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Let's solve each part of the question step-by-step.
1. Find the median from the following marks obtained by 9 students:
[tex]\[ 27, 28, 30, 18, 29, 16, 25, 23, 26 \][/tex]
- First, arrange the marks in ascending order:
[tex]\[ 16, 18, 23, 25, 26, 27, 28, 29, 30 \][/tex]
- Since there are 9 marks, the median will be the middle value, which is the 5th value in the sorted list:
[tex]\[ \text{Median} = 26 \][/tex]
2. Find the mean of numbers [tex]\(48, 38, 42, 38, 49, 62, 44\)[/tex]:
- First, sum the numbers:
[tex]\[ 48 + 38 + 42 + 38 + 49 + 62 + 44 = 321 \][/tex]
- Then, divide the sum by the number of values (7):
[tex]\[ \text{Mean} = \frac{321}{7} = 45.857142857142854 \][/tex]
3. Find the mode of the variables [tex]\(3, 4, 5, 6, 2, 3, 5, 6, 8, 3, 7, 9, 3\)[/tex]:
- The mode is the value that appears most frequently.
- Count the occurrences of each number:
- 3 appears 4 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
- The number that appears the most frequently is 3:
[tex]\[ \text{Mode} = 3 \][/tex]
4. Find the range of the given data: [tex]\(5, 10, 10, 20, 21, 25, 30\)[/tex]:
- The range is the difference between the maximum and minimum values.
- Maximum value = 30
- Minimum value = 5
[tex]\[ \text{Range} = 30 - 5 = 25 \][/tex]
5. Construct the mapping diagrams for the given relation [tex]\( \text{F:} \{(-4, 4), (-3, 3), (-1, 3), (0, 2)\} \)[/tex]:
- The mapping diagram for the given relation shows a set of ordered pairs where each element from the domain is mapped to an element in the co-domain:
[tex]\[ \text{Mapping Diagram} = \{-4: 4, -3: 3, -1: 3, 0: 2\} \][/tex]
6. Find the degree of the given polynomial [tex]\( x^2(y^5 + z^6) \)[/tex]:
- The degree of a polynomial is the highest degree of its monomials.
- In the polynomial [tex]\( x^2(y^5 + z^6) \)[/tex]:
- The degree of [tex]\( x^2 \cdot y^5 \)[/tex] is [tex]\(2 + 5 = 7\)[/tex]
- The degree of [tex]\( x^2 \cdot z^6 \)[/tex] is [tex]\(2 + 6 = 8\)[/tex]
- Therefore, the degree of the polynomial is:
[tex]\[ \text{Polynomial Degree} = 8 \][/tex]
7. Find the [tex]\( S_n \)[/tex] for the sequence: [tex]\( 1, 3, 9, 27, 81 \)[/tex]:
- To find [tex]\( S_n \)[/tex], sum the values of the sequence:
[tex]\[ S_n = 1 + 3 + 9 + 27 + 81 = 121 \][/tex]
8. If [tex]\( A = \{1, 2, 3\} \)[/tex] and [tex]\( B = \{a, b\} \)[/tex], find [tex]\( A \times B \)[/tex] and [tex]\( B \times A \)[/tex]:
- [tex]\( A \times B \)[/tex] is the set of all ordered pairs [tex]\((a, b)\)[/tex] where [tex]\( a \in A \)[/tex] and [tex]\( b \in B \)[/tex]:
[tex]\[ A \times B = \{(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)\} \][/tex]
- [tex]\( B \times A \)[/tex] is the set of all ordered pairs [tex]\((b, a)\)[/tex] where [tex]\( b \in B \)[/tex] and [tex]\( a \in A \)[/tex]:
[tex]\[ B \times A = \{(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)\} \][/tex]
Here is the complete answer to each part of the question.
1. Find the median from the following marks obtained by 9 students:
[tex]\[ 27, 28, 30, 18, 29, 16, 25, 23, 26 \][/tex]
- First, arrange the marks in ascending order:
[tex]\[ 16, 18, 23, 25, 26, 27, 28, 29, 30 \][/tex]
- Since there are 9 marks, the median will be the middle value, which is the 5th value in the sorted list:
[tex]\[ \text{Median} = 26 \][/tex]
2. Find the mean of numbers [tex]\(48, 38, 42, 38, 49, 62, 44\)[/tex]:
- First, sum the numbers:
[tex]\[ 48 + 38 + 42 + 38 + 49 + 62 + 44 = 321 \][/tex]
- Then, divide the sum by the number of values (7):
[tex]\[ \text{Mean} = \frac{321}{7} = 45.857142857142854 \][/tex]
3. Find the mode of the variables [tex]\(3, 4, 5, 6, 2, 3, 5, 6, 8, 3, 7, 9, 3\)[/tex]:
- The mode is the value that appears most frequently.
- Count the occurrences of each number:
- 3 appears 4 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
- The number that appears the most frequently is 3:
[tex]\[ \text{Mode} = 3 \][/tex]
4. Find the range of the given data: [tex]\(5, 10, 10, 20, 21, 25, 30\)[/tex]:
- The range is the difference between the maximum and minimum values.
- Maximum value = 30
- Minimum value = 5
[tex]\[ \text{Range} = 30 - 5 = 25 \][/tex]
5. Construct the mapping diagrams for the given relation [tex]\( \text{F:} \{(-4, 4), (-3, 3), (-1, 3), (0, 2)\} \)[/tex]:
- The mapping diagram for the given relation shows a set of ordered pairs where each element from the domain is mapped to an element in the co-domain:
[tex]\[ \text{Mapping Diagram} = \{-4: 4, -3: 3, -1: 3, 0: 2\} \][/tex]
6. Find the degree of the given polynomial [tex]\( x^2(y^5 + z^6) \)[/tex]:
- The degree of a polynomial is the highest degree of its monomials.
- In the polynomial [tex]\( x^2(y^5 + z^6) \)[/tex]:
- The degree of [tex]\( x^2 \cdot y^5 \)[/tex] is [tex]\(2 + 5 = 7\)[/tex]
- The degree of [tex]\( x^2 \cdot z^6 \)[/tex] is [tex]\(2 + 6 = 8\)[/tex]
- Therefore, the degree of the polynomial is:
[tex]\[ \text{Polynomial Degree} = 8 \][/tex]
7. Find the [tex]\( S_n \)[/tex] for the sequence: [tex]\( 1, 3, 9, 27, 81 \)[/tex]:
- To find [tex]\( S_n \)[/tex], sum the values of the sequence:
[tex]\[ S_n = 1 + 3 + 9 + 27 + 81 = 121 \][/tex]
8. If [tex]\( A = \{1, 2, 3\} \)[/tex] and [tex]\( B = \{a, b\} \)[/tex], find [tex]\( A \times B \)[/tex] and [tex]\( B \times A \)[/tex]:
- [tex]\( A \times B \)[/tex] is the set of all ordered pairs [tex]\((a, b)\)[/tex] where [tex]\( a \in A \)[/tex] and [tex]\( b \in B \)[/tex]:
[tex]\[ A \times B = \{(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)\} \][/tex]
- [tex]\( B \times A \)[/tex] is the set of all ordered pairs [tex]\((b, a)\)[/tex] where [tex]\( b \in B \)[/tex] and [tex]\( a \in A \)[/tex]:
[tex]\[ B \times A = \{(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)\} \][/tex]
Here is the complete answer to each part of the question.
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