### A-level H2 Mathematics (9758) Suggested Solutions (2017)

Here is the suggested solutions for H2 Mathematics (9758). They are all typed in LaTeX, so if it does not render, please leave a comment and let me know. Thank you.

The suggested solutions for H2 Mathematics (9740) is here.

Students of mine should obtain the modified A’levels Paper, and the solutions to the additional questions can be found here.

Year 2017

MF26

### Solutions to the modified A’levels Questions

Students of mine who have been diligently doing the modified TYS I sent them, and have difficulties with the questions that were added in to make the paper a full 3 hour paper, will find the following solutions helpful. Please try to do them in a single 3 hour seating, these are modified to cater to the 9758 syllabus…

The rest of the solutions (that are questions from the original TYS) can be found here.

2012/P1/Q10

2012/P2/Q2

20112/P2/Q7

2012/P2/Q7

### 2017 A-level H2 Mathematics (9758) Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

Paper 1

Paper 2

MF26

### 2017 A-level H2 Mathematics (9758) Paper 2 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

Numerical Answers (click the questions for workings/explanation)

Question 1: $2 \sqrt{15}; xy=6$
Question 2: $d = 1.5;~ r \approx 1.21 \text{~or~} r \approx -1.45;~n=42$
Question 3: $(\frac{1}{2}a, 0), (0,b);~ (a+1, 0,);~ (\frac{a+1}{2}, 0);~ (0, a), (b, 0);~ a = 1;~ gg(x) = x, x \in \mathbb{R}, x \neq 1 , ~ g^{-1}(x) = 1 - \frac{1}{1-x}, x \in \mathbb{R}, x \neq 1;~b= 2 \text{~or~}0$
Question 4: $15.1875;~ \frac{\pi}{2a(a-1)};~ b = \frac{1}{2} + \frac{1}{2}\sqrt{1-a+a^2}$
Question 5: $\frac{5}{12}, \frac{5}{14}, \frac{5}{28}, \frac{1}{21};~ \mathbb{E}(T) = \frac{20}{7}, \text{Var}(T) = \frac{75}{98};~ 0.238$
Question 6: $955514880;~ 1567641600;~ \frac{1001}{3876}$
Question 7: $31.8075, 0.245;~ p = 0.0139$; Do not reject $h_0$, Not necessary.
Question 8: Model (D); $a \approx 4.18, b \approx 74.0;~ r \approx 0.981$
Question 9: $0.632;~ 1.04 \times 10^{-4};~ 0.458;~ 0.421;~ 0.9408$
Question 10: $0.345;~ 0.612;~ \mu = 12.3, \sigma = 0.475;~ k \approx 55.7$

MF26

### 2017 A-level H2 Mathematics (9758) Paper 1 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

Numerical Answers (click the questions for workings/explanation)

Question 1: $ax + (2a - \frac{a^2}{2})x^2 + (\frac{a^3}{3} + 2a - a^2) x^3$; $a = 4$
Question 2: $x \textgreater \frac{1}{\sqrt{b}} + a$ or $x \textless a$
Question 3: $x = \pm \frac{1}{\sqrt{2}}$ ; Maximum point
Question 4: $a = 4, b =1$; translate the graph 4 units in negative y-direction and translate the graph 2 units in positive x-direction.
Question 5: $a = -1.5, b = 1.5, c = 7$; $x \approx -1.33$; $x \approx -0.145$ or $x \approx 1.15$
Question 6: $r = a + (\frac{d - a \cdot n}{b \cdot n}) b$
Question 7: $\frac{\text{sin}(2mx-2nx)}{4m-4n} - \frac{\text{sin}(2mx+2nx)}{4m+4n} + C$; $\pi$
Question 8: $z = -1 + 2i$ or $z = 2 - i$; $p =-6, q=-66$; $(w^2 - 2w+2)(w^2-4w+29)$
Question 9: $U_n = 2An - A +B$; $A = 3, B =-9$; $k=4$; $\frac{1}{4} (n^4 + 2n^3 + n^2)$ ; $e^x$
Question 10: $a = -4.4$; $R(1.5, 0.5, -1)$; $\frac{1}{2}\sqrt{10}$
Question 11: $\frac{dv}{dt}=c$; $v = 10t +4$; $v = \frac{1}{k}(10- 10 e^{-kt})$; $9.21s$

### Relevant materials

MF26

Firstly, to do well in this paper, student has to be quite intuitive, to be comfortable with the levels of unfamiliarity.

