# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-8/9

Exercises: Try Out - Set 8: Real exam simulation to sharpen your skills and build confidence. - Problem 9

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## Exercise 9

### Question

export const metadata = {
  title: "Problem 9",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

Line <InlineMath math="h" /> passes through point <InlineMath math="T(-1,3)" /> and has a gradient <InlineMath math="m" />. For <InlineMath math="h" /> to intersect the graph <InlineMath math="y = -x^2" /> at two distinct points, it must be that...


### Choices

- [ ] $$m > 6$$
- [ ] $$-2 < m < 6$$
- [ ] $$-6 < m < 2$$
- [ ] $$m \leq -2 \lor m \geq 6$$
- [x] $$m < -2 \lor m > 6$$

### Answer & Explanation

export const metadata = {
  title: "Solution for Problem 9",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

The equation of line <InlineMath math="h" /> is <InlineMath math="y = mx + c" />. Given <InlineMath math="y = 3" /> and <InlineMath math="x = -1" />, then:

<BlockMath math="\begin{aligned} 3 &= -m + c \\ c &= m + 3 \end{aligned}" />

Thus, the line equation can be written as <InlineMath math="y = mx + (m + 3)" />. Since line <InlineMath math="h" /> intersects the graph <InlineMath math="y = -x^2" /> at two distinct points, then:

<BlockMath math="\begin{aligned} mx + (m + 3) &= -x^2 \\ x^2 + mx + (m + 3) &= 0 \end{aligned}" />

For the line to intersect at two distinct points, the discriminant must satisfy <InlineMath math="D > 0" />.

<BlockMath math="\begin{aligned} m^2 - 4(m + 3) &> 0 \\ m^2 - 4m - 12 &> 0 \\ (m + 2)(m - 6) &> 0 \end{aligned}" />

The roots are <InlineMath math="m = -2" /> and <InlineMath math="m = 6" />. By testing intervals, we obtain the solution set:

<BlockMath math="m < -2 \lor m > 6" />


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