Standard +0.8 This question requires students to set up a quadratic equation from the intersection condition, apply the discriminant condition for two distinct roots (Δ > 0), and solve a quadratic inequality. While the individual techniques are standard A-level content, combining them in this context with the rational function and parameter 'a' requires solid algebraic manipulation and understanding of the discriminant condition, making it moderately challenging but not exceptional.
Allow \(\geqslant\). If no inequalities seen, M1 is implied by 2 correct final answers in \(a\) or \(x\)
\(a<0,\ a>\frac{8}{9}\) (or 0.889) OE
A1 A1
For final answers accept \(0>a>\frac{8}{9}\) but not \(\leqslant, \geqslant\)
## Question 2:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $ax+3a=-\frac{2}{x} \rightarrow ax^2+3ax+2\ (=0)$ | \*M1 | Rearrange into a 3-term quadratic |
| Apply $b^2-4ac>0$ SOI | DM1 | Allow $\geqslant$. If no inequalities seen, M1 is implied by 2 correct final answers in $a$ or $x$ |
| $a<0,\ a>\frac{8}{9}$ (or 0.889) OE | A1 A1 | For final answers accept $0>a>\frac{8}{9}$ but not $\leqslant, \geqslant$ |
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2 Find the set of values of $a$ for which the curve $y = - \frac { 2 } { x }$ and the straight line $y = a x + 3 a$ meet at two distinct points.\\
\hfill \mbox{\textit{CAIE P1 2017 Q2 [4]}}