# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-6

Exercises: Try Out - Set 6: 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 two lines <InlineMath math="y = 2x + 3" /> and <InlineMath math="y = ax + b" /> have an intersection point. Determine the intersection point!

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

1. <InlineMath math="a \neq 2" />
2. <InlineMath math="b = 3" />


### Choices

- [ ] Statement $$(1)$$ ALONE is sufficient, but statement $$(2)$$ alone is not sufficient.
- [ ] Statement $$(2)$$ ALONE is sufficient, but statement $$(1)$$ alone is not sufficient.
- [x] 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: "Problem 1 Solution",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

We are asked to determine the intersection point of two lines:

1. <InlineMath math="y = 2x + 3" />
2. <InlineMath math="y = ax + b" />

To determine a unique intersection point <InlineMath math="(x, y)" />, we must be able to solve this system of equations. Let's equate the two equations:

<MathContainer>
  <BlockMath math="2x + 3 = ax + b" />
  <BlockMath math="2x - ax = b - 3" />
  <BlockMath math="(2 - a)x = b - 3" />
</MathContainer>

From the equation above, the value of <InlineMath math="x" /> can be determined if and only if the coefficient of <InlineMath math="x" /> is not zero, i.e., <InlineMath math="2 - a \neq 0" /> or <InlineMath math="a \neq 2" />. If <InlineMath math="a \neq 2" />, then:

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

After finding <InlineMath math="x" />, we can find <InlineMath math="y" /> by substituting it into one of the equations. So, to determine the intersection point specifically, we need to know:

1. That <InlineMath math="a \neq 2" /> (so the lines intersect at one point).
2. The value of <InlineMath math="b" /> (to calculate the coordinate values).

#### Analysis of Statement 1

Statement <InlineMath math="(1)" /> provides the information <InlineMath math="a \neq 2" />. This guarantees that the two lines have different gradients (<InlineMath math="m_1 = 2" /> and <InlineMath math="m_2 = a \neq 2" />), so they definitely intersect at one point. However, we do not know the value of <InlineMath math="b" />. Without the value of <InlineMath math="b" />, we cannot determine the coordinates of the intersection point (values of <InlineMath math="x" /> and <InlineMath math="y" /> still depend on <InlineMath math="b" />). Therefore, statement <InlineMath math="(1)" /> alone is **not sufficient**.

#### Analysis of Statement 2

Statement <InlineMath math="(2)" /> provides the information <InlineMath math="b = 3" />. The second equation becomes <InlineMath math="y = ax + 3" />. We do not know the value of <InlineMath math="a" />.

- If <InlineMath math="a = 2" />, then the line becomes <InlineMath math="y = 2x + 3" />, which coincides with the first line (infinitely many intersection points).
- If <InlineMath math="a \neq 2" />, the lines intersect at one point.

Since <InlineMath math="a" /> is unknown, we cannot ensure there is a single intersection point, nor can we determine its coordinates. Therefore, statement <InlineMath math="(2)" /> alone is **not sufficient**.

#### Analysis of Both Statements Together

If we combine both statements:

1. <InlineMath math="a \neq 2" />
2. <InlineMath math="b = 3" />

We substitute into the equation to find <InlineMath math="x" />:

<MathContainer>
  <BlockMath math="x = \frac{3 - 3}{2 - a} = \frac{0}{2 - a} = 0" />
</MathContainer>

(Division by <InlineMath math="2 - a" /> is valid because <InlineMath math="a \neq 2" />). Then we find <InlineMath math="y" />:

<MathContainer>
  <BlockMath math="y = 2(0) + 3 = 3" />
</MathContainer>

We get a unique intersection point <InlineMath math="(0, 3)" />. Since we can determine the intersection point, both statements together are **sufficient**.

Therefore, **BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient**.


---

## Exercise 2

### Question

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

A piece of wood weighs <InlineMath math="90" /> grams. After being dried in the sun, it weighs <InlineMath math="60" /> grams. What is the percentage of water content in the wet wood?


### Choices

- [ ] $$80\%$$
- [ ] $$40\%$$
- [x] $$33\frac{1}{3}\%$$
- [ ] $$25\%$$
- [ ] $$20\%$$

### Answer & Explanation

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

Given:

- Initial weight (wet wood) = <InlineMath math="90" /> grams.
- Final weight (dry wood) = <InlineMath math="60" /> grams.

The water content in the wood is the difference between the initial weight and the final weight:

<MathContainer>
  <BlockMath math="\text{Water Weight} = 90 - 60 = 30 \text{ grams}" />
</MathContainer>

The percentage of water content is calculated relative to the initial weight (wet wood):

<MathContainer>
  <BlockMath math="\text{Percentage} = \frac{\text{Water Weight}}{\text{Initial Weight}} \times 100\%" />
  <BlockMath math="= \frac{30}{90} \times 100\%" />
  <BlockMath math="= \frac{1}{3} \times 100\%" />
  <BlockMath math="= 33\frac{1}{3}\%" />
</MathContainer>

Therefore, the water content in the wet wood is <InlineMath math="33\frac{1}{3}\%" />.


---

## Exercise 3

### Question

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

A family has <InlineMath math="3" /> children. It is known that at least one of the children is a girl. The probability that the family has <InlineMath math="2" /> girls and <InlineMath math="1" /> boy is


### Choices

- [ ] $$\frac{1}{2}$$
- [ ] $$\frac{1}{7}$$
- [ ] $$\frac{1}{4}$$
- [x] $$\frac{3}{7}$$
- [ ] $$\frac{8}{9}$$

### Answer & Explanation

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

Let <InlineMath math="B" /> be a boy and <InlineMath math="G" /> be a girl.

