### Vectors Question #3

This is a question a student sent me a few days back, and I shared with my class.

Find the Cartesian equation of the plane that is equidistant of the $xy$ plane and $xz$ plane.

The following should aid students to visualise.

Sidenote: I think Vectors is a very important topic for 9758 as its applications are wide. Students should do their best to understand the topic. I will share a few more applied questions next week when I have time.

### A little reminder to students doing Calculus now

When $\frac{dy}{dx} = 0$, it implies we have a stationary point.

To determine the nature of the stationary point, we can do either the first derivative test or the second derivative.

The first derivative test:

Students should write the actual values of $\alpha^-, \alpha, \alpha^+$ and $\frac{dy}{dx}$ in the table.

We use this under these two situations:
1. $\frac{d^2y}{dx^2}$ is difficult to solve for, that is, $\frac{dy}{dx}$ is tough to be differentiated
2. $\frac{d^2y}{dx^2} = 0$

The second derivative test:

Other things students should take note is concavity and drawing of the derivative graph.

### Vectors Question #2

If $c = |a| b + |b| a$, where $a$ , $b$ and $c$ are all non-zero vectors, show that $c$ bisects the angle between $a$ and $b$.

### Thoughts on A’levels H2 Mathematics 2016 Paper 2

I’ll keep this short since we are all busy. One thing about paper 1 we saw, there were many unknowns.

So topics which I think will come out…

Differentiation – I think a min/max problem will come out, possibly with r and h both not given and asked to express r in terms of h. But students should revise a on the properties of curves with differentiation; given a curve equation with an unknown, for instance $y=/frac{x^2+kx+1}{x-1}$, find the range of k such that there is stationary points.

Complex Number – Loci will definitely come out. I’m saying they will combine with trigonometry.

Integration – Modulus integration hasn’t really been tested. Else a question on Area/ Volume could be tested, and I’ll say they need students to do some
Conics too.

For statistics, my students should have gotten the h1 stats this year. And if it’s an indicator, then it should not be a struggle.

I expect PnC and probability to be combined. Conditional Probability in a poisson question should be tested too, so do revise it well. For hypothesis testing, students should be careful of their formula and read really carefully about the alternative hypothesis. Also, :9 know that the formulas for poison PDF and binompdf are both given in mf15. Lastly, know when to use CLT.

All the best!

### Random Questions from 2016 Prelims #12

HCI/1/6

A group of boys want to set up a camping tent. They lay down a rectangular tarp OABC on the horizontal ground with OA = 3 m and AB = 1.5 m and secure the points D and E vertically above O and B respectively, such that .

Assume that the tent takes the shape as shown above with 6 triangular surfaces and a rectangular base. The point O is taken as the origin and the unit vectors i, j and k are taken to be in the direction of , and respectively.

(i) Show that the line DE can be expressed as $r = 2k+\lambda(2i+j), \lambda \in \mathbb{R}$.

(ii) Find the Cartesian equation of the plane ADE.

(iii) Determine the acute angle between the planes ADE and OABC. Hence, or otherwise, find the acute angle between the planes ADE and CDE.

Note: Question can be made harder and trickier should Origin, O be placed in the center of the base OACB.

### 2016 A-level H1 Mathematics (8864) Paper 1 Suggested Solutions

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

As these workings and answers are rushed out asap, please pardon me for my mistakes and let me know if there is any typo. Many thanks.
I’ll try my best to attend to the questions as there is H2 Math Paper 2 coming up.

Question 1
(i)
$\frac{d}{dx} [2 \text{ln}(3x^2 +4)]$

$= \frac{12x}{3x^2+4}$

(ii)
$\frac{d}{dx} [\frac{1}{2(1-3x)^2}]$

$= \frac{6}{2(1-3x)^3}$

$= \frac{3}{(1-3x)^3}$

Question 2

$2e^{2x} \ge 9 - 3e^x$

$2u^2 + 3u - 9 \ge 0$

$(2u-3)(u+3) \ge 0$

$\Rightarrow u \le -3 \text{~or~} u \ge \frac{3}{2}$

$e^x \le -3$ (rejected since $e^x > 0$) or $e^x \ge \frac{3}{2}$

$\therefore x \ge \text{ln} \frac{3}{2}$

Question 3
(i)

(ii)
Using GC, required answer $= -1.606531 \approx -1.61$ (3SF)

