# Quod Erat Demonstrandum

## 2009/05/06

### 2009 CE Additional Mathematics Paper Section B

Filed under: Additional / Applied Mathematics,HKCEE — johnmayhk @ 6:04 下午

The following figure shows a park AED on a horizontal ground. The park is in the form of a right-angled triangle surrounded by a walking path with negligible width. Henry walks along the path at a constant speed. He starts from point A at 7:00 am. He reaches points B, C and D at 7:10 am, 7:15 am and 7:30 am respectively and returns to A via point E. The angles of elevation of H, the top of a tower outside the park, from A and D are 45$^o$ and 30$^o$ respectively. At point B, Henry is closest to the point K which is the projection of H on the ground. Let HK = h m.

(a) Express DK in terms of h. (1 mark)
(b) Show that AB = h$\sqrt{\frac{2}{3}}$ m. (3 marks)
(c) Find the angle of elevation of H from C correct to the nearest degree. (3 marks)
(d) Henry returns to A at 8:10 am. It is known that the area of the park is 9450 m$^2$.
(i) Find h.
(ii) A vertical pole of length 3 cm is located such that it is equidistant from A, D and E. Find the angle of elevation of H from the top of the pole correct to the nearest degree. (5 marks)

Q.14 Vector：大路，但最後一部分也有少許難度：

Let $\overrightarrow{AD} = \overrightarrow{p}, \overrightarrow{DH} = \overrightarrow{q}$. Suppose $H$ is the orthocentre of $\Delta ABC$, |$\overrightarrow{p}$| = 1 and |$\overrightarrow{q}$| = 2. Find $\overrightarrow{AE}$ in terms of $\overrightarrow{p}$ and $\overrightarrow{q}$, hence determine $\frac{AF}{FC}$.

Q.15 Differentiation and Integration：有些新意

The sphere is being pulled at a constant speed of $\frac{1}{4} cms^{-1}$. At the instant when h = 3, find the rate of change of (1) the depth of water (2) the distance between the top of the sphere and the water surface.

Q.16 Differentiation：小心表達

Given ABCD, PQRS are rectangles. Show that the area of PQRS is $\frac{(14 - x)(9 + x^2)}{3}$. Show that $0 \le x \le 2$ and $12 \le x \le 14$ and hence find the greatest area of PQRS.

Q.17 Coordinate geometry：題型舊（相信補習社及非補習社的授課員都捉不到路…），只要小心代數運算便可！

Prove that $AB^2 = \frac{(\sin\theta - 2\cos\theta)(\sin\theta + 14\cos\theta)}{9\cos^4\theta}$. Suppose $L_1$ touches $\Gamma$ at $R$, find possible slopes of $L_1$ and the corresponding lengths of $PR$. Suppose $L_2$ is a tangent to $\Gamma$ from $(1 , \frac{-1}{3})$, write down the slopes of $L_2$.

SECTION A：要盡量淺（取 30 分便可及格）
SECTION B：要盡量深（用來分出取 A 同學，但取 B,C,D 呢？）

## 7 則迴響 »

1. 18(b) 應該係sqrt(2/3)h m?
另外Q17我地應該要識做?

迴響 由 4E — 2009/05/06 @ 8:25 下午 | 回應

• thx and you should know how to solve Q.17 up to this stage

迴響 由 johnmayhk — 2009/05/07 @ 12:03 下午 | 回應

2. Q18 seems to be a general maths level question.
It doesn’t involve any special techniques learnt in A. Maths.

To some extent, this question is easier than the 3D trigonometry question in Maths paper this year because it can be solved just by repeated use of Pyth. thm and definition of tangent. You don’t even need to know the definition of sine, definition of consine, sine formula, cosine formula and Heron’s formula.

迴響 由 stupid girl — 2009/05/08 @ 11:54 下午 | 回應

3. 好奇一問
因為平時見人話Amath好難 但岩岩見到你話會考Amath 30分合格 即係110分入面拎到30分就100%合格 定只係今年係咁?

迴響 由 Science — 2009/05/21 @ 6:38 下午 | 回應

4. @ clever girl

Yes, you are right. But sometimes questions were set to test students integrated ‘abilities’ (esp. basic techniques) rather than ‘knowledge’. Like questions in mathematics competition (of course, the CE questions are not comparable to questions used in mathematics competition), it is likely that mathematics ‘knowledge’ (e.g. calculus) is NOT the key to solve problems, it should be the ability and mathematics sense. Sorry that I’m talking too much…

@ Science

I’m not working in HKEAA, but I am just told by some teachers that the passing mark of CE A.Math is *about* 30 (out of 110) in recent years.

迴響 由 johnmayhk — 2009/05/23 @ 5:39 下午 | 回應

5. 呀sir~第17題個分母係cos三次定係cos四次~?

迴響 由 Carmen — 2009/08/05 @ 12:30 下午 | 回應

• 四次，無打錯呀。

迴響 由 johnmayhk — 2009/08/07 @ 1:02 下午 | 回應