# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-6/13

Exercises: Try Out - Set 6: Real exam simulation to sharpen your skills and build confidence. - Problem 13

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## 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" />.


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