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RUMUS
FORMULAE |
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1.
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\(x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}\) |
15.
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\(y=\dfrac{u}{v},\dfrac{dy}{dx}=\dfrac{v\dfrac{du}{dx}-u\dfrac{dv}{dx}}{v^2}\) |
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2.
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\(a^m\times a^n=a^{m+n}\) |
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\(\dfrac{dy}{dx}=\dfrac{dy}{du}\times\dfrac{du}{dx}\)
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3.
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\(a^m\div a^n=a^{m-n}\) |
17.
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Luas di bawah lengkung / Area under the curve
\(=\int_a^b y\ dx\) atau (or) \(=\int_a^b x\ dy\)
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4.
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\((a^m)^n=a^{mn}\) |
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Isipadu kisaran / Volume of revolution
\(=\int_a^b\pi y^2\ dx\) atau (or) \(=\int_a^b \pi x^2\ dy\)
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5.
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\(\log_a{mn}=\log_a{m}+\log_a{n}\) |
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\(I=\dfrac{Q_1}{Q_0}\times 100\) |
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6.
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\(\log_a{\dfrac{m}{n}}=\log_a{m}-\log_a{n}\) |
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\(\bar{I}=\dfrac{\sum W_iI_i}{\sum W_i}\) |
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7.
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\(\log_a{m^n}=n\log_a{m}\) |
21.
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\({}^nP_r=\dfrac{n!}{(n-r)!}\) |
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8.
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\(\log_a{b}=\dfrac{\log_c{b}}{\log_c{a}}\) |
22.
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\({}^nC_r=\dfrac{n!}{(n-r)!r!}\) |
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9.
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\(T_n=a+(n+1)d\) |
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\(P(X=r)={}^nC_rp^rq^{n-r}, p+q=1\) |
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10.
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\(S_n=\dfrac{n}{2}[2a+(n-1)d]\) |
24.
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Min / Mean, \(\mu=np\)
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11.
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\(T_n=ar^{n-1}\) |
25.
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\(\sigma=\sqrt{npq}\) |
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12.
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\(S_n=\dfrac{a(r^n-1)}{r-1}=\dfrac{a(1-r^n)}{1-r},r\ne 1\) |
26.
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\(z=\dfrac{X-\mu}{\sigma}\)
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13.
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\(S_n=\dfrac{a}{r-1},|r|\lt1\) |
27.
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Panjang lengkok, \(s=j\theta\)
Arc length, \(s=r\theta\)
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14.
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\(y=uv,\dfrac{dy}{dx}=u\dfrac{dv}{dx}+v\dfrac{du}{dx}\) |
28.
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Luas sektor, \(L=\dfrac{1}{2}j^2\theta\)
Area of sector, \(A=\dfrac{1}{2}r^2\theta\)
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29.
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\(\sin^2{A}+\text{kos}^2A=1\)
\(\sin^2{A}+\cos^2{A}=1\)
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41.
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Titik yang membahagi suatu tembereng garis /
A point dividing a segment of a line
\((x,y)=\left( \dfrac{nx_1+mx_2}{m+n},\dfrac{ny_1+ny_2}{m+n} \right)\)
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30.
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\(\text{sek}^2A=1+\tan^2{A}\)
\(\sec^2{A}=1+\tan^2{A}\)
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42.
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Luas segi tiga / Area of triangle
\(=\dfrac{1}{2}|(x_1y_2+x_2y_3+x_3y_1)-(x_2y_1+x_3y_2+x_1y_3)|\)
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31.
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\(\text{kosek}^2A=1+\text{kot}^2A\)
\(\cosec^2A=1+\cot^2{A}\)
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43.
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\(|\utilde{r}|=\sqrt{x^2+y^2}\) |
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32.
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\(\sin{2A}=2\sin{A}\text{ kos } A\)
\(\sin{2A}=2\sin{A}\cos{A}\)
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44.
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\(\hat{r}=\dfrac{x\utilde{i}+y\utilde{j}}{\sqrt{x^2+y^2}}\) |
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33.
