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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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\(\log_a{mn}=\log_a{m}+\log_a{n}\) |
19.
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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 pemetaan bagi dua fungsi.
Diagram \(1\) shows the mapping for the two functions.
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Rajah \(1\) / Diagram \(1\)
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Diberi \(g^{-1}(x)=x-3\), cari
Given \(g^{-1}(x)=x-3\), find
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(a)
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nilai \(p\),
the value of \(p\),
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[ \(3\) markah / \(3\) marks ]
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(b)
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(i)
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\(f^2(x)\),
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[ \(2\) markah / \(2\) marks ]
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(ii)
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fungsi \(f^n(x)\) dalam sebutan \(n\) dan \(x\).
the function \(f^n(x)\) in terms of \(n\) and \(x\).
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[ \(2\) markah / \(2\) marks ]
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Jawapan / Answer :
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2.
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Diberi bahawa \(\sqrt{a+b\sqrt{2}}=\dfrac{7}{3-\sqrt{2}}\), di mana \(a\) dan \(b\) ialah pemalar.
It is given that \(\sqrt{a+b\sqrt{2}}=\dfrac{7}{3-\sqrt{2}}\), where \(a\) and \(b\) are constants.
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(a)
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Cari nilai \(a\) dan \(b\).
Find the value of \(a\) and \(b\).
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[ \(4\) markah / \(4\) marks ]
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(b)
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Seterusnya, selesaikan \(e^{2\ln{a}}+e^{2\ln b}\).
Hence, solve \(e^{2\ln{a}}+e^{2\ln b}\).
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[ \(2\) markah / \(2\) marks ]
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Jawapan / Answer :
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3.
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Selesaikan persamaan serentak berikut:
Solve the simultaneous equations:
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\(3x-y=2x^2+3y^2-5xy-14=2\)
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[ \(5\) markah / \(5\) marks ]
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Jawapan / Answer :
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4.
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Lengkung fungsi kuadratik \(f(x)=-(x-p)^2+3q\) menyilang paksi-\(x\) pada titik-titik \((-1,0)\) dan \((5,0)\). Garis lurus \(y=9\) adalah tangen kepada titik maksimum lengkung itu.
The curve of a quadratic function \(f(x)=-(x-p)^2+3q\) intersects the \(x\)-axis at points \((-1,0)\) and \((5,0)\). The straight line \(y=9\) is tangent to the maximum point of the curve.
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(a)
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Cari nilai \(p\) dan \(q\).
Find the value of \(p\) and \(q\).
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[ \(3\) markah / \(3\) marks ]
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(b)
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Seterusnya, lakar graf \(f(x)\) untuk \(0\le x\le 6\).
Hence, sketch the graph of \(f(x)\) for \(0\le x\le 6\).
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[ \(3\) markah / \(3\) marks ]
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(c)
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Tulis persamaan bagi lengkung, jika graf itu dipantulkan pada:
Write the equation of the curve, if the graph is reflected about:
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(i)
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paksi-\(x\)
\(x\)-axis
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[ \(1\) markah / \(1\) mark ]
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(ii)
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paksi-\(y\)
\(y\)-axis
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[ \(1\) markah / \(1\) mark ]
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Jawapan / Answer :
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5.
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Rajah \(2\) menunjukkan sisi empat \(AEFG\) dan segi tiga \(ABD\) dengan keadaan \(BD\) dan \(AG\) bersilang pada titik \(C\).
Diagram \(2\) shows quadrilateral \(AEFG\) and triangle \(ABD\) such that \(BD\) and \(AG\) intersect at point \(C\).
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Rajah \(2\) / Diagram \(2\)
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Diberi bahawa \(\overrightarrow{AB}=4\utilde{x}\), \(\overrightarrow{CB}=-\dfrac{1}{2}\utilde{y}\), \(\overrightarrow{GF}=\dfrac{3}{2}\utilde{y}\), \(\overrightarrow{EF}=3\utilde{x}\) dan nisbah \(BC:CD=2:3\).
Given that \(\overrightarrow{AB}=4\utilde{x}\), \(\overrightarrow{CB}=-\dfrac{1}{2}\utilde{y}\), \(\overrightarrow{GF}=\dfrac{3}{2}\utilde{y}\), \(\overrightarrow{EF}=3\utilde{x}\) and ratio \(BC:CD=2:3\).
