The curve $C$ has equation $y=\frac{a{{x}^{2}}-2ax+a-2}{x-2}$, $a\in \mathbb{R}$, $a\ne 0$.

Question

The curve $C$ has equation $y=\frac{a{{x}^{2}}-2ax+a-2}{x-2}$, $a\in \mathbb{R}$, $a\ne 0$. Find $\frac{\mathrm{d}y}{\mathrm{d}x}$.[2] Find the range of values of $a$ for which $C$ has no stationary points.[3] For $a=1$, Sketch the curve $C$, indicating the equations of asymptotes, coordinates of intersections with the axes and turning points, if any.[3] By sketching a suitable curve on the diagram in (iii), state the number of positive and negative roots of the equation ${{x}^{3}}-2{{x}^{2}}-x=k\left( x-2 \right)$ where $k>0$.[2]

Accepted Answer

(i) $\frac{\text{d}y}{\text{d}x}=a-\frac{a-2}{{{\left( x-2 \right)}^{2}}}$ (ii) $0< a <2$ (iv) From diagram, there are $2$ positive roots and $1$ negative root.

2024 ACJC Promo Q10

2024 ACJC Promo Q10

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