# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-7

Exercises: Try Out - Set 7: Real exam simulation to sharpen your skills and build confidence.

---

## Exercise 1

### Question

export const metadata = {
  title: "Problem 1",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

import { Graph } from "../graph";

Observe the curve below!

<Graph
  title="Parabola Graph"
  description={
    <span>
      A parabola intersecting the <InlineMath math="x" />-axis and the <InlineMath math="y" />-axis.
    </span>
  }
/>

The correct statements regarding the curve are

1. The intersection point of the curve with the <InlineMath math="x" />-axis is <InlineMath math="(4, 0)" />.
2. The intersection points of the curve with the <InlineMath math="y" />-axis are <InlineMath math="(0, -5)" /> and <InlineMath math="(0, 1)" />.
3. The axis of symmetry is <InlineMath math="y = -1" />.
4. The axis of symmetry is <InlineMath math="x = -3" />.


### Choices

- [ ] If $$(1)$$, $$(2)$$, and $$(3)$$ are correct.
- [x] If $$(1)$$ and $$(3)$$ are correct.
- [ ] If $$(2)$$ and $$(4)$$ are correct.
- [ ] If only $$(4)$$ is correct.
- [ ] If all are correct.

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 1",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

We will analyze each statement based on the given graph.

#### Analysis of Statement 1

Based on the graph, it is clear that the curve intersects the <InlineMath math="x" />-axis at the point <InlineMath math="(4, 0)" />.

Thus, statement <InlineMath math="(1)" /> is **correct**.

#### Analysis of Statement 2

Based on the graph, the curve intersects the <InlineMath math="y" />-axis at two points, namely at <InlineMath math="y = 3" /> and <InlineMath math="y = -5" />. So the intersection points are <InlineMath math="(0, 3)" /> and <InlineMath math="(0, -5)" />.

Statement <InlineMath math="(2)" /> states that the intersection points are <InlineMath math="(0, -5)" /> and <InlineMath math="(0, 1)" />. Since <InlineMath math="(0, 3) \neq (0, 1)" />, this statement does not match the graph.

Thus, statement <InlineMath math="(2)" /> is **incorrect**.

#### Analysis of Statement 3

The axis of symmetry for a horizontal parabola can be determined from the midpoint between the two <InlineMath math="y" />-intercepts (if known).

<MathContainer>
  <BlockMath math="y_p = \frac{y_1 + y_2}{2} = \frac{3 + (-5)}{2} = \frac{-2}{2} = -1" />
</MathContainer>

The axis of symmetry is obtained as the line <InlineMath math="y = -1" />.

Thus, statement <InlineMath math="(3)" /> is **correct**.

#### Analysis of Statement 4

Since the axis of symmetry is the horizontal line <InlineMath math="y = -1" />, the statement that the axis of symmetry is <InlineMath math="x = -3" /> (a vertical line) is clearly wrong.

Thus, statement <InlineMath math="(4)" /> is **incorrect**.

#### Conclusion

The correct statements are statements <InlineMath math="(1)" /> and <InlineMath math="(3)" />.


---

## Exercise 2

### Question

export const metadata = {
  title: "Question 2",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

Given <InlineMath math="y = -x^2 - 2x + 8" />, the equation of the tangent line at the point with abscissa <InlineMath math="-2" /> is...


### Choices

- [ ] $$y = 2x - 4$$
- [ ] $$y = -2x - 2$$
- [ ] $$y = -2x + 4$$
- [x] $$y = 2x + 12$$
- [ ] $$y = -2x + 12$$

### Answer & Explanation

export const metadata = {
  title: "Question 2 Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

Given the function <InlineMath math="y = -x^2 - 2x + 8" />. We want to find the equation of the tangent line at the point with abscissa <InlineMath math="x = -2" />.

#### Finding the Ordinate

Substitute <InlineMath math="x = -2" /> into the curve equation:

<MathContainer>
  <BlockMath math="y = -(-2)^2 - 2(-2) + 8" />
  <BlockMath math="y = -4 + 4 + 8" />
  <BlockMath math="y = 8" />
</MathContainer>

So, the point of tangency is <InlineMath math="(-2, 8)" />.

#### Finding the Gradient of the Tangent Line

The gradient of the tangent line <InlineMath math="m" /> is the value of the first derivative <InlineMath math="y'" /> at the abscissa of the point of tangency.

<MathContainer>
  <BlockMath math="y' = -2x - 2" />
</MathContainer>

Substitute <InlineMath math="x = -2" />:

<MathContainer>
  <BlockMath math="m = -2(-2) - 2" />
  <BlockMath math="m = 4 - 2" />
  <BlockMath math="m = 2" />
</MathContainer>

#### Determining the Equation of the Tangent Line

Use the point-slope form for the equation of a line passing through <InlineMath math="(x_1, y_1)" /> with gradient <InlineMath math="m" />:

<MathContainer>
  <BlockMath math="y - y_1 = m(x - x_1)" />
</MathContainer>

Substitute <InlineMath math="x_1 = -2" />, <InlineMath math="y_1 = 8" />, and <InlineMath math="m = 2" />:

<MathContainer>
  <BlockMath math="y - 8 = 2(x - (-2))" />
  <BlockMath math="y - 8 = 2(x + 2)" />
  <BlockMath math="y - 8 = 2x + 4" />
  <BlockMath math="y = 2x + 4 + 8" />
  <BlockMath math="y = 2x + 12" />
</MathContainer>

Thus, the equation of the tangent line is <InlineMath math="y = 2x + 12" />.


---

## Exercise 3

### Question

export const metadata = {
  title: "Question 3",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

Given the following <InlineMath math="4" /> numbers:

1. <InlineMath math="19" />
2. <InlineMath math="29" />
3. <InlineMath math="39" />
4. <InlineMath math="49" />

When these numbers are divided by <InlineMath math="5" />, the remainder is <InlineMath math="4" />, and when divided by <InlineMath math="4" />, the remainder is <InlineMath math="3" />. Which of the following is correct...


### Choices

- [ ] $$1, 2, \text{ and } 3$$
- [x] $$1 \text{ and } 3$$
- [ ] $$2 \text{ and } 4$$
- [ ] $$4 \text{ only}$$
- [ ] All correct

### Answer & Explanation

export const metadata = {
  title: "Question 3 Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

We will check each number to see if it satisfies the following two conditions:
1. When divided by <InlineMath math="5" />, the remainder is <InlineMath math="4" /> (<InlineMath math="n \equiv 4 \pmod 5" />).
2. When divided by <InlineMath math="4" />, the remainder is <InlineMath math="3" /> (<InlineMath math="n \equiv 3 \pmod 4" />).

