The curve $C$ has equation $y=\mathrm{f}\left( x \right)$ where \(\mathrm{f}\left( x \right

Question

The curve $C$ has equation $y=\mathrm{f}\left( x \right)$ where $\mathrm{f}\left( x \right)=\frac{a{{x}^{2}}+bx+c}{x+d}$, and $a$, $b$, $c$ and $d$ are constants, and $a\ne 0$. Given that $C$ has asymptote $y=x+1$, find the value of $a$ and show that $b=d+1$.[2] If $\mathrm{f}$ is an increasing function for all $x\in \mathbb{R}$, $x\ne -d$, show that $c\le d$.[2] It is further given that $c=1$ and $d=2$. Sketch $C$.[3] By sketching a suitable graph in the same diagram in part (b)(i), find the number of real roots to the equation ${{\left( \frac{{{x}^{2}}+3x+1}{x+2} \right)}^{2}}+4{{x}^{2}}=16$[2]

Accepted Answer

(a) $a=1$, $b=d+1$, $c\le d$ (b)(i) VA $x=-2$, OA $y=x+1$, $x$-intercepts $\left(\dfrac{-3\pm\sqrt5}{2},0\right)$, $y$-intercept $\left(0,\dfrac12\right)$ (b)(ii) $2$ real roots

2024 VJC P1 Q7

2024 VJC P1 Q7

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