Q1. Simple expansion using MF26. If you used it carefully, it should provide some guidance to Q9(c) actually.
Q2. Simple graphings, using secondary school modulus function knowledge.
Q3. Students have to know how to use $y = 5x$ to find back the y-coordinate.
Q4. (a) is even easier if you simply did long division.
Q5. Remainder Theorem from Secondary School for (i). (ii), students need to be alert that when the gradient is ALWAYS positive, the function is strictly increasing, not just increasing.
Q6. Interesting question, that is similar to the Specimen Paper.
Q7. Use of Factor Theorem form MF26 will make this integration much comfortable. By parts work too.
Q8. Standard complex number practice question.
Q9. Very interesting questions. Especially (c), but like mentioned a keen student who did Q1 well, will realise the sum to infinity is simply from MF26.
Q10. Standard vectors questions. Just read carefully and it will be manageable.
Q11. Simple DE too. For the terminal velocity, just need to read that its “after a long time”.

Overall, a manageable paper.
Now things that have yet to come out…
Reciprocal Graph, Area/ Volume, Parametric Equations, Min/Max Problem, APGP, Function, Integration Techniques, Complex Number (Polar Form, Modulus, Argument), Vectors (Planes, Ratio Theorem), Small angle approximation

### Integration By Parts

Today, I’ll share a little something about Integration by Parts. I want to share this because I observe that several students are over-reliant on the LIATE to perform Integration by Parts. This caused them to overlook/ appreciate its use.

Lets start by reviewing the formula for Integration by Parts

$\int v \frac{du}{dx} ~dx = uv - \int u \frac{dv}{dx} ~dx$

I like to share with students that Integration by Parts have two interesting facts.

1. It allows us to integrate expression that we cannot integrate. Eg. $\text{ln}x$ or inverse trigonometric functions. This is also closely related to how LIATE is established.

2. This is closely related to point 1. That is, instead of trying to integrate the expression, we are differentiating it instead. And that itself, is a very important aspect.

Next, I like to point out to students that LIATE is a general rule of thumb.
GENERAL – because it does not work 100% of the times. And today, I’ll use an example to illustrate how LIATE actually fails.

$\int \frac{te^t}{(t+1)^2} ~ dt$

Here we observe two terms $\frac{t}{(t+1)^2}$ and $e^t$. Going by LIATE, we should be differentiating $\frac{t}{(t+1)^2}$ and integrating $e^t$.

$\int t^2 e^{t} ~ dt$
$= \frac{t}{(t+1)^2} e^{t} - \int \frac{(1)(t+1)^2 - t(2)(t+1)}{(t+1)^4} e^t ~dt$

Now observe what happened here, after applying integration by parts, it got “worst”. We are stuck to integrating $\frac{(1)(t+1)^2 - t(2)(t+1)}{(t+1)^4}$ with an exponential… This should raise some question marks. But we did follow LIATE.

I’ll leave students to try out this on their own. And feel free to ask questions here. Have fun!

### In times of economic hardship, should a country still be expected to provide financial or material aid to others?

Students are expected to address the criteria of this question throughout the essay; it being in times of economic hardship. For a quality essay, the terms “should” and “expected” should be addressed as well. For instance, a country should still give aid, but perhaps not be expected at a time when its survival is in question, and it does not have a healthy budget balance.

