# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-5

Exercises: Try Out - Set 5: 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: "12/23/2025",
};

Given <InlineMath math="a_1 = -6" /> and <InlineMath math="a_2 = -8" />. If the recurrence relation satisfies the equation <InlineMath math="a_{n+1} = 2a_{n+2} + 3a_n" />, then the value of <InlineMath math="a_3 + 2a_4" /> is ...


### Choices

- [ ] $$39$$
- [ ] $$29$$
- [x] $$34$$
- [ ] $$24$$
- [ ] $$16$$

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 1",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

We are given the first two terms of the sequence, <InlineMath math="a_1 = -6" /> and <InlineMath math="a_2 = -8" />, and the recursive equation:

<MathContainer>
  <BlockMath math="a_{n+1} = 2a_{n+2} + 3a_n" />
</MathContainer>

Our goal is to find the value of <InlineMath math="a_3 + 2a_4" />. We will find the values of <InlineMath math="a_3" /> and <InlineMath math="2a_4" /> step by step.

#### Finding the Third Term

We use the recursive equation by substituting <InlineMath math="n = 1" />:

<MathContainer>
  <BlockMath math="a_{1+1} = 2a_{1+2} + 3a_1" />
  <BlockMath math="a_2 = 2a_3 + 3a_1" />
</MathContainer>

Substitute the known values of <InlineMath math="a_1" /> and <InlineMath math="a_2" />:

<MathContainer>
  <BlockMath math="-8 = 2a_3 + 3(-6)" />
  <BlockMath math="-8 = 2a_3 - 18" />
</MathContainer>

Move <InlineMath math="-18" /> to the left side:

<MathContainer>
  <BlockMath math="2a_3 = -8 + 18" />
  <BlockMath math="2a_3 = 10" />
  <BlockMath math="a_3 = 5" />
</MathContainer>

#### Finding Two Times the Fourth Term

Next, we use the recursive equation again with <InlineMath math="n = 2" /> to introduce the term <InlineMath math="a_4" />:

<MathContainer>
  <BlockMath math="a_{2+1} = 2a_{2+2} + 3a_2" />
  <BlockMath math="a_3 = 2a_4 + 3a_2" />
</MathContainer>

Substitute the values <InlineMath math="a_3 = 5" /> and <InlineMath math="a_2 = -8" />:

<MathContainer>
  <BlockMath math="5 = 2a_4 + 3(-8)" />
  <BlockMath math="5 = 2a_4 - 24" />
</MathContainer>

We can directly solve for <InlineMath math="2a_4" /> (since the question asks for <InlineMath math="a_3 + 2a_4" />):

<MathContainer>
  <BlockMath math="2a_4 = 5 + 24" />
  <BlockMath math="2a_4 = 29" />
</MathContainer>

#### Final Result

Now we sum the obtained values:

<MathContainer>
  <BlockMath math="a_3 + 2a_4 = 5 + 29 = 34" />
</MathContainer>

Thus, the value of <InlineMath math="a_3 + 2a_4" /> is 34.


---

## Exercise 2

### Question

export const metadata = {
  title: "Problem 2",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

The value of <InlineMath math="x" /> that satisfies the following equation is ...

<MathContainer>
  <BlockMath math="\begin{vmatrix} -6 & 3 \\ 2 & 4 \end{vmatrix} = \begin{vmatrix} 2x & 3 \\ -4x & -1 \end{vmatrix}" />
</MathContainer>


### Choices

- [ ] $$\frac{15}{2}$$
- [ ] $$\frac{9}{7}$$
- [ ] $$\frac{5}{2}$$
- [ ] $$-2$$
- [x] $$-3$$

### Answer & Explanation

export const metadata = {
    title: "Solution to Problem 2",
    authors: [{ name: "Nabil Akbarazzima Fatih" }],
    date: "12/23/2025",
};

We are given a determinant equation of <InlineMath math="2 \times 2" /> matrices. Recall that the formula for the determinant of a matrix <InlineMath math="\begin{vmatrix} a & b \\ c & d \end{vmatrix}" /> is <InlineMath math="ad - bc" />.

#### Calculating the Left Hand Side Determinant

First, we calculate the determinant of the matrix on the left side:

<MathContainer>
  <BlockMath math="\begin{vmatrix} -6 & 3 \\ 2 & 4 \end{vmatrix} = (-6)(4) - (3)(2)" />
  <BlockMath math="= -24 - 6" />
  <BlockMath math="= -30" />
</MathContainer>

#### Calculating the Right Hand Side Determinant

Next, we calculate the determinant of the matrix on the right side which contains the variable <InlineMath math="x" />:

<MathContainer>
  <BlockMath math="\begin{vmatrix} 2x & 3 \\ -4x & -1 \end{vmatrix} = (2x)(-1) - (3)(-4x)" />
  <BlockMath math="= -2x - (-12x)" />
  <BlockMath math="= -2x + 12x" />
  <BlockMath math="= 10x" />
</MathContainer>

#### Determining the Variable Value

Now we equate both determinant results according to the original equation:

<MathContainer>
  <BlockMath math="-30 = 10x" />
</MathContainer>

Divide both sides by 10 to find the value of <InlineMath math="x" />:

<MathContainer>
  <BlockMath math="x = \frac{-30}{10}" />
  <BlockMath math="x = -3" />
</MathContainer>

Thus, the value of <InlineMath math="x" /> that satisfies the equation is -3.


---

## Exercise 3

### Question

export const metadata = {
  title: "Problem 3",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

Given an integer <InlineMath math="n" />. If <InlineMath math="n" /> is divided by <InlineMath math="4" />, the remainder is <InlineMath math="2" />, and if the number is divided by <InlineMath math="6" />, the remainder is <InlineMath math="2" />. The possible value(s) of <InlineMath math="n" /> is/are ...

