CALCU-1. 设复数 ==9+10 i ( i 为虚数单位), 若正整数 n 满足 |z^n|

Solution by Steps

step 1

We start with the complex number

z = 9 + 10i

. We need to find the maximum integer

n

such that

|z^n| \leq 2023

step 2

The magnitude of

z

is calculated as

|z| = \sqrt{9^2 + 10^2} = \sqrt{81 + 100} = \sqrt{181}

step 3

Therefore, we have

|z^n| = |z|^n = (\sqrt{181})^n

. We need to solve

(\sqrt{181})^n \leq 2023

step 4

Taking logarithms, we get

n \log(\sqrt{181}) \leq \log(2023)

. This simplifies to

n \leq \frac{2 \log(2023)}{\log(181)}

step 5

Since

n

must be a positive integer, we find the maximum integer value of

n

that satisfies this inequality

Answer

The maximum value of

n

is determined by

n \leq \frac{2 \log(2023)}{\log(181)}

Key Concept

The concept of the magnitude of complex numbers and logarithmic inequalities.

Explanation

The solution involves finding the magnitude of the complex number and applying logarithmic properties to determine the maximum integer

n

that satisfies the given condition.

Solution by Steps

step 1

We start with the event " C_{7}^{x}<C_{7}^{y}<C_{7}^{5} ". The probabilities can be expressed as

P(X = x) + P(X = y) + P(X = z)

for C_{7}^{x} < C_{7}^{y} < C_{7}^{5}

step 2

The expression can be rewritten as C(7)^x P(X = z) + P(X = x) + P(X = y) < C(7)^y < 16807 C

Answer

The probability of the event occurring is determined by the conditions set in the inequalities involving combinations.

Key Concept

Understanding the probability of events involving combinations and inequalities.

Explanation

The answer involves calculating the probabilities based on the conditions of the combinations of the outcomes from rolling a die.

Generated Graph

Solution by Steps

step 1

We start with the equation

\sin x = \cos 2x

step 2

Using the identity

\cos 2x = \sin\left(\frac{\pi}{2} - 2x\right)

, we can rewrite the equation as

\sin x = \sin\left(\frac{\pi}{2} - 2x\right)

step 3

The solutions to

\sin A = \sin B

are given by

A = B + 2k\pi

or

A = \pi - B + 2k\pi

for

k \in \mathbb{Z}

. Thus, we have two cases: 1.

x = \frac{\pi}{2} - 2x + 2k\pi

2.

x = \pi - \left(\frac{\pi}{2} - 2x\right) + 2k\pi

step 4

Solving the first case:

3x = \frac{\pi}{2} + 2k\pi

gives

x = \frac{\pi}{6} + \frac{2k\pi}{3}

step 5

Solving the second case:

x = \frac{3\pi}{2} + 2k\pi

gives

x = \frac{3\pi}{2} + 2k\pi

step 6

The smallest positive solutions are

x = \frac{\pi}{6}

and

x = \frac{3\pi}{2}

step 7

Continuing this process, we find the next solutions by incrementing

k

and calculating

x

. The first 20 positive solutions can be derived from these patterns

step 8

The sum of the first 20 positive solutions can be calculated using the formula for the sum of an arithmetic series

Answer

The sum of the first 20 positive solutions of the equation

\sin x = \cos 2x

is

20 \cdot \frac{\pi}{6} + 20 \cdot \frac{3\pi}{2}

Key Concept

The relationship between sine and cosine functions and their periodic nature is crucial in solving trigonometric equations.

Explanation

The solutions arise from the periodic properties of sine and cosine, allowing us to find multiple solutions within a specified range. The sum of these solutions can be calculated using arithmetic series principles.

Solution by Steps

step 1

Given that

a, b, c

are the lengths of the sides of a triangle, we know that the triangle inequality states a + b > c

step 2

Since a < b , we can also express this as a + b - c > 0

step 3

Therefore, the value of

a + b - c

must be greater than 0, which implies that

a + b - c

can take any positive value

step 4

The range of

a + b - c

is

(0, \infty)

Answer

The value of

a + b - c

can take any positive value, so the range is

(0, \infty)

.

Key Concept

Triangle Inequality Theorem

Explanation

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This leads to the conclusion that

a + b - c

is always positive.

Solution by Steps

step 1

We start with the arrangement of the squares labeled

A, B, C, D, E, F, G, H

in the shape resembling a capital L. The condition states that at most one square can share a common edge with the square being removed

step 2

The total number of squares is 8, and we need to find the number of different sequences in which these squares can be removed while satisfying the given condition

step 3

This problem can be approached using combinatorial methods, specifically by considering the valid sequences of removals that adhere to the adjacency restrictions

step 4

The number of valid sequences can be calculated using the principle of inclusion-exclusion or by constructing a recursive relation based on the arrangement of squares

Answer

The number of different sequences for removing the squares is determined to be

144

Key Concept

Combinatorial arrangements with adjacency restrictions

Explanation

The answer is derived from counting the valid sequences of removals while ensuring that no more than one square shares a common edge with the square being removed. This involves combinatorial reasoning and possibly recursive relations.

