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 #2
avatar+1911 
0

We are asked to find the complex number \( z \) that satisfies the equation:

 

\[
(1+i)z - 2z^* = -11 + 25i
\]

 

where \( z^* \) denotes the conjugate of \( z \). Let \( z = x + yi \), where \( x \) and \( y \) are real numbers. Then, the conjugate \( z^* \) is given by:

 

\[
z^* = x - yi
\]

 

### Step 1: Substitute \( z = x + yi \) and \( z^* = x - yi \) into the equation

 

Substituting these into the equation, we get:

 

\[
(1+i)(x + yi) - 2(x - yi) = -11 + 25i
\]

 

### Step 2: Expand the terms

 

First, expand \( (1+i)(x + yi) \):

 

\[
(1+i)(x + yi) = 1 \cdot x + 1 \cdot yi + i \cdot x + i \cdot yi = x + yi + xi + i^2y = (x - y) + i(x + y)
\]

 

Next, expand \( -2(x - yi) \):

 

\[
-2(x - yi) = -2x + 2yi
\]

 

Substituting these into the equation:

 

\[
(x - y + 2y)i + (x - 2x - y) = -11 + 25i
\]

 

### Step 3: Combine like terms

 

Combining the real parts and imaginary parts, we have:

 

\[
(x - y - 2x) + (2y + x + 2yi) = -11 + 25i
\]

 

This simplifies to:

 

\[
-x - y + 2yi + (x - 2x) + (y + y)i = -11 + 25i
\]

 

### Step 4: Equate real and imaginary parts

 

Now, equate the real and imaginary parts of the equation:

 

For the real part:

 

\[
x - 2y = -11
\]

 

For the imaginary part:

 

\[
x + 2y = 25
\]

 

### Step 5: Solve the system of equations

 

We now solve the system of equations:

 

\[
x - 2y = -11
\]


\[
x + 2y = 25
\]

 

Add these two equations:

 

\[
(x - 2y) + (x + 2y) = -11 + 25
\]

 

This simplifies to:

 

\[
2x = 14
\]

 

So:

 

\[
x = 7
\]

 

Substitute \( x = 7 \) into one of the original equations:

 

\[
7 - 2y = -11
\]

 

Solve for \( y \):

\[
-2y = -18 \quad \Rightarrow \quad y = 9
\]

 

### Step 6: Write the complex number \( z \)

 

The complex number \( z \) is:

 

\[
z = x + yi = 7 + 9i
\]

 

Thus, the solution is:

 

\[
\boxed{7 + 9i}
\]

Aug 10, 2024
 #1
avatar+1911 
0

We can divide this problem into two cases:

 

Case 1: Each row has exactly one child from each family.

 

Choose one child from each family for the first row: there are 3 choices

.

Arrange the remaining two children in the first row (siblings can't be together): there are 2!=2 ways.

 

Arrange the second row similarly: 3⋅2=6​ ways total.

 

Case 2: One row has two children from the same family.

 

There are two subcases depending on the arrangement of siblings in the rows:

 

Subcase 2a: The first child in each row is from the same family.

 

Choose one pair of siblings: 3 choices

.

Arrange the siblings within the pair: 2 ways.

 

Arrange the remaining 4 children (2 from another pair and 2 from the third pair) in the second row: 4! ways.

 

Overcount: we've counted the arrangement as if sibling order matters within a pair (which it doesn't) twice (once for each sibling in the first row). So, we divide by 2!⋅2!.

 

Total: 2!⋅2!3⋅2⋅4!​=36 ways.

 

Subcase 2b: The first child in the first row is NOT a sibling of the first child in the second row.

 

Choose one pair to have their children occupy the third seats in each row: 3 choices.

 

Arrange the remaining 4 children in the first row: 4! ways.

 

Overcount: similar to subcase 2a, we divide by 2!⋅2! to account for sibling order not mattering.

 

Total: 2!⋅2!3⋅4!​=36 ways.

 

Since Cases 1 and 2 are mutually exclusive, to find the total number of arrangements, we simply add the number of arrangements from each case:

6 + 36 + 36 = 78​.

 

This gives us a final answer of 78.

Apr 24, 2024
 #1
avatar+1911 
+1

Since a rotation about point O maps A to B, B to C, and C to D, we know the following:

 

Angle AOD: This is given as 24 degrees.

 

Full Rotation: A full rotation around a point is 360 degrees.

 

Possible Rotations:

 

There are three possibilities for the rotation that maps A to B, B to C, and C to D, resulting in three possible measures for angle AOB:

 

Case 1: Single Rotation of 24 Degrees

 

In this case, the rotation about O maps A to B by 24 degrees clockwise, B to C by 24 degrees clockwise, and C to D by 24 degrees

clockwise, resulting in a total rotation of A to D of 24 + 24 + 24 = 72 degrees.

 

Since the full rotation is 360 degrees, the remaining angle for AOB must be 360 - 72 = 288 degrees.

 

However, angle measures are typically represented between 0 and 180 degrees. We can achieve this by subtracting a multiple of 180 (full rotations) from 288:

 

AOB = 288° - 180 = 108.

 

Case 2: Double Rotation (360 + 24 Degrees)

 

Here, the rotation about O maps A to B by a full rotation (360 degrees) followed by a 24-degree clockwise rotation. This effectively brings B back to its original position and then continues the rotation to C and D.

 

The total rotation for AOD in this case becomes 360 + 24 + 24 = 408 degrees.

 

Similar to case 1, we can adjust this to fit the 0-180 degree range:

 

AOB = 408° - 360° = 48°

 

Case 3: Triple Rotation (2 * 360 + 24 Degrees)

 

This case involves two full rotations followed by a 24-degree clockwise rotation from A to B. Again, the first two rotations effectively bring B back to its original position.

 

The total rotation for AOD becomes 2 * 360 + 24 = 744 degrees.

 

Adjusting for the 0-180 degree range:

 

AOB = 744° - 2 * 360° = 124°

 

Summary:

 

Therefore, the three possible degree measures for angle AOB, considering rotations between 0 and 180 degrees, are:

 

108° (Case 1)

 

48° (Case 2)

 

124° (Case 3)

Apr 14, 2024