1. <InlineMath math="18" />
2. <InlineMath math="24" />
3. <InlineMath math="36" />
4. <InlineMath math="38" />


### Choices

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

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 3",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

We are given two conditions for the integer <InlineMath math="n" />:
1. <InlineMath math="n \equiv 2 \pmod{4}" /> (remainder is <InlineMath math="2" /> when divided by <InlineMath math="4" />)
2. <InlineMath math="n \equiv 2 \pmod{6}" /> (remainder is <InlineMath math="2" /> when divided by <InlineMath math="6" />)

We will check each statement to see if the value of <InlineMath math="n" /> satisfies both conditions.

#### Checking Statement 1

Value <InlineMath math="n = 18" />.
- Divided by <InlineMath math="4" />: <InlineMath math="18 \div 4 = 4" /> remainder <InlineMath math="2" />. (Satisfied)
- Divided by <InlineMath math="6" />: <InlineMath math="18 \div 6 = 3" /> remainder <InlineMath math="0" />. (Not Satisfied)

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

#### Checking Statement 2

Value <InlineMath math="n = 24" />.
- Divided by <InlineMath math="4" />: <InlineMath math="24 \div 4 = 6" /> remainder <InlineMath math="0" />. (Not Satisfied)

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

#### Checking Statement 3

Value <InlineMath math="n = 36" />.
- Divided by <InlineMath math="4" />: <InlineMath math="36 \div 4 = 9" /> remainder <InlineMath math="0" />. (Not Satisfied)

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

#### Checking Statement 4

Value <InlineMath math="n = 38" />.
- Divided by <InlineMath math="4" />: <InlineMath math="38 \div 4 = 9" /> remainder <InlineMath math="2" />. (Satisfied)
- Divided by <InlineMath math="6" />: <InlineMath math="38 \div 6 = 6" /> remainder <InlineMath math="2" />. (Satisfied)

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

#### Conclusion

Of the four options, only statement <InlineMath math="(4)" /> satisfies both conditions.


---

## Exercise 4

### Question

export const metadata = {
  title: "Problem 4",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

For <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{1 - x^6}{1 - x^3}" /> | <InlineMath math="(1 + x)^3" /> |


### Choices

- [ ] $$P > Q$$
- [x] $$P < Q$$
- [ ] $$P = Q$$
- [ ] $$P + Q = 1$$
- [ ] The information provided is not sufficient to decide one of the options.

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 4",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

We will simplify the forms of <InlineMath math="P" /> and <InlineMath math="Q" /> first to compare their values.

#### Simplifying P Value

Given <InlineMath math="P = \frac{1 - x^6}{1 - x^3}" />.
Recall the difference of squares <InlineMath math="a^2 - b^2 = (a - b)(a + b)" />. We can view <InlineMath math="1 - x^6" /> as <InlineMath math="1^2 - (x^3)^2" />.

<MathContainer>
  <BlockMath math="P = \frac{(1 - x^3)(1 + x^3)}{1 - x^3}" />
  <BlockMath math="P = 1 + x^3" />
</MathContainer>

#### Simplifying Q Value

Given <InlineMath math="Q = (1 + x)^3" />.
We expand it using binomial expansion:

<MathContainer>
  <BlockMath math="Q = 1 + 3x + 3x^2 + x^3" />
</MathContainer>

#### Comparing P and Q

We find the difference between <InlineMath math="Q" /> and <InlineMath math="P" />:

<MathContainer>
  <BlockMath math="Q - P = (1 + 3x + 3x^2 + x^3) - (1 + x^3)" />
  <BlockMath math="Q - P = 3x + 3x^2" />
</MathContainer>

Given the condition <InlineMath math="0 < x < 1" />.
Since <InlineMath math="x" /> is positive (<InlineMath math="x > 0" />), then:
- <InlineMath math="3x > 0" />
- <InlineMath math="3x^2 > 0" />

Thus, their sum must be positive:
<MathContainer>
  <BlockMath math="3x + 3x^2 > 0" />
  <BlockMath math="Q - P > 0" />
  <BlockMath math="Q > P" />
</MathContainer>

Or it can be written as <InlineMath math="P < Q" />.

#### Conclusion

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


---

## Exercise 5

### Question

export const metadata = {
  title: "Problem 5",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

If <InlineMath math="f(x) = x^2 + 4x - 5" /> and <InlineMath math="g(x) = 2x - 1" />, then <InlineMath math="(g \circ f)(x)" /> is ...


### Choices

- [x] $$2x^2 + 8x - 11$$
- [ ] $$2x^2 + 8x - 6$$
- [ ] $$2x^2 + 8x - 9$$
- [ ] $$2x^2 + 4x - 6$$
- [ ] $$2x^2 + 4x - 9$$

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 5",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

We want to find the composite function <InlineMath math="(g \circ f)(x)" />.

#### Understanding Function Composition

The composite function <InlineMath math="(g \circ f)(x)" /> is defined as:

<MathContainer>
  <BlockMath math="(g \circ f)(x) = g(f(x))" />
</MathContainer>

This means we will substitute the entire function <InlineMath math="f(x)" /> into the variable <InlineMath math="x" /> in the function <InlineMath math="g(x)" />.

#### Function Substitution

Given:
- <InlineMath math="f(x) = x^2 + 4x - 5" />
- <InlineMath math="g(x) = 2x - 1" />

We substitute <InlineMath math="f(x)" /> into <InlineMath math="g(x)" />:

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

Since <InlineMath math="g(x) = 2x - 1" />, we replace every <InlineMath math="x" /> in <InlineMath math="g(x)" /> with <InlineMath math="(x^2 + 4x - 5)" />:

<MathContainer>
  <BlockMath math="(g \circ f)(x) = 2(x^2 + 4x - 5) - 1" />
</MathContainer>

#### Simplifying the Result

Now we distribute the multiplication and simplify:

<MathContainer>
  <BlockMath math="(g \circ f)(x) = (2 \cdot x^2) + (2 \cdot 4x) - (2 \cdot 5) - 1" />
  <BlockMath math="(g \circ f)(x) = 2x^2 + 8x - 10 - 1" />
  <BlockMath math="(g \circ f)(x) = 2x^2 + 8x - 11" />
</MathContainer>

#### Conclusion

Thus, the composite function is <InlineMath math="2x^2 + 8x - 11" />.