#### Checking Number 19

<MathContainer>
  <BlockMath math="19 = 3 \times 5 + 4 \quad (\text{remainder } 4)" />
  <BlockMath math="19 = 4 \times 4 + 3 \quad (\text{remainder } 3)" />
</MathContainer>

The number <InlineMath math="19" /> satisfies both conditions. (Correct)

#### Checking Number 29

<MathContainer>
  <BlockMath math="29 = 5 \times 5 + 4 \quad (\text{remainder } 4)" />
  <BlockMath math="29 = 7 \times 4 + 1 \quad (\text{remainder } 1)" />
</MathContainer>

The number <InlineMath math="29" /> does not satisfy the second condition because the remainder is <InlineMath math="1" /> when divided by <InlineMath math="4" />. (Incorrect)

#### Checking Number 39

<MathContainer>
  <BlockMath math="39 = 7 \times 5 + 4 \quad (\text{remainder } 4)" />
  <BlockMath math="39 = 9 \times 4 + 3 \quad (\text{remainder } 3)" />
</MathContainer>

The number <InlineMath math="39" /> satisfies both conditions. (Correct)

#### Checking Number 49

<MathContainer>
  <BlockMath math="49 = 9 \times 5 + 4 \quad (\text{remainder } 4)" />
  <BlockMath math="49 = 12 \times 4 + 1 \quad (\text{remainder } 1)" />
</MathContainer>

The number <InlineMath math="49" /> does not satisfy the second condition because the remainder is <InlineMath math="1" /> when divided by <InlineMath math="4" />. (Incorrect)

#### Conclusion

The numbers that satisfy the conditions are <InlineMath math="19" /> and <InlineMath math="39" /> (Statements <InlineMath math="1" /> and <InlineMath math="3" />).


---

## Exercise 4

### Question

export const metadata = {
  title: "Question 4",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

Given the following <InlineMath math="4" /> plane figures:

1. Right trapezoid.
2. Regular pentagon.
3. Equilateral triangle.
4. Kite (not a square).

How many of these plane figures have at most <InlineMath math="2" /> lines of symmetry and <InlineMath math="2" /> rotational symmetries?


### Choices

- [ ] $$0$$
- [ ] $$1$$
- [x] $$2$$
- [ ] $$3$$
- [ ] $$4$$

### Answer & Explanation

export const metadata = {
  title: "Question 4 Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

We will analyze the number of lines of symmetry and rotational symmetries for each plane figure. The requirement is to have **at most** <InlineMath math="2" /> lines of symmetry **and** <InlineMath math="2" /> rotational symmetries.

#### Analyzing Right Trapezoid

A right trapezoid has no axis of symmetry and no rotational symmetry of higher order (only order <InlineMath math="1" />, which is the initial position).

*   Lines of symmetry: <InlineMath math="0" />
*   Rotational symmetry: <InlineMath math="1" />

This figure satisfies the condition (since <InlineMath math="0 \le 2" /> and <InlineMath math="1 \le 2" />).

#### Analyzing Regular Pentagon

A regular pentagon has <InlineMath math="5" /> equal sides.

*   Lines of symmetry: <InlineMath math="5" />
*   Rotational symmetry: <InlineMath math="5" />

This figure does **not** satisfy the condition.

#### Analyzing Equilateral Triangle

An equilateral triangle has <InlineMath math="3" /> equal sides.

*   Lines of symmetry: <InlineMath math="3" />
*   Rotational symmetry: <InlineMath math="3" />

This figure does **not** satisfy the condition.

#### Analyzing Kite (Not a Square)

A kite has one main axis of symmetry.

*   Lines of symmetry: <InlineMath math="1" />
*   Rotational symmetry: <InlineMath math="1" />

This figure satisfies the condition.

#### Conclusion

The plane figures that satisfy the condition are:
1. Right trapezoid.
2. Kite (not a square).

Thus, there are <InlineMath math="2" /> plane figures that satisfy the condition.


---

## Exercise 5

### Question

export const metadata = {
  title: "Question 5",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

Given that <InlineMath math="a_n" /> is the <InlineMath math="n" />-th term of a sequence. If <InlineMath math="a_{n+1} = a_{n+2} + 2a_n" />, <InlineMath math="a_3 = -1" />, and <InlineMath math="a_5 = 3" />, then the value of <InlineMath math="a_2" /> is ...


### Choices

- [ ] $$-3$$
- [x] $$-1$$
- [ ] $$0$$
- [ ] $$1$$
- [ ] $$3$$

### Answer & Explanation

export const metadata = {
  title: "Question 5 Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

Given the recursive formula for the sequence:

<MathContainer>
  <BlockMath math="a_{n+1} = a_{n+2} + 2a_n" />
</MathContainer>

We are given <InlineMath math="a_3 = -1" /> and <InlineMath math="a_5 = 3" />. We need to find the value of <InlineMath math="a_2" />.

First, we find the value of <InlineMath math="a_4" /> by using the recursive formula and substituting <InlineMath math="n = 3" /> to relate <InlineMath math="a_3" />, <InlineMath math="a_4" />, and <InlineMath math="a_5" />.

<MathContainer>
  <BlockMath math="\begin{align*} a_{3+1} &= a_{3+2} + 2a_3 \\ a_4 &= a_5 + 2a_3 \end{align*}" />
</MathContainer>

Substitute the values <InlineMath math="a_5 = 3" /> and <InlineMath math="a_3 = -1" />:

<MathContainer>
  <BlockMath math="\begin{align*} a_4 &= 3 + 2(-1) \\ a_4 &= 3 - 2 \\ a_4 &= 1 \end{align*}" />
</MathContainer>

Next, we find the value of <InlineMath math="a_2" />. Since we now have the values of <InlineMath math="a_3" /> and <InlineMath math="a_4" />, we can use the recursive formula with <InlineMath math="n = 2" /> to relate <InlineMath math="a_2" />, <InlineMath math="a_3" />, and <InlineMath math="a_4" />.

<MathContainer>
  <BlockMath math="\begin{align*} a_{2+1} &= a_{2+2} + 2a_2 \\ a_3 &= a_4 + 2a_2 \end{align*}" />
</MathContainer>

Substitute the values <InlineMath math="a_3 = -1" /> and <InlineMath math="a_4 = 1" />:

<MathContainer>
  <BlockMath math="\begin{align*} -1 &= 1 + 2a_2 \\ -1 - 1 &= 2a_2 \\ -2 &= 2a_2 \\ a_2 &= -1 \end{align*}" />
</MathContainer>

Thus, the value of <InlineMath math="a_2" /> is <InlineMath math="-1" />.


