Let's go through the questions one by one from the image you've provided. I'll outline the steps required for each and solve them.

1. Find $d = \gcd(24, 175)$ and express $d = s \cdot 24 + t \cdot 175$, where $s, t \in \mathbb{Z}$. Hence, find $\text{LCM}(24, 175)$.

• Use the Euclidean algorithm to find $\gcd(24, 175)$.
• Then use the extended Euclidean algorithm to express $d = s \cdot 24 + t \cdot 175$.
• Finally, use the relation $\text{LCM}(a, b) = \frac{a \cdot b}{\gcd(a, b)}$.

2. Solve the equation $2 \otimes x = 5$ in $\mathbb{Z}_9$.

• This is a congruence equation $2x \equiv 5 \mod 9$.
• We need to find the multiplicative inverse of 2 modulo 9 and then solve for $x$.

3. Solve the system of congruences:

• $x \equiv 7 \pmod{9}$
• $x \equiv 3 \pmod{10}$
• $x \equiv 5 \pmod{11}$
• Use the Chinese Remainder Theorem to solve this system.

4. Differentiate the following functions:

• (a) $y = (2x^2)^3 - \frac{1}{2}(x^2 - 3)^2$
• (b) $y = (3x + 2)^3 (2x^2 - 4x - 1)^2$
• (c) $y = \frac{2x^2 - 1}{2x(x^2 + 1)^2}$

5. Find all relative extreme points and inflection points of $y = 2x^3 + 4x^2 - 5$. Sketch the curve, indicating all relative extreme points and inflection points.

• Differentiate $y$ to find critical points (first derivative).
• Use the second derivative to find inflection points and classify the critical points.

6. Given $y = \sqrt[3]{x}$, use the method of differentiation to approximate $\sqrt[3]{29}$.

• Differentiate $y$, use a suitable approximation near $x = 27$, and estimate $\sqrt[3]{29}$.

7. The volume of a spherical balloon, $V$ cm³, changes from $\frac{4}{3} \pi$ to $\frac{7}{3} \pi$. Find the approximate change in the radius.

• Use the volume formula for a sphere $V = \frac{4}{3} \pi r^3$ to find the change in radius.

I'll proceed to solve these step-by-step. Would you like me to focus on a specific question or provide a solution to all of them?