---

## Exercise 6

### Question

export const metadata = {
  title: "Problem 6",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

Given the following data:
<InlineMath math="2, 3, 5, 7, 8, 9, 11" />

Two identical natural numbers are added to the data, resulting in a new data set with an average of <InlineMath math="9" />.

| <InlineMath math="P" /> | <InlineMath math="Q" /> |
| :---: | :---: |
| The difference between the median of the new data and the mean of the new data | <InlineMath math="-12" /> |


### Choices

- [x] $$P > Q$$
- [ ] $$P < Q$$
- [ ] $$P = Q$$
- [ ] $$P + Q = 1$$
- [ ] The information provided is not sufficient to decide one of the options.

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 6",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

We will find the value of the added numbers, determine the new median, and then compare the values of <InlineMath math="P" /> and <InlineMath math="Q" />.

#### Finding the Added Numbers

Let the two added natural numbers be <InlineMath math="x" />.
The initial data consists of <InlineMath math="7" /> numbers: <InlineMath math="2, 3, 5, 7, 8, 9, 11" />.
Sum of initial data:
<MathContainer>
  <BlockMath math="2 + 3 + 5 + 7 + 8 + 9 + 11 = 45" />
</MathContainer>

After adding <InlineMath math="2" /> numbers of value <InlineMath math="x" />, the count of data becomes <InlineMath math="7 + 2 = 9" />.
The new average is given as <InlineMath math="9" />.

<MathContainer>
  <BlockMath math="\bar{x}_{new} = \frac{\text{Sum of Initial Data} + 2x}{\text{Count of New Data}}" />
  <BlockMath math="9 = \frac{45 + 2x}{9}" />
  <BlockMath math="81 = 45 + 2x" />
  <BlockMath math="2x = 81 - 45" />
  <BlockMath math="2x = 36" />
  <BlockMath math="x = 18" />
</MathContainer>

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

#### Determining the New Median

The new data after adding <InlineMath math="18" /> and <InlineMath math="18" /> is:
<InlineMath math="2, 3, 5, 7, 8, 9, 11, 18, 18" />

The data is already sorted. Since the number of data points <InlineMath math="n = 9" /> (odd), the median is the <InlineMath math="\frac{9+1}{2} = 5" />-th data point.

<MathContainer>
  <BlockMath math="\text{Median} = \text{5th Data} = 8" />
</MathContainer>

#### Calculating P Value

The value of <InlineMath math="P" /> is defined as the difference between the new median and the new mean.
The new mean is known to be <InlineMath math="9" />.

<MathContainer>
  <BlockMath math="P = \text{Median} - \text{Mean}" />
  <BlockMath math="P = 8 - 9" />
  <BlockMath math="P = -1" />
</MathContainer>

#### Comparing P and Q

Given <InlineMath math="Q = -12" />.

<MathContainer>
  <BlockMath math="P = -1" />
  <BlockMath math="Q = -12" />
</MathContainer>

Since <InlineMath math="-1 > -12" />, then:

<MathContainer>
  <BlockMath math="P > Q" />
</MathContainer>

#### Conclusion

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


---

## Exercise 7

### Question

export const metadata = {
  title: "Problem 7",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

If <InlineMath math="f(x) = 2x + 3" /> and <InlineMath math="3f^2(x) = f(x) + 7" /> are satisfied by <InlineMath math="x_1" /> and <InlineMath math="x_2" />, then the value of <InlineMath math="x_1 + x_2 =" /> ...


### Choices

- [ ] $$\frac{8}{3}$$
- [ ] $$-\frac{8}{3}$$
- [ ] $$\frac{17}{6}$$
- [x] $$-\frac{17}{6}$$
- [ ] $$\frac{19}{9}$$

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 7",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

We are given the function <InlineMath math="f(x) = 2x + 3" /> and the equation <InlineMath math="3f^2(x) = f(x) + 7" />.
We want to find the sum of the roots of the equation, which is <InlineMath math="x_1 + x_2" />.

#### Substituting the Function

The first step is to substitute <InlineMath math="f(x)" /> into the given equation.

<MathContainer>
  <BlockMath math="3(2x + 3)^2 = (2x + 3) + 7" />
</MathContainer>

#### Simplifying the Equation

Next, we expand the square and simplify the equation into the general quadratic form <InlineMath math="ax^2 + bx + c = 0" />.

<MathContainer>
  <BlockMath math="3(4x^2 + 12x + 9) = 2x + 10" />
  <BlockMath math="12x^2 + 36x + 27 = 2x + 10" />
</MathContainer>

Move all terms to the left side:

<MathContainer>
  <BlockMath math="12x^2 + 36x - 2x + 27 - 10 = 0" />
  <BlockMath math="12x^2 + 34x + 17 = 0" />
</MathContainer>

#### Using the Sum of Roots Formula

From the quadratic equation <InlineMath math="12x^2 + 34x + 17 = 0" />, we obtain the coefficients:
- <InlineMath math="a = 12" />
- <InlineMath math="b = 34" />
- <InlineMath math="c = 17" />

The sum of the roots of a quadratic equation can be calculated using the formula <InlineMath math="x_1 + x_2 = -\frac{b}{a}" />.

<MathContainer>
  <BlockMath math="x_1 + x_2 = -\frac{34}{12}" />
</MathContainer>

Simplify the fraction by dividing the numerator and denominator by <InlineMath math="2" />:

<MathContainer>
  <BlockMath math="x_1 + x_2 = -\frac{17}{6}" />
</MathContainer>

#### Conclusion

Thus, the value of <InlineMath math="x_1 + x_2" /> is <InlineMath math="-\frac{17}{6}" />.


