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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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\(a^m\div a^n=a^{m-n}\) |
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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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\(\log_a{m^n}=n\log_a{m}\) |
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\({}^nP_r=\dfrac{n!}{(n-r)!}\) |
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8.
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\(\log_a{b}=\dfrac{\log_c{b}}{\log_c{a}}\) |
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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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\(S_n=\dfrac{n}{2}[2a+(n-1)d]\) |
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Min / Mean, \(\mu=np\)
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11.
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\(T_n=ar^{n-1}\) |
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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
[64 markah]
[64 marks]
Jawab semua soalan.
Answer all questions.
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| 1. |
Diberi \(g(x)=\dfrac{ax+b}{cx+d}\), \(x\ne k\). |
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Given \(g(x)=\dfrac{ax+b}{cx+d}\), \(x\ne k\).
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(a)
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Ungkapkan \(k\) dalam sebutan \(c\) dan \(d\). Seterusnya, cari \(g^{-1}(x)\).
Express \(k\) in terms of \(c\) and \(d\). Hence, find \(g^{-1}(x)\).
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[ \(3\) markah / \(3\) marks ]
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(b)
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Dengan menggunakan fungsi \(g^{-1}(x)\) yang diperoleh, tentukan \(g^{-1}(x)\) bagi setiap fungsi berikut:
Using function \(g^{-1}(x)\) obtained, determine \(g^{-1}(x)\) for each of the following functions:
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(i)
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\(g(x)=\dfrac{7x-4}{2x-3}\), \(x\ne\dfrac{3}{2}\)
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(ii)
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\(g(x)=\dfrac{2x-3}{x-5}\), \(x\ne 5\)
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[ \(2\) markah / \(2\) marks ]
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(c)
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Jika \(c\ne 0\), apakah syarat ke atas \(a\) dan \(d\) supaya \(g=g^{-1}\)?
If \(c\ne 0\), what is the condition on \(a\) and \(d\) so that \(g=g^{-1}\)?
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[ \(1\) markah / \(1\) mark ]
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Jawapan / Answer :
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2.
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Cari julat nilai \(x\) bagi \(5\lt 2x^2+x+4\) dan \(2x^2+x+4\lt 10\).
Seterusnya selesaikan ketaksamaan \(5\lt 2x^2+x+4\lt 10\).
Find the range of values \(x\) for \(5\lt 2x^2+x+4\) and \(2x^2+x+4\lt 10\).
Hence, solve the inequality \(5\lt 2x^2+x+4\lt 10\).
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[ \(4\) markah / \(4\) marks ]
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Jawapan / Answer :
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3.
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(a)
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Tentukan sama ada persamaan berikut ialah sistem persamaan linear dalam tiga pemboleh ubah atau bukan.
Terangkan.
Determine whether the following equations are system of linear equations in three variables or not.
Explain.
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\(\begin{aligned} x-y+3z+6&=0 \\ 4xy+2z-y&=10 \\ 7y+3x-z&=8 \end{aligned}\) |
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[ \(1\) markah / \(1\) mark ]
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(b)
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Selesaikan sistem persamaan linear berikut dan jelaskan hasil dapatan.
Solve the following system of linear equations and explain the result of the findings.
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\(\begin{aligned} x-2y&=4 \\ 2x-3y+2z&=-2 \\ 4x-7y+2z&=6 \end{aligned}\) |
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[ \(4\) markah / \(4\) marks ]
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Jawapan / Answer :
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4.
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(a)
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Permudahkan / Simplify: |
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\(3^{-n}(15^n-27^n)\)
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(b)
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Ungkapkan \(h\) dalam sebutan \(a\) dan \(b\) jika penyelesaian bagi persamaan \(\sqrt{4x}-\sqrt{x}=\dfrac{3}{\sqrt{a}-\sqrt{b}}\), dengan keadaan \(a\) dan \(b\) merupakan pemalar ialah \(x=h(\sqrt{a}+\sqrt{b})^2\).
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Express \(h\) in terms of \(a\) and \(b\) if the solution to an equation \(\sqrt{4x}-\sqrt{x}=\dfrac{3}{\sqrt{a}-\sqrt{b}}\), such that \(a\) and \(b\) are constants is \(x=h(\sqrt{a}+\sqrt{b})^2\).
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(c)
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Diberi bahawa \(n=\ln{m}+1\) dengan keadaan \(m\) suatu nombor nisbah dan merupakan penyelesaian bagi persamaan \(5e^{n-3}=e^{-2}\), cari nilai \(m\).
Given that \(n=\ln{m}+1\), such that \(m\) is a rational number and is the solution to an equation \(5e^{n-3}=e^{-2}\), find the value of \(m\).
