If $b$ is a real number satisfying $b ^ { 4 } + \frac { 1 } { b ^ { 4 } } = 6$, find the value of $\left( b + \frac { i } { b } \right) ^ { 16 }$ where $i = \sqrt { - 1 }$.

1. If the set $A = \left\{ i , i ^ { 2 } , i ^ { 3 } , i ^ { 4 } \right\}$ (where $i$ is the imaginary unit), $B = \{ 1 , - 1 \}$, then $A \cap B$ equals
A. $\{ - 1 \}$
B. $\{ 1 \}$
C. $\{ 1 , - 1 \}$
D. $\phi$

1. Let i be the imaginary unit. $i ^ { 607 } = ( )$
A. i
B. $ - i$
C. $ 1$
D. $ - 1$

5. After the examination ends, please submit both this examination paper and the answer sheet.
I. Multiple Choice Questions: This section has 10 questions, each worth 5 points, for a total of 50 points. For each question, only one of the four options is correct.
1. Let $i$ be the imaginary unit. The conjugate of $\mathrm{i}^{607}$ is
A. $i$
B. $-i$
C. $1$
D. $-1$
2. In the ancient Chinese mathematical classic ``Mathematical Treatise in Nine Sections,'' there is a problem on ``grain and millet separation.'' A grain warehouse receives 1534 stones of rice. Upon inspection, the rice contains mixed millet. A sample of rice is taken, and among 254 grains, 28 are millet. Approximately how much millet is in this batch of rice?
A. 134 stones
B. 169 stones
C. 338 stones
D. 1365 stones
3. In the expansion of $(1+x)^n$, the binomial coefficients of the 4th term and the 8th term are equal. The sum of the binomial coefficients of odd-numbered terms is
A. $2^{12}$
B. $2^{11}$
C. $2^{10}$
D. $2^9$
4. Let $X \sim N(\mu_1, \sigma_1^2)$ and $Y \sim N(\mu_2, \sigma_2^2)$. The density curves of these two normal distributions are shown in the figure. Which of the following conclusions is correct?
A. $P(Y \geq \mu_2) \geq P(Y \geq \mu_1)$
B. $P(X \leq \sigma_2) \leq P(X \leq \sigma_1)$
C. For any positive number $t$, $P(X \leq t) \geq P(Y \leq t)$
D. For any positive number $t$, $P(X \geq t) \geq P(Y \geq t)$
[Figure]
Figure for Question 4
5. Let $a_1, a_2, \ldots, a_n \in \mathbf{R}$, $n \geq 3$. If $p$: $a_1, a_2, \ldots, a_n$ form a geometric sequence; $q$: $(a_1^2 + a_2^2 + \cdots + a_{n-1}^2)(a_2^2 + a_3^2 + \cdots + a_n^2) = (a_1a_2 + a_2a_3 + \cdots + a_{n-1}a_n)^2$, then
A. $p$ is a sufficient but not necessary condition for $q$
B. $p$ is a necessary but not sufficient condition for $q$
C. $p$ is a sufficient and necessary condition for $q$
D. $p$ is neither a sufficient nor a necessary condition for $q$

3. The main content of this test paper covers all content of the college entrance examination.

Section I

I. Multiple Choice Questions: This section contains 12 questions, each worth 5 points, totaling 60 points. For each question, only one of the four options is correct.
1. The conjugate of the complex number $z = \mathrm { i } ^ { 9 } ( - 1 - 2 \mathrm { i } )$ is
A. $2 + \mathrm { i }$
B. $2 - \mathrm { i }$
C. $- 2 + \mathrm { i }$
D. $- 2 - \mathrm { i }$
2. Let sets $A = \{ a , a + 1 \} , ~ B = \{ 1,2,3 \}$. If $A \cup B$ has 4 elements, then the set of possible values of $a$ is
A. $\{ 0 \}$
B. $\{ 0,3 \}$
C. $\{ 0,1,3 \}$
D. $\{ 1,2,3 \}$
3. For the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$, the length of the real axis and the focal distance are 2 and 4 respectively. The asymptote equations of hyperbola $C$ are
A. $y = \pm \frac { \sqrt { 3 } } { 3 } x$
B. $y = \pm \frac { 1 } { 3 } x$ C. $y = \pm \sqrt { 3 } x$
D. $y = \pm 3 x$