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\(\begin{aligned} \text{kos }2A&=\text{kos}^2A-\sin^2{A} \\ &=2\text{ kos}^2A-1 \\ &=1-2\sin^2{A} \end{aligned}\)
\(\begin{aligned} \cos{2A}&=\cos^2{A}-\sin^2{A} \\ &=2\cos^2{A}-1 \\ &=1-2\sin^2{A} \end{aligned}\)
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34.
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\(\tan{2A}=\dfrac{2\tan{A}}{1-\tan^2{A}}\) |
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35.
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\(\sin{(A\pm B)}=\sin{A}\text{ kos }B\pm \text{kos }A\sin{B}\)
\(\sin{(A\pm B)}=\sin{A}\cos{B}\pm \cos{A}\sin{B}\)
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36.
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\(\text{kos }(A\pm B)=\text{kos }A\text{ kos }B\mp \sin{A}\sin{B}\)
\(\cos{(A\pm B)}=\cos{A}\cos{B}\mp \sin{A}\sin{B}\)
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37.
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\(\tan{(A\pm B)}=\dfrac{\tan{A}\pm \tan{B}}{1\mp \tan{A}\tan{B}}\) |
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38.
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\(\dfrac{a}{\sin{A}}=\dfrac{b}{\sin{B}}=\dfrac{c}{\sin{C}}\) |
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39.
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\(a^2=b^2+c^2-2bc\text{ kos }A\)
\(a^2=b^2+c^2-2bc\cos{A}\)
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40.
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Luas segi tiga / Area of triangle
\(=\dfrac{1}{2}ab\sin{C}\)
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Bahagian A
Section A
[50 markah]
[50 marks]
Jawab semua soalan.
Answer all questions.
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1.
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Rajah \(1\) menunjukkan jarak lengkok ayunan ladung ringkas yang dilepaskan untuk bebas berayun.
Diagram \(1\) shows the length of the arc swing of a simple pendulum that is released to swing freely.
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Rajah \(1\) / Diagram \(1\)
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(a)
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Cari panjang lengkok, dalam cm, ayunan yang ke-\(10\).
Find the length of the arc, in cm, of \(10^\text{th}\) swing.
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[ \(3\) markah / \(3\) marks ]
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(b)
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Pada ayunan ke berapakah panjang lengkok kali pertama kurang daripada \(2\) cm?
On which swing is the length of the arc first less than \(2\) cm?
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[ \(2\) markah / \(2\) marks ]
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(c)
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Hitungkan jumlah jarak ayunan ladung itu, dalam cm, sebelum ia berhenti.
Calculate the total swing distance by the pendulum, in cm, before it stops.
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[ \(2\) markah / \(2\) marks ]
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Jawapan / Answer :
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2.
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(a)
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Rajah \(2\) menunjukkan sebuah segi empat tepat \(TUVW\) pada suatu satah Cartes.
Diagram \(2\) shows a rectangle \(TUVW\) on a Cartesian plane.
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Rajah \(2\) / Diagram \(2\)
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(i)
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Ungkapkan \(p\) dalam sebutan \(q\).
Express \(p\) in terms of \(q\).
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[ \(2\) markah / \(2\) marks ]
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(ii)
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Tunjukkan bahawa luas segi tiga \(TUV\) ialah \(p-\dfrac{5}{2}q+10\).
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Show that the area of triangle \(TUV\) is \(p-\dfrac{5}{2}q+10\).
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[ \(2\) markah / \(2\) marks ]
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(b)
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Seterusnya, cari koordinat \(V\) apabila diberi luas segi empat tepat \(TUVW\) ialah \(58\) unit\(^2\).
Hence, find the coordinates \(V\) when given that the area of the rectangle \(TUVW\) is \(58\) unit\(^2\).
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[ \(2\) markah / \(2\) marks ]
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Jawapan / Answer :
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3.
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(a)
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Diberi persamaan kuadratik, \(ax^2+bx+c=0\) dengan keadaan \(a\), \(b\) dan \(c\) ialah pemalar dan \(a\ne0\), mempunyai punca-punca \(\alpha\) dan \(\beta\).