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[ \(7\) markah / \(7\) marks ]
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Jawapan / Answer :
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5.
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(a)
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Pak Abu memulakan perniagaan roti canai dengan menjual sebanyak \(100\) keping sehari manakala Pak Kasim memulakan perniagaan roti canai dengan menjual sebanyak \(400\) keping sehari. Setiap hari jualan restoran Pak Abu meningkat sebanyak \(4\) keping manakala jualan restoran Pak Kasim meningkat \(2\) keping. Pada hari ke-\(n\) perniagaan, bilangan roti canai yang di jual di restoran Pak Abu kali pertama melebihi jualan roti canai di restoran Pak Kasim. Cari nilai \(n\).
Pak Abu started the roti canai business by selling \(100\) pieces a day while Pak Kasim started the roti canai business by selling \(400\) pieces a day. Every day the sales of Pak Abu's restaurant increased by \(4\) pieces while the sales of Pak Kasim's restaurant increased by \(2\) pieces. On the \(n^\text{th}\) day of business, the number of roti canai sold at Pak Abu's restaurant for the first time exceeded the sale of roti canai at Pak Kasim's restaurant. Find the value of \(n\).
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(b)
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Rajah \(1\) menunjukkan Alia Qistina bermain buaian yang dilepaskan dari suatu kedudukan tertentu.
Diagram \(1\) shows Alia Qistina plays a swing released from a specific position.
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Rajah \(1\) / Diagram \(1\)
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Jarak ayunan lengkap pertama ialah \(400\) cm dan jarak ayunan lengkap yang berikutnya ialah \(15\%\) kurang dari jarak sebelumnya. Cari bilangan ayunan lengkap jika hasil tambah jarak ayunan lengkap kurang daripada \(2434\) cm. Seterusnya, tentukan ayunan lengkap ke berapakah menghasilkan jarak ayunan lengkapnya kali pertama kurang daripada \(40\) cm.
The distance of the first complete oscillation is \(400\) cm and the distance of the next complete oscillation is \(15\%\) less than the previous distance. Find the number of complete oscillations if the sum of the complete oscillation distances is less than \(2434\) cm. Hence, determine the complete oscillation to what extent produces its complete oscillation distance the first time less than \(40\) cm.
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[ \(6\) markah / \(6\) marks ]
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Jawapan / Answer :
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6.
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Rajah \(2\) dan Rajah \(3\) menunjukkan graf garis lurus yang diperoleh dari suatu persamaan tak linear \(\dfrac{y}{x}=mx^2-nx\).
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Diagram \(2\) and Diagram \(3\) shows a straight line obtained by plotting from non-linear equation \(\dfrac{y}{x}=mx^2-nx\).
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Rajah \(2\) / Diagram \(2\)
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Rajah \(3\) / Diagram \(3\)
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Tunjukkan bahawa \(n=6m-18\).
Shows that \(n=6m-18\).
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[ \(4\) markah / \(4\) marks ]
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Jawapan / Answer :
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7.
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Encik Salleh mempunyai sebidang tanah berbentuk segi tiga. Dua sisi tanah tersebut mempunyai panjang \((2x+10)\) m dan \((5x-20)\) m masing-masing. Sudut di antara kedua-dua sisi tersebut ialah \(30^\circ\). Cari panjang kedua-dua sisi tanah tersebut dalam integer terhampir, jika luas tanah tersebut ialah \(1700\) m\(^2\).
Mr. Salleh has a triangular piece of land. The two sides of the land have lengths \((2x+10)\) m and \((5x-20)\) m respectively. The angle between the two sides is \(30^\circ\). Find the length of both sides of the land to the nearest integer, if the area of the land is \(1700\) m\(^2\).
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[ \(5\) markah / \(5\) marks ]
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Jawapan / Answer :
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8.
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Rajah \(4\) menunjukkan sebuah pasu bunga di mana permukaan sisi dalaman pasu itu boleh diwakili oleh persamaan \(y=ax^2\).
Diagram \(4\) shows a flower vase where the interior side surface of the vase can be represented by an equation \(y=ax^2\).
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Rajah \(4\) / Diagram \(4\)
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Dengan menggunakan ukuran yang diberi, cari isi padu pasu bunga itu dalam cm\(^3\). Tunjukkan jawapan dalam sebutan \(\pi\).
Using the measurements given, find the volume, in cm\(^3\) of the flower vase. Show the answer in term of \(\pi\).
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[ \(5\) markah / \(5\) marks ]
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Jawapan / Answer :
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9.
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Rajah \(5\) menunjukkan vektor \(\overrightarrow{OQ}\), \(\overrightarrow{OR}\) dan \(\overrightarrow{SO}\) di atas grid segi empat sama bersisi \(1\) unit.