The sample space for a family with <InlineMath math="3" /> children (without conditions) has <InlineMath math="2^3 = 8" /> possibilities:

<MathContainer>
  <BlockMath math="S_{total} = \{BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG\}" />
</MathContainer>

It is given that "at least one child is a girl". This means the event "all children are boys" (<InlineMath math="BBB" />) is impossible.

The new sample space (<InlineMath math="S" />) consists of all possibilities except <InlineMath math="BBB" />:

<MathContainer>
  <BlockMath math="S = \{BBG, BGB, BGG, GBB, GBG, GGB, GGG\}" />
</MathContainer>

The number of elements in the sample space is <InlineMath math="n(S) = 7" />.

The expected event (<InlineMath math="A" />) is having <InlineMath math="2" /> girls and <InlineMath math="1" /> boy.

From the sample space <InlineMath math="S" />, the outcomes that satisfy this condition are:

<MathContainer>
  <BlockMath math="A = \{BGG, GBG, GGB\}" />
</MathContainer>

The number of expected outcomes is <InlineMath math="n(A) = 3" />.

The probability of the event is:

<MathContainer>
  <BlockMath math="P(A) = \frac{n(A)}{n(S)} = \frac{3}{7}" />
</MathContainer>

Therefore, the probability that the family has <InlineMath math="2" /> girls and <InlineMath math="1" /> boy is <InlineMath math="\frac{3}{7}" />.


---

## Exercise 4

### Question

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

From the data <InlineMath math="4" />, <InlineMath math="5" />, <InlineMath math="5" />, <InlineMath math="2" />, <InlineMath math="3" />, <InlineMath math="1" />, <InlineMath math="6" />, which statements are true?

1. Median <InlineMath math="= 4" />.
2. Range <InlineMath math="= 6" />.
3. Range <InlineMath math="-" /> median <InlineMath math="= 1" />.
4. Mean <InlineMath math="= 4.5" />.


### Choices

- [ ] $$(1)$$, $$(2)$$, and $$(3)$$ ONLY are correct.
- [x] $$(1)$$ and $$(3)$$ ONLY are correct.
- [ ] $$(2)$$ and $$(4)$$ ONLY are correct.
- [ ] ONLY $$(4)$$ is correct.
- [ ] ALL choices are correct.

### Answer & Explanation

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

Sort the data from smallest to largest:

<MathContainer>
  <BlockMath math="1, 2, 3, 4, 5, 5, 6" />
</MathContainer>

Number of data points <InlineMath math="n = 7" />.

#### Statement 1 Analysis

The median is the <InlineMath math="\frac{n+1}{2} = \frac{8}{2} = 4" />-th value.
The <InlineMath math="4" />-th value is <InlineMath math="4" />.

<MathContainer>
  <BlockMath math="Me = 4" />
</MathContainer>

Statement <InlineMath math="1" /> is **CORRECT**.

#### Statement 2 Analysis

The range is the difference between the maximum and minimum values.

<MathContainer>
  <BlockMath math="Range = x_{max} - x_{min} = 6 - 1 = 5" />
</MathContainer>

Statement <InlineMath math="2" /> is **INCORRECT** (should be <InlineMath math="5" />, not <InlineMath math="6" />).

#### Statement 3 Analysis

Difference between range and median:

<MathContainer>
  <BlockMath math="Range - Me = 5 - 4 = 1" />
</MathContainer>

Statement <InlineMath math="3" /> is **CORRECT**.

#### Statement 4 Analysis

The mean (<InlineMath math="\bar{x}" />) is the sum of all data divided by the number of data points.

<MathContainer>
  <BlockMath math="\bar{x} = \frac{1 + 2 + 3 + 4 + 5 + 5 + 6}{7} = \frac{26}{7} = 3\frac{5}{7}" />
</MathContainer>

The value <InlineMath math="3\frac{5}{7}" /> is not equal to <InlineMath math="4.5" />.
Statement <InlineMath math="4" /> is **INCORRECT**.

#### Conclusion

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


---

## Exercise 5

### Question

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

If <InlineMath math="x" /> and <InlineMath math="y" /> are integers satisfying <InlineMath math="x^2 + y^2 = 169" />, then the possible values for <InlineMath math="x + y" /> are

1. <InlineMath math="\pm 7" />
2. <InlineMath math="\pm 10" />
3. <InlineMath math="\pm 13" />
4. <InlineMath math="\pm 14" />


### Choices

- [ ] $$(1)$$, $$(2)$$, and $$(3)$$ ONLY are correct.
- [x] $$(1)$$ and $$(3)$$ ONLY are correct.
- [ ] $$(2)$$ and $$(4)$$ ONLY are correct.
- [ ] ONLY $$(4)$$ is correct.
- [ ] ALL choices are correct.

### Answer & Explanation

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

Given the equation <InlineMath math="x^2 + y^2 = 169" /> where <InlineMath math="x" /> and <InlineMath math="y" /> are integers. The value <InlineMath math="169" /> is the square of <InlineMath math="13" /> (<InlineMath math="13^2 = 169" />). The integer pairs <InlineMath math="(x, y)" /> satisfying this circle equation can be found using Pythagorean triples or properties of perfect squares.