Given the quadratic equation, \(ax^2+bx+c=0\), where \(a\), \(b\) and \(c\) are constants and \(a\ne0\), has roots \(\alpha\) and \(\beta\).
Tunjukkan bahawa
Show that
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\(\alpha+\beta=-\dfrac{b}{a}\) dan / and \(\alpha\beta=\dfrac{c}{a}\)
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Seterusnya, cari hasil tambah punca dan hasil darab punca bagi persamaan \(\dfrac{2}{3}x^2=5x+2\).
Hence, find the sum of roots and the product of roots for the equation \(\dfrac{2}{3}x^2=5x+2\).
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[ \(4\) markah / \(4\) marks ]
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(b)
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Diberi bahawa \(f(x)=\dfrac{1}{x+2}+\dfrac{1}{x-3}\). Cari nilai-nilai yang mungkin bagi \(x\) jika \(f(x)=4\). Berikan jawapan anda dalam bentuk \(\dfrac{p\pm \sqrt{q}}{r}\) dengan \(p\), \(q\) dan \(r\) ialah integer positif.
Given that \(f(x)=\dfrac{1}{x+2}+\dfrac{1}{x-3}\). Find the possible values of \(x\) if \(f(x)=4\). Give your answer in the form of \(\dfrac{p\pm \sqrt{q}}{r}\) where \(p\), \(q\) and \(r\) are positive integers.
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[ \(3\) markah / \(3\) marks ]
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Jawapan / Answer :
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4.
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(a)
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Diberi bahawa persamaan suatu lengkung ialah \(y=x^3-12x\). Satu garis lurus menyentuh lengkung itu pada titik \(K(3,-9)\). Cari persamaan garis lurus itu.
It is given that the equation of a curve is \(y=x^3-12x\). A straight line touches the curve at point \(K(3,-9)\). Find the equation of the straight line.
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[ \(3\) markah / \(3\) marks ]
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(b)
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Diberi bahawa \(x+2y=4\). Cari nilai minimum bagi \(P\) jika \(P=x^2+xy-y^2\).
It is given that \(x+2y=4\). Find the minimum value of \(P\) if \(P=x^2+xy-y^2\).
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[ \(4\) markah / \(4\) marks ]
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Jawapan / Answer :
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5.
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(a)
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Diberi \(\log_2{3}=h\) dan \(\log_2{5}=k\), ungkapkan dalam sebutan \(h\) dan \(k\)
Given \(\log_2{3}=h\) and \(\log_2{5}=k\), express in terms of \(h\) and \(k\)
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(i)
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\(\log_2{3.6}\),
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(ii)
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\(\log_2{270}\).
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[ \(4\) markah / \(4\) mark ]
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(b)
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Selesaikan persamaan \(\sqrt{2}p-8=\sqrt{24}-\sqrt{8}p\).
Solve the equation \(\sqrt{2}p-8=\sqrt{24}-\sqrt{8}p\).
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[ \(3\) markah / \(3\) marks ]
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Jawapan / Answer :
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6.
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(a)
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Buktikan \(\text{kos }2x=1-2\sin^2{x}\).
Prove \(\cos{2x}=1-2\sin^2{x}\).
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[ \(2\) markah / \(2\) marks ]
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(b)
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Rajah \(3\) di bawah menunjukkan graf bagi suatu fungsi trigonometri untuk \(0\le x\le 2\pi\).
Diagram \(3\) below shows the graph of a trigonometric function for \(0\le x\le 2\pi\).
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Rajah \(3\) / Diagram \(3\)
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(i)
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Tuliskan persamaan graf fungsi trigonometri tersebut.
Write the equation of the graph of the trigonometric function.
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[ \(1\) markah / \(1\) mark ]
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(ii)
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Dengan menggunakan paksi yang sama, lakarkan graf bagi \(y=|\sin{x}|\) untuk \(0\le x\le 2\pi\), seterusnya cari bilangan penyelesaiannya.