---

## Exercise 8

### Question

export const metadata = {
  title: "Problem 8",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

To complete the construction of a bridge, <InlineMath math="24" /> workers are needed and it is targeted to be finished in <InlineMath math="35" /> days. After <InlineMath math="8" /> days of work, half of the workers stop working. To ensure the bridge construction is completed, additional time of ... days is needed.


### Choices

- [ ] $$54$$
- [ ] $$38$$
- [ ] $$28$$
- [ ] $$14$$
- [x] $$19$$

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 8",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

We can solve this problem using the concept of inverse proportion or total workload (man-days).

#### Calculating Workload

The total planned workload is:
<MathContainer>
  <BlockMath math="\text{Total Workload} = 24 \text{ workers} \times 35 \text{ days}" />
</MathContainer>

The work is divided into two phases:
1. First phase: <InlineMath math="24" /> workers work for <InlineMath math="8" /> days.
2. Second phase: Half of the workers stop, so <InlineMath math="24 / 2 = 12" /> workers remain working for <InlineMath math="P" /> days.

The equation is:

<MathContainer>
  <BlockMath math="24 \times 35 = (24 \times 8) + (12 \times P)" />
</MathContainer>

#### Finding the Value of P

We solve the equation above to find <InlineMath math="P" /> (the duration for the remaining project).

<MathContainer>
  <BlockMath math="840 = 192 + 12P" />
  <BlockMath math="840 - 192 = 12P" />
  <BlockMath math="648 = 12P" />
  <BlockMath math="P = \frac{648}{12}" />
  <BlockMath math="P = 54" />
</MathContainer>

So, the time needed to complete the remaining work with <InlineMath math="12" /> workers is <InlineMath math="54" /> days.

#### Calculating Additional Time

Based on the provided solution, the additional time is calculated as the difference between the duration of the remaining work (<InlineMath math="P" />) and the total initial target (<InlineMath math="35" /> days).

<MathContainer>
  <BlockMath math="\text{Additional days} = 54 - 35 = 19 \text{ days}" />
</MathContainer>

Thus, <InlineMath math="19" /> additional days are needed.


---

## Exercise 9

### Question

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

export const metadata = {
  title: "Problem 9",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

Given the following graph:

<QuestionGraph title="Function Graph" description="Inequality System and Graph." />

Given <InlineMath math="4" /> statements:

1. Point <InlineMath math="A (0, -1)" />.
2. Point <InlineMath math="B (1, 0)" />.
3. Point <InlineMath math="C (2, 0)" />.
4. Point <InlineMath math="D (2, 2)" />.

The correct statement is ...


### Choices

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

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 9",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

We verify each statement based on the functions provided in the graph:
Exponential function: <InlineMath math="y = 2^x - 2" />
Linear function: <InlineMath math="y = -2x + 4" />

#### Checking Point A

Point <InlineMath math="A" /> is the y-intercept of the exponential curve (<InlineMath math="x=0" />).
<MathContainer>
  <BlockMath math="y = 2^0 - 2 = 1 - 2 = -1" />
</MathContainer>
So, <InlineMath math="A(0, -1)" />. Statement <InlineMath math="(1)" /> is **TRUE**.

#### Checking Point B

Point <InlineMath math="B" /> is the x-intercept of the exponential curve (<InlineMath math="y=0" />).
<MathContainer>
  <BlockMath math="0 = 2^x - 2 \Rightarrow 2^x = 2 \Rightarrow x = 1" />
</MathContainer>
So, <InlineMath math="B(1, 0)" />. Statement <InlineMath math="(2)" /> is **TRUE**.

#### Checking Point C

Point <InlineMath math="C" /> is the x-intercept of the linear line (<InlineMath math="y=0" />).
<MathContainer>
  <BlockMath math="0 = -2x + 4 \Rightarrow 2x = 4 \Rightarrow x = 2" />
</MathContainer>
So, <InlineMath math="C(2, 0)" />. Statement <InlineMath math="(3)" /> is **TRUE**.

#### Checking Point D

Point <InlineMath math="D" /> is the intersection point or a point on the line. The statement claims <InlineMath math="D(2, 2)" />.
If we test coordinate <InlineMath math="(2, 2)" /> into the line equation <InlineMath math="y = -2x + 4" />:
<MathContainer>
  <BlockMath math="y = -2(2) + 4 = -4 + 4 = 0" />
</MathContainer>
The result is <InlineMath math="y = 0" />, not <InlineMath math="2" />. Thus, point <InlineMath math="(2, 2)" /> does not lie on the line <InlineMath math="y = -2x + 4" />.
Statement <InlineMath math="(4)" /> is **FALSE**.

Conclusion: Statements <InlineMath math="(1)" />, <InlineMath math="(2)" />, and <InlineMath math="(3)" /> are correct.


---

## Exercise 10

### Question

export const metadata = {
  title: "Problem 10",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

Given the following plane figures:

1. Regular hexagon.
2. Isosceles right triangle.
3. Right trapezoid.
4. Non-square rectangle.

How many figures have the same number of lines of symmetry as the order of rotational symmetry?


### Choices

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

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 10",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

We analyze the lines of symmetry (reflectional symmetry) and rotational symmetry order for each given plane figure:

1. **Regular Hexagon**
   - Lines of symmetry: <InlineMath math="6" /> (lines through vertices and midpoints of sides).
   - Rotational symmetry order: <InlineMath math="6" /> (angles of rotation <InlineMath math="60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ, 360^\circ" />).
   - Conclusion: <InlineMath math="6 = 6" />. Figure <InlineMath math="(1)" /> satisfies the condition.

2. **Isosceles Right Triangle**
   - Lines of symmetry: <InlineMath math="1" /> (altitude from the right angle to the hypotenuse).
   - Rotational symmetry order: <InlineMath math="1" /> (only the initial position <InlineMath math="360^\circ" />).
   - Conclusion: <InlineMath math="1 = 1" />. Figure <InlineMath math="(2)" /> satisfies the condition.