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(a)
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Ungkapkan dalam sebutan \(\utilde{x}\) dan \(\utilde{y}\) :
Express in terms of \(\utilde{x}\) and \(\utilde{y}\) :
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(i)
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\(\overrightarrow{AC}\),
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(ii)
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\(\overrightarrow{AD}\).
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[ \(3\) markah / \(3\) marks ]
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(b)
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Diberi \(\overrightarrow{AG}\) ialah \(\lambda\) kali ganda \(\overrightarrow{AC}\) dan \(\overrightarrow{AE}\) ialah \(\mu\) kali ganda \(\overrightarrow{AD}\).
Ungkapkan vektor \(\overrightarrow{AF}\) dalam sebutan
Given \(\overrightarrow{AG}\) is \(\lambda\) times of \(\overrightarrow{AC}\) and \(\overrightarrow{AE}\) is \(\mu\) times of \(\overrightarrow{AD}\).
Express vector \(\overrightarrow{AF}\) in terms of
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(i)
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\(\lambda\), \(\utilde{x}\) dan / and \(\utilde{y}\),
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(ii)
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\(\mu\), \(\utilde{x}\) dan / and \(\utilde{y}\). |
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Seterusnya, cari nilai \(\lambda\) dan nilai \(\mu\).
Hence, find the value of \(\lambda\) and of \(\mu\).
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[ \(5\) markah / \(5\) marks ]
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Jawapan / Answer :
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6.
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(a)
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Diberi pembezaan peringkat pertama bagi \(3x(x^2+5)^4\) ialah \(3(x^2+5)^3(hx^2+5)\). Cari nilai \(h\).
Given the first derivative of \(3x(x^2+5)^4\) is \(3(x^2+5)^3(hx^2+5)\). Find the value of \(h\).
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[ \(3\) markah / \(3\) marks ]
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(b)
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Rajah \(3\) menunjukkan sebuah kon terbalik dengan diameter tapak \(12\) cm dan tinggi \(18\) cm. Diberi tinggi air dalam kon berkenaan ialah \(h\) cm dan jejari permukaan air ialah \(r\) cm.
Diagram \(3\) shows an inverted cone with a base diameter of \(12\) cm and a height of \(18\) cm. Given the height of water in the cone is \(h\) cm and the radius of water surface is \(r\) cm.
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Rajah \(3\) / Diagram \(3\)
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Air mengalir keluar melalui lubang kecil di bahagian bucu bawah kon berkenaan.
Hitung perubahan kecil bagi tinggi air jika isipadu air menyusut daripada \(8\pi\) cm\(^3\) kepada \(7.5\pi\) cm\(^3\).
The water leaks out through a small hole at the tip of the cone.
Calculate the small changes in the height of water when the volume decreases from \(8\pi\) cm\(^3\) to \(7.5\pi\) cm\(^3\).
\(\left[V_\text{kon/cone}=\dfrac{1}{3}\pi r^2h\right]\)
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[ \(5\) markah / \(5\) marks ]
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Jawapan / Answer :
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7.
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Hasil tambah \(n\) sebutan pertama suatu janjang aritmetik diberi oleh \(S_n=an^2+bn\).
The sum of the first \(n\) terms of an arithmetic progression is given by \(S_n=an^2+bn\).
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(a)
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Ungkapkan \(T_n\) dalam bentuk termudah, dalam sebutan \(a\), \(b\) dan \(n\).
Express \(T_n\) in simplest form, in terms of \(a\), \(b\) and \(n\).
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[ \(2\) markah / \(2\) marks ]
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(b)
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Diberi bahawa \(S_4=44\) dan \(S_8=152\). Cari nilai \(a\) dan \(b\).
Given that \(S_4=44\) and \(S_8=152\). Find the values of \(a\) and \(b\).
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[ \(3\) markah / \(3\) marks ]
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(c)
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Cari beza sepunya
Find the common different
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[ \(3\) markah / \(3\) 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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Dalam Rajah \(4\), \(OAB\) ialah sebuah segi tiga. Diberi \(\overrightarrow{OP}=\dfrac{2}{3}\overrightarrow{OA}\), \(\overrightarrow{AB}=2\overrightarrow{AQ}\), \(\overrightarrow{OR}=\dfrac{4}{5}\overrightarrow{OQ}\), \(\overrightarrow{OA}=9h\) dan \(\overrightarrow{OB}=4k\).
