Q1
5 marksInequalitiesSet Operations Using Inequality-Defined SetsView
Given sets $A = \{ - 1,0,1,2 \} , B = \left\{ x \mid x ^ { 2 } \leqslant 1 \right\}$ , then $A \cap B =$ A. $\{ - 1,0,1 \}$ B. $\{ 0,1 \}$ C. $\{ - 1,1 \}$ D. $\{ 0,1,2 \}$
Q2
5 marksComplex Numbers ArithmeticSolving Equations for Unknown Complex NumbersView
If $z ( 1 + \mathrm { i } ) = 2 \mathrm { i }$ , then $z =$ A. $- 1 - \mathrm { i }$ B. $- 1 + \mathrm { i }$ C. $1 - \mathrm { i }$ D. $1 + \mathrm { i }$
Journey to the West, Romance of the Three Kingdoms, Water Margin, and Dream of the Red Chamber are treasures of classical Chinese literature, collectively known as the Four Great Classical Novels of China. To understand the reading situation of these four classics among students in a school, a random survey was conducted of 100 students. Among them, 90 students had read either Journey to the West or Dream of the Red Chamber, 80 students had read Dream of the Red Chamber, and 60 students had read both Journey to the West and Dream of the Red Chamber. The estimated value of the ratio of the number of students who have read Journey to the West to the total number of students in the school is A. 0.5 B. 0.6 C. 0.7 D. 0.8
Q4
5 marksBinomial Theorem (positive integer n)Find a Specific Coefficient in a Product of Binomial/Polynomial ExpressionsView
The coefficient of $x ^ { 3 }$ in the expansion of $\left( 1 + 2 x ^ { 2 } \right) ( 1 + x ) ^ { 4 }$ is A. 12 B. 16 C. 20 D. 24
Q5
5 marksGeometric Sequences and SeriesFinite Geometric Sum and Term RelationshipsView
A geometric sequence $\left\{ a _ { n } \right\}$ with all positive terms has the sum of its first 4 terms equal to 15, and $a _ { 5 } = 3 a _ { 3 } + 4 a _ { 1 }$ . Then $a _ { 3 } =$ A. 16 B. 8 C. 4 D. 2
Q6
5 marksTangents, normals and gradientsDetermine unknown parameters from tangent conditionsView
The tangent line to the curve $y = a \mathrm { e } ^ { x } + x \ln x$ at the point $( 1 , a \mathrm { e } )$ has equation $y = 2 x + b$ . Then A. $a = \mathrm { e } , b = - 1$ B. $a = \mathrm { e } , b = 1$ C. $a = \mathrm { e } ^ { - 1 } , b = 1$ D. $a = \mathrm { e } ^ { - 1 } , b = - 1$
Q7
5 marksCurve SketchingIdentifying the Correct Graph of a FunctionView
The graph of the function $y = \frac { 2 x ^ { 3 } } { 2 ^ { x } + 2 ^ { - x } }$ on $[ - 6,6 ]$ is approximately A. [graph A] B. [graph B] C. [graph C] D. [graph D]
Q8
5 marksVectors: Lines & PlanesCoplanarity and Relative Position of PlanesView
As shown in the figure, point $N$ is the center of square $ABCD$, $\triangle ECD$ is an equilateral triangle, plane $ECD \perp$ plane $ABCD$, and $M$ is the midpoint of segment $ED$. Then A. $BM = EN$, and lines $BM$ and $EN$ are intersecting lines B. $BM \neq EN$, and lines $BM$ and $EN$ are intersecting lines C. $BM = EN$, and lines $BM$ and $EN$ are skew lines D. $BM \neq EN$, and lines $BM$ and $EN$ are skew lines
Q9
5 marksSequences and SeriesAlgorithmic/Computational Implementation for Sequences and SeriesView
Executing the flowchart on the right, if the input $\varepsilon$ is 0.01, then the output value of $s$ equals A. $2 - \frac { 1 } { 2 ^ { 4 } }$ B. $2 - \frac { 1 } { 2 ^ { 5 } }$ C. $2 - \frac { 1 } { 2 ^ { 6 } }$ D. $2 - \frac { 1 } { 2 ^ { 7 } }$
