Yishun Junior College H2 Math papers, with video solutions

1. By using the substitution \(x=3\sec \theta \), evaluate \(\displaystyle\displaystyle \displaystyle\int\nolimits_{3\sqrt{2}}^{6}{\frac{3x+1}{\sqrt{{{x}^{2}}-9}}\mathrm{d}x}\) exactly.[5]
2. 
   The diagram shows an ellipse with equation \(\frac{{{x}^{2}}}{16}+\frac{{{\left( y-2 \right)}^{2}}}{4}=1\).
   1. Find the area of the shaded region, giving your answer correct to \(3\) decimal places.[2]
   2. Find the exact volume of the solid generated when the shaded region is rotated \(180{}^\circ \) about the \(y\)-axis.[4]

The curve $C$ has equation $y=\frac{{{x}^{2}}}{x-2}$.

1. Find the equation(s) of the asymptote(s) of $C$.

   [1]

2. Sketch the curve $C$, labelling the equation(s) of its asymptote(s) and coordinates of any axial intercepts and turning points.

   [2]

3. Hence find the range of values of $k$ for which the equation ${{x}^{2}}=k\left( {{x}^{2}}-4 \right)$ has no real roots.