The possible pairs are:

1. If one variable is <InlineMath math="0" />, then the other is <InlineMath math="\pm 13" />. Pairs: <InlineMath math="(13, 0), (-13, 0), (0, 13), (0, -13)" />. Possible values for <InlineMath math="x + y" />:
   <MathContainer>
     <BlockMath math="\pm 13 + 0 = \pm 13" />
   </MathContainer>

2. If both variables are non-zero, we look for combinations of squares summing to <InlineMath math="169" />. The Pythagorean triple involving <InlineMath math="13" /> as the hypotenuse is <InlineMath math="(5, 12, 13)" /> because <InlineMath math="5^2 + 12^2 = 25 + 144 = 169" />. Possible pairs <InlineMath math="(x, y)" /> (including negative values): <InlineMath math="(\pm 5, \pm 12)" /> and <InlineMath math="(\pm 12, \pm 5)" />. Possible values for <InlineMath math="x + y" /> from these combinations:
   <MathContainer>
     <BlockMath math="\begin{aligned} 5 + 12 &= 17 \\ 5 + (-12) &= -7 \\ -5 + 12 &= 7 \\ -5 + (-12) &= -17 \end{aligned}" />
   </MathContainer>
   Thus, <InlineMath math="x + y" /> can be <InlineMath math="\pm 17" /> or <InlineMath math="\pm 7" />.

The set of all possible values for <InlineMath math="x + y" /> is <InlineMath math="\{\pm 13, \pm 17, \pm 7\}" />.

#### Statement 1 Analysis

The value <InlineMath math="\pm 7" /> is in the solution set. Statement <InlineMath math="1" /> is **CORRECT**.

#### Statement 2 Analysis

The value <InlineMath math="\pm 10" /> is **NOT** in the solution set. Statement <InlineMath math="2" /> is **INCORRECT**.

#### Statement 3 Analysis

The value <InlineMath math="\pm 13" /> is in the solution set. Statement <InlineMath math="3" /> is **CORRECT**.

#### Statement 4 Analysis

The value <InlineMath math="\pm 14" /> is **NOT** in the solution set. Statement <InlineMath math="4" /> is **INCORRECT**.

#### Conclusion

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


---

## Exercise 6

### Question

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

Given functions <InlineMath math="f(x) = 2x - 3" /> and <InlineMath math="(g \circ f)(x) = 4x^2 - 12x + 10" />. The value of <InlineMath math="g(1)" /> is ....


### Choices

- [ ] $$1$$
- [x] $$2$$
- [ ] $$3$$
- [ ] $$4$$
- [ ] $$5$$

### Answer & Explanation

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

We are given the function <InlineMath math="f(x) = 2x - 3" /> and the composite function <InlineMath math="(g \circ f)(x) = 4x^2 - 12x + 10" />. We are asked to find the value of <InlineMath math="g(1)" />.

The definition of a composite function is <InlineMath math="(g \circ f)(x) = g(f(x))" />. Let <InlineMath math="u = f(x) = 2x - 3" />. We rearrange this equation to express <InlineMath math="x" /> in terms of <InlineMath math="u" />:

<MathContainer>
  <BlockMath math="2x = u + 3" />
  <BlockMath math="x = \frac{u + 3}{2}" />
</MathContainer>

Substitute <InlineMath math="x" /> into the equation <InlineMath math="(g \circ f)(x)" />:

<MathContainer>
  <BlockMath math="g(u) = 4\left(\frac{u + 3}{2}\right)^2 - 12\left(\frac{u + 3}{2}\right) + 10" />
  <BlockMath math="g(u) = 4\left(\frac{u^2 + 6u + 9}{4}\right) - 6(u + 3) + 10" />
  <BlockMath math="g(u) = (u^2 + 6u + 9) - (6u + 18) + 10" />
  <BlockMath math="g(u) = u^2 + 6u - 6u + 9 - 18 + 10" />
  <BlockMath math="g(u) = u^2 + 1" />
</MathContainer>

Thus, we have <InlineMath math="g(x) = x^2 + 1" />.

Then the value of <InlineMath math="g(1)" /> is:

<MathContainer>
  <BlockMath math="g(1) = 1^2 + 1 = 1 + 1 = 2" />
</MathContainer>

So, the value of <InlineMath math="g(1)" /> is <InlineMath math="2" />.


---

## Exercise 7

### Question

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

The value of <InlineMath math="x" /> that satisfies <InlineMath math="\begin{vmatrix} 3x & x \\ 2 & x \end{vmatrix} = \begin{vmatrix} 3 & 1 \\ 1 & 2 \end{vmatrix}" /> is ....


### Choices

- [ ] $$5 \text{ and } -3$$
- [ ] $$-5 \text{ and } 3$$
- [x] $$\frac{5}{3} \text{ and } -1$$
- [ ] $$-\frac{5}{3} \text{ and } 1$$
- [ ] $$5 \text{ and } -1$$

### Answer & Explanation

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

We are given the following determinant equation:

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

The first step is to calculate the determinant of the matrices on the left and right sides. Recall that the determinant of a matrix <InlineMath math="\begin{pmatrix} a & b \\ c & d \end{pmatrix}" /> is <InlineMath math="ad - bc" />.

Determinant of the left side:

<MathContainer>
  <BlockMath math="\begin{vmatrix} 3x & x \\ 2 & x \end{vmatrix} = (3x)(x) - (x)(2) = 3x^2 - 2x" />
</MathContainer>

Determinant of the right side:

<MathContainer>
  <BlockMath math="\begin{vmatrix} 3 & 1 \\ 1 & 2 \end{vmatrix} = (3)(2) - (1)(1) = 6 - 1 = 5" />
</MathContainer>

Now we equate both results:

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

Move <InlineMath math="5" /> to the left side to form a quadratic equation:

<MathContainer>
  <BlockMath math="3x^2 - 2x - 5 = 0" />
</MathContainer>

We factor this quadratic equation. We look for two numbers that multiply to <InlineMath math="3 \times (-5) = -15" /> and sum to <InlineMath math="-2" />. These numbers are <InlineMath math="-5" /> and <InlineMath math="3" />.