3. **Right Trapezoid**
   - Lines of symmetry: <InlineMath math="0" /> (no axis of symmetry).
   - Rotational symmetry order: <InlineMath math="1" /> (only the initial position <InlineMath math="360^\circ" />).
   - Conclusion: <InlineMath math="0 \neq 1" />. Figure <InlineMath math="(3)" /> does not satisfy the condition.

4. **Non-square Rectangle**
   - Lines of symmetry: <InlineMath math="2" /> (vertical and horizontal lines through the center).
   - Rotational symmetry order: <InlineMath math="2" /> (angles of rotation <InlineMath math="180^\circ" /> and <InlineMath math="360^\circ" />).
   - Conclusion: <InlineMath math="2 = 2" />. Figure <InlineMath math="(4)" /> satisfies the condition.

Thus, the figures that have the same number of lines of symmetry as the order of rotational symmetry are figures <InlineMath math="(1)" />, <InlineMath math="(2)" />, and <InlineMath math="(4)" />. There are <InlineMath math="3" /> figures.


---

## Exercise 11

### Question

export const metadata = {
  title: "Problem 11",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

There are <InlineMath math="700" /> students on a campus, <InlineMath math="350" /> students choose sports activities, and <InlineMath math="450" /> students choose arts activities. If <InlineMath math="P" /> is the maximum number of students who participate in both and <InlineMath math="Q" /> is the minimum number of students who participate in both, then the difference between <InlineMath math="P" /> and <InlineMath math="Q" /> is ... people.


### Choices

- [ ] $$350$$
- [x] $$250$$
- [ ] $$150$$
- [ ] $$100$$
- [ ] $$75$$

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 11",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

Given:
- Total students <InlineMath math="= 700" />.
- Students choosing sports <InlineMath math="= 350" />.
- Students choosing arts <InlineMath math="= 450" />.

We want to find the value of <InlineMath math="P" /> (maximum intersection) and <InlineMath math="Q" /> (minimum intersection).

1. **Finding <InlineMath math="P" /> (Maximum Intersection)**
   The maximum number of students participating in both occurs if one set is a subset of the other (or as many members of the smaller set as possible are included in the larger set).
   <MathContainer>
     <BlockMath math="P = \min(350, 450) = 350 \text{ students}" />
   </MathContainer>

2. **Finding <InlineMath math="Q" /> (Minimum Intersection)**
   The minimum number of students participating in both occurs if the distribution of students is as spread out as possible so that the union of the two sets is maximized (equal to the total universe, which is <InlineMath math="700" />).
   Using the Inclusion-Exclusion Principle:
   <MathContainer>
     <BlockMath math="\text{Intersection} = (\text{Sports} + \text{Arts}) - \text{Total}" />
     <BlockMath math="Q = (350 + 450) - 700" />
     <BlockMath math="Q = 800 - 700 = 100 \text{ students}" />
   </MathContainer>

3. **Finding the Difference Between <InlineMath math="P" /> and <InlineMath math="Q" />**
   <MathContainer>
     <BlockMath math="P - Q = 350 - 100 = 250 \text{ students}" />
   </MathContainer>

Thus, the difference between <InlineMath math="P" /> and <InlineMath math="Q" /> is <InlineMath math="250" /> students.


---

## Exercise 12

### Question

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

export const metadata = {
  title: "Problem 12",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

Consider the graph below!

<QuestionGraph
  title="Parabola Graph"
  description={
    <>
      A parabola opening to the left intersecting the <InlineMath math="y" />-axis at <InlineMath math="4" /> and <InlineMath math="-2" />, and the <InlineMath math="x" />-axis at <InlineMath math="8" />.
    </>
  }
/>

The correct statements are:

1. One of the intersection points with the <InlineMath math="y" />-axis is <InlineMath math="(0, -2)" />.
2. <InlineMath math="f(y) = -(y-1)^2 + 9" />.
3. Axis of symmetry <InlineMath math="y = 1" />.
4. Intersection point with the <InlineMath math="x" />-axis is <InlineMath math="(8, 0)" />.


### Choices

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

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 12",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

We analyze each statement based on the graph:

1. **Statement <InlineMath math="(1)" />**: The graph intersects the <InlineMath math="y" />-axis at <InlineMath math="4" /> and <InlineMath math="-2" />. So the intersection points are <InlineMath math="(0, 4)" /> and <InlineMath math="(0, -2)" />. Statement <InlineMath math="(1)" /> is **CORRECT**.

2. **Statement <InlineMath math="(2)" />**:
   Since the graph intersects the <InlineMath math="y" />-axis at <InlineMath math="y = 4" /> and <InlineMath math="y = -2" />, the equation is of the form:
   <MathContainer>
     <BlockMath math="x = a(y - 4)(y + 2)" />
   </MathContainer>
   The graph passes through the point <InlineMath math="(8, 0)" /> (x-intercept). Substitute into the equation:
   <MathContainer>
     <BlockMath math="8 = a(0 - 4)(0 + 2)" />
     <BlockMath math="8 = a(-4)(2)" />
     <BlockMath math="8 = -8a \implies a = -1" />
   </MathContainer>
   So the function equation is:
   <MathContainer>
     <BlockMath math="x = -(y - 4)(y + 2)" />
     <BlockMath math="x = -(y^2 - 2y - 8)" />
     <BlockMath math="x = -y^2 + 2y + 8" />
   </MathContainer>
   Converting to vertex form:
   <MathContainer>
     <BlockMath math="x = -(y^2 - 2y + 1 - 1 - 8)" />
     <BlockMath math="x = -((y - 1)^2 - 9)" />
     <BlockMath math="x = -(y - 1)^2 + 9" />
   </MathContainer>
   Or in function notation <InlineMath math="f(y)" />:
   <MathContainer>
     <BlockMath math="f(y) = -(y - 1)^2 + 9" />
   </MathContainer>
   Statement <InlineMath math="(2)" /> is **CORRECT**.