Let $z = 1 + a i$, be a complex number, $a > 0$, such that $z ^ { 3 }$ is a real number. Then, the sum $1 + z + z ^ { 2 } + \ldots + z ^ { 11 }$ is equal to :
(1) $1365 \sqrt { 3 } i$
(2) $- 1365 \sqrt { 3 } i$
(3) $- 1250 \sqrt { 3 } i$
(4) $1250 \sqrt { 3 } i$

If $z = \frac { \sqrt { 3 } } { 2 } + \frac { i } { 2 }$ $( i = \sqrt { - 1 } )$, then $1 + i z + z ^ { 5 } + i z ^ { 8 }$ is equal to:
(1) - 1
(2) 1
(3) 0
(4) $- 1 + 2 i ^ { 9 }$

If $\alpha$ and $\beta$ are the distinct roots of the equation $x ^ { 2 } + ( 3 ) ^ { \frac { 1 } { 4 } } x + 3 ^ { \frac { 1 } { 2 } } = 0$, then the value of $\alpha ^ { 96 } \left( \alpha ^ { 12 } - 1 \right) + \beta ^ { 96 } \left( \beta ^ { 12 } - 1 \right)$ is equal to:
(1) $56 \times 3 ^ { 25 }$
(2) $56 \times 3 ^ { 24 }$
(3) $52 \times 3 ^ { 24 }$
(4) $28 \times 3 ^ { 25 }$

Let $\alpha$ and $\beta$ be the roots of the equation $x ^ { 2 } + ( 2 i - 1 ) = 0$. Then, the value of $\left| \alpha ^ { 8 } + \beta ^ { 8 } \right|$ is equal to
(1) 50
(2) 250
(3) 1250
(4) 1550

If $z = 2 + 3 i$, then $z ^ { 5 } + ( \bar { z } ) ^ { 5 }$ is equal to:
(1) 244
(2) 224
(3) 245
(4) 265

Let $z_1$ and $z_2$ be two complex numbers such that $z_1 + z_2 = 5$ and $z_1^3 + z_2^3 = 20 + 15i$. Then $|z_1^4 + z_2^4|$ equals
(1) $30\sqrt{3}$
(2) $75\sqrt{2}$
(3) $15\sqrt{15}$
(4) $25\sqrt{3}$

Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } - x + 2 = 0$ with $\operatorname { Im } ( \alpha ) > \operatorname { Im } ( \beta )$. Then $\alpha ^ { 6 } + \alpha ^ { 4 } + \beta ^ { 4 } - 5 \alpha ^ { 2 }$ is equal to

If the set $R = \{ ( a , b ) : a + 5 b = 42 , a , b \in \mathbb { N } \}$ has $m$ elements and $\sum _ { n = 1 } ^ { m } \left( 1 - i ^ { n ! } \right) = x + i y$, where $i = \sqrt { - 1 }$, then the value of $m + x + y$ is
(1) 12
(2) 4
(3) 8
(4) 5

Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } - a x - b = 0$ with $\operatorname { Im } ( \alpha ) < \operatorname { Im } ( \beta )$. Let $P _ { n } = \alpha ^ { n } - \beta ^ { n }$. If $\mathrm { P } _ { 3 } = - 5 \sqrt { 7 } i , \mathrm { P } _ { 4 } = - 3 \sqrt { 7 } i , \mathrm { P } _ { 5 } = 11 \sqrt { 7 } i$ and $\mathrm { P } _ { 6 } = 45 \sqrt { 7 } i$, then $\left| \alpha ^ { 4 } + \beta ^ { 4 } \right|$ is equal to

The function f on the set of complex numbers is
$$f ( z ) = \sum _ { k = 0 } ^ { 101 } z ^ { k }$$
is defined in this form. Accordingly, what is the value of f(i)?
A) $1 + i$
B) $1 - \mathrm { i }$
C) i
D) - i
E) 1

$$( 1 + i ) ^ { 4 } \cdot \left( 2 - \frac { 2 } { i } \right) ^ { 2 }$$
What is the result of this operation?
A) $4 i$
B) 16
C) $- 32 i$
D) - 8
E) 12

$\frac { \left( 1 - i ^ { 2 } \right) \cdot \left( 1 - i ^ { 6 } \right) \cdot \left( 1 - i ^ { 10 } \right) } { ( 1 - i ) \cdot \left( 1 - i ^ { 3 } \right) \cdot \left( 1 - i ^ { 5 } \right) }$\ What is the result of this operation?\ A) 1\ B) 2\ C) $2 + 2 i$\ D) $2 + 2 i$\ E) $1 + 2 i$