<MathContainer>
  <BlockMath math="3x^2 - 5x + 3x - 5 = 0" />
  <BlockMath math="x(3x - 5) + 1(3x - 5) = 0" />
  <BlockMath math="(x + 1)(3x - 5) = 0" />
</MathContainer>

Thus, the solutions are:

<MathContainer>
  <BlockMath math="x + 1 = 0 \Rightarrow x = -1" />
  <BlockMath math="3x - 5 = 0 \Rightarrow x = \frac{5}{3}" />
</MathContainer>

So, the values of <InlineMath math="x" /> that satisfy the equation are <InlineMath math="\frac{5}{3}" /> and <InlineMath math="-1" />.


---

## Exercise 8

### Question

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

If <InlineMath math="W(P@Q\#R) = 2P^2 - Q + 3\sqrt{R}" />, then the value of <InlineMath math="W(5@8\#4) =" /> ....


### Choices

- [ ] $$46$$
- [x] $$48$$
- [ ] $$54$$
- [ ] $$56$$
- [ ] $$58$$

### Answer & Explanation

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

Given the function operation definition <InlineMath math="W(P@Q\#R) = 2P^2 - Q + 3\sqrt{R}" />.

We are asked to find the value of <InlineMath math="W(5@8\#4)" />.

Based on the position of variables in the function definition, we can identify the following values:
- <InlineMath math="P = 5" />
- <InlineMath math="Q = 8" />
- <InlineMath math="R = 4" />

Substitute these values into the function equation:

<MathContainer>
  <BlockMath math="W(5@8\#4) = 2(5)^2 - 8 + 3\sqrt{4}" />
</MathContainer>

Perform the calculation step by step:
1. Calculate the square: <InlineMath math="5^2 = 25" />
2. Calculate the square root: <InlineMath math="\sqrt{4} = 2" />

Then the equation becomes:

<MathContainer>
  <BlockMath math="W(5@8\#4) = 2(25) - 8 + 3(2)" />
  <BlockMath math="W(5@8\#4) = 50 - 8 + 6" />
  <BlockMath math="W(5@8\#4) = 42 + 6" />
  <BlockMath math="W(5@8\#4) = 48" />
</MathContainer>

So, the value of <InlineMath math="W(5@8\#4)" /> is <InlineMath math="48" />.


---

## Exercise 9

### Question

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

Consider the following plane figures:
1. Isosceles right triangle.
2. Right trapezoid (not a rectangle).
3. Rectangle (not a square).
4. Rhombus (not a square).

How many figures have the number of rotational symmetries equal to the number of folding symmetries (lines of symmetry)?


### Choices

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

### Answer & Explanation

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

Let's analyze the number of lines of symmetry (folding symmetry) and rotational symmetries for each given plane figure:

1. **Isosceles right triangle**
   - Lines of symmetry: <InlineMath math="1" /> (altitude from the right angle to the hypotenuse).
   - Rotational symmetries: <InlineMath math="1" /> (rotation of <InlineMath math="360^\circ" />).
   - **Conclusion**: The numbers are equal.

2. **Right trapezoid**
   - Lines of symmetry: <InlineMath math="0" />.
   - Rotational symmetries: <InlineMath math="1" /> (rotation of <InlineMath math="360^\circ" />).
   - **Conclusion**: The numbers are not equal.

3. **Rectangle** (not a square)
   - Lines of symmetry: <InlineMath math="2" /> (vertical and horizontal axes).
   - Rotational symmetries: <InlineMath math="2" /> (rotation of <InlineMath math="180^\circ" /> and <InlineMath math="360^\circ" />).
   - **Conclusion**: The numbers are equal.

4. **Rhombus** (not a square)
   - Lines of symmetry: <InlineMath math="2" /> (both diagonals).
   - Rotational symmetries: <InlineMath math="2" /> (rotation of <InlineMath math="180^\circ" /> and <InlineMath math="360^\circ" />).
   - **Conclusion**: The numbers are equal.

The figures that have the same number of rotational symmetries and lines of symmetry are:
1. Isosceles right triangle.
2. Rectangle.
3. Rhombus.

So, there are <InlineMath math="3" /> figures that satisfy the condition.


---

## Exercise 10

### Question

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

Given <InlineMath math="a_1 = 8" />, <InlineMath math="a_2 = -5" />, and <InlineMath math="a_{n+1} = a_{n+2} - 4a_n" />. The value of <InlineMath math="a_3 + a_4" /> is ....


### Choices

- [ ] $$10$$
- [ ] $$20$$
- [ ] $$24$$
- [x] $$34$$
- [ ] $$44$$

### Answer & Explanation

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

Given the recursive equation:

<MathContainer>
  <BlockMath math="a_{n+1} = a_{n+2} - 4a_n" />
</MathContainer>

We can rearrange the equation to find <InlineMath math="a_{n+2}" />:

<MathContainer>
  <BlockMath math="a_{n+2} = a_{n+1} + 4a_n" />
</MathContainer>

**Step 1**: Find the value of <InlineMath math="a_3" /> by substituting <InlineMath math="n = 1" />. Given <InlineMath math="a_1 = 8" /> and <InlineMath math="a_2 = -5" />.

