C D$. If we randomly select 3 points from $O , A , B , C , D$, the probability that the 3 points are collinear is A. $\frac { 1 } { 5 }$ B. $\frac { 2 } { 5 }$ C. $\frac { 1 } { 2 }$ D. $\frac { 4 } { 5 }$
A study group at a school conducted seed germination experiments at 20 different temperature conditions to investigate the relationship between the germination rate $y$ of a certain crop seed and temperature $x$ (in ${}^{\circ}\mathrm{C}$). From the experimental data $\left( x _ { i } , y _ { i } \right) ( i = 1,2 , \cdots , 20 )$, a scatter plot was obtained. Based on this scatter plot, between $10^{\circ}\mathrm{C}$ and $40^{\circ}\mathrm{C}$, which of the following four regression equation types is most suitable as the regression equation type for the germination rate $y$ and temperature $x$? A. $y = a + b x$ B. $y = a + b x ^ { 2 }$ C. $y = a + b \mathrm { e } ^ { x }$ D. $y = a + b \ln x$
Given the circle $x ^ { 2 } + y ^ { 2 } - 6 x = 0$ , the minimum length of the chord cut by this circle from a line passing through the point $( 1,2 )$ is A. 1 B. 2 C. 3 D. 4
The function $f ( x ) = \cos \left( \omega x + \frac { \pi } { 6 } \right)$ has a graph on $[ - \pi , \pi ]$ as shown. The minimum positive period of $f ( x )$ is A. $\frac { 10 \pi } { 9 }$ B. $\frac { 7 \pi } { 6 }$ C. $\frac { 4 \pi } { 3 }$ D. $\frac { 3 \pi } { 2 }$
Let $\left\{ a _ { n } \right\}$ be a geometric sequence with $a _ { 1 } + a _ { 2 } + a _ { 3 } = 1 , a _ { 2 } + a _ { 3 } + a _ { 4 } = 2$ , then $a _ { 6 } + a _ { 7 } + a _ { 8 } =$ A. 12 B. 24 C. 30 D. 32
Let $F _ { 1 } , F _ { 2 }$ be the two foci of the hyperbola $C : x ^ { 2 } - \frac { y ^ { 2 } } { 3 } = 1$ , $O$ be the origin, and point $P$ on $C$ with $| O P | = 2$ . The area of $\triangle P F _ { 1 } F _ { 2 }$ is A. $\frac { 7 } { 2 }$ B. 3 C. $\frac { 5 } { 2 }$ D. 2
Let $A , B , C$ be three points on the surface of sphere $O$, and $\odot O _ { 1 }$ be the circumcircle of $\triangle A B C$. If the area of $\odot O _ { 1 }$ is $4 \pi$ and $A B = B C = A C = O O _ { 1 }$ , then the surface area of sphere $O$ is A. $64 \pi$ B. $48 \pi$ C. $36 \pi$ D. $32 \pi$
If $x , y$ satisfy the constraint conditions $\left\{ \begin{array} { l } 2 x + y - 2 \leqslant 0 , \\ x - y - 1 \geqslant 0 , \\ y + 1 \geqslant 0 , \end{array} \right.$ then the maximum value of $z = x + 7 y$ is $\_\_\_\_$
Let vectors $\boldsymbol { a } = ( 1 , - 1 ) , \boldsymbol { b } = ( m + 1,2 m - 4 )$ . If $\boldsymbol { a } \perp \boldsymbol { b }$ , then $m =$ $\_\_\_\_$.
The sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { n + 2 } + ( - 1 ) ^ { n } a _ { n } = 3 n - 1$ . The sum of the first 16 terms is 540. Then $a _ { 1 } =$ $\_\_\_\_$.
In $\triangle A B C$, the sides opposite to angles $A , B , C$ are $a , b , c$ respectively. Given $B = 150 ^ { \circ }$ , (1) If $a = \sqrt { 3 } c , b = 2 \sqrt { 7 }$ , find the area of $\triangle A B C$ ; (2) If $\sin A + \sqrt { 3 } \sin C = \frac { \sqrt { 2 } } { 2 }$ , find $C$ .
As shown in the figure, $D$ is the apex of the cone, $O$ is the center of the base of the cone, $\triangle A B C$ is an equilateral triangle inscribed in the base, and $P$ is a point on $D O$ with $\angle A P C = 90 ^ { \circ }$ . (1) Prove that plane $P A B \perp$ plane $P A C$ ; (2) Given $D O = \sqrt { 2 }$ and the lateral surface area of the cone is $\sqrt { 3 } \pi$ , find the volume of the triangular pyramid $P - A B C$ .
Given the function $f ( x ) = \mathrm { e } ^ { x } - a ( x + 2 )$ . (1) When $a = 1$ , discuss the monotonicity of $f ( x )$ ; (2) If $f ( x )$ has two zeros, find the range of values for $a$ .
Let $A , B$ be the left and right vertices of the ellipse $E : \frac { x ^ { 2 } } { a ^ { 2 } } + y ^ { 2 } = 1 ( a > 1 )$ respectively, $G$ be the upper vertex of $E$ , and $\overrightarrow { A G } \cdot \overrightarrow { G B } = 8$ . $P$ is a moving point on the line $x = 6$ , the other intersection point of $P A$ with $E$ is $C$ , and the other intersection point of $P B$ with $E$ is $D$ . (1) Find the equation of $E$ ; (2) Prove that the line $C D$ passes through a fixed point.
Q22
10 marksParametric curves and Cartesian conversionView
[Elective 4-4: Coordinate Systems and Parametric Equations] (10 points) In the rectangular coordinate system $x O y$, the parametric equation of curve $C _ { 1 }$ is $\left\{ \begin{array} { l } x = \cos ^ { k } t , \\ y = \sin ^ { k } t \end{array} \right.$ ($t$ is the parameter). Establishing a polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of curve $C _ { 2 }$ is $$4 \rho \cos \theta - 16 \rho \sin \theta + 3 = 0$$ (1) When $k = 1$ , what type of curve is $C _ { 1 }$? (2) When $k = 4$ , find the rectangular coordinates of the common points of $C _ { 1 }$ and $C _ { 2 }$ .