By using the same axis, sketch the graph of \(y=|\sin{x}|\) for \(0\le x\le 2\pi\), then find the number of solutions.
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[ \(3\) markah / \(3\) marks ]
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Jawapan / Answer :
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7.
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Rajah \(4\) menunjukkan sebuah bongkah pepejal yang terdiri daripada sebuah kon tegak terletak di atas sebuah silinder berjejari \(r\) cm. Panjang sendeng kon itu ialah \(2r\) cm dan isi padu silinder itu ialah \(81\pi\) cm\(^3\).
Diagram \(4\) shows a solid block consisting of a right cone which is located above a cylinder of radius \(r\) cm. The length of slant of the cone is \(2r\) cm and the volume of the cylinder is \(81\pi\) cm\(^3\).
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Rajah \(4\) / Diagram \(4\)
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(a)
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Tunjukkan bahawa luas permukaan bongkah itu, \(L\) cm\(^2\) diberi oleh \(L=3\pi \left( r^2+\dfrac{54}{r}\right)\).
Show that the surface area of the block, \(L\) cm\(^2\) is given by \(L=3\pi \left( r^2+\dfrac{54}{r}\right)\).
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[ \(3\) markah / \(3\) marks ]
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(b)
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Hitung nilai minimum bagi luas permukaan bongkah itu, dalam sebutan \(\pi\).
Calculate the minimum value for the surface area of the block, in terms of \(\pi\).
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[ \(3\) markah / \(3\) marks ]
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(c)
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Diberi bahawa \(L\) bertambah dengan kadar \(63\pi\) cm\(^2\)s\(^{-1}\), cari kadar pertambahan jejari ketika jejarinya ialah \(6\) cm.
It is given that \(L\) is increasing at a rate of \(63\pi\) cm\(^2\)s\(^{-1}\), find the increasing rate of radius when the radius is \(6\) cm.
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[ \(2\) markah / \(2\) marks ]
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(d)
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Cari perubahan kecil \(L\) apabila \(r\) menokok daripada \(6\) cm kepada \(6.002\) cm.
Find the small change in \(L\) when \(r\) increases from \(6\) cm to \(6.002\) cm.
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[ \(2\) markah / \(2\) marks ]
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Jawapan / Answer :
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Bahagian B
Section B
[30 markah]
[30 marks]
Jawab mana-mana tiga soalan dari bahagian ini.
Answer any three questions in this section.
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8.
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(a)
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Rajah \(5\) menunjukkan sebahagian lengkung \(y=27-(x-2)^3\) yang bersilang dengan paksi-\(x\) pada titik \(P\). Cari,
Diagram \(5\) shows part of the curve \(y=27-(x-2)^3\) intersecting the \(x\)-axis at point \(P\). Find,
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(i)
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koordinat \(P\),
coordinates of \(P\),
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[ \(2\) markah / \(2\) marks ]
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(ii)
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luas rantau berlorek.
the area of the shaded region.
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[ \(4\) markah / \(4\) marks ]
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Rajah \(5\) / Diagram \(5\)
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(b)
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Rajah \(6\) menunjukkan rantau berlorek yang dibatasi lengkung \(y=\sqrt{3x+4}\), garis \(x=k\), dan \(x=4\). Apabila rantau berlorek diputarkan \(360^\circ\) pada paksi-\(x\), isi padu yang terjana adalah \(26\pi\) unit\(^3\). Cari nilai \(k\).
Diagram \(6\) shows the shaded region bounded by the curve \(y=\sqrt{3x+4}\), the line \(x=k\), and \(x=4\). When the shaded region is rotated through \(360^\circ\) about the \(x\)-axis, the volume generated is \(26\pi\) unit\(^3\). Find the value of \(k\).
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[ \(4\) markah / \(4\) marks ]
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Rajah \(6\) / Diagram \(6\)
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9.
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(a)
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Buktikan identiti \(\sin{(x+\dfrac{\pi}{6})}-\sin{(x-\dfrac{\pi}{6})}=\text{kos }x\).