<MathContainer>
  <BlockMath math="a_{1+2} = a_{1+1} + 4a_1" />
  <BlockMath math="a_3 = a_2 + 4a_1" />
  <BlockMath math="a_3 = -5 + 4(8)" />
  <BlockMath math="a_3 = -5 + 32" />
  <BlockMath math="a_3 = 27" />
</MathContainer>

**Step 2**: Find the value of <InlineMath math="a_4" /> by substituting <InlineMath math="n = 2" />.

<MathContainer>
  <BlockMath math="a_{2+2} = a_{2+1} + 4a_2" />
  <BlockMath math="a_4 = a_3 + 4a_2" />
  <BlockMath math="a_4 = 27 + 4(-5)" />
  <BlockMath math="a_4 = 27 - 20" />
  <BlockMath math="a_4 = 7" />
</MathContainer>

**Step 3**: Calculate the sum <InlineMath math="a_3 + a_4" />.

<MathContainer>
  <BlockMath math="a_3 + a_4 = 27 + 7" />
  <BlockMath math="a_3 + a_4 = 34" />
</MathContainer>

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


---

## Exercise 11

### Question

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

The ratio of a miniature of an object to its actual size is as follows. Miniature size : actual size = <InlineMath math="5 : 2500" /> If there is a rectangular object with an area of <InlineMath math="540\text{ m}^2" /> and a length of <InlineMath math="30\text{ m}" />, then the width of the miniature of the object is ... cm.


### Choices

- [ ] $$0.9$$
- [ ] $$1.9$$
- [ ] $$2.3$$
- [ ] $$2.6$$
- [x] $$3.6$$

### Answer & Explanation

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

**Step 1**: Calculate the actual width of the object. Given the area of the rectangle is <InlineMath math="540\text{ m}^2" /> and the length is <InlineMath math="30\text{ m}" />.

<MathContainer>
  <BlockMath math="\text{Width} = \frac{\text{Area}}{\text{Length}}" />
  <BlockMath math="\text{Width} = \frac{540}{30} = 18\text{ m}" />
</MathContainer>

Convert to cm:

<MathContainer>
  <BlockMath math="18\text{ m} = 18 \times 100\text{ cm} = 1800\text{ cm}" />
</MathContainer>

**Step 2**: Calculate the width of the miniature using the given ratio. The ratio of miniature size to actual size is <InlineMath math="5 : 2500" /> or <InlineMath math="\frac{5}{2500}" />.

<MathContainer>
  <BlockMath math="\text{Miniature width} = \frac{5}{2500} \times \text{Actual width}" />
  <BlockMath math="\text{Miniature width} = \frac{5}{2500} \times 1800" />
  <BlockMath math="\text{Miniature width} = \frac{1}{500} \times 1800" />
  <BlockMath math="\text{Miniature width} = \frac{1800}{500}" />
  <BlockMath math="\text{Miniature width} = 3.6\text{ cm}" />
</MathContainer>

So, the width of the miniature object is <InlineMath math="3.6\text{ cm}" />.


---

## Exercise 12

### Question

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

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

The value of <InlineMath math="a" /> in the figure below is ...

<Graph title="Triangle" description="Geometric illustration with labeled angles." />


### Choices

- [x] $$20^\circ$$
- [ ] $$30^\circ$$
- [ ] $$40^\circ$$
- [ ] $$50^\circ$$
- [ ] $$80^\circ$$

### Answer & Explanation

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

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

<Graph title="Triangle" description="Geometric illustration with labeled angles." />

Consider the small triangle at the bottom. The sum of angles in a triangle is <InlineMath math="180^\circ" />. Given the top vertex angle is <InlineMath math="100^\circ" />. So, the sum of the two base angles is:

<MathContainer>
  <BlockMath math="x + y = 180^\circ - 100^\circ = 80^\circ" />
</MathContainer>

Next, consider the large triangle. Its base angles are <InlineMath math="(x+y)" /> and <InlineMath math="(y+x)" />. Since <InlineMath math="x+y = 80^\circ" />, both base angles of the large triangle are <InlineMath math="80^\circ" />. The sum of angles in the large triangle is <InlineMath math="180^\circ" />:

<MathContainer>
  <BlockMath math="a + (x+y) + (y+x) = 180^\circ" />
  <BlockMath math="a + 80^\circ + 80^\circ = 180^\circ" />
  <BlockMath math="a + 160^\circ = 180^\circ" />
  <BlockMath math="a = 20^\circ" />
</MathContainer>

Thus, the value of <InlineMath math="a" /> is <InlineMath math="20^\circ" />.


---

## Exercise 13

### Question

export const metadata = {
  title: "Problem 13",
  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^4}{1-x^2}" /> | <InlineMath math="2+x" /> |


### Choices

- [ ] $$P > Q$$
- [x] $$P < Q$$
- [ ] $$P = Q$$
- [ ] $$P + Q = 3$$
- [ ] Cannot be determined.

### Answer & Explanation

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

We will analyze quantities <InlineMath math="P" /> and <InlineMath math="Q" /> separately.

#### Analyzing Quantity P

The expression for <InlineMath math="P" /> can be simplified as follows:

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

Given <InlineMath math="0 < x < 1" />. Thus:

<MathContainer>
  <BlockMath math="0 < x^2 < 1" />
  <BlockMath math="1 < 1+x^2 < 2" />
  <BlockMath math="1 < P < 2" />
</MathContainer>

#### Analyzing Quantity Q

Given <InlineMath math="Q = 2+x" />.
Since <InlineMath math="0 < x < 1" />, then:

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

#### Comparing P and Q

From the analysis above, we have:
*   <InlineMath math="1 < P < 2" />
*   <InlineMath math="2 < Q < 3" />

It is clear that the maximum value of <InlineMath math="P" /> is less than the minimum value of <InlineMath math="Q" />.

