### H2 Math Mon 2pm

This page contains all questions and answers asked by students from this class. The most recent questions will be at the top.

MF26

Vectors Q7 [Homework]
(i)
$\vec{OL} = \begin{bmatrix}2\\ 7\\ -1\end{bmatrix}$
$\vec{OM} = \begin{bmatrix}9\\ 0\\ -8\end{bmatrix}$
Using ratio theorem, $\vec{OP} = \frac{2\vec{OM}+5\vec{OL}}{7} = \begin{bmatrix}4\\ 5\\ -3\end{bmatrix}$
Since $\vec{OP}$ is perpendicular to $\begin{bmatrix}4\\ 1\\ q\end{bmatrix}$
$\Rightarrow \begin{bmatrix}4\\ 5\\ -3\end{bmatrix} \bullet \begin{bmatrix}4\\ 1\\ q\end{bmatrix} = 0$
$q = 7$

(ii)
To be a parallelogram, $\vec{OM} = \vec{LN} = \vec{ON} - \vec{OL}$
$\vec{ON} =\begin{bmatrix}11\\ 7\\ -9\end{bmatrix}$
$Area = |\vec{OM} \times \vec{OL}|$
$= |\begin{bmatrix}56\\ -7\\ 63\end{bmatrix}|$
$= \sqrt{7154} = 7 \sqrt{146} units^2$

(iii)
Let $\vec{OQ} = \begin{bmatrix}x\\ y\\ 0\end{bmatrix}$
Since $|\vec{OQ}| = |\vec{OP}|$
$\sqrt{x^2 + y^2} = \sqrt{50}$ — (1)
$\begin{bmatrix}x\\ y\\ 0\end{bmatrix} \bullet \begin{bmatrix}1\\ 0\\ 0\end{bmatrix} = |\begin{bmatrix}x\\ y\\ 0\end{bmatrix} | |\begin{bmatrix}1\\ 0\\ 0\end{bmatrix} | \mathrm{cos} \theta$ — (2)
Solving, $x = \sqrt{50} \mathrm{cos} \theta = 5 \sqrt{2} \mathrm{cos} \theta$
$y = \sqrt{50} \mathrm{sin} \theta = 5 \sqrt{2} \mathrm{sin} \theta$
$\Rightarrow \vec{OQ} = \begin{bmatrix}{5 \sqrt{2} \mathrm{cos} \theta}\\ {5 \sqrt{2} \mathrm{sin} \theta}\\ 0\end{bmatrix}$

Vectors Q8 [Homework]
(i)
$\vec{OA} = \begin{bmatrix}-5\\ -2\\ 3\end{bmatrix}$
$\vec{OC} = \begin{bmatrix}5\\ 2\\ 6\end{bmatrix}$
$\vec{AC} = \vec{OC} - \vec{OA} = \begin{bmatrix}5\\ 2\\ 6\end{bmatrix} - \begin{bmatrix}-5\\ -2\\ 3\end{bmatrix} = \begin{bmatrix}10\\ 4\\ 3\end{bmatrix}$
$l: r = \begin{bmatrix}5\\ 2\\ 6\end{bmatrix} + \lambda \begin{bmatrix}10\\ 4\\ 3\end{bmatrix}, \lambda \in \mathbb{R}$

(ii)
Let R be the top of the vertical pillar,
$l_{QR}: r = \begin{bmatrix}15\\ 6\\ 0\end{bmatrix} + \mu \begin{bmatrix}0\\ 0\\ 1\end{bmatrix}, \mu \in \mathbb{R}$
Since R is collinear with A and C, R is the intersection of line AC and QR.
$\begin{bmatrix}{5 + 10 \mu}\\ {2 + 4 \mu}\\ {6 + 3 \mu}\end{bmatrix} = \begin{bmatrix}15\\ 6\\ {\mu}\end{bmatrix}$
$\Rightarrow \lambda = 1, \mu = 9$
$\vec{OR} = \begin{bmatrix}15\\ 6\\ 9\end{bmatrix}$, and the height is 9m.

(iii)
$\vec{OD} = \begin{bmatrix}-5\\ 2\\ 6\end{bmatrix}$
$\vec{AD} = \vec{OD} - \vec{OA} = \begin{bmatrix}0\\ 4\\ 3\end{bmatrix}$
$\vec{AX} = (\vec{AD} \bullet \frac{\begin{bmatrix}10\\ 4\\ 3\end{bmatrix}}{| \begin{bmatrix}10\\ 4\\ 3\end{bmatrix}|}) \frac{\begin{bmatrix}10\\ 4\\ 3\end{bmatrix}}{| \begin{bmatrix}10\\ 4\\ 3\end{bmatrix}|}$
$= (\begin{bmatrix}0\\ 4\\ 3\end{bmatrix} \bullet \frac{\begin{bmatrix}10\\ 4\\ 3\end{bmatrix}}{\sqrt{125}}) \frac{\begin{bmatrix}10\\ 4\\ 3\end{bmatrix}}{\sqrt{125}}$
$= \frac{25}{125} \begin{bmatrix}10\\ 4\\ 3\end{bmatrix}$
$= \begin{bmatrix}2\\ 0.8\\ 0.6\end{bmatrix}$
$\vec{OX} = \vec{OA} + \vec{AX} = \begin{bmatrix}-3\\ 1.2\\ 3.6\end{bmatrix}$

