2025 CJC P1 Q11
2025 CJC P1 Q11
14 marks
Show that \(\frac{3x^2}{(x+1)(3x^2+x+1)}\) can be expressed as
\(\frac{A}{x+1} + \frac{Bx+C}{3x^2+x+1}\),
where \(A\), \(B\) and \(C\) are constants to be determined.
[2]- Hence, find \(\displaystyle \displaystyle\int \frac{3x^2}{(x+1)(3x^2+x+1)} \, \mathrm{d}x\).[4]
- The region \(R\) is bounded by the curve \(y = \frac{3x^2}{(x+1)(3x^2+x+1)}\), the line \(x=4\) and the \(x\)-axis.
- Sketch the graph of \(y = \frac{3x^2}{(x+1)(3x^2+x+1)}\) for \(x \ge 0\), giving the coordinates of any intercepts with the axes and the equations of any asymptotes. Shade the region \(R\) on your sketch.[3]
- Find the exact area of \(R\).[3]
- Find the volume of the solid generated when \(R\) is rotated through \(2\pi\) radians about the \(x\)-axis.[2]
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Answer:(a)(i) \(A=1,\ B=0,\ C=-1\) (ii) \(\ln|x+1|-\dfrac2{\sqrt{11}}\tan^{-1}\!\left(\dfrac{6x+1}{\sqrt{11}}\right)+C\) (b)(ii) \(\ln5+\dfrac2{\sqrt{11}}\left[\tan^{-1}\!\left(\dfrac1{\sqrt{11}}\right)-\tan^{-1}\!\left(\dfrac{25}{\sqrt{11}}\right)\right]\) (iii) \(0.715\text{ units}^3\)
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