A curve $C$ has equation $y = ax+b+\frac{b-2a}{x+2}$, where $a$ and $b$ are real consta

Question

A curve $C$ has equation $y = ax+b+\frac{b-2a}{x+2}$, where $a$ and $b$ are real constants such that $a > 0$, $a \neq \frac{1}{2}b$ and $x \neq -2$. Given that $C$ has stationary points, use differentiation to find the relationship between $a$ and $b$.[3] It is now given that $a = 1$ and $b = 3$. Prove algebraically that $y$ cannot lie between $-1$ and $3$.[3] Sketch $C$, stating the equations of any asymptotes and the coordinates of any axial intercepts and turning points.[2]

Accepted Answer

(a) $b-2a>0$; (b) $y\le-1$ or $y\ge3$; (c) asymptotes $x=-2$, $y=x+3$, stationary points $(-3,-1)$, $(-1,3)$

2025 RVHS P1 Q9

2025 RVHS P1 Q9

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