https://file.aiquickdraw.com/mathbot/file/1726741009225hxy43dii.jpg

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?

The original question submitted by the user is a set of seven mathematical problems covering GCD, modular arithmetic, systems of congruences, differentiation, extrema, and volume-related calculations.

This page provides solutions to advanced mathematical problems involving the GCD using the Euclidean algorithm, solving systems of congruences via the Chinese Remainder Theorem, differentiating complex functions, and calculating volume changes in spherical objects. Perfect for students at the college level.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation