answer:Let's define the variables for the Venn Diagram: - a: Number of students taking exactly Music and Art. - b: Number of students taking exactly Music and Drama. - c: Number of students taking all three classes (Music, Art, and Drama). - d: Number of students taking exactly Art and Drama. We are given: - Total students taking at least two classes: 11. - Total students taking Music: 15. - Total students taking Art: 14. - Total students taking Drama: 13. From the information, we know: [ a + b + c + d = 11 ] Calculating students taking only one class: 1. **Students taking only Music**: [ text{Total Music} - (a + b + c) = 15 - (a + b + c) ] Using a + b + c + d = 11, we find a + b + c = 11 - d. Thus: [ 15 - (11 - d) = 4 + d ] 2. **Students taking only Art**: [ text{Total Art} - (a + c + d) = 14 - (a + c + d) ] Using a + b + c + d = 11, we find a + c + d = 11 - b. Thus: [ 14 - (11 - b) = 3 + b ] 3. **Students taking only Drama**: [ text{Total Drama} - (b + c + d) = 13 - (b + c + d) ] Using a + b + c + d = 11, we find b + c + d = 11 - a. Thus: [ 13 - (11 - a) = 2 + a ] Total students taking exactly one class: [ (4 + d) + (3 + b) + (2 + a) = 15 ] [ 9 + a + b + d = 15 ] Using a + b + c + d = 11, substitute 11 - c for a + b + d: [ 9 + (11 - c) = 15 ] [ 20 - c = 15 ] [ c = 5 ] Thus, the number of students taking all three classes is 5. The final answer is boxed{textbf{(D)} 5}

answer:We need to express the decimal number 14 as a sum of powers of 2. The powers of 2 that sum up to 14 are: - 2^3 = 8 - 2^2 = 4 - 2^1 = 2 - 2^0 = 0 (not needed here since we don't need the unit place to make 14) Thus, 14 = 1cdot 2^3 + 1cdot 2^2 + 1cdot 2^1 + 0cdot 2^0. Hence, the binary representation of fourteen is: 14 = boxed{1110_2}

answer:Let's break it down by the days we know: - Monday: Billy ate 2 apples. - Tuesday: He ate twice as many as Monday, so 2 * 2 = 4 apples. - Friday: He ate half of what he ate on Monday, so 2 / 2 = 1 apple. - Thursday: He ate four times as many as Friday, so 1 * 4 = 4 apples. Now let's add up what we know so far: Monday + Tuesday + Thursday + Friday = 2 + 4 + 4 + 1 = 11 apples. Billy ate 20 apples in total this week, so to find out how many he ate on Wednesday, we subtract the apples eaten on the other days from the total: 20 - 11 = 9 apples. So, Billy ate boxed{9} apples on Wednesday.

answer:(1) When a=1, the function becomes f(x)=ln x+x+frac{2}{x}+3. The derivative of the function is f'(x)=frac{1}{x}+1-frac{2}{x^2}. Therefore, f'(2)=1 and f(2)=ln 2+6. The equation of the tangent line is: y-(ln 2+6)=1(x-2), which simplifies to boxed{x-y+ln 2+4=0}. (2) The derivative of the function is f'(x)=frac{1}{x}+a-frac{1+a}{x^2}=frac{(ax+a+1)(x-1)}{x^2}. When a=0, f'(x)=frac{x-1}{x^2}. In this case, f(x) is decreasing on (0,1) and increasing on (1,+infty). When aneq 0, solving f'(x)=0 gives x_1=1 and x_2=-1-frac{1}{a}. bullet If -frac{1}{2} < a < 0, then -1-frac{1}{a} > 1. In this case, f(x) is decreasing on (0,1) and (-1-frac{1}{a},+infty), and increasing on (1,-1-frac{1}{a}). bullet If a > 0, then -1-frac{1}{a} < 0. In this case, f(x) is decreasing on (0,1) and increasing on (1,+infty). In summary: bullet When -frac{1}{2} < a < 0, f(x) is decreasing on (0,1) and (-1-frac{1}{a},+infty), and increasing on (1,-1-frac{1}{a}). bullet When ageqslant 0, f(x) is decreasing on (0,1) and increasing on (1,+infty).