Q10
5 marksConic sectionsTriangle or Quadrilateral Area and Perimeter with FociView
The right focus of the hyperbola $C : \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 2 } = 1$ is $F$. Point $P$ is on one of the asymptotes of $C$, and $O$ is the origin. If $| PO | = | PF |$, then the area of $\triangle PFO$ is A. $\frac { 3 \sqrt { 2 } } { 4 }$ B. $\frac { 3 \sqrt { 2 } } { 2 }$ C. $2 \sqrt { 2 }$ D. $3 \sqrt { 2 }$
Q11
5 marksLaws of LogarithmsCompare or Order Logarithmic ValuesView
Let $f ( x )$ be an even function with domain $\mathbf { R }$ that is monotonically decreasing on $( 0 , + \infty )$. Then A. $f \left( \log _ { 3 } \frac { 1 } { 4 } \right) > f \left( 2 ^ { - \frac { 3 } { 2 } } \right) > f \left( 2 ^ { - \frac { 2 } { 3 } } \right)$ B. $f \left( \log _ { 3 } \frac { 1 } { 4 } \right) > f \left( 2 ^ { - \frac { 2 } { 3 } } \right) > f \left( 2 ^ { - \frac { 3 } { 2 } } \right)$ C. $f \left( 2 ^ { - \frac { 3 } { 2 } } \right) > f \left( 2 ^ { - \frac { 2 } { 3 } } \right) > f \left( \log _ { 3 } \frac { 1 } { 4 } \right)$ D. $f \left( 2 ^ { - \frac { 2 } { 3 } } \right) > f \left( 2 ^ { - \frac { 3 } { 2 } } \right) > f \left( \log _ { 3 } \frac { 1 } { 4 } \right)$
Let the function $f ( x ) = \sin \left( \omega x + \frac { \pi } { 5 } \right) ( \omega > 0 )$. It is known that $f ( x )$ has exactly 5 zeros on $[ 0,2 \pi ]$. The following are four conclusions: (1) $f ( x )$ has exactly 3 local maximum points on $( 0,2 \pi )$ (2) $f ( x )$ has exactly 2 local minimum points on $( 0,2 \pi )$ (3) $f ( x )$ is monotonically increasing on $\left( 0 , \frac { \pi } { 10 } \right)$ (4) The range of $\omega$ is $\left[ \frac { 12 } { 5 } , \frac { 29 } { 10 } \right)$ The numbers of all correct conclusions are A. (1)(4) B. (2)(3) C. (1)(2)(3) D. (1)(3)(4)
Q13
5 marksVectors Introduction & 2DAngle or Cosine Between VectorsView
Given that $\boldsymbol { a } , \boldsymbol { b }$ are unit vectors and $\boldsymbol { a } \cdot \boldsymbol { b } = 0$ , if $\boldsymbol { c } = 2 \boldsymbol { a } - \sqrt { 5 } \boldsymbol { b }$ , then $\cos \langle \boldsymbol { a } , \boldsymbol { c } \rangle =$ \_\_\_\_\_\_.
Q14
5 marksArithmetic Sequences and SeriesCompute Partial Sum of an Arithmetic SequenceView
Let $S _ { n }$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$. If $a _ { 1 } \neq 0 , a _ { 2 } = 3 a _ { 1 }$ , then $\frac { S _ { 10 } } { S _ { 5 } } =$ \_\_\_\_\_\_.
Q15
5 marksConic sectionsTriangle or Quadrilateral Area and Perimeter with FociView
Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 20 } = 1$ , and $M$ be a point on $C$ in the first quadrant. If $\triangle M F _ { 1 } F _ { 2 }$ is an isosceles triangle, then the coordinates of $M$ are \_\_\_\_\_\_.
Q23
10 marksInequalitiesOptimization Subject to an Algebraic ConstraintView
Let $x, y, z \in \mathbf{R}$ and $x + y + z = 1$. (1) Find the minimum value of $(x-1)^2 + (y+1)^2 + (z+1)^2$; (2) If $(x-2)^2 + (y-1)^2 + (z-a)^2 \geqslant \frac{1}{3}$ holds for all $x, y, z$ satisfying $x + y + z = 1$, prove that $a \leqslant -3$ or $a \geqslant -1$.