---

## Exercise 6

### Question

export const metadata = {
  title: "Question 6",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

Given the matrix <InlineMath math="A = \begin{pmatrix} 2 & x-2 \\ -1 & 1 \end{pmatrix}" />. If <InlineMath math="\det(A) = 8" />, then the value of <InlineMath math="x" /> is ...


### Choices

- [ ] $$4$$
- [ ] $$2$$
- [ ] $$0$$
- [ ] $$-2$$
- [x] $$8$$

### Answer & Explanation

export const metadata = {
  title: "Question 6 Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

Given the matrix <InlineMath math="A = \begin{pmatrix} 2 & x-2 \\ -1 & 1 \end{pmatrix}" />. We need to find the value of <InlineMath math="x" /> if <InlineMath math="\det(A) = 8" />.

The determinant of a <InlineMath math="2 \times 2" /> matrix is calculated using the formula <InlineMath math="ad - bc" />.

<MathContainer>
  <BlockMath math="\det(A) = (2)(1) - (x-2)(-1)" />
</MathContainer>

We know that <InlineMath math="\det(A) = 8" />, so:

<MathContainer>
  <BlockMath math="\begin{align*} 8 &= 2 - (-(x-2)) \\ 8 &= 2 + (x-2) \\ 8 &= 2 + x - 2 \\ 8 &= x \end{align*}" />
</MathContainer>

Thus, the value of <InlineMath math="x" /> is <InlineMath math="8" />.


---

## Exercise 7

### Question

export const metadata = {
  title: "Question 7",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

Given <InlineMath math="f(x) = x + 1" /> and <InlineMath math="(f(x))^2 = 5f(x) - 4" /> are satisfied by <InlineMath math="x_1" /> or <InlineMath math="x_2" />. What is the value of <InlineMath math="x_1 + x_2" />?


### Choices

- [ ] $$-6$$
- [ ] $$-3$$
- [ ] $$0$$
- [x] $$3$$
- [ ] $$6$$

### Answer & Explanation

export const metadata = {
  title: "Question 7 Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

We are given the equation <InlineMath math="(f(x))^2 = 5f(x) - 4" />. We can rearrange it into a quadratic equation in terms of <InlineMath math="f(x)" />:

<MathContainer>
  <BlockMath math="(f(x))^2 - 5f(x) + 4 = 0" />
</MathContainer>

This equation is satisfied by <InlineMath math="x_1" /> and <InlineMath math="x_2" />. Based on the sum of roots formula for quadratic equations (<InlineMath math="x_1 + x_2 = -\frac{b}{a}" />), the sum of the values of <InlineMath math="f(x)" /> satisfying the equation is:

<MathContainer>
  <BlockMath math="f(x_1) + f(x_2) = -\frac{-5}{1} = 5" />
</MathContainer>

Given <InlineMath math="f(x) = x + 1" />. We substitute this into the equation above:

<MathContainer>
  <BlockMath math="\begin{align*} (x_1 + 1) + (x_2 + 1) &= 5 \\ x_1 + x_2 + 2 &= 5 \\ x_1 + x_2 &= 5 - 2 \\ x_1 + x_2 &= 3 \end{align*}" />
</MathContainer>

Thus, the value of <InlineMath math="x_1 + x_2" /> is <InlineMath math="3" />.


---

## Exercise 8

### Question

export const metadata = {
  title: "Question 8",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

In a village with a population of <InlineMath math="600" /> people, there are <InlineMath math="450" /> people who raise goats and <InlineMath math="300" /> people who raise chickens. What is the sum of the maximum number of people who raise both animals and the minimum number of people who raise both animals?


### Choices

- [ ] $$900$$
- [ ] $$600$$
- [x] $$450$$
- [ ] $$300$$
- [ ] $$150$$

### Answer & Explanation

export const metadata = {
  title: "Question 8 Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

We have two groups:
- People raising goats: <InlineMath math="K = 450" />
- People raising chickens: <InlineMath math="A = 300" />
- Total population: <InlineMath math="S = 600" />

#### Maximum Number of People Raising Both

The maximum number of people raising both animals is the size of the smaller group. This happens if all members of the smaller group are also members of the larger group.

<MathContainer>
  <BlockMath math="\text{Maximum} = \min(K, A) = \min(450, 300) = 300" />
</MathContainer>

#### Minimum Number of People Raising Both

The minimum number of people raising both animals occurs when the two groups are spread out as much as possible, minimizing their intersection. The formula is:

<MathContainer>
  <BlockMath math="\text{Minimum} = (K + A) - S" />
</MathContainer>

Substitute the known values:

<MathContainer>
  <BlockMath math="\begin{align*} \text{Minimum} &= (450 + 300) - 600 \\ &= 750 - 600 \\ &= 150 \end{align*}" />
</MathContainer>

#### Sum of Maximum and Minimum

We are asked to find the sum of these maximum and minimum values.

<MathContainer>
  <BlockMath math="\text{Total} = 300 + 150 = 450" />
</MathContainer>

Thus, the sum of the maximum and minimum number of people raising both animals is <InlineMath math="450" />.


---

## Exercise 9

### Question

export const metadata = {
  title: "Problem 9",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

A code consists of <InlineMath math="2" /> vowels and <InlineMath math="2" /> digits in an alternating sequence, starting with a letter. The formation of the code follows these rules:

1. The first letter is not <InlineMath math="A" />.
2. The first digit is an odd number.
3. The second letter is the letter <InlineMath math="O" />.
4. The second digit is chosen from <InlineMath math="0, 1, 2, 4, \text{ and } 6" />.

How many codes can be formed if no letters or digits can be repeated?


### Choices

- [x] $$72$$
- [ ] $$96$$
- [ ] $$120$$
- [ ] $$144$$
- [ ] $$168$$

### Answer & Explanation

export const metadata = {
  title: "Solution for Problem 9",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/24/2025",
};

We need to form a code with the pattern **Letter - Digit - Letter - Digit**.

Let the slots for the code be:
<MathContainer>
  <BlockMath math="\underline{L_1} \times \underline{D_1} \times \underline{L_2} \times \underline{D_2}" />
</MathContainer>

Available sets:
- Vowels: <InlineMath math="\{A, I, U, E, O\}" /> (<InlineMath math="5" /> letters).
- Odd digits for <InlineMath math="D_1" />: <InlineMath math="\{1, 3, 5, 7, 9\}" /> (<InlineMath math="5" /> digits).
- Choice digits for <InlineMath math="D_2" />: <InlineMath math="\{0, 1, 2, 4, 6\}" /> (<InlineMath math="5" /> digits).

Rules and Constraints:
1. <InlineMath math="L_1 \neq A" />.
2. <InlineMath math="L_2 = O" />.
3. No repetition of letters or digits is allowed.

