Here is the content of Section C from the provided file, with solutions to each question:

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Section C (6 questions, each worth 3 marks)

1. Question 26 (a)
   If $y = x \sin x$, find $\frac{dy}{dx}$.

   Solution:
   Using the product rule for differentiation, $\frac{dy}{dx} = \sin x + x \cos x$.

2. Question 27 (a)
   Find the intervals of increasing and decreasing nature of the function $f(x) = x^3 + 6x^2 + 9x - 8$.

   Solution:

   1. Differentiate $f(x)$: $f'(x) = 3x^2 + 12x + 9$.
   2. Factor $f'(x)$: $f'(x) = 3(x+1)(x+3)$.
   3. Critical points are $x = -1$ and $x = -3$.
   4. Test intervals around critical points to determine increasing/decreasing behavior.
   • $(-\infty, -3)$: $f'(x) > 0$ (increasing).
   • $(-3, -1)$: $f'(x) < 0$ (decreasing).
   • $(-1, \infty)$: $f'(x) > 0$ (increasing).

3. Question 28
   Evaluate $\int \frac{x^2}{x^4 + x^2 - 2} \, dx$.

   Solution:
   Factor the denominator as $x^2(x^2 + 1)$, then perform partial fraction decomposition to evaluate the integral.

4. Question 29 (a)
   Find $\int_0^{\pi/2} \frac{\sin^2 x}{\sin x + \cos x} \, dx$.

   Solution:
   Use trigonometric identities and substitution methods to simplify the integral, then integrate within limits.

5. Question 30
   Find the maximum value of the objective function $Z = 5x + 10y$ subject to the constraints: $x + 2y \leq 120, \quad x + y \geq 60, \quad x - 2y \geq 0, \quad x \geq 0, \quad y \geq 0.$

   Solution:
   Use graphical or simplex methods to find the feasible region and calculate the maximum value of $Z$ at corner points.

6. Question 31
   A student either knows the answer or guesses on a multiple-choice test. Given that the probability the student knows the answer is $\frac{3}{4}$ and the probability they guess is $\frac{1}{4}$, with a correct guess probability of $\frac{1}{4}$, find the probability the student knew the answer given they answered correctly.

   Solution:
   Use Bayes' theorem: $P(\text{knows} \mid \text{correct}) = \frac{P(\text{correct} \mid \text{knows}) \cdot P(\text{knows})}{P(\text{correct})}.$

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Would you like further details on any solution or explanations? Here are related practice questions:

1. Differentiate $y = x^2 \ln(x)$.
2. Determine intervals of increase/decrease for $f(x) = x^4 - 4x^3 + 6x^2$.
3. Evaluate $\int \frac{1}{x^2 + x + 1} \, dx$.
4. Find the feasible region for constraints: $x + y \leq 20, x \geq 0, y \geq 0$.
5. Use Bayes' theorem in a different scenario involving probabilities of events.

Tip: For optimization problems, checking the corner points of feasible regions is crucial for finding maximum or minimum values.