Prove the identities \(\sin{(x+\dfrac{\pi}{6})}-\sin{(x-\dfrac{\pi}{6})}=\cos{x}\).
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[ \(2\) markah / \(2\) marks ]
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(b)
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Lakarkan graf \(y=3\text{ kos }2x+2\) bagi \(0\lt x\lt \pi\). Seterusnya, tentukan bilangan penyelesaian bagi persamaan trigonometri \(3\pi\text{ kos }2x=8x-\pi\).
Sketch the graph \(y=3\cos{2x}+2\) for \(0\lt x\lt \pi\). Then, determine the number of solutions for trigonometric equations \(3\pi \cos{2x}=8x-\pi\).
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[ \(5\) markah / \(5\) marks ]
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(c)
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Cari nilai \(x\) yang tercangkum di antara \(0^\circ\) dengan \(360^\circ\) yang memuaskan persamaan \(\sin{2x}+\text{kos }x=0\).
Find the values of \(x\) that range from \(0^\circ\) to \(360^\circ\) that satisfy the following equations \(\sin{2x}+\cos{x}=0\).
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[ \(3\) markah / \(3\) marks ]
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Jawapan / Answer :
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10.
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Gunakan kertas graf untuk menjawab soalan ini.
Use graph paper to answer this question.
Pemboleh ubah \(x\) dan \(y\) dihubungkan oleh persamaan \(y=mn^{-x}\), dengan keadaan \(m\) dan \(n\) ialah pemalar. Nilai-nilai pemboleh ubah, \(x\) dan \(y\) yang diperoleh daripada eksperimen ditunjukkan dalam Jadual \(1\). Salah satu daripada nilai \(y\) tersalah catat.
The variables \(x\) and \(y\) are related by the equation \(y=mn^{-x}\), where \(m\) and \(n\) are constants. The values of variables \(x\) and \(y\) obtained from an experiment shown in Table \(1\). One of the values of \(y\) is wrongly recorded.
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\(1\) |
\(2\) |
\(3\) |
\(4\) |
\(5\) |
\(6\) |
\(7\) |
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\(56.2\) |
\(29.9\) |
\(25.1\) |
\(8.91\) |
\(6.31\) |
\(3.35\) |
\(1.78\) |
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Jadual \(1\) / Table \(1\)
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(a)
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Plot \(\log_{10}{y}\) melawan \(x\), menggunakan skala \(2\) cm kepada \(1\) unit pada paksi-\(x\) dan \(2\) cm kepada \(0.2\) unit pada paksi-\(\log_{10}{y}\). Seterusnya lukis garis lurus penyuaian terbaik.
Plot \(\log_{10}{y}\) against \(x\), using a scale of \(2\) cm to \(1\) unit on the \(x\)-axis and \(2\) cm to \(0.2\) unit on the \(\log_{10}{y}\)-axis. Hence, draw the line of best fit.
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[ \(4\) markah / \(4\) marks ]
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(b)
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Gunakan graf anda di soalan \(10\)(a) untuk mencari
Use the graph in question \(10\)(a) to find
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(i)
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nilai \(y\) yang tersalah catat dan nyatakan nilai \(y\) yang betul.
The value of \(y\) that is wrongly recorded and state the correct value of \(y\).
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[ \(1\) markah / \(1\) mark ]
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(ii)
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nilai \(m\) dan \(n\).
the value of \(m\) and \(n\).
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[ \(5\) markah / \(5\) marks ]
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Jawapan / Answer :
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(a)
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Rujuk graf di muka surat sebelah.
Refer graph on the next page.
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(b)
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Graf untuk soalan \(10\)(a)
Graph for question \(10\)(a) |
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11.
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(a)
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Didapati bahawa \(20\%\) daripada murid sekolah rendah di daerah Muar mengalami kekurangan zat makanan.
It is found that \(20\%\) of primary school students in Muar district suffer from malnutrition.