Since <InlineMath math="D_1" /> must be odd and <InlineMath math="D_2" /> is chosen from a specific set, there is an overlap at the digit <InlineMath math="1" />. We need to split the calculation into cases based on the value of <InlineMath math="D_2" />.

#### Case 1 If The Second Digit Is 1
If we choose the digit <InlineMath math="1" /> for position <InlineMath math="D_2" />:
1. <InlineMath math="L_2" /> (Second letter): Must be <InlineMath math="O" /> (<InlineMath math="1" /> way).
2. <InlineMath math="D_2" /> (Second digit): Must be <InlineMath math="1" /> (<InlineMath math="1" /> way).
3. <InlineMath math="L_1" /> (First letter): Vowel other than <InlineMath math="A" /> and other than <InlineMath math="O" /> (since <InlineMath math="O" /> is used in <InlineMath math="L_2" />).
   - Choices: <InlineMath math="\{I, U, E\}" /> (<InlineMath math="3" /> ways).
4. <InlineMath math="D_1" /> (First digit): Odd digit other than <InlineMath math="1" /> (since <InlineMath math="1" /> is used in <InlineMath math="D_2" />).
   - Choices: <InlineMath math="\{3, 5, 7, 9\}" /> (<InlineMath math="4" /> ways).

Total ways for Case <InlineMath math="1" />:
<MathContainer>
  <BlockMath math="3 \times 4 \times 1 \times 1 = 12 \text{ ways}" />
</MathContainer>

#### Case 2 If The Second Digit Is Not 1
If we choose a digit other than <InlineMath math="1" /> for position <InlineMath math="D_2" /> (which means it is even):
1. <InlineMath math="L_2" /> (Second letter): Must be <InlineMath math="O" /> (<InlineMath math="1" /> way).
2. <InlineMath math="D_2" /> (Second digit): Chosen from <InlineMath math="\{0, 2, 4, 6\}" /> (<InlineMath math="4" /> ways).
3. <InlineMath math="L_1" /> (First letter): Vowel other than <InlineMath math="A" /> and other than <InlineMath math="O" />.
   - Choices: <InlineMath math="\{I, U, E\}" /> (<InlineMath math="3" /> ways).
4. <InlineMath math="D_1" /> (First digit): Odd digit <InlineMath math="\{1, 3, 5, 7, 9\}" />. Since <InlineMath math="D_2" /> is even, there is no conflict. All odd digits can be chosen.
   - Choices: <InlineMath math="5" /> ways.

Total ways for Case <InlineMath math="2" />:
<MathContainer>
  <BlockMath math="3 \times 5 \times 1 \times 4 = 60 \text{ ways}" />
</MathContainer>

#### Total Combinations
Sum the results from both cases:
<MathContainer>
  <BlockMath math="12 + 60 = 72" />
</MathContainer>

Therefore, the number of codes that can be formed is <InlineMath math="72" />.


---

## Exercise 10

### Question

export const metadata = {
  title: "Problem 10",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

Given an ordered data set <InlineMath math="4, 4, a, b, \text{ and } c" />. The average of these five numbers is <InlineMath math="8" />. The largest data value is known to be <InlineMath math="3" /> times the median value. Which of the following relationships is correct between quantities <InlineMath math="P" /> and <InlineMath math="Q" /> based on the information provided?

| <InlineMath math="P" /> | <InlineMath math="Q" /> |
| :---: | :---: |
| Median | <InlineMath math="4" /> |


### Choices

- [x] $$P > Q$$
- [ ] $$P < Q$$
- [ ] $$P = Q$$
- [ ] $$PQ = 32$$
- [ ] Cannot be determined

### Answer & Explanation

export const metadata = {
  title: "Solution for Problem 10",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

The given ordered data set is <InlineMath math="4, 4, a, b, c" />. Since the data is already ordered and there are an odd number of values (<InlineMath math="5" /> values), the median is the <InlineMath math="3" />rd value, which is <InlineMath math="a" />.

The largest data value is <InlineMath math="c" />, and it is <InlineMath math="3" /> times the median, so:
<MathContainer>
  <BlockMath math="c = 3a" />
</MathContainer>

The average of the five numbers is <InlineMath math="8" />:
<MathContainer>
  <BlockMath math="\frac{4 + 4 + a + b + c}{5} = 8" />
  <BlockMath math="8 + a + b + c = 40" />
  <BlockMath math="a + b + c = 32" />
</MathContainer>

Substitute <InlineMath math="c = 3a" /> into the equation:
<MathContainer>
  <BlockMath math="a + b + 3a = 32" />
  <BlockMath math="b + 4a = 32" />
  <BlockMath math="b = 32 - 4a" />
</MathContainer>

Since the data is ordered, <InlineMath math="4 \le a \le b \le c" /> holds true. We will find the bounds for the value of <InlineMath math="a" />.

#### Upper Bound of a
From the inequality <InlineMath math="a \le b" />:
<MathContainer>
  <BlockMath math="a \le 32 - 4a" />
  <BlockMath math="5a \le 32" />
  <BlockMath math="a \le 6.4" />
</MathContainer>

#### Lower Bound of a
From the inequality <InlineMath math="b \le c" />:
<MathContainer>
  <BlockMath math="32 - 4a \le 3a" />
  <BlockMath math="32 \le 7a" />
  <BlockMath math="a \ge \frac{32}{7}" />
  <BlockMath math="a \ge 4.57" />
</MathContainer>

Note also that <InlineMath math="a \ge 4" /> (from the data order). Since <InlineMath math="4.57 > 4" />, the stronger condition is <InlineMath math="a \ge 4.57" />.

#### Conclusion
The range of values for <InlineMath math="a" /> is <InlineMath math="4.57 \le a \le 6.4" />.

The value of <InlineMath math="P" /> is the median (<InlineMath math="a" />) and the value of <InlineMath math="Q" /> is <InlineMath math="4" />.
Since the minimum value of <InlineMath math="a" /> is <InlineMath math="4.57" />, which is greater than <InlineMath math="4" />, it is certain that <InlineMath math="P > Q" />.

Therefore, the correct relationship is <InlineMath math="P > Q" />.


---

## Exercise 11

### Question

export const metadata = {
  title: "Problem 11",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

Given <InlineMath math="0 < x < 1" />. Which of the following relationships is correct between quantities <InlineMath math="P" /> and <InlineMath math="Q" /> based on the information provided?

| <InlineMath math="P" /> | <InlineMath math="Q" /> |
| :---: | :---: |
| <InlineMath math="\frac{4^x - 1}{2^x - 1}" /> | <InlineMath math="\frac{2^x + 1}{2^x}" /> |


### Choices

- [x] $$P > Q$$
- [ ] $$P < Q$$
- [ ] $$P = Q$$
- [ ] $$PQ = 1$$
- [ ] Cannot be determined

### Answer & Explanation

export const metadata = {
  title: "Solution for Problem 11",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

We will simplify the form of <InlineMath math="P" /> first.
Recall that <InlineMath math="4^x = (2^2)^x = (2^x)^2" />. Thus, the form <InlineMath math="4^x - 1" /> is a difference of squares which can be factored into <InlineMath math="(2^x - 1)(2^x + 1)" />.

