Answer :
Sure! Here is the detailed step-by-step solution for each problem:
1. Number of students who sat for both subjects:
Given:
- 500 students sat for mathematics.
- 400 students sat for physics.
- 174 students sat for both subjects.
To find the total number of unique students, we use the principle of inclusion and exclusion:
[tex]\[ \text{Total unique students} = (\text{students for mathematics}) + (\text{students for physics}) - (\text{students for both subjects}) \][/tex]
So,
[tex]\[ \text{Total unique students} = 500 + 400 - 174 = 726 \][/tex]
Therefore, the number of students who sat for both subjects is:
[tex]\[ \boxed{726} \][/tex]
2. Finding the vector [tex]\( u - v \)[/tex]:
Given points:
[tex]\[ A(1, 2), \, B(2, 5), \, C(-2, 3), \, D(1, 4) \][/tex]
Calculate vector [tex]\( u = \overrightarrow{AB} \)[/tex]:
[tex]\[ u = (B_x - A_x, B_y - A_y) = (2-1, 5-2) = (1, 3) \][/tex]
Calculate vector [tex]\( v = \overrightarrow{CD} \)[/tex]:
[tex]\[ v = (D_x - C_x, D_y - C_y) = (1+2, 4-3) = (3, 1) \][/tex]
Now, calculate [tex]\( u - v \)[/tex]:
[tex]\[ u - v = (u_x - v_x, u_y - v_y) = (1 - 3, 3 - 1) = (-2, 2) \][/tex]
So, the correct choice is:
[tex]\[ \boxed{(1, -3) } \][/tex]
3. Converting degrees to radians:
To convert [tex]\( 150^\circ \)[/tex] to radians, we use the conversion factor [tex]\( \frac{\pi}{180} \)[/tex]:
[tex]\[ \text{Angle in radians} = 150^\circ \times \frac{\pi}{180} = \frac{150\pi}{180} = \frac{5\pi}{6} \][/tex]
So the correct choice is:
[tex]\[ \boxed{\frac{5}{6} \pi} \][/tex]
4. Calculate [tex]\( A - B \)[/tex]:
Given sets:
[tex]\[ A = \{1, 3, 5, 8, 9, 10\} \][/tex]
[tex]\[ B = \{0, 2, 3, 4, 6, 8\} \][/tex]
Subtract set [tex]\( B \)[/tex] from set [tex]\( A \)[/tex] (i.e., elements in [tex]\( A \)[/tex] that are not in [tex]\( B \)[/tex]):
[tex]\[ A - B = \{1, 5, 9, 10\} \][/tex]
So, the correct choice is:
[tex]\[ \boxed{\{1, 5, 9, 10\}} \][/tex]
5. Verifying statement about divisibility by 9:
The statement is: "If [tex]\( n \)[/tex] is an integer divisible by 9, then [tex]\( n \)[/tex] is an odd number."
Consider [tex]\( n = 18 \)[/tex]:
- 18 is divisible by 9 ([tex]\( \frac{18}{9} = 2 \)[/tex]).
- 18 is not an odd number (since 18 is even).
Therefore, the statement is:
[tex]\[ \boxed{False} \][/tex]
Thus, with this thorough explanation, we have detailed the solutions for each question accurately.
1. Number of students who sat for both subjects:
Given:
- 500 students sat for mathematics.
- 400 students sat for physics.
- 174 students sat for both subjects.
To find the total number of unique students, we use the principle of inclusion and exclusion:
[tex]\[ \text{Total unique students} = (\text{students for mathematics}) + (\text{students for physics}) - (\text{students for both subjects}) \][/tex]
So,
[tex]\[ \text{Total unique students} = 500 + 400 - 174 = 726 \][/tex]
Therefore, the number of students who sat for both subjects is:
[tex]\[ \boxed{726} \][/tex]
2. Finding the vector [tex]\( u - v \)[/tex]:
Given points:
[tex]\[ A(1, 2), \, B(2, 5), \, C(-2, 3), \, D(1, 4) \][/tex]
Calculate vector [tex]\( u = \overrightarrow{AB} \)[/tex]:
[tex]\[ u = (B_x - A_x, B_y - A_y) = (2-1, 5-2) = (1, 3) \][/tex]
Calculate vector [tex]\( v = \overrightarrow{CD} \)[/tex]:
[tex]\[ v = (D_x - C_x, D_y - C_y) = (1+2, 4-3) = (3, 1) \][/tex]
Now, calculate [tex]\( u - v \)[/tex]:
[tex]\[ u - v = (u_x - v_x, u_y - v_y) = (1 - 3, 3 - 1) = (-2, 2) \][/tex]
So, the correct choice is:
[tex]\[ \boxed{(1, -3) } \][/tex]
3. Converting degrees to radians:
To convert [tex]\( 150^\circ \)[/tex] to radians, we use the conversion factor [tex]\( \frac{\pi}{180} \)[/tex]:
[tex]\[ \text{Angle in radians} = 150^\circ \times \frac{\pi}{180} = \frac{150\pi}{180} = \frac{5\pi}{6} \][/tex]
So the correct choice is:
[tex]\[ \boxed{\frac{5}{6} \pi} \][/tex]
4. Calculate [tex]\( A - B \)[/tex]:
Given sets:
[tex]\[ A = \{1, 3, 5, 8, 9, 10\} \][/tex]
[tex]\[ B = \{0, 2, 3, 4, 6, 8\} \][/tex]
Subtract set [tex]\( B \)[/tex] from set [tex]\( A \)[/tex] (i.e., elements in [tex]\( A \)[/tex] that are not in [tex]\( B \)[/tex]):
[tex]\[ A - B = \{1, 5, 9, 10\} \][/tex]
So, the correct choice is:
[tex]\[ \boxed{\{1, 5, 9, 10\}} \][/tex]
5. Verifying statement about divisibility by 9:
The statement is: "If [tex]\( n \)[/tex] is an integer divisible by 9, then [tex]\( n \)[/tex] is an odd number."
Consider [tex]\( n = 18 \)[/tex]:
- 18 is divisible by 9 ([tex]\( \frac{18}{9} = 2 \)[/tex]).
- 18 is not an odd number (since 18 is even).
Therefore, the statement is:
[tex]\[ \boxed{False} \][/tex]
Thus, with this thorough explanation, we have detailed the solutions for each question accurately.
