The curve $C$ has equation $y = \frac{x^2 + ax + b}{x + 1}$, where $a$ and $b$ are cons

Question

The curve $C$ has equation $y = \frac{x^2 + ax + b}{x + 1}$, where $a$ and $b$ are constants. It is given that $C$ has a minimum point at $(2,0)$. Show that $a = -4$ and $b = 4$.[3] Sketch the graph of $C$, indicating clearly the equations of any asymptotes and the coordinates of any turning points and axial intercepts.[3] By adding an appropriate graph to your sketch in $\mathbf{(ii)}$, deduce the number of real roots of the equation $$x^3 + ax^2 + (b + 1)x + 1 = 0.$$[2]

Accepted Answer

(i) $a=-4,\ b=4$ (shown). (ii) Asymptotes $x=-1$, $y=x-5$; maximum $(-4,-12)$, minimum $(2,0)$; intercepts $(0,4)$, $(2,0)$. (iii) $1$ real root.

2024 JPJC Promo Q5

2024 JPJC Promo Q5

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