<MathContainer>
  <BlockMath math="P = \frac{4^x - 1}{2^x - 1}" />
  <BlockMath math="P = \frac{(2^x - 1)(2^x + 1)}{2^x - 1}" />
  <BlockMath math="P = 2^x + 1" />
</MathContainer>

Next, let's look at the form of <InlineMath math="Q" />:
<MathContainer>
  <BlockMath math="Q = \frac{2^x + 1}{2^x}" />
</MathContainer>

We can see the relationship between <InlineMath math="P" /> and <InlineMath math="Q" /> by substituting <InlineMath math="P = 2^x + 1" /> into the equation for <InlineMath math="Q" />:
<MathContainer>
  <BlockMath math="Q = \frac{P}{2^x}" />
  <BlockMath math="P = Q \cdot 2^x" />
</MathContainer>

Given <InlineMath math="0 < x < 1" />. Let's analyze the value of <InlineMath math="2^x" /> in that range:
- If <InlineMath math="x > 0" />, then <InlineMath math="2^x > 2^0 \implies 2^x > 1" />.
- If <InlineMath math="x < 1" />, then <InlineMath math="2^x < 2^1 \implies 2^x < 2" />.

Thus, <InlineMath math="1 < 2^x < 2" />.

Back to the equation <InlineMath math="P = Q \cdot 2^x" />:
Since <InlineMath math="2^x > 1" /> and <InlineMath math="Q" /> is positive (because <InlineMath math="2^x > 0" />), multiplying <InlineMath math="Q" /> by a number greater than <InlineMath math="1" /> will result in a value greater than <InlineMath math="Q" /> itself.

<MathContainer>
  <BlockMath math="P = Q \cdot (\text{number} > 1)" />
  <BlockMath math="P > Q" />
</MathContainer>

Therefore, the correct relationship is <InlineMath math="P > Q" />.


---

## Exercise 12

### Question

export const metadata = {
  title: "Problem 12",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

Given an operation on the set of integers defined by the rule <InlineMath math="\beta(w \oplus x \ominus y \otimes z) = 2w + x \cdot y - (-z)" />. The value of <InlineMath math="\beta(2 \oplus 3 \ominus 4 \otimes (-2))" /> is ...


### Choices

- [x] $$14$$
- [ ] $$18$$
- [ ] $$13$$
- [ ] $$9$$
- [ ] $$7$$

### Answer & Explanation

export const metadata = {
  title: "Solution for Problem 12",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

Given the operation rule:
<MathContainer>
  <BlockMath math="\beta(w \oplus x \ominus y \otimes z) = 2w + x \cdot y - (-z)" />
</MathContainer>

We are asked to find the value of:
<MathContainer>
  <BlockMath math="\beta(2 \oplus 3 \ominus 4 \otimes (-2))" />
</MathContainer>

By comparing variable positions, we get:
- <InlineMath math="w = 2" />
- <InlineMath math="x = 3" />
- <InlineMath math="y = 4" />
- <InlineMath math="z = -2" />

Substitute these values into the formula:
<MathContainer>
  <BlockMath math="\beta(2 \oplus 3 \ominus 4 \otimes (-2)) = 2(2) + 3 \cdot 4 - (-(-2))" />
  <BlockMath math="= 4 + 12 - (2)" />
  <BlockMath math="= 16 - 2" />
  <BlockMath math="= 14" />
</MathContainer>

Therefore, the value of the operation is <InlineMath math="14" />.


---

## Exercise 13

### Question

import { Graph } from "../graph";

export const metadata = {
  title: "Problem 13",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

<Graph
  title="Triangle ABC"
  description={
    <span>
      Visualization of Triangle <InlineMath math="ABC" /> with points <InlineMath math="P, Q," /> and <InlineMath math="R" />.
    </span>
  }
/>

Triangle <InlineMath math="ABC" /> is an isosceles triangle with <InlineMath math="AC = BC" />. Determine the measure of angle <InlineMath math="\angle BPC" />!

Decide whether statements <InlineMath math="(1)" /> and <InlineMath math="(2)" /> below are sufficient to answer the question!

1. <InlineMath math="\angle CAB = 40^\circ" />.
2. <InlineMath math="PC" /> is the angle bisector of triangle <InlineMath math="ABC" />.


### Choices

- [ ] Statement $$(1)$$ ALONE is sufficient, but statement $$(2)$$ ALONE is not sufficient.
- [x] Statement $$(2)$$ ALONE is sufficient, but statement $$(1)$$ ALONE is not sufficient.
- [ ] BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- [ ] EACH statement ALONE is sufficient.
- [ ] Statements $$(1)$$ and $$(2)$$ TOGETHER are NOT sufficient.

### Answer & Explanation

export const metadata = {
  title: "Solution for Problem 13",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

We are asked to determine the measure of angle <InlineMath math="\angle BPC" /> in an isosceles triangle <InlineMath math="ABC" /> with <InlineMath math="AC = BC" />.

#### Analysis of Statement 1

Given <InlineMath math="\angle CAB = 40^\circ" />. Since <InlineMath math="\triangle ABC" /> is isosceles with <InlineMath math="AC = BC" />, the base angles are equal:

<MathContainer>
  <BlockMath math="\angle CBA = \angle CAB = 40^\circ" />
</MathContainer>

We can calculate the vertex angle <InlineMath math="\angle ACB" />:

<MathContainer>
  <BlockMath math="\angle ACB = 180^\circ - (40^\circ + 40^\circ) = 100^\circ" />
</MathContainer>

However, this information does not specify the position of point <InlineMath math="P" /> on side <InlineMath math="AB" />. Point <InlineMath math="P" /> can be anywhere along the segment <InlineMath math="AB" />. Consequently, the measure of <InlineMath math="\angle BPC" /> varies depending on the position of <InlineMath math="P" />. Therefore, statement <InlineMath math="(1)" /> ALONE is not sufficient to answer the question.

#### Analysis of Statement 2

Given <InlineMath math="PC" /> is the angle bisector of triangle <InlineMath math="ABC" />. In this context, <InlineMath math="PC" /> is the bisector of angle <InlineMath math="C" /> intersecting side <InlineMath math="AB" /> at point <InlineMath math="P" />.

