The curve $C$ has equation $y = \frac{3x^2 + ax + 3}{x + 1}$, where $a$ is a constant, \(

Question

The curve $C$ has equation $y = \frac{3x^2 + ax + 3}{x + 1}$, where $a$ is a constant, $a \neq 6$. Find the set of values of $a$ such that $C$ does not cut the $x$-axis.[2] It is given that the equation of $C$ is $y = \frac{3x^2 - 6x + 3}{x + 1}$. Sketch $C$ and give the equations of any asymptotes. Also, state the coordinates of any points where $C$ crosses the axes and of any turning points.[4] Let $k$ be a positive constant. By sketching a suitable graph in the same diagram in part (b), find an inequality satisfied by $k$ such that the equation $$k^2(x - 1)^2 + \left(\frac{3x^2 - 6x + 3}{x + 1} + 12\right)^2 = k^2$$ has 2 real and distinct roots.[2]

Accepted Answer

(a) ${\frac{1}{3}\pi x^2 \sqrt{R^2 - x^2}}$ (b) ${9V \frac{\mathrm{d}V}{\mathrm{d}x} = \pi^2 \left(2R^2 x^3 - 3x^5ight)}$ or ${9V \left(\frac{\mathrm{d}V}{\mathrm{d}x}ight) = \pi^2 x^3 \left(2R^2 - 3x^2ight) 	ext{-----} (*)}$

2023 VJC Promo Q10

2023 VJC Promo Q10