Diagram \(5\) shows vectors \(\overrightarrow{OQ}\), \(\overrightarrow{OR}\) and \(\overrightarrow{SO}\) on a square grid with sides of \(1\) unit.
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Rajah \(5\) / Diagram \(5\)
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(a)
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Cari \(|-\overrightarrow{OQ}|\).
Find \(|-\overrightarrow{OQ}|\).
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[ \(2\) markah / \(2\) marks ]
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(b)
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Diberi bahawa \(\overrightarrow{OQ}=\utilde{q}\) dan \(\overrightarrow{OR}=\utilde{r}\), ungkapkan \(\overrightarrow{RQ}\) dalam sebutan \(\utilde{q}\) dan \(\utilde{r}\).
Given that \(\overrightarrow{OQ}=\utilde{q}\) and \(\overrightarrow{OR}=\utilde{r}\), express \(\overrightarrow{RQ}\) in terms of \(\utilde{q}\) and \(\utilde{r}\).
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[ \(2\) markah / \(2\) marks ]
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Jawapan / Answer :
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10.
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Rajah \(6\) menunjukkan sebuah separa bulatan berpusat \(O\) dan jejari \(10\) cm.
Diagram \(6\) shows a semi circle with centre \(O\) and radius of \(10\) cm.
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Rajah \(6\) / Diagram \(6\)
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(a)
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Cari panjang lengkok \(BC\) dalam sebutan \(\alpha\).
Find the length of arc \(BC\) in terms of \(\alpha\).
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[ \(2\) markah / \(2\) marks ]
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(b)
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Diberi bahawa panjang lengkok \(AB\) adalah sama dengan jejari separa bulatan tersebut, tentukan nilai \(\theta\) dalam radian. Seterusnya, cari luas kawasan berlorek.
[ Guna \(\pi=3.142\) ]
It is given that the length of arc \(AB\) is equal to the radius of semi circle, state the value of \(\theta\) in radians. Hence, find the area of the shaded region.
[ Use \(\pi=3.142\) ]
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[ \(4\) markah / \(4\) marks ]
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Jawapan / Answer :
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11.
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Rajah \(7\) menunjukkan pelan bagi sebuah kawasan kilang di Banting, Selangor.
Diagram \(7\) shows a factory plan area in Banting, Selangor.
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Rajah \(7\) / Diagram \(7\)
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Trapezium \(PQRS\) merupakan kawasan untuk PAD \(4\). Diberi jarak \(PS\), \(TS\), \(QR\) dan \(PQ\) masing-masing ialah \(600\) m, \(150\) m, \(900\) m dan
\(400\) m.
Trapezium \(PQRS\) is the area for PAD \(4\). Given the distances \(PS\), \(TS\), \(QR\) and \(PQ\) are \(600\) m, \(150\) m, \(900\) m and \(400\) m respectively.
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Dengan menggunakan \(\utilde{v}\) untuk mewakili \(50\) m pada aras \(PS\) dan \(\utilde{u}\) untuk mewakili \(100\) m pada arah \(PQ\), ungkapkan dalam sebutan \(\utilde{v}\) dan \(\utilde{u}\) bagi \(\overrightarrow{PR}\).
By using \(\utilde{v}\) to represent \(50\) m in the direction of \(PS\) and \(\utilde{u}\) to represent \(100\) m in the direction of \(PQ\), express in terms of \(\utilde{v}\) and \(\utilde{u}\) for \(\overrightarrow{PR}\).
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[ \(2\) markah / \(2\) marks ]
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(b)
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Titik \(X\) berada di dalam kawasan PAD \(4\) tersebut dengan keadaan \(\overrightarrow{TX}=m\overrightarrow{PQ}\) dan \(m\) ialah pemalar.
Point \(X\) located in the area of PAD \(4\) such that \(\overrightarrow{TX}=m\overrightarrow{PQ}\) and \(m\) is a constant.
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Ungkapkan \(\overrightarrow{TX}\) dalam sebutan \(m\) dan \(\utilde{u}\).
Express \(\overrightarrow{TX}\) in terms of \(m\) and \(\utilde{u}\).
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Seterusnya, jika titik \(P\), \(X\) dan \(R\) segaris, cari jarak \(TX\).
Hence, if the points \(P\), \(X\) and \(R\) are collinear, find the distance of \(TX\).
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[ \(5\) markah / \(5\) marks ]
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Jawapan / Answer :
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12.
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Sebuah balang mengandungi dua jenis guli berwarna merah dan biru dengan nisbah \(5:9\). Lapan biji guli dikeluarkan secara rawak satu demi satu dengan pengembalian.