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(i)
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Jika jumlah murid yang mengalami kekurangan zat makanan adalah \(50\), hitung jumlah keseluruhan murid sekolah rendah di daerah tersebut.
If the number of students suffering from malnutrition is \(50\), calculate the total number of primary school students in the district.
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(ii)
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Jika \(10\) orang murid dari daerah ini dipilih secara rawak, cari kebarangkalian bahawa tepat empat orang murid mengalami kekurangan zat makanan.
If \(10\) students from the district are selected at random, find the probability that exactly four of them are malnutrition.
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[ \(4\) markah / \(4\) marks ]
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(b)
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Jisim bayi yang dilahirkan di sebuah hospital bertaburan normal dengan min \(3.1\) kg dan varians \(0.09\) kg.
The masses of babies born in a hospital are normally distributed with a mean of \(3.1\) kg and variance \(0.09\) kg.
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(i)
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Cari kebarangkalian bahawa seorang bayi yang lahir di hospital itu mempunyai jisim antara \(2.9\) kg dengan \(3.3\) kg.
Find the probability that a baby born in that hospital has a mass between \(2.9\) kg and \(3.3\) kg.
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(ii)
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Jika \(25\%\) daripada bayi yang dilahirkan di hospital tersebut dikategorikan sebagai kurang berat, cari jisim maksimum bagi kategori ini.
If \(25\%\) of babies born in that hospital are categorised as underweight, find the maximum mass of this category.
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[ \(6\) markah / \(6\) marks ]
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Jawapan / Answer :
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Bahagian C
Section C
[20 markah]
[20 marks]
Jawab mana-mana dua soalan dari bahagian ini.
Answer any two questions in this section.
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12.
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Jadual \(2\) menunjukkan indeks harga bagi empat komponen perbelanjaan pelancong ke Negeri Sarawak pada tahun \(2019\) dan tahun \(2022\) dengan menggunakan tahun \(2012\) sebagai tahun asas dan pemberat yang sepadan.
Table \(2\) shows the price index of the four components of tourist's expenditure to the State of Sarawak in the year \(2019\) and the year \(2022\) using the year \(2012\) as the base year and the corresponding weightage.
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Komponen / Perbelanjaan
Expenditure / Component |
Indeks harga pada \(2019\) berasaskan \(2012\)
Price index in \(2019\) based on \(2012\) |
Perubahan indeks harga tahun \(2019\) kepada \(2022\)
The changes price index in \(2019\) to \(2022\) |
Pemberat
Weightage |
Makanan dan minuman
Food and drinks |
\(q\) |
Meningkat \(10\%\)
Increase by \(10\%\) |
\(2p\) |
Membeli-belah
Shopping |
\(121\) |
Tidak berubah
Unchanged |
\(6\) |
Penginapan
Accomodation |
\(112\) |
Tidak berubah
Unchanged |
\(3\) |
Tiket aktiviti
Activity's ticket |
\(140\) |
Meningkat \(10\%\)
Increase by \(10\%\) |
\(p\) |
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Jadual \(2\) / Table \(2\)
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(a)
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Diberi harga komponen makanan dan minuman pada tahun \(2019\) dan tahun \(2012\) masing-masing ialah RM \(100\) dan RM \(80\). Cari nilai \(q\).
Given the prices of component food and drinks in the year \(2019\) and \(2012\) are RM \(100\) and RM \(80\) respectively. Find the value of \(q\).
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[ \(2\) markah / \(2\) marks ]
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(b)
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(i)
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Hitung indeks harga bagi komponen makanan dan minuman dan tiket aktiviti pada tahun \(2022\) berasaskan tahun \(2012\),
Calculate the price index of components of food and drinks and of activity's ticket in the year \(2022\) based on the year \(2012\),
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(ii)
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Seterusnya, cari nilai \(p\), jika indeks gubahan bagi harga komponen-komponen itu pada tahun \(2022\) berasaskan \(2012\) ialah \(128\).
Hence, find the value of \(p\), if the composite index for the prices of these components in the year \(2022\) based on the year \(2012\) is \(128\).
