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1. In a certain school, 500 students sat for mathematics, and 400 students sat for physics in the model examination. If the number of students who sat for both subjects is 174, then which of the following gives the number of students who sat for the two subjects?

A. 900
B. 726
C. 326
D. 226

2. Consider the points [tex]\(A(1,2), B(2,5), C(-2,3)\)[/tex], and [tex]\(D(1,4)\)[/tex] in the [tex]\(xy\)[/tex]-plane. If [tex]\(\mathbf{u} = \overrightarrow{AB}\)[/tex] and [tex]\(\mathbf{v} = \overrightarrow{CD}\)[/tex], then which one of the following is equal to [tex]\(\mathbf{u} - \mathbf{v}\)[/tex]?

A. [tex]\((2,-2)\)[/tex]
B. [tex]\((1,-3)\)[/tex]
C. [tex]\((3,-1)\)[/tex]
D. [tex]\((-2,2)\)[/tex]

3. The radian measure of an angle of [tex]\(150^{\circ}\)[/tex] is equal to:

A. [tex]\(\frac{2}{3} \pi\)[/tex]
B. [tex]\(\frac{\pi}{8}\)[/tex]
C. [tex]\(\frac{5}{6} \pi\)[/tex]
D. [tex]\(\frac{\sqrt{3}}{3} \pi\)[/tex]

4. Let [tex]\(A=\{1,3,5,8,9,10\}\)[/tex] and [tex]\(B=\{0,2,3,4,6,8\}\)[/tex]. Which one of the following is true?

A. [tex]\(B - A = \{0,2,4,6\}\)[/tex]
B. [tex]\(A - B = \{1,5,8,9,10\}\)[/tex]
C. [tex]\(A - B = \{0,2,4,6,10\}\)[/tex]
D. [tex]\(B - A = \{1,5,9,10\}\)[/tex]

5. Which of the following statements is true?

A. If [tex]\(n\)[/tex] is an integer divisible by 9, then [tex]\(n\)[/tex] is an odd number.



Answer :

GinnyAnswer
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.