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


---

## Exercise 14

### Question

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

If <InlineMath math="g(x) = 3x + 2" /> and <InlineMath math="2g^2(x) - g(x^2) - 5g(x) = 7" /> are satisfied by <InlineMath math="x_1" /> and <InlineMath math="x_2" />, then the value of <InlineMath math="x_1 + x_2" /> is ...


### Choices

- [x] $$-\frac{3}{5}$$
- [ ] $$\frac{1}{5}$$
- [ ] $$\frac{4}{3}$$
- [ ] $$\frac{2}{3}$$
- [ ] $$\frac{3}{5}$$

### Answer & Explanation

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

Given the function <InlineMath math="g(x) = 3x + 2" />. We will substitute this function into the given equation.

<MathContainer>
  <BlockMath math="2g^2(x) - g(x^2) - 5g(x) - 7 = 0" />
</MathContainer>

First, determine the forms of <InlineMath math="g^2(x)" /> and <InlineMath math="g(x^2)" />:

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

Substitute these forms into the initial equation:

<MathContainer>
  <BlockMath math="2(9x^2 + 12x + 4) - (3x^2 + 2) - 5(3x + 2) - 7 = 0" />
  <BlockMath math="(18x^2 + 24x + 8) - (3x^2 + 2) - (15x + 10) - 7 = 0" />
</MathContainer>

Group like terms:

<MathContainer>
  <BlockMath math="(18 - 3)x^2 + (24 - 15)x + (8 - 2 - 10 - 7) = 0" />
  <BlockMath math="15x^2 + 9x - 11 = 0" />
</MathContainer>

This quadratic equation has roots <InlineMath math="x_1" /> and <InlineMath math="x_2" />.
The sum of the roots of a quadratic equation <InlineMath math="ax^2 + bx + c = 0" /> is <InlineMath math="x_1 + x_2 = -\frac{b}{a}" />.

<MathContainer>
  <BlockMath math="x_1 + x_2 = -\frac{9}{15}" />
  <BlockMath math="x_1 + x_2 = -\frac{3}{5}" />
</MathContainer>

Thus, the value of <InlineMath math="x_1 + x_2" /> is <InlineMath math="-\frac{3}{5}" />.


---

## Exercise 15

### Question

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

The average of three numbers is <InlineMath math="24" />. If one of the numbers is replaced by the number <InlineMath math="14" />, the average of the three numbers decreases by <InlineMath math="3" />.

| <InlineMath math="P" /> | <InlineMath math="Q" /> |
| :---: | :---: |
| The number being replaced | <InlineMath math="42" /> |


### Choices

- [ ] $$P > Q$$
- [x] $$P < Q$$
- [ ] $$P = Q$$
- [ ] $$P = 2Q$$
- [ ] Cannot determine the relationship between P and Q

### Answer & Explanation

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

We will find the value of the replaced number (quantity <InlineMath math="P" />) first.

#### Analyzing Quantity P

Let the three numbers be <InlineMath math="a, b," /> and <InlineMath math="c" />.
The initial average is given as <InlineMath math="24" />:

<MathContainer>
  <BlockMath math="\frac{a + b + c}{3} = 24" />
  <BlockMath math="a + b + c = 72" />
</MathContainer>

Suppose the number <InlineMath math="c" /> is replaced by the number <InlineMath math="14" />.
The new average decreases by <InlineMath math="3" /> from the initial average, so the new average becomes <InlineMath math="24 - 3 = 21" />.

<MathContainer>
  <BlockMath math="\frac{a + b + 14}{3} = 21" />
  <BlockMath math="a + b + 14 = 63" />
  <BlockMath math="a + b = 63 - 14" />
  <BlockMath math="a + b = 49" />
</MathContainer>

We substitute the value <InlineMath math="a + b = 49" /> into the initial equation to find the value of <InlineMath math="c" />:

<MathContainer>
  <BlockMath math="(a + b) + c = 72" />
  <BlockMath math="49 + c = 72" />
  <BlockMath math="c = 72 - 49" />
  <BlockMath math="c = 23" />
</MathContainer>

Thus, the replaced number is <InlineMath math="23" />.
So, <InlineMath math="P = 23" />.

#### Comparing P and Q

Given <InlineMath math="Q = 42" />.

<MathContainer>
  <BlockMath math="P = 23" />
  <BlockMath math="Q = 42" />
</MathContainer>

It is clear that <InlineMath math="23 < 42" />.

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


---

## Exercise 16

### Question

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

To complete the construction of a building, <InlineMath math="15" /> people are needed to work for <InlineMath math="35" /> days. However, the building owner wants the building to be finished in <InlineMath math="3" /> weeks. To finish exactly as desired by the building owner, ... additional workers are needed.


### Choices

- [ ] $$30$$
- [ ] $$25$$
- [ ] $$20$$
- [ ] $$15$$
- [x] $$10$$

### Answer & Explanation

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

This problem involves inverse proportion. If the number of workers increases, the time required decreases, and vice versa.

Given:
*   <InlineMath math="15" /> people can complete the work in <InlineMath math="35" /> days.
*   The work is desired to be completed in <InlineMath math="3" /> weeks <InlineMath math="= 3 \times 7 = 21" /> days.

Let <InlineMath math="P" /> be the total number of workers needed to complete the work in <InlineMath math="21" /> days.