3. **Statement <InlineMath math="(3)" />**:
   The axis of symmetry of a horizontal parabola is <InlineMath math="y_p" /> (ordinate of the vertex).
   From the form <InlineMath math="x = a(y - y_p)^2 + x_p" />, we get <InlineMath math="y_p = 1" />.
   Or from the midpoint of the roots:
   <MathContainer>
     <BlockMath math="y = \frac{4 + (-2)}{2} = \frac{2}{2} = 1" />
   </MathContainer>
   So the axis of symmetry is <InlineMath math="y = 1" />. Statement <InlineMath math="(3)" /> is **CORRECT**.

4. **Statement <InlineMath math="(4)" />**:
   The intersection with the <InlineMath math="x" />-axis occurs when <InlineMath math="y = 0" />.
   From the graph, it clearly intersects at <InlineMath math="x = 8" />.
   By calculation: <InlineMath math="x = -(0 - 1)^2 + 9 = -1 + 9 = 8" />.
   So the intersection point is <InlineMath math="(8, 0)" />. Statement <InlineMath math="(4)" /> is **CORRECT**.

Conclusion: All statements <InlineMath math="(1)" />, <InlineMath math="(2)" />, <InlineMath math="(3)" />, and <InlineMath math="(4)" /> are correct.


---

## Exercise 13

### Question

export const metadata = {
  title: "Problem 13",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

The ratio of the length between the drawing and the original is as follows:
drawing size : original size = <InlineMath math="3 : 600" />.
If there is a rectangular field with an area of <InlineMath math="384\text{ m}^2" /> and a length of <InlineMath math="32\text{ m}" />, then the width of the field in the drawing is ... cm.


### Choices

- [ ] $$12$$
- [x] $$6$$
- [ ] $$5$$
- [ ] $$4$$
- [ ] $$3$$

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 13",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

First, we find the actual width of the field.
Given the area of the field <InlineMath math="L = 384\text{ m}^2" /> and length <InlineMath math="p = 32\text{ m}" />.
Then the actual width (<InlineMath math="l" />) is:
<MathContainer>
  <BlockMath math="l = \frac{L}{p}" />
  <BlockMath math="l = \frac{384}{32} = 12\text{ m}" />
</MathContainer>

Next, we convert the actual width to cm to match the drawing unit.
<MathContainer>
  <BlockMath math="12\text{ m} = 12 \times 100\text{ cm} = 1200\text{ cm}" />
</MathContainer>

Finally, we calculate the width of the field in the drawing using the scale ratio <InlineMath math="3 : 600" />.
<MathContainer>
  <BlockMath math="\text{Drawing width} = \frac{3}{600} \times \text{Actual width}" />
  <BlockMath math="\text{Drawing width} = \frac{3}{600} \times 1200" />
  <BlockMath math="\text{Drawing width} = \frac{1}{200} \times 1200" />
  <BlockMath math="\text{Drawing width} = 6\text{ cm}" />
</MathContainer>

So, the width of the field in the drawing is <InlineMath math="6\text{ cm}" />.


---

## Exercise 14

### Question

export const metadata = {
  title: "Problem 14",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

Given: <InlineMath math="y = -2x^2 - 5x + 7" />, the equation of the tangent line at the point with abscissa <InlineMath math="2" /> is...


### Choices

- [ ] $$-13x - y - 15 = 0$$
- [ ] $$13x - y - 15 = 0$$
- [x] $$13x + y - 15 = 0$$
- [ ] $$-13x + y - 15 = 0$$
- [ ] $$13x + y - 37 = 0$$

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 14",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

Given the quadratic function <InlineMath math="y = -2x^2 - 5x + 7" />. We need to find the equation of the tangent line at the point with abscissa <InlineMath math="x_1 = 2" />.

First, we find the ordinate (<InlineMath math="y_1" />) of the tangent point by substituting <InlineMath math="x_1 = 2" /> into the curve equation:
<MathContainer>
  <BlockMath math="y_1 = -2(2)^2 - 5(2) + 7" />
  <BlockMath math="y_1 = -2(4) - 10 + 7" />
  <BlockMath math="y_1 = -8 - 10 + 7" />
  <BlockMath math="y_1 = -11" />
</MathContainer>
So, the tangent point is <InlineMath math="(2, -11)" />.

Second, we find the gradient of the tangent line (<InlineMath math="m" />) using the first derivative (<InlineMath math="y'" />):
<MathContainer>
  <BlockMath math="y' = -4x - 5" />
</MathContainer>
Substitute <InlineMath math="x = 2" /> into the first derivative to get the gradient:
<MathContainer>
  <BlockMath math="m = -4(2) - 5" />
  <BlockMath math="m = -8 - 5" />
  <BlockMath math="m = -13" />
</MathContainer>

Third, we determine the tangent line equation using the formula <InlineMath math="y - y_1 = m(x - x_1)" />:
<MathContainer>
  <BlockMath math="y - (-11) = -13(x - 2)" />
  <BlockMath math="y + 11 = -13x + 26" />
</MathContainer>
Move all terms to the left side to get the form <InlineMath math="Ax + By + C = 0" />:
<MathContainer>
  <BlockMath math="13x + y + 11 - 26 = 0" />
  <BlockMath math="13x + y - 15 = 0" />
</MathContainer>

So, the equation of the tangent line is <InlineMath math="13x + y - 15 = 0" />.


---

## Exercise 15

### Question

export const metadata = {
  title: "Problem 15",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

An equation is expressed in the form <InlineMath math="\begin{vmatrix} x^2 & x \\ 2 & 1 \end{vmatrix} = 8" />. The possible values of <InlineMath math="x" /> are...