In Diagram \(4\), \(OAB\) is a triangle. Given that \(\overrightarrow{OP}=\dfrac{2}{3}\overrightarrow{OA}\), \(\overrightarrow{AB}=2\overrightarrow{AQ}\), \(\overrightarrow{OR}=\dfrac{4}{5}\overrightarrow{OQ}\), \(\overrightarrow{OA}=9h\) and \(\overrightarrow{OB}=4k\).
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Rajah \(4\) / Diagram \(4\)
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(a)
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Ungkapkan dalam sebutan \(h\) dan/atau \(k\).
Express, in terms of \(h\) and/or \(k\).
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(i)
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\(\overrightarrow{PB}\)
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(ii)
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\(\overrightarrow{OQ}\)
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[ \(3\) markah / \(3\) marks ]
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(b)
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Seterusnya, buktikan bahawa titik \(P\), \(R\) dan \(B\) adalah segaris.
Hence, prove that points \(P\), \(R\) and \(B\) are collinear.
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[ \(4\) markah / \(4\) marks ]
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(c)
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Diberi luas \(PAB\) ialah \(12\) cm\(^2\), cari luas segi tiga \(OAB\).
Given the area of triangle \(PAB\) is \(12\) cm\(^2\), find the area of \(OAB\).
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[ \(3\) markah / \(3\) marks ]
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Jawapan / Answer :
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9.
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Gunakan kertas graf untuk menjawab soalan ini.
Use paper graph to answer this question.
Jadual \(1\) menunjukkan nilai-nilai bagi \(2\) pemboleh ubah \(x\) dan \(y\) yang diperoleh dari suatu eksperimen. Pemboleh ubah \(x\) dan \(y\) dihubungkan oleh persamaan \(y=rs^{2x-1}\), dengan keadaan \(r\) dan \(s\) ialah pemalar.
Table \(1\) shows the values of two variables, \(x\) and \(y\) obtained from an experiment. The variables \(x\) and \(y\) are related by the equation \(y=rs^{2x-1}\), where \(r\) and \(s\) are constants.
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| \(x\) |
\(1\) |
\(2\) |
\(3\) |
\(4\) |
\(5\) |
\(6\) |
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\(0.24\) |
\(0.35\) |
\(0.5\) |
\(0.72\) |
\(1.03\) |
\(1.49\) |
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Jadual \(1\) / Table \(1\)
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(a)
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Berdasarkan Jadual \(1\), bina jadual bagi nilai-nilai \(\log_{10}y\) dan \((2x-1)\).
Based on Table \(1\), construct a table for the values of \(\log_{10}y\) and \((2x-1)\).
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[ \(2\) markah / \(2\) marks ]
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(b)
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Plot \(\log_{10}y\) melawan \((2x-1)\) menggunakan skala \(2\) cm kepada \(0.1\) unit pada paksi-\(\log_{10}y\) dan \(2\) cm kepada \(2\) unit pada paksi-\((2x-1)\). Seterusnya lukis garis lurus penyuaian terbaik.
Plot \(\log_{10}y\) against \((2x-1)\) using a scale of \(2\) cm to \(0.1\) unit on the \(\log_{10}y\)-axis and \(2\) cm to \(2\) unit on the \((2x-1)\)-axis. Hence, draw the line of best fit.
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[ \(3\) markah / \(3\) marks ]
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(c)
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Gunakan graf di (b) untuk mencari nilai
Use the graph in (b) to find the value of
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(i)
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\(r\)
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(ii)
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\(s\)
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(iii)
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\(\log_{10}y\) bila/when \(x=2.5\)
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[ \(5\) markah / \(5\) marks ]
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Jawapan / Answer :
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Graf untuk soalan \(9\)(b)
Graph for question \(9\)(b) |
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10.
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(a)
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Taburan kebarangkalian bagi satu pemboleh ubah rawak diskret \(X=\{1,2,3,4\}\) diberi oleh \(P(X=r)=m(r+1)^2\) bagi setiap nilai \(r\).
The probability distribution for a discrete random variable \(X=\{1,2,3,4\}\) is given by \(P(X=r)=m(r+1)^2\) for each \(r\).
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(i)
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Cari nilai \(m\).
Find the value of \(m\).
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(ii)
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Lukis satu graf bagi taburan kebarangkalian \(X\).
Draw a graph for the probability distribution of \(X\).
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[ \(4\) markah / \(4\) marks ]
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Jawapan / Answer :
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(b)
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Jadual \(2\) menunjukkan markah purata yang diperoleh sekumpulan pelajar bagi satu ujian Matematik Tambahan. Markah-markah itu diberikan gred \(A\), \(B\) dan \(C\).