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[ \(5\) markah / \(5\) marks ]
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(c)
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(i)
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Komponen manakah menunjukkan perbelanjaan paling penting bagi pelancong Negeri Sarawak? Berikan justifikasi anda.
Which component shows the most important expenditure for Sarawak State tourists? Give your justification.
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(ii)
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Pada tahun \(2025\), indeks harga bagi komponen di (c)(i) berasaskan tahun \(2012\) dijangka menjadi \(151.25\). Kirakan peratus peningkatan indeks harga komponen tersebut pada tahun \(2025\) berasaskan tahun \(2022\).
In the year \(2025\), the price index for the component in (c)(i) based on the year \(2012\) is predicted to be \(151.25\). Calculate the percentage increase in the price index of the component in the year \(2025\) based on the year \(2022\).
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[ \(3\) markah / \(3\) marks ]
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Jawapan / Answer :
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13.
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Rajah \(7\) menunjukkan sebuah sisi empat \(ABCD\). Diberi bahawa \(\angle{ADC}\) ialah sudut cakah.
Diagram \(7\) shows a quadrilateral of \(ABCD\). Given that \(\angle{ADC}\) is an obtuse angle.
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Rajah \(7\) / Diagram \(7\)
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(a)
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Tanpa sebarang pengiraan, nyatakan titik yang paling jauh dari titik \(A\). Berikan sebab kepada jawapan anda.
Without any calculation, state the point, which is furthest from point \(A\). Give reason for your answer.
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[ \(1\) markah / \(1\) mark ]
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(b)
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Kirakan
Calculate
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(i)
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\(\angle{ADC}\),
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(ii)
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luas kawasan, dalam cm\(^2\), sisi empat \(ABCD\).
area of region, in cm\(^2\), the quadrilateral \(ABCD\).
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[ \(6\) markah / \(6\) marks ]
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(c)
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Garis \(CD\) dipanjangkan kepada titik \(E\) dengan keadaan titik \(E\) ialah titik yang berada pada jarak yang paling dekat dari \(A\) ke \(CD\).
Line \(CD\) is extended to point \(E\) such as point \(E\) is a point that lies at a shortest distance from \(A\) to \(CD\).
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(i)
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Pada Rajah \(7\), tandakan titik \(E\).
On Diagram \(7\), mark point \(E\).
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(ii)
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Cari jarak terdekat \(E\) dari \(A\) ke \(CD\).
Find the shortest distance of \(E\) from \(A\) to \(CD\).
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[ \(3\) markah / \(3\) marks ]
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Jawapan / Answer :
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14.
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Bengkel Perabot Ros menghasilkan dua jenis kerusi, \(P\) dan \(Q\). Penghasilan setiap kerusi melibatkan dua proses, iaitu membuat dan mengecat.
Jadual \(3\) menunjukkan tempoh masa yang diambil untuk proses membuat dan mengecat seunit kerusi \(P\) dan seunit kerusi \(Q\).
Ros workshop produces two types of chair, \(P\) and \(Q\). The production of each type of chair involves two processes, making and painting.
Table \(3\) shows the time taken to make and paint a unit of chair of type \(P\) and a unit of chair of type \(Q\).
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Kerusi
Chair |
Tempoh masa membuat (minit)
Time taken in making (minutes) |
Tempoh masa mengecat (minit)
Time taken in painting (minutes) |
| \(P\) |
\(45\) |
\(50\) |
| \(Q\) |
\(30\) |
\(70\) |
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Jadual \(3\) / Table \(3\)
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Jumlah masa maksimum untuk membuat kedua-dua kerusi adalah \(5.5\) jam dan masa untuk mengecat kedua-dua kerusi adalah sekurang-kurangnya \(5\) jam \(50\) minit. Nisbah bilangan kerusi \(P\) kepada bilangan kerusi \(Q\) adalah tidak melebihi \(4:5\). Bengkel Perabot Ros menghasilkan \(x\) unit kerusi \(P\) dan \(y\) unit kerusi \(Q\) sehari.
