Let $z = \frac { 1 - \mathrm { i } } { 1 + \mathrm { i } } + 2 \mathrm { i }$, then $| z | =$ A. 0 B. $\frac { 1 } { 2 }$ C. 1 D. $\sqrt { 2 }$
Q4
5 marksConic sectionsEccentricity or Asymptote ComputationView
Given an ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { 4 } = 1$ with one focus at $( 2,0 )$, then the eccentricity of $C$ is A. $\frac { 1 } { 3 }$ B. $\frac { 1 } { 2 }$ C. $\frac { \sqrt { 2 } } { 2 }$ D. $\frac { 2 \sqrt { 2 } } { 3 }$
Q6
5 marksTangents, normals and gradientsDetermine unknown parameters from tangent conditionsView
Let $f ( x ) = x ^ { 3 } + ( a - 1 ) x ^ { 2 } + a x$. If $f ( x )$ is an odd function, then the equation of the tangent line to $y = f ( x )$ at the point $( 0,0 )$ is A. $y = - 2 x$ B. $y = - x$ C. $y = 2 x$ D. $y = x$
Q7
5 marksVectors Introduction & 2DExpressing a Vector as a Linear CombinationView
In $\triangle A B C$, $AD$ is the median to side $BC$, and $E$ is the midpoint of $AD$. Then $\overrightarrow { E B } =$ A. $\frac { 3 } { 4 } \overrightarrow { A B } - \frac { 1 } { 4 } \overrightarrow { A C }$ B. $\frac { 1 } { 4 } \overrightarrow { A B } + \frac { 3 } { 4 } \overrightarrow { A C }$ C. $\frac { 3 } { 4 } \overrightarrow { A B } + \frac { 1 } { 4 } \overrightarrow { A C }$ D. $\frac { 1 } { 4 } \overrightarrow { A B } + \frac { 3 } { 4 } \overrightarrow { A C }$
Given the function $f ( x ) = 2 \cos ^ { 2 } x - \sin ^ { 2 } x + 2$, then A. The minimum positive period of $f ( x )$ is $\pi$, and the maximum value is 3 B. The minimum positive period of $f ( x )$ is $\pi$, and the maximum value is 4 C. The minimum positive period of $f ( x )$ is $2 \pi$, and the maximum value is 3 D. The minimum positive period of $f ( x )$ is $2 \pi$, and the maximum value is 4
Q10
5 marksVectors 3D & LinesVolume of a 3D SolidView
In rectangular parallelepiped $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$, $A B = B C = 2$, and the angle between $A C _ { 1 }$ and plane $B B _ { 1 } C _ { 1 } C$ is $30 ^ { \circ }$. Then the volume of the rectangular parallelepiped is A. 8 B. $6 \sqrt { 2 }$ C. $8 \sqrt { 2 }$ D. $8 \sqrt { 3 }$
Q11
5 marksAddition & Double Angle FormulaeMulti-Step Composite Problem Using IdentitiesView
The vertex of angle $\alpha$ is at the origin, its initial side coincides with the positive $x$-axis, and two points on its terminal side are $A ( 1 , a )$ and $B ( 2 , b )$. If $\cos 2 \alpha = \frac { 2 } { 3 }$, then $| a - b | =$ A. $\frac { 1 } { 5 }$ B. $\frac { \sqrt { 5 } } { 5 }$ C. $\frac { 2 \sqrt { 5 } } { 5 }$ D. (incomplete)
Q12
5 marksComposite & Inverse FunctionsDetermine Domain or Range of a Composite FunctionView
Let $f ( x ) = \begin{cases} 2 ^ { -x } & x \leq 0 \\ 1 & x > 0 \end{cases}$. Then the range of $x$ satisfying $f ( x + 1 ) < f ( 2 x )$ is A. $( - \infty , - 1 ]$ B. $( 0 , + \infty )$ C. $( - 1,0 )$ D. $( - \infty , 0 )$
Q13
5 marksLaws of LogarithmsDetermine Parameters of a Logarithmic FunctionView
Given the function $f ( x ) = \log _ { 2 } \left( x ^ { 2 } + a \right)$. If $f ( 3 ) = 1$, then $a = $ \_\_\_\_
Q14
5 marksInequalitiesLinear Programming (Optimize Objective over Linear Constraints)View
If $x , y$ satisfy the constraints $\left\{ \begin{array} { l } x - 2 y - 2 \leq 0 , \\ x - y + 1 \geq 0 , \\ y \leq 0 , \end{array} \right.$ then the maximum value of $z = 3 x + 2 y$ is \_\_\_\_
Q15
5 marksCirclesChord Length and Chord PropertiesView
The line $y = x + 1$ intersects the circle $x ^ { 2 } + y ^ { 2 } + 2 y - 3 = 0$ at points $A$ and $B$. Then $| A B | = $ \_\_\_\_
Q16
5 marksSine and Cosine RulesDetermine an angle or side from a trigonometric identity/equationView
In $\triangle A B C$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given $b \sin C + c \sin B = 4 a \sin B \sin C$ and $b ^ { 2 } + c ^ { 2 } - a ^ { 2 } = 8$, then the area of $\triangle A B C$ is \_\_\_\_
Q17
12 marksGeometric Sequences and SeriesGeometric Sequence from Recurrence IdentificationView
Given a sequence $\{ a _ { n } \}$ satisfying $a _ { 1 } = 1$ and $n a _ { n - 1 } = 2 ( n + 1 ) a _ { n }$. Let $b _ { n } = \frac { a _ { n } } { n }$. (1) Find $b _ { 1 } , b _ { 2 } , b _ { 3 }$; (2) Determine whether the sequence $\{ b _ { n } \}$ is a geometric sequence and explain the reasoning; (3) Find the general term formula for $\{ a _ { n } \}$.