Table \(2\) shows the average marks obtained by students for an Additional Mathematics test. The marks are graded \(A\), \(B\) and \(C\).
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| Gred / Grade |
\(A\) |
\(B\) |
\(C\) |
| Markah purata / Average mark |
\(x\ge m\) |
\(60.0\le x\lt m\) |
\(n\le x\lt 60.0\) |
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Jadual \(2\) / Table \(2\)
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Markah purata itu mempunyai taburan normal dengan min \(59.7\) markah dan sisihan piawai \(11.2\) markah. Didapati bahawa \(10\%\) daripada pelajar itu mendapat Gred \(A\) dan \(15\%\) daripada pelajar mendapat Gred \(C\) bagi ujian itu. Cari nilai \(m\) dan \(n\).
The average marks are distributed normally with a mean of \(59.7\) marks and a standard deviation of \(11.2\) marks. It is found that \(10\%\) of the students obtained Grade \(A\) and \(15\%\) of the students obtained Grade \(C\) for this test. Find the value of \(m\) and \(n\).
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[ \(6\) markah / \(6\) marks ]
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Jawapan / Answer :
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11.
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Rajah \(5\) di bawah menunjukkan titik \(0\) adalah asalan. Lengkung \(y=4-x^2\) bersilang pada garis lurus \(y=2x+4\) pada titik-titik \((0,4)\) dan \((-2,0)\).
Diagram \(5\) below shows point \(0\) is the origin. The curve \(y=4-x^2\) intersect the line \(y=2x+4\) at the points \((0,4)\) and \((-2,0)\).
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Rajah \(5\) / Diagram \(5\)
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Hitung
Calculate
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(a)
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luas rantau berlorek,
the area of the shaded region,
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[ \(5\) markah / \(5\) marks ]
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(b)
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isipadu yang dijanakan dalam sebutan \(\pi\) apabila rantau berlorek dikisarkan \(360^\circ\) pada paksi-\(y\).
the volume generated, in term of \(\pi\) when the shaded region is revolved through \(360^\circ\) about the \(y\)-axis.
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[ \(5\) markah / \(5\) 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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Dalam Rajah \(6\), \(AB=8.4\) cm, \(BC=10.2\) cm, \(DA=10\) cm, \(\angle{BCD}=76^\circ25'\) dan \(\angle{DBC}=47^\circ\).
In Diagram \(6\), \(AB=8.4\) cm, \(BC=10.2\) cm, \(DA=10\) cm, \(\angle{BCD}=76^\circ25'\) and \(\angle{DBC}=47^\circ\).
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Rajah \(6\) / Diagram \(6\)
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(a)
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Hitung panjang dalam cm, bagi \(DB\).
Calculate the length, in cm, of \(DB\).
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[ \(3\) markah / \(3\) marks ]
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(b)
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\(\angle{ADB}\).
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[ \(2\) markah / \(2\) marks ]
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(c)
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Lakar sebuah \(\triangle B'D'C'\) yang mempunyai bentuk berbeza daripada \(\triangle{BDC}\) dengan keadaan \(C'D'=CD\), \(B'D'=BD\) dan \(\angle{C'B'D'}=\angle{CBD}\). Seterusnya nyatakan sudut \(B'C'D'\).
Sketch \(\triangle B'D'C'\) which has a different shape from \(\triangle{BDC}\) such that \(C'D'=CD\), \(B'D'=BD\) and \(\angle{C'B'D'}=\angle{CBD}\). Hence state the angle of \(B'C'D'\).
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[ \(2\) markah / \(2\) marks ]
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(d)
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Luas sisi empat \(ABCD\).
Area of quadrilateral \(ABCD\).
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[ \(3\) markah / \(3\) marks ]
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Jawapan / Answer :
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13.
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Jadual \(3\) menunjukkan harga, indeks harga dan komposisi berat bagi empat bahan utama, \(A\), \(B\), \(C\) dan \(D\) untuk membuat sejenis roti.
Table \(3\) shows the prices, the price indices and the composition by weight for the four main ingredients, \(A\), \(B\), \(C\) and \(D\) used in making a type of bread.
