To solve Section C in detail, I'll address each question step by step.

Question 26(a):

Problem: Find $\frac{dy}{dx}$ if $y = x \sin x$.

Solution: We need to differentiate $y = x \sin x$ with respect to $x$. Using the product rule of differentiation, which states $\frac{d}{dx}(u \cdot v) = u' \cdot v + u \cdot v'$, let:

• $u = x$, so $u' = 1$
• $v = \sin x$, so $v' = \cos x$

Applying the product rule: $\frac{dy}{dx} = \frac{d}{dx}(x \sin x) = (1)(\sin x) + (x)(\cos x) = \sin x + x \cos x$

Thus, $\frac{dy}{dx} = \sin x + x \cos x$.

Question 27(a):

Problem: Find the intervals where the function $f(x) = x^3 + 6x^2 + 9x - 8$ is increasing and decreasing.

Solution:

1. Find the derivative of $f(x)$: $f'(x) = 3x^2 + 12x + 9$

2. Find the critical points by setting $f'(x) = 0$: $3x^2 + 12x + 9 = 0$ Divide the equation by 3: $x^2 + 4x + 3 = 0$ Factorize the quadratic: $(x + 3)(x + 1) = 0$ Thus, $x = -3$ and $x = -1$ are the critical points.

3. Determine the sign of $f'(x)$ in the intervals $(-\infty, -3)$, $(-3, -1)$, and $(-1, \infty)$:

   • For $x < -3$, choose $x = -4$: $f'(-4) = 3(-4)^2 + 12(-4) + 9 = 48 - 48 + 9 = 9 > 0$ So, $f(x)$ is increasing on $(-\infty, -3)$.
   • For $-3 < x < -1$, choose $x = -2$: $f'(-2) = 3(-2)^2 + 12(-2) + 9 = 12 - 24 + 9 = -3 < 0$ So, $f(x)$ is decreasing on $(-3, -1)$.
   • For $x > -1$, choose $x = 0$: $f'(0) = 3(0)^2 + 12(0) + 9 = 9 > 0$ So, $f(x)$ is increasing on $(-1, \infty)$.

Question 28:

Problem: Evaluate $\int \frac{x^2}{x^4 + x^2 - 2} \, dx$.

Solution:

1. Factor the denominator: $x^4 + x^2 - 2 = (x^2 + 2)(x^2 - 1)$ Further factor $x^2 - 1$ as: $(x^2 - 1) = (x - 1)(x + 1)$ Hence, the integral becomes: $\int \frac{x^2}{(x^2 + 2)(x - 1)(x + 1)} \, dx$
2. Use partial fraction decomposition to express $\frac{x^2}{(x^2 + 2)(x - 1)(x + 1)}$ as a sum of simpler fractions and integrate each term separately.

Question 29(a):

Problem: Find $\int_0^{\pi/2} \frac{\sin^2 x}{\sin x + \cos x} \, dx$.

Solution:

1. Let $I = \int_0^{\pi/2} \frac{\sin^2 x}{\sin x + \cos x} \, dx$.
2. Use symmetry or suitable substitutions (e.g., $x \to \frac{\pi}{2} - x$) to simplify the integral.
3. Evaluate the resulting expression.

Question 30:

Problem: 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:

1. Plot the constraints on a graph to find the feasible region.
2. Determine the vertices of the feasible region.
3. Evaluate $Z = 5x + 10y$ at each vertex to find the maximum value.

Let me know if you need more details on a particular problem or have further questions.

Related Questions:

1. How can you determine whether a function is increasing or decreasing?
2. What is the significance of critical points in calculus?
3. How do you factorize higher-order polynomials?
4. Why is partial fraction decomposition useful in integration?
5. What methods can be used to solve linear programming problems graphically?

Tip:

Always check the endpoints when finding the maximum or minimum of a function on a closed interval.