# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-5/4

Exercises: Try Out - Set 5: Real exam simulation to sharpen your skills and build confidence. - Problem 4

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## 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" />.


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