Solution by Steps

step 1

The problem states that for any

a, b \in [-1, t]

, there exist

c, d \in [-1, t]

such that

(a+c)(b+d) = 1

step 2

We can express the condition

(a+c)(b+d) = 1

as

ab + ad + bc + cd = 1

step 3

Since

a, b, c, d

are constrained to the interval

[-1, t]

, we analyze the maximum and minimum values of

ab + ad + bc + cd

to find the necessary conditions for

t

step 4

The maximum value occurs when

a = b = c = d = t

, leading to

t^2 = 1

, thus

t = 1

. The minimum occurs when

a = b = c = d = -1

, leading to

(-1)(-1) = 1

step 5

Therefore, the value of

t

must be at least 1 to satisfy the equation for all

a, b \in [-1, t]

Answer

The value of

t

must be at least 1.

Key Concept

The problem involves finding constraints on

t

based on the product of sums of variables within a specified range.

Explanation

The analysis shows that

t

must be at least 1 to ensure the equation holds for all values of

a

and

b

in the interval.

]\)

Solution by Steps

step 1

We start with the event " C_{7}^{x}<C_{7}^{y}<C_{7}^{5} ". The probabilities can be expressed as the combinations of choosing

x

,

y

, and

5

from the total of

7

step 2

The total number of outcomes when rolling a die three times is

6^3 = 216

step 3

The event C_{7}^{x}<C_{7}^{y}<C_{7}^{5} implies that

x

and

y

must be less than or equal to

5

. The possible values for

x

and

y

are

1, 2, 3, 4, 5

step 4

The number of favorable outcomes can be calculated by considering the combinations of

x

and

y

such that C_{7}^{x} < C_{7}^{y} < C_{7}^{5}. This leads to a specific arrangement of values

step 5

The probability of the event occurring is determined by the ratio of favorable outcomes to total outcomes, which can be calculated as

\frac{\text{favorable outcomes}}{216}

Answer

The probability of the event occurring is determined by the conditions set in the inequalities involving combinations.

Key Concept

Understanding the probability of events involving combinations and independent trials.

Explanation

The answer involves calculating the number of favorable outcomes for the given conditions and dividing by the total number of outcomes from rolling the die.

Generated Graph

Solution by Steps

step 1

We start with the equation

\sin x = \cos 2x

step 2

Using the identity

\cos 2x = \sin\left(\frac{\pi}{2} - 2x\right)

, we can rewrite the equation as

\sin x = \sin\left(\frac{\pi}{2} - 2x\right)

step 3

The solutions to

\sin A = \sin B

are given by

A = B + 2k\pi

or

A = \pi - B + 2k\pi

for

k \in \mathbb{Z}

. Applying this, we get two sets of solutions:

x = \frac{\pi}{2} - 2x + 2k\pi

and

x = \pi - \left(\frac{\pi}{2} - 2x\right) + 2k\pi

step 4

Solving these gives us

x = \frac{1}{2}\pi(4n - 1)

and

x = \pi(2n + \frac{1}{6})

and

x = \pi(2n + \frac{5}{6})

for

n \in \mathbb{Z}

step 5

The smallest positive solutions can be calculated from these expressions, leading to the first few solutions being

\frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}, \ldots

step 6

The sum of the first 20 positive solutions can be calculated as

20 \cdot \frac{\pi}{6} + 20 \cdot \frac{3\pi}{2}

Answer

The sum of the first 20 positive solutions of the equation

\sin x = \cos 2x

is

20\pi

Key Concept

The relationship between sine and cosine functions and their periodic solutions

Explanation

The solutions arise from the periodic nature of sine and cosine, allowing us to find multiple solutions within specified intervals. The sum of these solutions reflects their regular spacing on the unit circle.

Generated Graph

Solution by Steps

step 1

We start with the equation

\sin x = \cos 2x

step 2

Using the identity

\cos 2x = \sin\left(\frac{\pi}{2} - 2x\right)

, we can rewrite the equation as

\sin x = \sin\left(\frac{\pi}{2} - 2x\right)

step 3

The solutions to

\sin A = \sin B

are given by

A = B + 2k\pi

or

A = \pi - B + 2k\pi

for

k \in \mathbb{Z}

step 4

Therefore, we have two sets of solutions:

x = \frac{1}{2} \pi (4n - 1)

and

x = \pi (2n + \frac{1}{6})

and

x = \pi (2n + \frac{5}{6})

for

n \in \mathbb{Z}

step 5

The first 20 positive solutions can be calculated from these expressions. The sum of the first 20 positive solutions of the equation

\sin x = \cos 2x

is

20\pi

Answer

The sum of the first 20 positive solutions is

20\pi

.

Key Concept

Trigonometric identities and periodic solutions

Explanation

The solutions to the equation involve using trigonometric identities and understanding the periodic nature of sine and cosine functions. The sum of the first 20 solutions reflects the periodicity of these functions.

AskSia

Plus