The maximum total time for making both chairs is \(5.5\) hours and the time for painting both chairs is at least \(5\) hours and \(50\) minutes. The ratio of the number of chair \(P\) to the number of chair \(Q\) is not more than \(4:5\). Ros Workshop produces \(x\) unit of chair \(P\) and \(y\) unit of chair \(Q\) per day.
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(a)
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Tulis tiga ketaksamaan, selain daripada \(x\ge0\) dan \(y\ge0\), yang memenuhi semua kekangan di atas.
Write three inequalities, other than \(x\ge0\) and \(y\ge0\), which satisfy all the above constraints.
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[ \(3\) markah / \(3\) marks ]
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(b)
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Menggunakan skala \(2\) cm kepada \(1\) unit kerusi pada kedua-dua paksi, bina dan lorek rantau \(\textbf{R}\) yang memenuhi semua kekangan di atas.
Using a scale of \(2\) cm to a unit of chair on both axes, construct and shade the region \(\textbf{R}\) which satisfies all the above constraints.
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[ \(3\) markah / \(3\) marks ]
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(c)
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Gunakan graf yang dibina di \(14\)(b) untuk menjawab soalan-soalan berikut:
Use the graph constructed in \(14\)(b) to answer the following questions:
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(i)
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Julat bilangan kerusi \(Q\) jika bilangan kerusi \(P\) yang dihasilkan ialah \(3\) unit sehari.
The range of chair \(Q\) if \(3\) units of chair \(P\) are produced per day.
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(ii)
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Keuntungan maksimum sehari jika keuntungan yang diperoleh daripada satu unit kerusi \(P\) dan satu unit kerusi \(Q\) masing-masing ialah RM \(16\) dan RM \(10\).
The maximum profit per day if the profit obtained from a unit of chair \(P\) and a unit of chair \(Q\) are RM \(16\) and RM \(10\) respectively.
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[ \(4\) markah / \(4\) marks ]
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Jawapan / Answer :
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Graf untuk soalan \(14\)(b)
Graph for question \(14\)(b) |
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15.
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Penyelesaian secara lakaran graf tidak diterima.
Solution by graph sketching is not accepted.
Suatu zarah bergerak di sepanjang garis lurus dan melalui satu titik tetap \(O\). Halajunya, \(v\) ms\(^{-1}\), \(t\) saat selepas melalui \(O\) diberi oleh \(v=t^2-pt+15\), dengan keadaan \(p\) ialah pemalar. Pecutan zarah itu ialah \(4\) ms\(^{-2}\) apabila \(t=6\) s. Zarah itu berhenti seketika di titik \(A\) dan titik \(B\) di sepanjang garis lurus itu.
A particle moves along a straight line from a fixed point \(O\). Its velocity, \(v\) ms\(^{-1}\), \(t\) seconds after point \(O\) is given by \(v=t^2-pt+15\), where \(p\) is a constant. Given that the acceleration of the particle is \(4\) ms\(^{-2}\) when \(t=6\) s. The particle stops at points \(A\) and \(B\) along the straight line.
[Anggap pergerakan ke kanan adalah positif]
[Assume it's motion to the right is positive]
Cari
Find
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(a)
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(i)
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nilai \(p\),
the value of \(p\),
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[ \(2\) markah / \(2\) marks ]
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(ii)
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masa, dalam saat, apabila zarah berada di \(A\) dan \(B\),
the time, in seconds, when the particle is at \(A\) and \(B\),
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[ \(2\) markah / \(2\) marks ]
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(b)
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Lakarkan graf halaju-masa zarah bagi \(0\le t\le 5\). Seterusnya tentukan jumlah jarak, dalam m, yang dilalui oleh zarah itu dalam \(5\) saat yang pertama.
Sketch the velocity-time graph of the particle for \(0\le t\le 5\). Hence, find the total distance, in m, travelled by the particle in the first \(5\) seconds.
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[ \(6\) markah / \(6\) marks ]
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Jawapan / Answer :
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KERTAS PEPERIKSAAN TAMAT
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