In an isosceles triangle with <InlineMath math="AC = BC" />, the angle bisector drawn from the vertex (the angle between the equal sides) to the base has special properties. It is also:

1. An altitude (perpendicular to the base).
2. A median (bisects the base).

Since <InlineMath math="PC" /> is an altitude, <InlineMath math="PC \perp AB" />. Thus, the angle formed is a right angle:

<MathContainer>
  <BlockMath math="\angle BPC = 90^\circ" />
</MathContainer>

We obtain a definite value for <InlineMath math="\angle BPC" />. Therefore, statement <InlineMath math="(2)" /> ALONE is sufficient to answer the question.

#### Conclusion

Statement <InlineMath math="(1)" /> is not sufficient, while statement <InlineMath math="(2)" /> is sufficient. Thus, **statement <InlineMath math="(2)" /> ALONE is sufficient, but statement <InlineMath math="(1)" /> ALONE is not sufficient**.


---

## Exercise 14

### Question

import { Graph } from "../graph";

export const metadata = {
  title: "Problem 14",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

Given <InlineMath math="f(x) = 5 - x" /> and <InlineMath math="g(x) = 2^x - 1" />. Determine which of the following statements are correct:

1. <InlineMath math="g(x)" /> is a linear line with a gradient of <InlineMath math="2" />.
2. <InlineMath math="f(x)" /> and <InlineMath math="g(x)" /> intersect at <InlineMath math="x > 0" />.
3. <InlineMath math="f(x)" /> is above <InlineMath math="g(x)" /> for all values of <InlineMath math="x" />.
4. The graphs of <InlineMath math="f(x)" /> and <InlineMath math="g(x)" /> intersect at <InlineMath math="(2, 3)" />.

<Graph title="Function Graph" description={<>Graph of functions <InlineMath math="f(x)" /> and <InlineMath math="g(x)" />.</>} />


### Choices

- [ ] $$(1), (2), \text{ and } (3) \text{ are correct}.$$
- [ ] $$(1) \text{ and } (3) \text{ are correct}.$$
- [x] $$(2) \text{ and } (4) \text{ are correct}.$$
- [ ] $$\text{Only } (4) \text{ is correct}.$$
- [ ] $$\text{All are correct}.$$

### Answer & Explanation

export const metadata = {
  title: "Solution for Problem 14",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

We will analyze each statement one by one based on the functions <InlineMath math="f(x) = 5 - x" /> and <InlineMath math="g(x) = 2^x - 1" />.

#### Analysis of Statement 1

Statement: <InlineMath math="g(x)" /> is a linear line with a gradient of <InlineMath math="2" />.

The function <InlineMath math="g(x) = 2^x - 1" /> is an exponential function, not a linear function. A linear function has the general form <InlineMath math="y = mx + c" />, whereas this is an exponential form. Therefore, statement <InlineMath math="(1)" /> is **incorrect**.

#### Analysis of Statement 2

Statement: <InlineMath math="f(x)" /> and <InlineMath math="g(x)" /> intersect at <InlineMath math="x > 0" />.

To find the intersection point, we equate the two functions:

<MathContainer>
  <BlockMath math="f(x) = g(x)" />
  <BlockMath math="5 - x = 2^x - 1" />
</MathContainer>

If we substitute <InlineMath math="x = 2" />:

<MathContainer>
  <BlockMath math="5 - 2 = 3" />
  <BlockMath math="2^2 - 1 = 4 - 1 = 3" />
</MathContainer>

Both functions have the same value of <InlineMath math="3" /> when <InlineMath math="x = 2" />. Since <InlineMath math="2 > 0" />, statement <InlineMath math="(2)" /> is **correct**.

#### Analysis of Statement 3

Statement: <InlineMath math="f(x)" /> is above <InlineMath math="g(x)" /> for all values of <InlineMath math="x" />.

Let's check a value <InlineMath math="x > 2" />, for example <InlineMath math="x = 3" />:

<MathContainer>
  <BlockMath math="f(3) = 5 - 3 = 2" />
  <BlockMath math="g(3) = 2^3 - 1 = 8 - 1 = 7" />
</MathContainer>

Here it is seen that <InlineMath math="g(3) > f(3)" />, which means the curve <InlineMath math="g(x)" /> is above <InlineMath math="f(x)" />. Therefore, the statement that <InlineMath math="f(x)" /> is always above <InlineMath math="g(x)" /> is incorrect. Statement <InlineMath math="(3)" /> is **incorrect**.

#### Analysis of Statement 4

Statement: The graphs of <InlineMath math="f(x)" /> and <InlineMath math="g(x)" /> intersect at <InlineMath math="(2, 3)" />.

As proven in the analysis of statement <InlineMath math="(2)" />, both graphs intersect when <InlineMath math="x = 2" /> and yield the value <InlineMath math="y = 3" />. Thus, the intersection point is <InlineMath math="(2, 3)" />. Statement <InlineMath math="(4)" /> is **correct**.

#### Conclusion

The correct statements are <InlineMath math="(2)" /> and <InlineMath math="(4)" />.


---

## Exercise 15

### Question

export const metadata = {
  title: "Problem 15",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

Joko's water tank can hold <InlineMath math="86" /> liters of water. Unfortunately, the tank is cracked, causing a leak of between <InlineMath math="0.3" /> liters and <InlineMath math="0.7" /> liters per hour. At <InlineMath math="07.00" /> AM, Joko filled his tank completely. Between <InlineMath math="07.00 - 11.00" />, there was no refilling or water usage. Which of the following relationships between quantities <InlineMath math="P" /> and <InlineMath math="Q" /> is correct based on the given information?

| <InlineMath math="P" /> | <InlineMath math="Q" /> |
| :---: | :--- |
| <InlineMath math="83.1" /> liters | Volume of water at <InlineMath math="11.00" /> |


### Choices

- [ ] $$P > Q$$
- [x] $$P < Q$$
- [ ] $$P = Q$$
- [ ] $$P + Q = 80$$
- [ ] Cannot be determined

### Answer & Explanation

export const metadata = {
  title: "Explanation for Problem 15",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

Given:
- Initial water volume = <InlineMath math="86" /> liters.
- Leakage per hour (<InlineMath math="x" />) is between <InlineMath math="0.3" /> and <InlineMath math="0.7" /> liters. This means <InlineMath math="0.3 < x < 0.7" />.
- The observation time is from <InlineMath math="07.00" /> to <InlineMath math="11.00" />, which is <InlineMath math="4" /> hours.

