### Solutions to December Holiday Revision

Hopefully you find your holidays meaningful. Take the time to unwind and relax. Also, if you have completed the Set A and Set B. The solutions can be found here.

If you do have any questions, please WhatsApp me. ðŸ™‚

Solutions to Set A

Solutions to Set B

Relevant Materials:Â MF26

### Solutions to Set B

Hopefully, you guys have started on the Set B. You will find the following solutions useful. Click on the question. Please do attempt them during this December Holidays. ðŸ™‚

If you do have any questions, please WhatsApp me. ðŸ™‚

Relevant Materials:Â MF26

### Solutions to Set A

Hopefully, you guys have started on the Set A. You will find the following solutions useful. Click on the question. Please do attempt them during this December Holidays. ðŸ™‚

If you do have any questions, please WhatsApp me. ðŸ™‚

Relevant Materials:Â MF26

### 2017 A-level H1 Mathematics (8865) 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:
Question 2:
Question 3:
Question 4:
Question 5:
Question 6: $\mu = 1.69, \sigma^2 = 0.0121$
Question 7: $0.254; 0.194; 0.908$
Question 8: $40320; 0.0142; \frac{1}{4}$
Question 9: $\text{r}=0.978; a=0.182, b=2.56$; \$293
Question 10: $0.0336; \bar{y}=0.64, s^2 = 0.0400$; Sufficient evidence.
Question 11: $\frac{48+x}{80+x}, \frac{32+x}{80+x}; x= 16; \frac{25}{32}; \frac{7}{16}; \frac{341}{8930}$
Question 12: $0.773; 0.0514; 0.866; 0.362$

MF26

### 2017 A-level H1 General Paper (8807) Paper 1

Sharing the A-level H1 General Paper (8807) Paper 1…

1. ‘The past is not dead; it is not even past.’ Discuss.
2. Can the use of animals for scientific research ever be justified?
3. In your society, to what extent is it acceptable for public money to be used for the acquisition of works of art?
4. ‘Rehabilitation, not punishment, should be the purpose of the justice system.’ Discuss.
5. Is regulation of the press desirable?
6. Do events, rather than politicians, shape the future?
7. How far is science fiction becoming fact?
8. Examine the role of music in establishing a national identity in your society.
9. To what extent are people judged more by their physical appearance than by their abilities?
10. ‘Practical ability is just as important as intellectual skills.’ How far is this true in your society?
11. Assess the view that attempts to control climate change can never be truly effective.
12. The quality of written language is being destroyed by social media.’ What is your view?
13.

### 2017 A-level H2 Mathematics (9740) 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: 2
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: $(\frac{1}{a}, \frac{1}{ae}); \frac{1}{a^2}$
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: $\frac{dv}{dt}=c$; $v = 10t +4$; $v = \frac{1}{k}(10- 10 e^{-kt})$

### Relevant materials

MF26

To be honest, this paper is really the same as the H2 Mathematics (9758). They just rephrased everything. You can see for yourself here.

### Interesting Trigonometry Question

We know how A’levels like to combine a few topics across. We also know how bad trigonometry can be. I was coaching my International Bachelorette (IB) class last week and came across a fairly interesting question. It tests students on their abilities to manage double angle identities. Not a tough question, but definitely good practice ðŸ™‚
FYI: All double angle identities are found in MF26.
Here it is…

(a) Show that $(1 + i \text{tan} \theta)^n + (1 - i \text{tan} \theta)^n = \frac{2 \text{cos} n \theta}{ \text{cos}^n \theta}$, $\text{cos} \theta \neq 0$.

(b)
(i) Use the double angle identity $\text{tan} 2 \theta = \frac{2 \text{tan} \theta}{1 - \text{tan}^2 \theta}$ to show that $\text{tan} \frac{\pi}{8} = \sqrt{2} - 1$.

(ii) Show that $\text{cos} 4x = 8 \text{cos}^4 x - 8 \text{cos}^2 x + 1$.

(iii) Hence find the value of $\int^{\frac{\pi}{8}}_0 {\frac{2 \text{cos} 4x}{ \text{cos}^2 x}}~dx$ exactly.

### 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) 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