<MathContainer>
  <BlockMath math="\text{Workers}_1 \times \text{Time}_1 = \text{Workers}_2 \times \text{Time}_2" />
  <BlockMath math="15 \times 35 = P \times 21" />
  <BlockMath math="525 = 21P" />
  <BlockMath math="P = \frac{525}{21}" />
  <BlockMath math="P = 25" />
</MathContainer>

So, a total of <InlineMath math="25" /> workers are needed.

The question asks for the **additional** workers.
The initial number of workers is <InlineMath math="15" />.

<MathContainer>
  <BlockMath math="\text{Additional} = \text{Total workers} - \text{Initial workers}" />
  <BlockMath math="\text{Additional} = 25 - 15 = 10 \text{ people}" />
</MathContainer>

Thus, <InlineMath math="10" /> additional workers are needed.


---

## Exercise 17

### Question

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

Given an integer <InlineMath math="m" />, if <InlineMath math="m" /> is divided by <InlineMath math="7" /> the remainder is <InlineMath math="3" />, and if <InlineMath math="m" /> is divided by <InlineMath math="3" /> the remainder is <InlineMath math="2" />. Possible values for <InlineMath math="m" /> are
1. <InlineMath math="52" />
2. <InlineMath math="59" />
3. <InlineMath math="42" />
4. <InlineMath math="38" />


### Choices

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

### Answer & Explanation

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

Given the conditions for integer <InlineMath math="m" />:
*   <InlineMath math="m \equiv 3 \pmod{7}" /> (If divided by <InlineMath math="7" />, remainder is <InlineMath math="3" />)
*   <InlineMath math="m \equiv 2 \pmod{3}" /> (If divided by <InlineMath math="3" />, remainder is <InlineMath math="2" />)

Let's check each statement:

#### Analysis of Statement 1
Value <InlineMath math="m = 52" />. Divided by <InlineMath math="7" />: <InlineMath math="52 = 7 \times 7 + 3" />. Remainder <InlineMath math="3" /> (Satisfied). Divided by <InlineMath math="3" />: <InlineMath math="52 = 17 \times 3 + 1" />. Remainder <InlineMath math="1" /> (**Not satisfied**, should be <InlineMath math="2" />). Thus, statement <InlineMath math="1" /> is **incorrect**.

#### Analysis of Statement 2
Value <InlineMath math="m = 59" />. Divided by <InlineMath math="7" />: <InlineMath math="59 = 8 \times 7 + 3" />. Remainder <InlineMath math="3" /> (Satisfied). Divided by <InlineMath math="3" />: <InlineMath math="59 = 19 \times 3 + 2" />. Remainder <InlineMath math="2" /> (Satisfied). Thus, statement <InlineMath math="2" /> is **correct**.

#### Analysis of Statement 3
Value <InlineMath math="m = 42" />. Divided by <InlineMath math="7" />: <InlineMath math="42 = 6 \times 7 + 0" />. Remainder <InlineMath math="0" /> (**Not satisfied**, should be <InlineMath math="3" />). Divided by <InlineMath math="3" />: <InlineMath math="42 = 14 \times 3 + 0" />. Remainder <InlineMath math="0" /> (**Not satisfied**, should be <InlineMath math="2" />). Thus, statement <InlineMath math="3" /> is **incorrect**.

#### Analysis of Statement 4
Value <InlineMath math="m = 38" />. Divided by <InlineMath math="7" />: <InlineMath math="38 = 5 \times 7 + 3" />. Remainder <InlineMath math="3" /> (Satisfied). Divided by <InlineMath math="3" />: <InlineMath math="38 = 12 \times 3 + 2" />. Remainder <InlineMath math="2" /> (Satisfied). Thus, statement <InlineMath math="4" /> is **correct**.

Since statements <InlineMath math="2" /> and <InlineMath math="4" /> are correct, the correct answer is **If <InlineMath math="(2)" /> and <InlineMath math="(4)" /> are correct**.


---

## Exercise 18

### Question

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

In a district, a survey was conducted on <InlineMath math="300" /> teenagers. There are <InlineMath math="135" /> people who participate in sports activities, and <InlineMath math="255" /> people who participate in nature lover activities. If <InlineMath math="A" /> is the minimum number of teenagers who participate in both and <InlineMath math="B" /> is the maximum number of teenagers who participate in both. Then the difference between <InlineMath math="B" /> and <InlineMath math="A" /> is ... people.


### Choices

- [ ] $$90$$
- [ ] $$80$$
- [ ] $$70$$
- [ ] $$65$$
- [x] $$45$$

### Answer & Explanation

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

Given:
*   Total teenagers (<InlineMath math="S" />) = <InlineMath math="300" />
*   Sports activities (<InlineMath math="O" />) = <InlineMath math="135" />
*   Nature lover activities (<InlineMath math="C" />) = <InlineMath math="255" />

We are asked to find the difference between the maximum value (<InlineMath math="B" />) and the minimum value (<InlineMath math="A" />) of the intersection of these two sets (<InlineMath math="O \cap C" />).

