gaokao 2015 Q5
Powers of i or Complex Number Integer Powers
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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 $q$