### Choices

- [x] $$4 \text{ or } -2$$
- [ ] $$-4 \text{ or } 2$$
- [ ] $$-2 \text{ or } 3$$
- [ ] $$2 \text{ or } -3$$
- [ ] $$3 \text{ or } 8$$

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 15",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

Given the matrix determinant equation <InlineMath math="\begin{vmatrix} x^2 & x \\ 2 & 1 \end{vmatrix} = 8" />.

Recall that the determinant of a <InlineMath math="2 \times 2" /> matrix is the difference between the product of the main diagonal and the secondary diagonal: <InlineMath math="\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc" />.

So, we expand the equation:
<MathContainer>
  <BlockMath math="(x^2)(1) - (x)(2) = 8" />
  <BlockMath math="x^2 - 2x = 8" />
</MathContainer>

Move all terms to the left side to form a quadratic equation:
<MathContainer>
  <BlockMath math="x^2 - 2x - 8 = 0" />
</MathContainer>

Factor the quadratic equation. We look for two numbers that multiply to <InlineMath math="-8" /> and add up to <InlineMath math="-2" />. These numbers are <InlineMath math="-4" /> and <InlineMath math="2" />.
<MathContainer>
  <BlockMath math="(x - 4)(x + 2) = 0" />
</MathContainer>

The zeros are:
<MathContainer>
  <BlockMath math="x - 4 = 0 \Rightarrow x = 4" />
  <BlockMath math="x + 2 = 0 \Rightarrow x = -2" />
</MathContainer>

So, the possible values of <InlineMath math="x" /> are <InlineMath math="4" /> or <InlineMath math="-2" />.


---

## Exercise 16

### Question

export const metadata = {
  title: "Problem 16",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

The following numbers form a series <InlineMath math="72, x, 65, 60, 30, 27, y, 8" />. The values of <InlineMath math="x" /> and <InlineMath math="y" /> respectively are...


### Choices

- [ ] $$81 \text{ and } 10$$
- [x] $$65 \text{ and } 9$$
- [ ] $$66 \text{ and } 11$$
- [ ] $$68 \text{ and } 12$$
- [ ] $$68 \text{ and } 8$$

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 16",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

The given number series is <InlineMath math="72, x, 65, 60, 30, 27, y, 8" />. Let's analyze the pattern:

<MathContainer>
  <BlockMath math="72 \xrightarrow{-7} 65" />
  <BlockMath math="65 \xrightarrow{\div 1} 65" />
  <BlockMath math="65 \xrightarrow{-5} 60" />
  <BlockMath math="60 \xrightarrow{\div 2} 30" />
  <BlockMath math="30 \xrightarrow{-3} 27" />
  <BlockMath math="27 \xrightarrow{\div 3} 9" />
  <BlockMath math="9 \xrightarrow{-1} 8" />
</MathContainer>

The pattern involves alternating operations of subtraction and division.
1. Subtraction by decreasing odd numbers: <InlineMath math="-7, -5, -3, -1" />.
2. Division by increasing integers: <InlineMath math="\div 1, \div 2, \div 3" />.

Based on this pattern:
- The value of <InlineMath math="x" /> is obtained from <InlineMath math="72 - 7 = 65" />.
- The value of <InlineMath math="y" /> is obtained from <InlineMath math="27 \div 3 = 9" />.

So, the values of <InlineMath math="x" /> and <InlineMath math="y" /> respectively are <InlineMath math="65" /> and <InlineMath math="9" />.


---

## Exercise 17

### Question

export const metadata = {
  title: "Question 17",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

Andi's money is four times Budi's money. If Andi donates Rp4,000.00 and Budi spends Rp1,000.00, then the remainder of Andi's money is equal to the remainder of Budi's money plus Cici's money. If Cici's money <InlineMath math="= C" />, then Andi's initial money is


### Choices

- [ ] $$C + \text{Rp}4,000.00$$
- [ ] $$2C + \text{Rp}4,000.00$$
- [x] $$\frac{4}{3}C + \text{Rp}4,000.00$$
- [ ] $$\frac{3}{2}C + \text{Rp}1,000.00$$
- [ ] $$2C + \text{Rp}2,000.00$$

### Answer & Explanation

export const metadata = {
    title: "Answer to Question 17",
    authors: [{ name: "Nabil Akbarazzima Fatih" }],
    date: "12/23/2025",
};

Let Andi's initial money be <InlineMath math="A" />, Budi's money be <InlineMath math="B" />, and Cici's money be <InlineMath math="C" />.

Given that Andi's money is four times Budi's money, we can write the equation:

<MathContainer>
  <BlockMath math="A = 4B \implies B = \frac{1}{4}A" />
</MathContainer>

If Andi donates Rp4,000.00, his remaining money is <InlineMath math="A - 4,000" />.

If Budi spends Rp1,000.00, his remaining money is <InlineMath math="B - 1,000" />.

It is known that the remainder of Andi's money is equal to the remainder of Budi's money plus Cici's money. The equation is:

<MathContainer>
  <BlockMath math="A - 4,000 = (B - 1,000) + C" />
</MathContainer>

Next, substitute <InlineMath math="B = \frac{1}{4}A" /> into the equation to find the value of <InlineMath math="A" />:

<MathContainer>
  <BlockMath math="A - 4,000 = \left(\frac{1}{4}A - 1,000\right) + C" />
  <BlockMath math="A - \frac{1}{4}A = 4,000 - 1,000 + C" />
  <BlockMath math="\frac{3}{4}A = 3,000 + C" />
</MathContainer>

Multiply both sides by <InlineMath math="\frac{4}{3}" />:

<MathContainer>
  <BlockMath math="A = \frac{4}{3}(3,000 + C)" />
  <BlockMath math="A = 4,000 + \frac{4}{3}C" />
  <BlockMath math="A = \frac{4}{3}C + 4,000" />
</MathContainer>

So, Andi's initial money is <InlineMath math="\frac{4}{3}C + \text{Rp}4,000.00" />.


