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Numerical Answers (workings/explanations are after the numerical answers.)
Question 1: 
Question 2: 
Question 3: 
Question 4: 
Question 5: 
Question 6: 
Question 7: 
Question 8: 
Question 9: 
Question 10: 
Question 11: 
Question 12: 
, and 
, using elimination, we have
.
![= - \frac{1}{6} \big[ \cot 3x \big]_{\pi /18}^{\pi / 9}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-5e7d4f6023d261e79de8a8bc2f67326a_l3.png "Rendered by QuickLaTeX.com")
and 
. To solve for intersections, we can use the GC to plot the graphs.
and 
.
parallel to the y axis. Lastly, translate the graph a units in the positive y direction.
since f is a self inverse function.
since f is a self inverse function and 2023 is an odd number.
, then 
and 
.
![= \big[ - \frac{1}{2} x^{-2} \cdot \ln x \big]_1^3 - \int_1^3 - \frac{1}{2} x^{-2} \cdot \frac{1}{x} dx](http://theculture.sg/wp-content/ql-cache/quicklatex.com-ab68ceeec81a025eecdf6209ab1de69e_l3.png "Rendered by QuickLaTeX.com")
![= - \frac{\ln 3}{18} + \frac{1}{2} \big[ - \frac{1}{2}x^{-2} \bigg]_1^3](http://theculture.sg/wp-content/ql-cache/quicklatex.com-cffe646e3504f3ffa7ad99334d8ed2ee_l3.png "Rendered by QuickLaTeX.com")
![= - \bigg[ \ln (x^2 + 2x + 1) + \frac{3}{x+1} \bigg]_0^{0.5} + \bigg[ \ln (x^2 + 2x + 1) + \frac{3}{x+1} \bigg]_{0.5}^{2}](http://theculture.sg/wp-content/ql-cache/quicklatex.com-555261cdb5e9c5d759cb9a4640f2bdf1_l3.png "Rendered by QuickLaTeX.com")
, then 
.
.![\frac{\sin \frac{\pi}{3} [ 1 - (- \cos \frac{\pi}{3} )^7]}{1 + \cos \frac{\pi}{3} }](http://theculture.sg/wp-content/ql-cache/quicklatex.com-f0f3adcc77ca6dd1b27e871b79b26611_l3.png "Rendered by QuickLaTeX.com")
![= \frac{0.5 \sqrt{3} [ 1 - (- 0.5 )^7]}{1 + 0.5 }](http://theculture.sg/wp-content/ql-cache/quicklatex.com-48c51d8fa9b5c62b358224776f382b60_l3.png "Rendered by QuickLaTeX.com")
since there are no stationary points.
, then 
.
, and use the GC to plot the graph of 
since it is a scale factor of a parallel to the y axis, while x is invariant. The coordinates of axial intercepts are 
, 
and 
. The asymptotes are 
and 
.
.
. Thus, the range is 
or 
.
or 
. 
![\bigg[ \begin{pmatrix}200\\20\\-15\end{pmatrix} + \lambda \begin{pmatrix} -936\\-72\\21 \end{pmatrix} \bigg] \cdot \begin{pmatrix}5\\1\\2\end{pmatrix} = 2560](http://theculture.sg/wp-content/ql-cache/quicklatex.com-d9952dc942f062600303a2ebf57a5c5f_l3.png "Rendered by QuickLaTeX.com")
.
(1 d.p.)
, where 
is a positive constant.
, where 
is an arbitrary constant.
, then 
, then 
, where 
is an arbitrary constant.
, we see that as 
, then 
. Thus, the population will increase and tend to 500.
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Your ans for sum of first 7 terms is wrong, it should be 129/128sqrt(3)
Hi Mr Teng, could you send me a softcopy of the question paper please? Thank you 🙂