Given sets $A = \{ 1,3,5,7 \} , B = \{ 2,3,4,5 \}$, then $A \cap B =$ A. $\{ 3 \}$ B. $\{ 5 \}$ C. $\{ 3,5 \}$ D. $\{ 1,2,3,4,5,7 \}$
Q3
5 marksCurve SketchingIdentifying the Correct Graph of a FunctionView
The graph of the function $f ( x ) = \frac { \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } } { x ^ { 2 } }$ is approximately (see options A, B, C, D in the figures).
Vectors $\boldsymbol { a } , \boldsymbol { b }$ satisfy $| \boldsymbol { a } | = 1 , \boldsymbol { a } \cdot \boldsymbol { b } = - 1$, then $\boldsymbol { a } \cdot ( 2 \boldsymbol { a } - \boldsymbol { b } ) =$ A. 4 B. 3 C. 2 D. 0
From 2 male students and 3 female students, 2 people are selected to participate in community service. The probability that both selected are female students is A. 0.6 B. 0.5 C. 0.4 D. 0.3
Q6
5 marksConic sectionsEccentricity or Asymptote ComputationView
The hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ has eccentricity $\sqrt { 3 }$. Its asymptotes are A. $y = \pm \sqrt { 2 } x$ B. $y = \pm \sqrt { 3 } x$ C. $y = \pm \frac { \sqrt { 2 } } { 2 } x$ D. $y = \pm \frac { \sqrt { 3 } } { 2 } x$
Q7
5 marksSine and Cosine RulesFind a side length using the cosine ruleView
In $\triangle A B C$, $\cos \frac { C } { 2 } = \frac { \sqrt { 5 } } { 5 } , B C = 1 , A C = 5$, then $A B =$ A. $4 \sqrt { 2 }$ B. $\sqrt { 30 }$ C. $\sqrt { 29 }$ D. $2 \sqrt { 5 }$
Q9
5 marksVectors 3D & LinesMCQ: Angle Between Skew LinesView
In the rectangular prism $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$, $E$ is the midpoint of edge $C C _ { 1 }$. The tangent of the angle between skew lines $A E$ and $C D$ is A. $\frac { \sqrt { 2 } } { 2 }$ B. $\frac { \sqrt { 3 } } { 2 }$ C. $\frac { \sqrt { 5 } } { 2 }$ D. $\frac { \sqrt { 7 } } { 2 }$
If $f ( x ) = \cos x - \sin x$ is decreasing on $[ 0 , a ]$, then the maximum value of $a$ is A. $\frac { \pi } { 4 }$ B. $\frac { \pi } { 2 }$ C. $\frac { 3 \pi } { 4 }$ D. $\pi$
Q11
5 marksConic sectionsEccentricity or Asymptote ComputationView
Let $F _ { 1 } , F _ { 2 }$ be the two foci of ellipse $C$. $P$ is a point on $C$. If $P F _ { 1 } \perp P F _ { 2 }$ and $\angle P F _ { 2 } F _ { 1 } = 60 ^ { \circ }$, then the eccentricity of $C$ is A. $1 - \frac { \sqrt { 3 } } { 2 }$ B. $2 - \sqrt { 3 }$ C. $\frac { \sqrt { 3 } - 1 } { 2 }$ D. $\sqrt { 3 } - 1$
Q12
5 marksSequences and SeriesEvaluation of a Finite or Infinite SumView
Let $f ( x )$ be an odd function with domain $( - \infty , + \infty )$, satisfying $f ( 1 - x ) = f ( 1 + x )$ and $f ( 1 ) = 2$. Then $f ( 1 ) + f ( 2 ) + f ( 3 ) + \cdots + f ( 50 ) =$ A. $- 50$ B. $0$ C. $2$ D. $50$
Q13
5 marksTangents, normals and gradientsFind tangent line equation at a given pointView
The equation of the tangent line to the curve $y = 2 \ln x$ at the point $( 1,0 )$ is \_\_\_\_.
Q14
5 marksInequalitiesLinear Programming (Optimize Objective over Linear Constraints)View
Given the constraints $\left\{ \begin{array} { l } x + 2 y - 5 \geq 0 , \\ x - 2 y + 3 \geq 0 , \\ x - 5 \leq 0 , \end{array} \right.$ the minimum value of $z = x + y$ is \_\_\_\_.
Q15
5 marksAddition & Double Angle FormulaeAddition/Subtraction Formula EvaluationView
Q17
12 marksArithmetic Sequences and SeriesMulti-Part Structured Problem on APView
Let $S _ { n }$ be the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$. Given $a _ { 1 } = - 7 , S _ { 3 } = - 15$. (1) Find the general term formula for $\left\{ a _ { n } \right\}$; (2) Find $S _ { n }$ and the minimum value of $S _ { n }$.
The figure below is a line graph of environmental infrastructure investment $y$ (in units of 100 million yuan) from 2000 to 2016 in a certain region. To predict the environmental infrastructure investment in 2018 for this region, two linear regression models for $y$ and time variable $t$ were established. Based on data from 2000 to 2016 (time variable $t$ takes values $1,2 , \cdots , 17$ respectively), Model (1) was established: $\hat { y } = - 30.4 + 13.5 t$. Based on data from 2010 to 2016 (time variable $t$ takes values $1,2 , \cdots , 7$ respectively), Model (2) was established: $\hat { y } = 99 + 17.5 t$. (1) Using each of these two models, find the predicted value of environmental infrastructure investment for 2018 in this region; (2) Which model's prediction do you think is more reliable? Explain your reasoning.
Q19
12 marksVectors: Lines & PlanesDistance Computation (Point-to-Plane or Line-to-Line)View
As shown in the figure, in the triangular pyramid $P - A B C$, $A B = B C = 2 \sqrt { 2 }$, $P A = P B = P C = A C = 4$, and $O$ is the midpoint of $A C$. (1) Prove: $P O \perp$ plane $A B C$; (2) If point $M$ is on edge $B C$ such that $M C = 2 M B$, find the distance from point $C$ to plane $P O M$.
Q20
12 marksConic sectionsCircle-Conic Interaction with Tangency or IntersectionView
Let the parabola $C : y ^ { 2 } = 4 x$ have focus $F$. A line $l$ through $F$ with slope $k ( k > 0 )$ intersects $C$ at points $A , B$, with $| A B | = 8$. (1) Find the equation of $l$; (2) Find the equation of the circle passing through points $A , B$ and tangent to the directrix of $C$.
Q21
12 marksStationary points and optimisationDetermine intervals of increase/decrease or monotonicity conditionsView
Given the function $f ( x ) = \frac { 1 } { 3 } x ^ { 3 } - a \left( x ^ { 2 } + x + 1 \right)$. (1) When $a = 3$, find the monotonic intervals of $f ( x )$; (2) Prove: $f ( x )$ has exactly one zero.
Q22
10 marksParametric curves and Cartesian conversionView
[Elective 4-4: Polar Coordinates and Parametric Equations] In the rectangular coordinate system $x O y$, the parametric equation of curve $C$ is $\left\{ \begin{array} { l } x = 2 \cos \theta , \\ y = 4 \sin \theta \end{array} \right.$ ($\theta$ is the parameter). The parametric equation of line $l$ is $\left\{ \begin{array} { l } x = 1 + t \cos \alpha , \\ y = 2 + t \sin \alpha \end{array} \right.$ ($t$ is the parameter). (1) Find the rectangular coordinate equations of $C$ and $l$; (2) If the midpoint of the line segment obtained by the intersection of curve $C$ and line $l$ is $( 1,2 )$, find the slope of $l$.