(iii)
When $x = 0.5, y = 0.35653066$

$y - 0.35653066 = \frac{-1}{-1.606531}(x-0.5)$

$y = 0.622459 x +0.04530106$

$y = 0.622 x + 0.0453$ (3SF)

(iv)

$\int_0^k e^{-x} -x^2 dx$

$= -e^{-x} - \frac{x^3}{3} \bigl|_0^k$

$= -e^{-k} - \frac{k^3}{3} + 1$

Question 4
(i)
$y = 1 + 6x - 3x^2 -4x^3$

$\frac{dy}{dx} = 6 - 6x - 12x^2$

Let $\frac{dy}{dx} = 0$

$6 - 6x - 12x^2 = 0$

$1 - x - 2x^2 = 0$

$2x^2 + x - 1 = 0$

$x = -1 or \frac{1}{2}$

When $x =-1, y = -4$

When $x = \frac{1}{2}, y = 2.75$

Coordinates $= (-1, -4) \text{~or~} (0.5, 2.75)$

(ii)
$\frac{dy}{dx} = 6 - 6x - 12x^2$

$\frac{d^2y}{dx^2} = - 6 - 24x$

When $x = -1, \frac{d^2y}{dx^2} = 18 > 0$. So $(-1, -4)$ is a minimum point.

When $x = 0.5, \frac{d^2y}{dx^2} = -18 \textless 0$. So $(0.5, 2.75)$ is a maximum point.

(iii)

x-intercept $= (-1.59, 0) \text{~and~} (1, 0) \text{~and~} (-0.157, 0)$

(iv)
Using GC, $\int_0.5^1 y dx = 0.9375$

Question 5
(i)
Area of ABEDFCA $= \frac{1}{2}(2x)(2x)\text{sin}60^{\circ} - \frac{1}{2}(y)(y)\text{sin}60^{\circ}$

$2\sqrt{3} = \sqrt{3}x^2 - \frac{\sqrt{3}}{4}y^2$

$2 = x^2 - \frac{y^2}{4}$

$4x^2 - y^2 =8$

(ii)

Perimeter $= 10$

$4x+2y + (2x-y) = 10$

$6x + y = 10$

$y = 10 - 6x$

$4x^2 - (10-6x)^2 = 8$

$4x^2 - 100 +120x -36x^2 = 8$

$32x^2 -120x+108=0$

$x=2.25 \text{~or~} 1.5$

When $x=2.25, y = -3.5$ (rejected since $y >0$)

When $x=1.5, y = 1$

Question 6
(i)
The store manager has to survey $\frac{1260}{2400} \times 80 = 42$ male students and $\frac{1140}{2400} \times 80 = 38$ female students in the college. He will do random sampling to obtain the required sample.

(ii)
Stratified sampling will give a more representative results of the students expenditure on music annually, compared to simple random sampling.

(iii)
Unbiased estimate of population mean, $\bar{x} = \frac{\sum x}{n} = \frac{312}{80} = 3.9$

Unbiased estimate of population variance, $s^2 = \frac{1}{79}[1328 - \frac{312^2}{80}] = 1.40759 \approx 1.41$

Question 7
(i)
[Venn diagram to be inserted]

(ii)
(a)
$\text{P}(A \cup B) = \text{P}(A) + \text{P}(B) - \text{P}(A \cap B) = 0.8$
(b)
$\text{P}(A \cup B) - \text{P}(A \cap B) = 0.75$

(iii)
$\text{P}(A | B')$

$= \frac{\text{P}(A \cap B')}{\text{P}(B')}$

$= \frac{\text{P}(A) - \text{P}(A \cap B)}{1 - \text{P}(B)}$

$= \frac{0.6 - 0.05}{1-0.25}$

$= \frac{11}{15}$

Question 8
(i)
Required Probability $= \frac{1}{2} \times \frac{3}{10} \times \frac{2}{9} = \frac{1}{30}$

(ii)
Find the probability that we get same color. then consider the complement.

Required Probability $= 1 - \frac{1}{30} - \frac{1}{2} \times \frac{5}{10} \times \frac{4}{9} - \frac{1}{2} \times \frac{2}{10} \times \frac{1}{9} - \frac{1}{2} \times \frac{4}{6} \times \frac{3}{5} - \frac{1}{2} \times \frac{2}{6} \times \frac{1}{5} = \frac{11}{18}$

(iii)
$\text{P}(\text{Both~balls~are~red} | \text{same~color})$

$= \frac{\text{P}(\text{Both~balls~are~red~and~same~color})}{\text{P}(\text{same~color})}$

$= \frac{\text{P}(\text{Both~balls~are~red})}{\text{P}(\text{same~color})}$

$= \frac{1/30}{7/18}$

$= \frac{3}{35}$

Question 9
(i)
Let X denote the number of batteries in a pack of 8 that has a life time of less than two years.

