## 1. Preparation for the Final Exam

As the final exam approaches, it is very important for 12th grade students to prepare themselves well. One of the subjects that is often a bugbear for students is elective mathematics. Here are some examples of elective mathematics questions and answers to help students prepare themselves.

**Example Questions:**

- Calculate the integral of the function \( f(x) = 3x^2 + 2x + 5 \).
- A triangle has side lengths a = 5, b = 6, and c = 7. Calculate the area of the triangle.
- A trapezoid has two parallel sides measuring 8 cm and 12 cm, and a height of 5 cm. Calculate the area of the trapezoid.

## 2. Answers to Class 12 Mathematics Interest Questions

To help students prepare for the Grade 12 Mathematics elective exam, here are the answers to the sample questions above.

**Answer to Question 1:**

To calculate the integral of the function \( f(x) \), we use the following integral formula: \( \int f(x) dx = \frac{1}{n+1}x^{n+1} + C \). Thus, the integral of \( f(x) = 3x^2 + 2x + 5 \) is:

Integral \( f(x) = 3x^2 + 2x + 5 \) = \( \frac{3}{3}x^3 + \frac{2}{2}x^2 + 5x + C = x^3 + x^2 + 5x + C \).

**Answer to Question 2:**

To calculate the area of a triangle, we use the formula for the area of a triangle given by \( \frac{1}{2} \times a \times t \). With the known side lengths and height, the area of the triangle is:

Area of triangle = \( \frac{1}{2} \times 5 \times 6 = 15 \) area units.

**Answer to Question 3:**

To calculate the area of a trapezoid, we use the trapezoid area formula, which is \( \frac{1}{2} \times (a + b) \times t \). With the length of the parallel sides and the known height, the area of the trapezoid is:

Area of trapezoid = \( \frac{1}{2} \times (8 + 12) \times 5 = 50 \) area units.

## 3. Tips for Successfully Solving Mathematics Problems

To successfully complete the 12th grade elective math questions, there are several tips that need to be considered. Here are the tips:

**– Understand the Basic Concepts:**

Before working on math problems, make sure you understand the basic concepts first. Without a strong understanding of the concepts, you will have difficulty solving more complex problems.

**– Practice diligently:**

The key to success in math is to practice diligently. The more you practice, the more familiar you will be with various types of math problems.

**– Use Formulas Wisely:**

When working on math problems, make sure you use formulas that are relevant to the problem. Don't forget to understand how to use the formulas.

## 4. Class 12 Mathematics Practice Questions

Here are some Grade 12 math practice questions to help you prepare for the final exam:

**Question 1:**

Calculate the integral of the function \( g(x) = 4x^3 + 2x^2 + 3 \).

**Question 2:**

It is known that a circle has a radius of 7 cm. Calculate the area of the circle.

**Question 3:**

A cone has a base radius of 6 cm and a height of 10 cm. Calculate the volume of the cone.

## 5. Discussion of Answers to Mathematics Practice Questions

**Answer to Question 1:**

The integral of the function \( g(x) = 4x^3 + 2x^2 + 3 \) is:

Integral \( g(x) = \frac{4}{4}x^4 + \frac{2}{3}x^3 + 3x + C = x^4 + \frac{2}{3}x^3 + 3x + C \).

**Answer to Question 2:**

The area of a circle can be calculated using the formula for the area of a circle, namely \( \pi \times r^2 \). With the known radius, the area of the circle is:

Area of a circle = \( 3.14 \times 7^2 = 153.86 \) cm\(^2\).

**Answer to Question 3:**

The volume of a cone can be calculated using the cone volume formula, namely \( \frac{1}{3} \times \pi \times r^2 \times t \). With the known radius and height, the volume of the cone is:

Volume kerucut = \( \frac{1}{3} \times 3.14 \times 6^2 \times 10 = 376 \) cm\(^3\).

By understanding the questions and answers of grade 12 mathematics electives and practicing diligently, it is hoped that students can face the final exam with more confidence. Happy studying and good luck!