A jar contains two types of marbles red and blue with ratio \(5:9\). Eight marbles are taken out randomly one by one with replacement.
Cari kebarangkalian bahawa
Find the probability that
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(a)
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tepat \(4\) biji guli berwarna biru yang dipilih,
exactly \(4\) blue marbles are selected,
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[ \(3\) markah / \(3\) marks ]
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(b)
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lebih daripada \(3\) biji guli berwarna merah yang dipilih.
more than \(3\) red marbles are selected.
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[ \(2\) markah / \(2\) marks ]
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Jawapan / Answer :
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Bahagian B
Section B
[16 markah]
[16 marks]
Bahagian ini mengandungi tiga soalan. Jawab dua soalan.
This section contains three questions. Answer two questions.
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13.
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Rajah \(8\) menunjukkan graf fungsi mutlak \(f(x)\) bagi domain \(0\le x\le 4\).
Diagram \(8\) shows the graph of absolute function \(f(x)\) for the domain \(0\le x\le 4\).
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Rajah \(8\) / Diagram \(8\)
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(a)
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Nyatakan nilai \(k\).
State the value of \(k\).
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[ \(1\) markah / \(1\) mark ]
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(b)
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Cari pintasan-\(x\) bagi graf tersebut.
Find the \(x\)-intercepts of the graph.
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[ \(3\) markah / \(3\) marks ]
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(c)
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Cari domain bagi \(f(x)\le1\).
Find the domain of \(f(x)\le1\).
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[ \(2\) markah / \(2\) marks ]
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(d)
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Cari nilai \(x\) yang dipetakan kepada diri sendiri dalam domain yang diberi.
Find the value of \(x\) that is mapped onto itself in the given domain.
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[ \(2\) markah / \(2\) marks ]
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Jawapan / Answer :
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14.
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Rajah \(9\) menunjukkan sebuah taman berbentuk sektor \(AOB\) bagi sebuah bulatan dengan pusat \(O\). Jejari dan sudut sektor itu masing-masing ialah \(R\) m dan \(2\theta\). Dalam taman ini, satu tapak berbentuk bulat akan ditanam dengan bunga ros. Diberi jejari tapak ini ialah \(r\) m.
Diagram \(9\) shows a garden in the form of sector \(AOB\) of a circle with a centre \(O\). The radius and angle of the sector are \(R\) m and \(2\theta\) respectively. In the garden, a circular plot is to be planted with roses. Given that the radius of the plot is \(r\) m.
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Rajah \(9\) / Diagram \(9\)
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(a)
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Tunjukkan bahawa \(R=r\left( 1+\dfrac{1}{\sin{\theta}} \right)\).
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Show that \(R=r\left( 1+\dfrac{1}{\sin{\theta}} \right)\).
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[ \(2\) markah / \(2\) marks ]
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(b)
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Diberi bahawa \(\theta=30^\circ\), cari nisbah bahagian taman yang akan ditanam dengan bunga ros.
Given that \(\theta=30^\circ\), find the ratio of the garden that is to be planted with roses.
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[ \(3\) markah / \(3\) marks ]
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(c)
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Seterusnya, hitung panjang pagar yang diperlukan untuk memagari keseluruhan kawasan berlorek jika \(R=15\).
Hence, calculate the length of fencing required to fence along the shaded region when \(R=15\).
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[ \(3\) markah / \(3\) marks ]
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Jawapan / Answer :
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15.
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Fifi, seorang ahli kimia mempunyai tiga jenis larutan. Larutan yang pertama mengandungi \(10\%\) asid, larutan kedua mengandungi \(40\%\) asid manakala larutan yang ketiga pula mengandungi \(60\%\) asid.
Pada suatu hari, Fifi ingin membuat larutan yang menggunakan tiga jenis larutan. Fifi ingin menyediakan \(500\) liter campuran larutan yang mengandungi \(45\%\) asid. Bekalan larutan yang mengandungi \(10\%\) asid adalah dua kali bekalan yang mengandungi \(40\%\) asid. Bolehkah anda mencadangkan berapakah isipadu setiap larutan yang perlu digunakan oleh Fifi?
Fifi, a chemist, has three types of solutions. The first solution contain \(10\%\) of acid, the second solution \(40\%\) of acid and the third solution \(60\%\) of acid.
One day, Fifi intends to prepare a solution from three types of solutions. Fifi intends to prepare \(500\) litres of mixed solution with an acid concentration of \(45\%\). The available amount of \(10\%\) acid solution is twice the amount of \(40\%\) acid solution. How much of each solution would you suggest Fifi use?
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[ \(8\) markah / \(8\) marks ]
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Jawapan / Answer :
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KERTAS PEPERIKSAAN TAMAT
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