$X \sim B(8, 0.6)$
(a)
Required Probability $=\text{P}(X=8) = 0.01679616 \approx 0.0168$

(b)
Required Probability $=\text{P}(X \ge 4) = 1 - \text{P}(X \le 3) = 0.8263296 \approx 0.826$

(ii)
Let Y denote the number of packs of batteries, out of 4 packs, that has at least half of the batteries having a lifetime of less than two years.

$Y \sim B(4, 0.8263296)$

Required Probability $=\text{P}(Y \le 2) = 0.1417924285 \approx 0.142$

(iii)
Let W denote the number of batteries out of 80 that has a life time of less than two years.

$W \sim B(80, 0.6)$

Since n is large, $np = 48 > 5, n(1-p)=32 >5$

$W \sim N(48, 19.2)$ approximately

Required Probability

$= \text{P}(w \ge 40)$

$= \text{P}(w > 39.5)$ by continuity correction

$= 0.9738011524$
$\approx 0.974$

Question 10
(i)
Let X be the top of speed of cheetahs.
Let $\mu$ be the population mean top speed of cheetahs.

$H_0: \mu = 95$

$H_1: \mu \neq 95$

Under $H_0, \bar{X} \sim N(95, \frac{4.1^2}{40})$

Test Statistic, $Z = \frac{\bar{X}-\mu}{\frac{4.1}{\sqrt{40}}} \sim N(0,1)$

Using GC, $p=0.0449258443 \textless 0.05 \Rightarrow H_0$ is rejected.

(ii)
$H_0: \mu = 95$

$H_1: \mu > 95$

For $H_0$ to be not rejected,

$\frac{\bar{x}-95}{\frac{4.1}{\sqrt{40}}} \textless 1.644853626$

$\bar{x} \textless 96.06 \approx 96.0$ (round down to satisfy the inequality)

$\therefore \{ \bar{x} \in \mathbb{R}^+: \bar{x} \textless 96.0 \}$

Question 11
(i)
[Sketch to be inserted]

(ii)
Using GC, $r = 0.9030227 \approx 0.903$ (3SF)

(iii)
Using GC, $y = 0.2936681223 x - 1.88739083$

$y = 0.294 x - 1.89$ (3SF)

(iv)
When $x = 16.9, y = 3.0756 \approx 3.08$(3SF)

Time taken $= 3.08$ minutes

Estimate is reliable since $x = 16.9$ is within the given data range and $|r|=0.903$ is close to 1.

(v)
Using GC, $r = 0.5682278 \approx 0.568$ (3SF)

(vi)
The answers in (ii) is more likely to represent since $|r|=0.903$ is close to 1. This shows a strong positive linear correlation between x and y.

Question 12
Let X, Y denotes the mass of the individual biscuits and its empty boxes respectively.

$X \sim N(20, 1.1^2)$

$Y \sim N(5, 0.8^2)$

(i)
$\text{P}(X \textless 19) = 0.181651 \approx 0.182$

(ii)
$X_1 + ... + X_12 + Y \sim N(245, 15.16)$

$\text{P}(X_1 + ... + X_12 + Y > 248) = 0.2205021 \approx 0.221$

(iii)
$0.6X \sim N(12, 0.66^2)$

$0.2Y \sim N(1, 0.16^2)$

Let $A=0.6X$ and $B = 0.2Y$

$A_1+...+A_12+B \sim N(145, 5.2528)$

$\text{P}(142 \textless A_1+...+A_12+B \textless 149) = 0.864257 \approx 0.864$

### 2016 A-level H2 Mathematics (9740) Paper 2 Suggested Solutions

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

Numerical Answers (click the questions for workings/explanation)