Financial aid- capital/loans/money

Material aid- manpower/distribution of basic goods and necessities

• Provision of aid during economic hard times could be a political statement and commitment to the recipient country, helping to foster greater political relations in the long term. Giving of aid should be expected especially if the recipient country needs it more than the donor country. Such circumstances could be when the recipient country is facing civil war and there is urgent need of aid to cease the fighting etc. Another possible instance could be during times of natural disasters/emergencies. Also, for most of the donor country, aid takes up a small portion of their budget, hence it should not affect the current economy severely even if the country continues to give out aid during hard times. Aid accounts for 0.5% of the US budget yearly.
• Governments should be responsible and accountable to their citizens first especially during hard times, hence aid should be allocated domestically rather than elsewhere. The dollar votes of the citizens and their voices are important, and it is only right that countries should be concerned with their own self-preservation before others. After all, an economically prosperous country will then be able to contribute more to the international community, rather than a slow and stagnate economy that is facing difficult times.
• Perhaps a country should look at other means of support, rather than the provision of aid during bad times. It is presumptuous to assume that aid helps to alleviate the problems faced by the recipient countries. Often, aid may actually harm local industries and foster this sense of self-entitlement and dependency on the donor nations.
• Legal obligations under international law could possibly bind countries to continue giving aid to another country. A short term economic difficulty does not suffice to repudiate this commitment, especially when contracts have been signed beforehand, and the donor country may risk ruining their legitimacy and international standing.

In dire situations, countries should still continue to give aid to one another. However, expectations to scale down in terms of aid is definitely reasonable and justifiable.

### 2016 A Level H2 Physics (9646) Paper 1 Suggested Solutions

All solutions here are SUGGESTED. Casey will hold no liability for any errors. Comments are entirely personal opinions.

1. B
2. C
3. B
4. D
5. D
6. C
7. D
8. D
9. B
10. C
11. A
12. D
13. D
14. C
15. C
16. A
17. D
18. A
19. B
20. D
21. Question 21 is a flawed question. When unpolarised light goes through a polarizer, the I is halved while the A is reduced by a factor of root 2. But based on the information Cambridge provides, the answer is C.
22. D
23. D
24. C
25. C
26. B
27. C
28. A
29. B
30. B
31. A
32. C
33. D
34. B
35. B
36. B
37. D
38. C
39. C
40. C

Note to all: Casey will not respond to most of the comments as he is busy. You may contact him by SMS at  +65 9474 5005 if you have a burning question.

Feel free to explain the answers, if you are confident. Many thanks.

### How far has modern technology made it unnecessary for individuals to possess mathematical skills?

1. How far has modern technology made it unnecessary for individuals to possess mathematical skills?

Argument 1: Modern technology has provided us with tools that automate mathematical operations for us, making possessing mathematical skills redundant.

Elaboration: All we have to do are to insert the data we want to be processed into our modern technological tools, and their software automates the required operations and calculations for us, allowing us to attain the necessary processed data effortlessly.

Example: Calculators has made skills such as the ability simple mental calculations mostly redundant in everyday life. Online free tools such as desmos and Wolframalpha. Paid tools such as MatLab, Maple and R.

Link: Modern technology has provided us with tools that make certain mathematical skills such as doing simple mental calculations seemingly redundant.

(Counter) Argument 2: We still ultimately require mathematical knowledge and skills to utilize these technological tools

Elaboration: To say that possessing mathematical skills is unnecessary altogether would be too much of a far-fetched statement

Example: We still require intermediate knowledge on math to effectively operate a Graphic Calculator for more complex math problems. Basic knowledge for syntax used in programs in Graphic Calculator, or algorithm is needed.

Link: Mathematical skills are still required to allow modern technology to operate in our favor.

(Counter) Argument 3: Recognizing patterns, reasoning and logical thinking are important life skills which are largely mathematical.

Elaboration: These life skills are largely inculcated into our children via an education in mathematics.

Example: Number or shape pattern recognition is a skill inculcated into primary school children via math education. Over the years, this analytical skill is honed and eventually applied to solve industry-relevant problems such as market trends. Famous Hedge-fund owner James Simmons, only hires Mathematicians and Physicists, who do not need any business or banking background, to work as bankers in his company, Renaissance Technologies. The top Quantitative Analyst in the world has a doctorate in Physics and to quote him, in research he deals with 23 variables, but now in life and finance, he just has 3 to handle.

Link: Mathematical skills are a prerequisite to develop higher level cognitive and analytical skills in order for us to excel in society.

(Counter) Argument 4: Being equipped with the most basic Mathematical skills provides us with much greater convenience as compared to having sole reliance on technological tools.