We calculate the range of water leaked over <InlineMath math="4" /> hours:

<MathContainer>
  <BlockMath math="4(0.3) < 4x < 4(0.7)" />
  <BlockMath math="1.2 < 4x < 2.8" />
</MathContainer>

So, the total water leaked is between <InlineMath math="1.2" /> liters and <InlineMath math="2.8" /> liters.

Next, we calculate the remaining water volume (<InlineMath math="Q" />) at <InlineMath math="11.00" />. The volume <InlineMath math="Q" /> is the initial volume minus the total leakage:

<MathContainer>
  <BlockMath math="Q = 86 - 4x" />
</MathContainer>

Since the total leakage is between <InlineMath math="1.2" /> and <InlineMath math="2.8" /> liters, the range for <InlineMath math="Q" /> is:
- Lower bound of <InlineMath math="Q" /> (maximum leakage): <InlineMath math="86 - 2.8 = 83.2" /> liters.
- Upper bound of <InlineMath math="Q" /> (minimum leakage): <InlineMath math="86 - 1.2 = 84.8" /> liters.

Thus, <InlineMath math="83.2 < Q < 84.8" />.

Now we compare <InlineMath math="P" /> and <InlineMath math="Q" />:
- <InlineMath math="P = 83.1" /> liters.
- The minimum value of <InlineMath math="Q" /> is greater than <InlineMath math="83.2" /> liters.

Since <InlineMath math="83.1 < 83.2" />, the value of <InlineMath math="P" /> is always smaller than the minimum possible value of <InlineMath math="Q" />.

Therefore, <InlineMath math="P < Q" />.


---

## Exercise 16

### Question

export const metadata = {
  title: "Problem 16",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

It is known that the current ratio of Farhat's salary to Abas's salary is <InlineMath math="2 : 3" />. If Farhat and Abas both receive a salary increase of <InlineMath math="40,000" /> rupiah, the ratio of their salaries becomes <InlineMath math="5 : 7" />. What is Abas's current salary?


### Choices

- [ ] $$120,000$$ rupiah
- [ ] $$160,000$$ rupiah
- [ ] $$200,000$$ rupiah
- [x] $$240,000$$ rupiah
- [ ] $$280,000$$ rupiah

### Answer & Explanation

export const metadata = {
  title: "Explanation for Problem 16",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

Let Farhat's salary be <InlineMath math="F" /> and Abas's salary be <InlineMath math="A" />.

The initial ratio of their salaries is given as <InlineMath math="2 : 3" />:

<MathContainer>
  <BlockMath math="F : A = 2 : 3 \Rightarrow F = \frac{2}{3}A" />
</MathContainer>

After both receive a salary increase of <InlineMath math="40,000" /> rupiah, the ratio becomes <InlineMath math="5 : 7" />. To simplify calculations, we will work in thousands (removing the last 3 zeros temporarily), so the increase is <InlineMath math="40" />.

<MathContainer>
  <BlockMath math="(F + 40) : (A + 40) = 5 : 7" />
</MathContainer>

We can write this equation as:
<MathContainer>
  <BlockMath math="\frac{F + 40}{A + 40} = \frac{5}{7}" />
</MathContainer>

Cross-multiply the equation:
<MathContainer>
  <BlockMath math="7(F + 40) = 5(A + 40)" />
</MathContainer>

Substitute <InlineMath math="F = \frac{2}{3}A" /> into the equation:
<MathContainer>
  <BlockMath math="7\left(\frac{2}{3}A + 40\right) = 5(A + 40)" />
  <BlockMath math="\frac{14}{3}A + 280 = 5A + 200" />
</MathContainer>

Group the terms containing <InlineMath math="A" /> on one side and the constants on the other:

<MathContainer>
  <BlockMath math="280 - 200 = 5A - \frac{14}{3}A" />
  <BlockMath math="80 = \frac{15}{3}A - \frac{14}{3}A" />
  <BlockMath math="80 = \frac{1}{3}A" />
</MathContainer>

Multiply both sides by 3 to find the value of <InlineMath math="A" />:
<MathContainer>
  <BlockMath math="A = 80 \times 3" />
  <BlockMath math="A = 240" />
</MathContainer>

Since we used units of thousands, Abas's actual salary is <InlineMath math="240,000" /> rupiah.

Thus, Abas's current salary is <InlineMath math="240,000" /> rupiah.


---

## Exercise 17

### Question

export const metadata = {
  title: "Problem 17",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

Given <InlineMath math="4" /> numbers as follows:
1. <InlineMath math="18" />
2. <InlineMath math="38" />
3. <InlineMath math="58" />
4. <InlineMath math="78" />

If these numbers are divided by <InlineMath math="5" />, the remainder is <InlineMath math="3" />, and if divided by <InlineMath math="4" />, the remainder is <InlineMath math="2" />. Among the four numbers, which one is correct?


### Choices

- [ ] $$1, 2, 3$$
- [ ] $$1, 3$$
- [ ] $$2, 4$$
- [ ] $$4$$ only
- [x] all

### Answer & Explanation

export const metadata = {
  title: "Explanation for Problem 17",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

We will check each number to see if it satisfies both conditions: divided by <InlineMath math="5" /> gives a remainder of <InlineMath math="3" /> AND divided by <InlineMath math="4" /> gives a remainder of <InlineMath math="2" />.

#### Analysis of Statement <InlineMath math="1" />

Number: <InlineMath math="18" />.

- Divided by <InlineMath math="5" />: <InlineMath math="18 = 3 \times 5 + 3" />. Remainder is <InlineMath math="3" />. (Satisfied)
- Divided by <InlineMath math="4" />: <InlineMath math="18 = 4 \times 4 + 2" />. Remainder is <InlineMath math="2" />. (Satisfied)

So, the number <InlineMath math="18" /> is correct.

#### Analysis of Statement <InlineMath math="2" />

Number: <InlineMath math="38" />.

- Divided by <InlineMath math="5" />: <InlineMath math="38 = 7 \times 5 + 3" />. Remainder is <InlineMath math="3" />. (Satisfied)
- Divided by <InlineMath math="4" />: <InlineMath math="38 = 9 \times 4 + 2" />. Remainder is <InlineMath math="2" />. (Satisfied)

So, the number <InlineMath math="38" /> is correct.

#### Analysis of Statement <InlineMath math="3" />

Number: <InlineMath math="58" />.

- Divided by <InlineMath math="5" />: <InlineMath math="58 = 11 \times 5 + 3" />. Remainder is <InlineMath math="3" />. (Satisfied)
- Divided by <InlineMath math="4" />: <InlineMath math="58 = 14 \times 4 + 2" />. Remainder is <InlineMath math="2" />. (Satisfied)

So, the number <InlineMath math="58" /> is correct.

#### Analysis of Statement <InlineMath math="4" />

Number: <InlineMath math="78" />.