Question 1: $0.0251 \text{~m/min}$
Question 2: $\frac{x^2}{2} \text{sin}nx + \frac{2x}{n^2} \text{cos}nx - \frac{2}{n^3} \text{sin}nx + C;~ a = 2 \text{~or~} 6;~ \pi (\frac{2}{5} - \text{ln} \frac{3}{\sqrt{5}})$
Question 3: $a + \frac{3}{4} - \text{sin}a - \text{cos}a + \frac{1}{4} \text{cos}2a;~ \frac{1}{4}(\pi + 1)^2$
Question 4: $z = 2.63 + 1.93i, ~ 3.37 + 0.0715i ;~8^{\frac{1}{6}}e^{i(-\frac{\pi}{12})}, ~8^{\frac{1}{6}}e^{i(-\frac{3 \pi}{4})}, ~8^{\frac{1}{6}}e^{i(\frac{7\pi}{12})};~ n=7$
Question 5: $\frac{11}{42};~ \frac{4}{11};~ \frac{4}{1029}$
Question 6: $60;~ 10;~ \{ \bar{x} \in \mathbb{R}, 0 \textless \bar{x} \le 34.8 \};~ \{ \alpha \in \mathbb{R}, 0 \textless \alpha \le 8.68 \}$
Question 7: $24;~ 576;~ \frac{1}{12};~ \frac{5}{12}$
Question 8: $r = -0.980;~ c = -17.5;~ d = 91.8;~ y = 85.9$
Question 9: $a = 7.41;~ p = 0.599 \text{~or~} 0.166;~ 0.792$
Question 10: $0.442;~ 0.151 ;~ 0.800;~ \lambda = 1.85$

MF15

### 2016 A-level H2 Mathematics (9740) Paper 1 Suggested Solutions

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

As these workings and answers are rushed out asap, please pardon me for my mistakes and let me know if there is any typo. Many thanks. As these workings and answers are rushed out asap, please pardon me for my mistakes and let me know if there is any typo. Many thanks.

I’ll try my best to attend to the questions as there is H2 Math Paper 2 coming up and I have a lot of prepare. I’ll try my best to attend to the questions as there is H2 Math Paper 2 coming up and I have a lot of prepare.

Side note: I think the paper is tedious, but definitely manageable. Hard ones, could have been q3, q7a, 10, 11ib. So if you did your tutorials and past year papers well, with proper time management and no careless, 70 is manageable. To get an A, you need to fight for that 30 marks which really test you on your comprehension skills and precisions. And these questions should distinct the students who deserve an A.

Thoughts about Paper 2.

Numerical Answers (click the questions for workings/explanation)

Question 1: $x \textless -2, \text{~or~} \frac{1}{3} \textless x \textless 4$
Question 2: $0, -\text{ln}2;~ y=2, y = -x\text{ln}2 + \frac{\pi \text{ln}2}{2} + 1;~ (\frac{\pi}{2} - \frac{1}{\text{ln}2}, 2)$
Question 3: $m = b,~ l = a, ~ kl^4 + m = c$
Question 4: $0.74;~ \frac{b(0.74^n)}{0.26}$
Question 5: $\begin{pmatrix}{2b}\\{4b-4a}\\{-2a}\end{pmatrix};~ \pm \frac{1}{6\sqrt{2}};~ 3$
Question 6: $4,~ 14,~ 44;~ \frac{1}{4}n(n+1)(n^2+n+2)+2$
Question 7: $2+3i;~ a=3,~ k=-30$
Question 8: $f(x)=1+2ax+2a^2x^2+\frac{8}{3}a^3x^3 ;~ \text{tan}2x = 2x + \frac{8}{3}x^3$
Question 9: $y = 5-5e^{-2t}; x = 5t + \frac{5}{2}e^{-2t} - \frac{5}{2};~ x = 5t^2 + 20\text{sin}\frac{t}{2}-10t;~ 1.47s,~ 1.05s$
Question 10: $f^{-1}(x) = (x-1)^2, x \in \mathbb{R}, x \ge 1;~ x = 2.62;~6,~ 8,~ 9;~\text{No}$
Question 11: $t = -\frac{5}{9},~ \lambda = -\frac{8}{9}, ~ \mu = \frac{19}{18};~ -2x+y +2z=35 \text{~or~} -2x+y+2z = -37;~a = 4.5$

MF15

### June Revision Exercise

You can find the solutions of all ten sets of the June Revision Exercise we did in class.

Have fun!

June Revision Exercise 1
June Revision Exercise 2
June Revision Exercise 3
June Revision Exercise 4
June Revision Exercise 5
June Revision Exercise 6
June Revision Exercise 7
June Revision Exercise 8
June Revision Exercise 9
June Revision Exercise 10