Elaboration: Simple day-to-day activities largely involve the use of mathematical skills and would ironically take a longer time and effort if we were to fully rely on technological tools. Moreover, it would be a hassle if we were to require such tools to physically be with us all the time as compared to solving problems on the spot with our mathematical knowledge.

Example: Mental calculations would greatly aid us in buying and selling as opposed to solely relying on a calculator even for simple operations. The absolute necessity of having a calculator around with you would often times become a hassle in that case. The basic skill of doing rough mental calculations and pattern recognition would enable us to have greater control over our finances in estimating and analyzing our spending habits and saving progress.

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### ‘Everyone has an opinion, but not everyone’s opinion is of equal value.’ What is your view?

‘Everyone has an opinion, but not everyone’s opinion is of equal value.’ What is your view?

Define opinion and its purpose.

An opinion is a view or judgment formed about something, not necessarily based on fact or knowledge. Opinions are predicated on the individual’s current understanding and analysis of perceived phenomena, and its dynamic nature brings value and meaning to the uniqueness of individuality and colorful context to otherwise mundane events.
Thesis statement

While everyone has an opinion and every opinion should be given fair amount consideration in order for society to best function and cater to the interest of every individual as much as possible, the harsh truth remains that not everyone’s opinion is of equal value. Firstly, opinions predicated upon evidence are more reliable to derive inspiration from compared to one which manifests purely from emotions and personal bias. Secondly, psychological bias renders the notion that everyone’s opinion holds equal value and be given the same amount of consideration to be impractical. Hence, everyone has an opinion and is indeed entitled to have one, but not everyone’s opinion is of equal value for society.

(Counter) Argument 1:
Every opinion should be equally considered in order for society to best function and cater to the interest of every individual as much as possible

Elaboration: The conscious societal effort to hold everyone’s opinion with equal regard protects the interest of every individual, especially of those from minority groups. If we were to disregard opinions which do not interest the majority of us, it can lead to immoral and biased behavior, policies and consequences.

Example: Severe racism against blacks in the US in the past was due to the majority group’s absolute lack of consideration of opinion and interests of the perceived minority group.

Link: Every opinion should be carefully regarded in order for society to not neglect the interest of certain groups in the midst of societal progress.

(Counter) Argument 2:
Every individual is unique and as valuable, and hence every opinion carries a perspective that has their own distinct and equal value which should not be ignored.

Elaboration: Based on the premise that “all lives are equal”, all opinions should be equally considered as every opinion represents a distinct and unique individual who is as valuable as another.

Evidence: In a democratic government, every vote from every eligible individual is considered and carries the same weight, acknowledging the notion that every individual citizen is of equal value.

Point: Every opinion matters as every individual is unique.

Argument 3:
Informed opinions hold more meaningful value

Elaboration: “You are not entitled to your opinion. You are entitled to your informed opinion. No one is entitled to be ignorant.” – Harlan Ellison

Opinions predicated upon evidence are more reliable to derive inspiration from compared to one which manifests purely from emotions and personal bias. Opinions direct societal progress, and hence informed opinions lead to more informed decisions which would allow society to function and develop in its best interest.

Example: In the political context, a deep understanding of politics and the local political scene would compel the informed citizen to support the most capable political party. This informed opinion leads to be more logical and rational political vote which brings great positive value to the country, as opposed to having disgruntled citizens angrily voting for political parties which capitalize on emotions without decent consideration for practicality.

Link: Informed opinions is of greater value than uninformed ones.

Argument 4:
Personal bias renders the notion of everyone’s opinion holding equal value impractical

Elaboration: Opinions generate the greatest amount of real tangible value when they resonate with the masses. And the extent of how well an opinion is received by the masses is influenced by psychological bias driven by factors such as age, the perceived credibility and prestige of the opinion-bearer and how well that opinion aligns with our own personal beliefs.

Example: In the campaign for gender equality, it is evident that Emma Watson’s efforts hold greater influence and are held in higher regard than most people who have been essentially advocating and saying the same thing. This is the reason why charities tend to tap onto the prestige of celebrities to amplify their impact.

Link: It is impossible for everyone’s opinion to be genuinely held with equal regard due to personal bias.

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