Sure, let's solve these step by step.
### Step 1: Calculate the Mean
To find the mean of the numbers [tex]\(48, 38, 42, 38, 49, 62, 44\)[/tex]:
1. Add up all the numbers:
[tex]\[ 48 + 38 + 42 + 38 + 49 + 62 + 44 = 321 \][/tex]
2. Divide the sum by the number of values:
[tex]\[ \text{Mean} = \frac{321}{7} \approx 45.857142857142854 \][/tex]
So, the mean is approximately [tex]\(45.86\)[/tex].
### Step 2: Find the Mode
To find the mode of the variables [tex]\(3, 4, 5, 6, 2, 3, 5, 6, 8, 3, 7, 9\)[/tex]:
1. List out the frequencies of each number:
- 3 appears 3 times
- 4 appears 1 time
- 5 appears 2 times
- 6 appears 2 times
- 2 appears 1 time
- 8 appears 1 time
- 7 appears 1 time
- 9 appears 1 time
2. Identify the number that appears most frequently:
[tex]\[ \text{Mode} = 3 \][/tex]
So, the mode is [tex]\(3\)[/tex].
### Step 3: Find the Range
To find the range of the given data [tex]\(5, 10, 10, 20, 21, 25, 30\)[/tex]:
1. Identify the minimum and maximum values:
- Minimum = 5
- Maximum = 30
2. Subtract the minimum value from the maximum value:
[tex]\[ \text{Range} = 30 - 5 = 25 \][/tex]
So, the range is [tex]\(25\)[/tex].
### Step 4: Construct the Mapping Diagrams
For the given relation [tex]\( \text{F: }\{(-4,4),(-3,3),(-1,3),(0,2)\} \)[/tex]:
We can represent this as a dictionary (mapping):
[tex]\[ \text{Mapping Diagrams} = \{-4: 4, -3: 3, -1: 3, 0: 2\} \][/tex]
### Step 5: Find the Degree of the Polynomial
For the polynomial [tex]\( x^2(y^5 + z^6) \)[/tex]:
1. Identify the degrees of individual terms:
- [tex]\(x^2 y^5\)[/tex] has a degree of [tex]\(2 + 5 = 7\)[/tex]
- [tex]\(x^2 z^6\)[/tex] has a degree of [tex]\(2 + 6 = 8\)[/tex]
2. The degree of the polynomial is the highest degree of its terms:
[tex]\[ \text{Degree} = 8 \][/tex]
So, the degree is [tex]\(8\)[/tex].
### Step 6: Find [tex]\( S_n \)[/tex] for the Sequence
For the sequence [tex]\(1, 3, 9, 27, 81\)[/tex]:
1. Add up the terms of the sequence:
[tex]\[ S_n = 1 + 3 + 9 + 27 + 81 = 121 \][/tex]
So, [tex]\( S_n \)[/tex] is [tex]\(121\)[/tex].
### Step 7: Find the Cartesian Products [tex]\( A \times B \)[/tex] and [tex]\( B \times A \)[/tex]
For sets [tex]\(A = \{1, 2, 3\}\)[/tex] and [tex]\(B = \{a, b\}\)[/tex]:
1. Cartesian product [tex]\(A \times B\)[/tex]:
[tex]\[ A \times B = \{(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)\} \][/tex]
2. Cartesian product [tex]\(B \times A\)[/tex]:
[tex]\[ B \times A = \{(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)\} \][/tex]
So, [tex]\( A \times B = \{(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)\} \)[/tex] and [tex]\( B \times A = \{(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)\} \)[/tex].
These are the step-by-step solutions to the given problem.