Vectors Q9 [Homework]
(i)
$\vec{AB} = \begin{bmatrix}-4\\ 5\\ 3\end{bmatrix}$
$\vec{AC} = \begin{bmatrix}1\\ -3\\ 6\end{bmatrix}$
Normal of $\pi_1, ~n_1=\begin{bmatrix}-4\\ 5\\ 3\end{bmatrix} \times \begin{bmatrix}1\\ -3\\ 6\end{bmatrix} = \begin{bmatrix}-21\\ -21\\ -7\end{bmatrix} = -7 \begin{bmatrix}3\\ 3\\ 1\end{bmatrix}$
$\pi_1: r \bullet \begin{bmatrix}3\\ 3\\ 1\end{bmatrix} = \begin{bmatrix}5\\ -1\\ 0\end{bmatrix} \bullet \begin{bmatrix}3\\ 3\\ 1\end{bmatrix} = 12$

(ii)
Let $\theta$ be the acute angle
$\theta - \mathrm{cos}^{-1} |\frac{\begin{bmatrix}3\\ 3\\ 1\end{bmatrix} \bullet \begin{bmatrix}1\\ -1\\ 1\end{bmatrix}}{\sqrt{19}} \sqrt{3}|$
$\theta = 82.4 ^{\circ}$

(iii)
$3x + 3 y + z = 12$ — (1)
$x - y + z = 1$ — (2)

Using GC, $l: r = \begin{bmatrix}2.5\\ 1.5\\ 0\end{bmatrix} + \lambda \begin{bmatrix}-2\\ 1\\ 3\end{bmatrix}, \lambda \in \mathbb{R}$

(iv)
Let $n_3$ be the normal of $\pi_3$
Length of projection $= |\vec{AB} \times n_3|$
$= \frac{1}{\sqrt{26}} |\begin{bmatrix}4\\ -5\\ 3\end{bmatrix} \times \begin{bmatrix}5\\ -1\\ 0\end{bmatrix}| = 15\sqrt{\frac{3}{26}}$

(v)
Required distance $= \frac{1}{\sqrt{3}} + \frac{2}{\sqrt{3}} = \sqrt{3} units$

(vi)
Let normal of $\pi_4 = n_4 = \begin{bmatrix}-2\\ 1\\ 3\end{bmatrix} \times \begin{bmatrix}1\\ -1\\ 1\end{bmatrix} = \begin{bmatrix}4\\ 5\\ 1\end{bmatrix}$
$\pi_4: r \bullet \begin{bmatrix}4\\ 5\\ 1\end{bmatrix} = 4k+6$
If $\pi_1, \pi_2 \mathrm{~and~} \pi_4$ intersect at l,n$\begin{bmatrix}2.5\\ 1.5\\ 0\end{bmatrix}$ lies on $pi_4$
$\Rightarrow \begin{bmatrix}2.5\\ 1.5\\ 0\end{bmatrix} \bullet \begin{bmatrix}4\\ 5\\ 1\end{bmatrix} = 4k+6$
$k = \frac{23}{8}$

### Consider the argument that the main purpose of television should be to educate rather than simply to entertain.

Consider the argument that the main purpose of television should be to educate rather than simply to entertain.

This is a comparison question on the role of television- whether it should be of an educational role or an entertainment role.  So what is exactly an educational or even entertainment role?

Define- Educate: to impart values and knowledge to society. Entertain- to allow audiences to relieve stress, rewind and enjoy after a day of work.

Educational role is more important than entertainment

1)Television is often the main vehicle of transmission of important news that the government would like the public to know.  In critical situations, such as the spread of diseases such as SARS and MERs, the government has to reach out to the wider public to disseminate news in the fastest possible time.  They have to resort to more traditional medium of transmission such as the television as not everyone has access to social media.  The media also plays an important role during war time and national emergency to disseminate news such as places to go to for shelter and food rations.

2)Besides being an important transmission of news, television has the social obligation to impart values to the society. These values being imparted are often state- endorsed for social stability. But one must note that censorship is strongly evident, due to the fact that most governments control the media and that the government decides for the people what they should consume and what they should not. E.g. Shows and dramas in Singapore are usually focused on family-centric themes of filial piety and heterosexual marriages.  Any themes dealing with homosexuality or politically divisive news are being censored.  A recent film that was being censored by our government includes “To Singapore, with Love”.

If television focuses on entertainment, it may lead to negative impacts on adolescences’ views on sexual attitudes and expectations, which may destabilize the institution of marriage and family.  Lots of entertainment programs such as “sex and the city”, “desperate housewives”  and even “sweet sixteen” that are gaining attention, but what is the value of watching such programs if our moral values and attitudes get eroded?

3)Educational programs help people to broaden their horizons and perspectives about other nations.  They serve as a bridge in helping people understand and catch a glimpse of other people’s lifestyles and values, even if they cannot afford to travel to know more about the world.  Such education programs include National Geographic and even news channel like BBC. Having global citizens who are well tuned to world happenings is definitely a cutting-edged skill that countries need today.

Entertainment is more important than educational purposes

1)Entertainment role should not be underestimated given its ability to allow the masses to unwind and enjoy programs after a hard day of work. It is a cheap form of entertainment that families can bond with one another.

2) Entertainment is essential as it brings in advertising revenues for the media industries and creates job for the local industry. Programs focusing on entertainment rather than education usually have higher viewership which will translate into greater profits.

Here are just some reasons on the role of the television- in terms of entertainment and education. But of course, do note that the television’s role goes far beyond these. Can we fit in other roles of the television in entertainment or even education categories? Where does the role of citizen journalism fit in? Do let me know.

Students should note that by the nature of the subject, there is several other possible pointers too, so feel free to discuss freely below!