---

## Exercise 18

### Question

export const metadata = {
  title: "Question 18",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

An order of <InlineMath math="y" /> dozen clothes can be completed by <InlineMath math="x" /> convection workers in <InlineMath math="20" /> days. If the order must be completed in <InlineMath math="5" /> days, which of the following is the correct relationship between quantity <InlineMath math="P" /> and <InlineMath math="Q" /> based on the information provided?

| <InlineMath math="P" /> | <InlineMath math="Q" /> |
| :---: | :---: |
| Number of workers | <InlineMath math="4x" /> |


### Choices

- [ ] $$P > Q$$
- [ ] $$P < Q$$
- [x] $$P = Q$$
- [ ] $$P + Q = 12x$$
- [ ] The information provided is not sufficient to decide on one of the four choices above

### Answer & Explanation

export const metadata = {
  title: "Answer to Question 18",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

This problem falls into the category of **Inverse Proportion**. If the number of workers increases, the time required to complete the work decreases, and vice versa.

Given:

- <InlineMath math="x" /> workers can complete the order in <InlineMath math="20" /> days.
- The order must be completed in <InlineMath math="5" /> days with <InlineMath math="P" /> workers.

The relationship between the number of workers and the work time can be written as:

<MathContainer>
  <BlockMath math="\text{Number of workers}_1 \times \text{Time}_1 = \text{Number of workers}_2 \times \text{Time}_2" />
</MathContainer>

Substitute the known values:

<MathContainer>
  <BlockMath math="x \cdot 20 = P \cdot 5" />
  <BlockMath math="20x = 5P" />
</MathContainer>

Then the value of <InlineMath math="P" /> is:

<MathContainer>
  <BlockMath math="P = \frac{20x}{5}" />
  <BlockMath math="P = 4x" />
</MathContainer>

It is known that <InlineMath math="Q = 4x" />.

Therefore:

<MathContainer>
  <BlockMath math="P = 4x \quad \text{and} \quad Q = 4x" />
  <BlockMath math="P = Q" />
</MathContainer>

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


---

## Exercise 19

### Question

export const metadata = {
  title: "Question 19",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

Given <InlineMath math="f(x) = 2x - 3" />, <InlineMath math="g(x) = x^2 + 4x - 1" />, and <InlineMath math="(g \circ f)(x) = ax^2 + bx + c" />, then the value of <InlineMath math="a + b + c =" />


### Choices

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

### Answer & Explanation

export const metadata = {
  title: "Answer to Question 19",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

We are given the functions <InlineMath math="f(x) = 2x - 3" /> and <InlineMath math="g(x) = x^2 + 4x - 1" />. We need to find the value of <InlineMath math="a + b + c" /> from the composite function <InlineMath math="(g \circ f)(x) = ax^2 + bx + c" />.

First, let's determine the composite function <InlineMath math="(g \circ f)(x)" />:

<MathContainer>
  <BlockMath math="(g \circ f)(x) = g(f(x))" />
  <BlockMath math="= g(2x - 3)" />
  <BlockMath math="= (2x - 3)^2 + 4(2x - 3) - 1" />
</MathContainer>

Expand the squared term and the product:

<MathContainer>
  <BlockMath math="= (4x^2 - 12x + 9) + (8x - 12) - 1" />
</MathContainer>

Combine like terms:

<MathContainer>
  <BlockMath math="= 4x^2 - 12x + 8x + 9 - 12 - 1" />
  <BlockMath math="= 4x^2 - 4x - 4" />
</MathContainer>

From the form <InlineMath math="ax^2 + bx + c" />, we obtain:

<MathContainer>
  <BlockMath math="a = 4, \quad b = -4, \quad c = -4" />
</MathContainer>

So the value of <InlineMath math="a + b + c" /> is:

<MathContainer>
  <BlockMath math="a + b + c = 4 + (-4) + (-4)" />
  <BlockMath math="= 4 - 4 - 4" />
  <BlockMath math="= -4" />
</MathContainer>

Thus, the value of <InlineMath math="a + b + c = -4" />.


---

## Exercise 20

### Question

export const metadata = {
  title: "Question 20",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

The result of:

<MathContainer>
  <BlockMath math="\frac{3x}{x^2 - 4} - \frac{2}{x - 2}" />
</MathContainer>

is...


### Choices

- [ ] $$\frac{2x + 12}{x^2 - 4}$$
- [ ] $$\frac{x - 12}{x^2 - 4}$$
- [ ] $$\frac{-x + 12}{x^2 - 4}$$
- [ ] $$\frac{-2x}{x^2 - 4}$$
- [x] $$\frac{x - 4}{x^2 - 4}$$

### Answer & Explanation

export const metadata = {
  title: "Answer to Question 20",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

We are asked to simplify the following algebraic fraction:

<MathContainer>
  <BlockMath math="\frac{3x}{x^2 - 4} - \frac{2}{x - 2}" />
</MathContainer>

Notice the denominator of the first fraction, <InlineMath math="x^2 - 4" />. This can be factored into <InlineMath math="(x - 2)(x + 2)" />.

Thus, we make the denominators of both fractions the same, which is <InlineMath math="x^2 - 4" /> or <InlineMath math="(x - 2)(x + 2)" />:

<MathContainer>
  <BlockMath math="= \frac{3x}{x^2 - 4} - \frac{2(x + 2)}{(x - 2)(x + 2)}" />
  <BlockMath math="= \frac{3x}{x^2 - 4} - \frac{2(x + 2)}{x^2 - 4}" />
</MathContainer>

Now we can combine the numerators:

<MathContainer>
  <BlockMath math="= \frac{3x - 2(x + 2)}{x^2 - 4}" />
  <BlockMath math="= \frac{3x - 2x - 4}{x^2 - 4}" />
  <BlockMath math="= \frac{x - 4}{x^2 - 4}" />
</MathContainer>

So, the simplified result is <InlineMath math="\frac{x - 4}{x^2 - 4}" />.


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