All solutions here are suggested. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions
Numerical Answers (workings/explanations are after the numerical answers.)
Question 1: 
Question 2: 
Question 3: 
Question 4: 
Question 5: 
Question 6: Smallest 
, Translate the graph of 
units in the negative 
direction
Question 7: 
Question 8: 
Question 9: 
is maximum; 
Question 10: ![\frac{dN}{dt} = kN, k = \frac{1}{2}\ln3, t = 4.19; N = \frac{2}{\ln 3} [(50 \ln 3 - d) e^{0.5 \ln 3 t} + d]; d > 50 \ln3; t = 5.35](https://theculture.sg/wp-content/ql-cache/quicklatex.com-ee1138aed6ad37c92bd378e42ca54a13_l3.png "Rendered by QuickLaTeX.com")
Question 11: 
![\bigg[ \frac{a}{4}x^4 + \frac{b}{3}x^3 + \frac{c}{2}x^2 + dx \bigg]_0^1 = 6](https://theculture.sg/wp-content/ql-cache/quicklatex.com-d427376fa3d909595321352d744d9423_l3.png "Rendered by QuickLaTeX.com")
.
and 
.
.
![(1+z)^4 + (1 + z^*)^4 = 2 + 4(z + z^*) + 6[z^2 + (z^2)^*] + 4[z^3 + (z^3)^*] + [z^4 + (z^4)^*]](https://theculture.sg/wp-content/ql-cache/quicklatex.com-27c10a3f8d1a1dc580632bae9c9ec76e_l3.png "Rendered by QuickLaTeX.com")
![(1+z)^4 + (1 + z^*)^4 = 2 + 4[2Re(z)] + 6[2Re(z^2)] + 4[2Re(z^3)] + 2Re(z^4)](https://theculture.sg/wp-content/ql-cache/quicklatex.com-0a0d7cccef49831889f0af84d62ed588_l3.png "Rendered by QuickLaTeX.com")
![(1+z)^4 + (1 + z^*)^4 = 2 + 4[2\cos \frac{\pi}{4}] + 12\cos \frac{\pi}{2} + 8\cos \frac{3\pi}{4} + 4\cos \pi](https://theculture.sg/wp-content/ql-cache/quicklatex.com-e8d096c734756febeea51bab2bff34b3_l3.png "Rendered by QuickLaTeX.com")
. 
.
and 
, thus the minimum is when 
and thus 
and thus we should translate the graph of 
is perpendicular to direction of 
, the lines are both perpendicular.
and 
.
and 
represents any position vector lying on 
and 
values for will give the intersection that this common perpendicular cuts 
is a maximum.
![= \frac{\pi}{2} \bigg[ \frac{1}{2} e^{2x} + \frac{1}{4} e^{2x} \cos 2x + \frac{1}{4} e^{2x} \sin 2x \bigg]_0^{\pi /2}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-ff7400f5c46c18ba31d5e6189009e19e_l3.png "Rendered by QuickLaTeX.com")
where 
since 
minutes using GC.
where 
.![N = \frac{2}{\ln 3}[(50 \ln 3 - d)e^{kt} + d]](https://theculture.sg/wp-content/ql-cache/quicklatex.com-557d22dca20e346d42a582b486125e6f_l3.png "Rendered by QuickLaTeX.com")
![N = \frac{2}{\ln 3}[(50 \ln 3 - d)e^{0.5 \ln 3 t} + d]](https://theculture.sg/wp-content/ql-cache/quicklatex.com-72c3cac0218ce61d9600ef86a5e3a865_l3.png "Rendered by QuickLaTeX.com")
![\therefore \quad N = \frac{2}{\ln 3} [ (50 \ln 3 - d)3^{0.5t} + d]](https://theculture.sg/wp-content/ql-cache/quicklatex.com-e2767b857eccdd620642a8c23744ac42_l3.png "Rendered by QuickLaTeX.com")
.![N = \frac{2}{\ln 3}[(50 \ln 3 - 58)e^{kt} + 58] = 0](https://theculture.sg/wp-content/ql-cache/quicklatex.com-ca75459948093df305a7ca6e4dcde707_l3.png "Rendered by QuickLaTeX.com")
![PS = a[2\pi - \beta - \sin (2\pi - \beta)] - a(\beta - \sin \beta)](https://theculture.sg/wp-content/ql-cache/quicklatex.com-7d7d8f273f156a8b1f90695e8d3600c5_l3.png "Rendered by QuickLaTeX.com")
![= a^2 \bigg[ \frac{3}{2} \theta - 2\sin \theta + \frac{1}{4} \sin 2 \theta \bigg]_{\beta}^{2\pi - \beta}](https://theculture.sg/wp-content/ql-cache/quicklatex.com-6e28171547d62f8588f63392da6ceb0d_l3.png "Rendered by QuickLaTeX.com")
![= a^2 \bigg[ \frac{3}{2} (2\pi - \beta)- 2\sin (2\pi - \beta) + \frac{1}{4} \sin 2(2\pi - \beta) - ( \frac{3}{2} \beta - 2\sin \beta + \frac{1}{4} \sin 2 \beta) \bigg]](https://theculture.sg/wp-content/ql-cache/quicklatex.com-4f44a420a252896c77ff552aeaa37844_l3.png "Rendered by QuickLaTeX.com")
![= a^2 \bigg[ 3\pi -\frac{3}{2} \beta + 2\sin \beta - \frac{1}{4} \sin \beta - \frac{3}{2} \beta + 2\sin \beta - \frac{1}{4} \sin 2 \beta) \bigg]](https://theculture.sg/wp-content/ql-cache/quicklatex.com-6135b51f1d53adea7d6a751437be99af_l3.png "Rendered by QuickLaTeX.com")
![= a^2 [ 3\pi - 3 \beta + 4 \sin \beta - \frac{1}{2} \sin 2 \beta ]](https://theculture.sg/wp-content/ql-cache/quicklatex.com-5bcedade1852bab951550e958f7a5bfd_l3.png "Rendered by QuickLaTeX.com")
.
.
, where 
, 
.Things that have yet to come out… Sequences and series/ definite integral/ functions, and more… All the best for paper 2!
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question 6 should be -2a in positive x-axis
yes, indeed. typed wrongly.
can u do h1 math pls!!
sure! if I do get the time 🙂 All the best for the remaining papers!
yes pls do H1 math there are no suggested answers anywhere 🙁
Hello, I will finish up the pure math section soon.
http://theculture.sg/2021/11/2021-a-level-h1-mathematics-8865-paper-1-suggested-solutions/
Hello, I will finish up the pure math section soon.
http://theculture.sg/2021/11/2021-a-level-h1-mathematics-8865-paper-1-suggested-solutions/
Q4: Should
be 
instead?
yes, you are right. I believe it is below in the solutions. I probably missed it out in the final numeric section. Will add in. Thank you for pointing it out.
do u hv the qns ppr
so sorry, I don’t as I believe there is copyright laws involved. 🙁 But I believe reddit has it circulating.
Is it possible to state the marks allocation for each part
Hi, I believe you can go reddit and find the question paper circulating and it should help. I can’t share it as there are copyright laws involved.
11.b what if i did the integration correctly but put the limits wrongly? how manyy marks would i lose?
I would say a maximum of 3 marks since it is a 6 marks question.
Is there ecf for stats
there is no ecf in mathematics, simply method and answer mark.
If I didn’t simplify my expression in q2, how much marks I lose?