- Divided by <InlineMath math="5" />: <InlineMath math="78 = 15 \times 5 + 3" />. Remainder is <InlineMath math="3" />. (Satisfied)
- Divided by <InlineMath math="4" />: <InlineMath math="78 = 19 \times 4 + 2" />. Remainder is <InlineMath math="2" />. (Satisfied)

So, the number <InlineMath math="78" /> is correct.

In conclusion, all four numbers are correct.


---

## Exercise 18

### Question

export const metadata = {
  title: "Problem 18",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

Given <InlineMath math="4" /> plane figures as follows:

1. Isosceles trapezoid.
2. Rectangle (not a square).
3. Equilateral triangle.
4. Kite (not a square).

How many plane figures have a number of lines of symmetry that is not equal to their order of rotational symmetry?


### Choices

- [x] $$0$$
- [ ] $$1$$
- [ ] $$2$$
- [ ] $$3$$
- [ ] $$4$$

### Answer & Explanation

export const metadata = {
  title: "Explanation for Problem 18",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

We will determine the number of lines of symmetry and the order of rotational symmetry for each given plane figure.

#### Analysis of Isosceles Trapezoid

- **Lines of Symmetry**: Has <InlineMath math="1" /> line of symmetry (vertical line in the middle).
- **Rotational Symmetry**: Order <InlineMath math="1" /> (returns to original position only after <InlineMath math="360^\circ" /> rotation).

The numbers are equal (<InlineMath math="1 = 1" />).

#### Analysis of Rectangle (Not a Square)

- **Lines of Symmetry**: Has <InlineMath math="2" /> lines of symmetry (horizontal and vertical).
- **Rotational Symmetry**: Order <InlineMath math="2" /> (rotates <InlineMath math="180^\circ" /> and <InlineMath math="360^\circ" />).

The numbers are equal (<InlineMath math="2 = 2" />).

#### Analysis of Equilateral Triangle

- **Lines of Symmetry**: Has <InlineMath math="3" /> lines of symmetry.
- **Rotational Symmetry**: Order <InlineMath math="3" /> (rotates <InlineMath math="120^\circ" />, <InlineMath math="240^\circ" />, and <InlineMath math="360^\circ" />).

The numbers are equal (<InlineMath math="3 = 3" />).

#### Analysis of Kite (Not a Square)

- **Lines of Symmetry**: Has <InlineMath math="1" /> line of symmetry (main diagonal).
- **Rotational Symmetry**: Order <InlineMath math="1" /> (returns to original position only after <InlineMath math="360^\circ" /> rotation).

The numbers are equal (<InlineMath math="1 = 1" />).

#### Conclusion

All plane figures have the same number of lines of symmetry and rotational symmetry order.
Therefore, the number of plane figures where the reflectional and rotational symmetries are **not equal** is <InlineMath math="0" />.


---

## Exercise 19

### Question

export const metadata = {
  title: "Problem 19",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

Given <InlineMath math="a_n" /> is the n-th term of a sequence of numbers. If <InlineMath math="a_n = a_{n+1} + 3a_{n+3}" />, <InlineMath math="a_1 = -1" />, and <InlineMath math="a_2 = 2" />, then the value of <InlineMath math="a_4 + a_1" /> is


### Choices

- [x] $$-2$$
- [ ] $$-1$$
- [ ] $$0$$
- [ ] $$1$$
- [ ] $$2$$

### Answer & Explanation

export const metadata = {
  title: "Explanation for Problem 19",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

We are given a recursive formula for a sequence of numbers:

<MathContainer>
  <BlockMath math="a_n = a_{n+1} + 3a_{n+3}" />
</MathContainer>

We also know that <InlineMath math="a_1 = -1" /> and <InlineMath math="a_2 = 2" />. We are asked to find the value of <InlineMath math="a_4 + a_1" />.

#### Determining the Value of the Fourth Term

To find <InlineMath math="a_4" />, we can use the given equation by substituting <InlineMath math="n = 1" />:

<MathContainer>
  <BlockMath math="\begin{aligned} a_1 &= a_{1+1} + 3a_{1+3} \\ a_1 &= a_2 + 3a_4 \end{aligned}" />
</MathContainer>

Next, we substitute the values <InlineMath math="a_1 = -1" /> and <InlineMath math="a_2 = 2" /> into the equation:

<MathContainer>
  <BlockMath math="\begin{aligned} -1 &= 2 + 3a_4 \\ 3a_4 &= -1 - 2 \\ 3a_4 &= -3 \\ a_4 &= \frac{-3}{3} \\ a_4 &= -1 \end{aligned}" />
</MathContainer>

#### Calculating the Sum

Now that we have found <InlineMath math="a_4 = -1" />, we can calculate <InlineMath math="a_4 + a_1" />:

<MathContainer>
  <BlockMath math="\begin{aligned} a_4 + a_1 &= (-1) + (-1) \\ &= -2 \end{aligned}" />
</MathContainer>

So, the value of <InlineMath math="a_4 + a_1" /> is <InlineMath math="-2" />.


---

## Exercise 20

### Question

export const metadata = {
  title: "Problem 20",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

Given matrix <InlineMath math="A = \begin{pmatrix} 2 & -2 \\ x-1 & 1 \end{pmatrix}" />. If <InlineMath math="\det(A) = 4" />, then the value of <InlineMath math="x" /> is


### Choices

- [x] $$2$$
- [ ] $$1$$
- [ ] $$0$$
- [ ] $$-1$$
- [ ] $$-2$$

### Answer & Explanation

export const metadata = {
  title: "Explanation for Problem 20",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

We are given matrix <InlineMath math="A" /> as follows:

<MathContainer>
  <BlockMath math="A = \begin{pmatrix} 2 & -2 \\ x-1 & 1 \end{pmatrix}" />
</MathContainer>

It is known that the determinant of matrix <InlineMath math="A" /> is <InlineMath math="4" />. The determinant formula for a <InlineMath math="2 \times 2" /> matrix like <InlineMath math="\begin{pmatrix} a & b \\ c & d \end{pmatrix}" /> is <InlineMath math="ad - bc" />.

#### Calculating the Determinant

We will calculate the determinant of matrix <InlineMath math="A" /> and set it equal to <InlineMath math="4" />:

<MathContainer>
  <BlockMath math="\begin{aligned} \det(A) &= (2)(1) - (-2)(x-1) \\ 4 &= 2 - (-2x + 2) \\ 4 &= 2 + 2x - 2 \\ 4 &= 2x \\ x &= \frac{4}{2} \\ x &= 2 \end{aligned}" />
</MathContainer>

So, the value of <InlineMath math="x" /> is <InlineMath math="2" />.


---