#### Finding Minimum Value A
The minimum intersection occurs when the union of the two sets is maximized, which equals the total universe (<InlineMath math="300" />). This means everyone participates in at least one activity.
<MathContainer>
  <BlockMath math="n(O \cup C) = n(O) + n(C) - n(O \cap C)" />
  <BlockMath math="300 = 135 + 255 - A" />
  <BlockMath math="300 = 390 - A" />
  <BlockMath math="A = 390 - 300" />
  <BlockMath math="A = 90" />
</MathContainer>

#### Finding Maximum Value B
The maximum intersection occurs when the smaller set is a subset of the larger set. In this case, all sports participants also participate in nature lover activities.
<MathContainer>
  <BlockMath math="B = \min(n(O), n(C))" />
  <BlockMath math="B = \min(135, 255)" />
  <BlockMath math="B = 135" />
</MathContainer>

#### Calculating the Difference
<MathContainer>
  <BlockMath math="B - A = 135 - 90" />
  <BlockMath math="B - A = 45" />
</MathContainer>

Thus, the difference between <InlineMath math="B" /> and <InlineMath math="A" /> is <InlineMath math="45" /> people.


---

## Exercise 19

### Question

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

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

Given the following graph:

<Graph title="Function Graph" description="Graph of linear and exponential functions." />

1. point <InlineMath math="A(0, -2)" />
2. point <InlineMath math="B(0, 6)" />
3. point <InlineMath math="C(-3, 0)" />
4. point <InlineMath math="D(-2, 2)" />

The correct statements are


### 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 for Problem 19",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

We will check the validity of each statement based on the given graph. The graph consists of the line <InlineMath math="y = 2x + 6" /> and the exponential curve <InlineMath math="y = (1/2)^x - 3" />.

#### Analysis of Statement 1

Point A is the y-intercept of the curve <InlineMath math="y = (1/2)^x - 3" />. Substitute <InlineMath math="x = 0" />:

<BlockMath math="y = \left(\frac{1}{2}\right)^0 - 3 = 1 - 3 = -2" />

So, point A is <InlineMath math="(0, -2)" />. Statement <InlineMath math="1" /> is **correct**.

#### Analysis of Statement 2

Point B is the y-intercept of the line <InlineMath math="y = 2x + 6" />. Substitute <InlineMath math="x = 0" />:

<BlockMath math="y = 2(0) + 6 = 6" />

So, point B is <InlineMath math="(0, 6)" />. Statement <InlineMath math="2" /> is **correct**.

#### Analysis of Statement 3

Point C is the x-intercept of the line <InlineMath math="y = 2x + 6" />. Substitute <InlineMath math="y = 0" />:

<BlockMath math="\begin{aligned} 2x + 6 &= 0 \\ 2x &= -6 \\ x &= -3 \end{aligned}" />

So, point C is <InlineMath math="(-3, 0)" />. Statement <InlineMath math="3" /> is **correct**.

#### Analysis of Statement 4

Statement <InlineMath math="4" /> states that point D is <InlineMath math="(-2, 2)" />. On the graph, D is the intersection point of the line and the curve. We check if the point <InlineMath math="(-2, 2)" /> lies on both functions.

For the line <InlineMath math="y = 2x + 6" />:

<BlockMath math="y = 2(-2) + 6 = -4 + 6 = 2" />

Point <InlineMath math="(-2, 2)" /> is on the line.

For the curve <InlineMath math="y = (1/2)^x - 3" />:

<BlockMath math="y = \left(\frac{1}{2}\right)^{-2} - 3 = 2^2 - 3 = 4 - 3 = 1" />

Since <InlineMath math="y = 1 \neq 2" />, point <InlineMath math="(-2, 2)" /> is not on the curve. Therefore, D is not <InlineMath math="(-2, 2)" />. Statement <InlineMath math="4" /> is **incorrect**.

Thus, the correct statements are <InlineMath math="1" />, <InlineMath math="2" />, and <InlineMath math="3" />.


---

## Exercise 20

### Question

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

If <InlineMath math="f(x) = 5 - 4x^2" /> and <InlineMath math="g(x) = \frac{3x - 5}{2}" />, then <InlineMath math="(f \circ g)(x) =" /> ...


### Choices

- [ ] $$-9x^2 + 30x + 30$$
- [x] $$-9x^2 + 30x - 20$$
- [ ] $$-9x^2 - 30x - 20$$
- [ ] $$-9x^2 + 30$$
- [ ] $$-9x^2 - 20$$

### Answer & Explanation

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

Given the functions:

<MathContainer>
  <BlockMath math="f(x) = 5 - 4x^2" />
  <BlockMath math="g(x) = \frac{3x - 5}{2}" />
</MathContainer>

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

By definition, <InlineMath math="(f \circ g)(x) = f(g(x))" />. This means we substitute the entire function <InlineMath math="g(x)" /> into the variable <InlineMath math="x" /> in function <InlineMath math="f(x)" />.

<MathContainer>
  <BlockMath math="\begin{aligned} (f \circ g)(x) &= f\left(\frac{3x - 5}{2}\right) \\ &= 5 - 4\left(\frac{3x - 5}{2}\right)^2 \end{aligned}" />
</MathContainer>

Next, expand the squared fraction:

<MathContainer>
  <BlockMath math="\begin{aligned} &= 5 - 4\left(\frac{(3x - 5)^2}{2^2}\right) \\ &= 5 - 4\left(\frac{9x^2 - 30x + 25}{4}\right) \end{aligned}" />
</MathContainer>

Notice that the <InlineMath math="4" /> in the numerator and denominator cancel out:

<MathContainer>
  <BlockMath math="\begin{aligned} &= 5 - (9x^2 - 30x + 25) \end{aligned}" />
</MathContainer>

Distribute the negative sign:

<MathContainer>
  <BlockMath math="\begin{aligned} &= 5 - 9x^2 + 30x - 25 \\ &= -9x^2 + 30x - 20 \end{aligned}" />
</MathContainer>

Thus, <InlineMath math="(f \circ g)(x) = -9x^2 + 30x - 20" />.


---