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Bahan
Ingredient |
Harga (RM) per kg
Price (RM) per kg |
Indeks harga pada tahun
\(2012\) berasaskan tahun \(2011\)
Price index in the year \(2012\)
based on the year \(2011\) |
Komposisi
mengikut berat
Composition by
weight |
Tahun \(2011\)
Year \(2011\) |
Tahun \(2012\)
Year \(2012\) |
| \(A\) |
\(8.00\) |
\(12.00\) |
\(x\) |
\(3.85\) kg |
| \(B\) |
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\(140\) |
\(3.25\) kg |
| \(C\) |
\(4.00\) |
\(5.00\) |
\(125\) |
\(700\) g |
| \(D\) |
\(y\) |
\(4.00\) |
\(80\) |
\(2.2\) kg |
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Jadual \(3\) / Table \(3\)
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(a)
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Cari nilai-nilai bagi \(x\) dan \(y\).
Find the values of \(x\) and \(y\).
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[ \(3\) markah / \(3\) marks ]
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(b)
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Harga bahan \(B\) pada tahun \(2012\) adalah RM \(1.00\) lebih daripada harganya pada tahun \(2011\). Hitungkan harga bahan \(B\) pada tahun \(2012\).
The price of ingredient \(B\) in year \(2012\) is RM \(1.00\) more than its price in year \(2011\). Calculate the price of ingredient \(B\) in year \(2012\).
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[ \(2\) markah / \(2\) marks ]
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(c)
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(i)
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Hitung peratus perubahan bagi kos membuat roti pada tahun \(2012\) berbanding tahun \(2011\).
Calculate the percentage change for cost of making the bread in the year \(2012\) compared to the year \(2011\).
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[ \(3\) markah / \(3\) marks ]
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(ii)
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Kos untuk membuat roti itu pada tahun \(2011\) ialah RM \(4\ 072\). Hitung kos yang sepadan pada tahun \(2013\) jika kos semua bahan utama meningkat \(15\%\) dari tahun \(2012\) ke tahun \(2013\).
The cost of making the bread in the year \(2011\) was RM \(4\ 072\). Calculate the corresponding cost in the year \(2013\) if the cost for all main ingredients increased by \(15\%\) from year \(2012\) to year \(2013\).
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[ \(2\) markah / \(2\) marks ]
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Jawapan / Answer :
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14.
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Gunakan kertas graf yang disediakan untuk menjawab soalan ini.
Use the graph paper provided to answer this question.
Sebuah institusi pengajian tinggi menawarkan kursus dalam bidang teknologi maklumat dan kejuruteraan. Bilangan pelajar dalam bidang teknologi maklumat ialah \(x\) dan dalam bidang kejuruteraan ialah \(y\).
An institution of higher learning offers undergraduate course in information technologies and engineering. The number of information technologies student is \(x\) and engineering student is \(y\).
Enrolmen bagi dua kursus itu adalah berdasarkan kekangan berikut:
The enrolment of these two courses are based on the following constraints:
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\(\text{I}\)
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Jumlah bilangan pelajar bagi dua kursus itu adalah selebih-lebihnya \(450\).
The total number of students for these two courses is at most \(450\).
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\(\text{II}\)
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Bilangan pelajar kejuruteraan sekurang-kurangnya \(50\) orang lebih daripada bilangan pelajar teknologi maklumat.
The number of engineering students exceeds the number of information technologies student by at least \(50\).
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\(\text{III}\)
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Bilangan pelajar kejuruteraan tidak melebihi tiga kali bilangan pelajar teknologi maklumat.
The number of engineering student is not more than three times the number of information technologies student.
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(a)
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Tulis tiga ketaksamaan, selain \(x\ge 0\) dan \(y\ge 0\), yang memenuhi semua kekangan yang diberi.
Write three inequalities, other than \(x\ge 0\) and \(y\ge 0\), which satisfy all the given constraints.
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[ \(3\) markah / \(3\) marks ]
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(b)
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Dengan menggunakan skala \(2\) cm kepada \(50\) orang pelajar pada kedua-dua paksi, bina dan lorek rantau \(R\) yang memenuhi semua kekangan di atas.
Using a scale of \(2\) cm to \(50\) students on both axes, construct and shade the region \(R\) that satisfies all the given constraints.
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[ \(3\) markah / \(3\) marks ]
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(c)
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Dengan menggunakan graf anda di \(14\)(b), cari
Use your graph in \(14\)(b), find
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(i)
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julat bilangan pelajar dalam bidang teknologi maklumat.
the range of the number of information technologies student.
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(ii)
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yuran pengajian minimum yang diperoleh setiap tahun akademik, jika terdapat \(125\) orang pelajar teknologi maklumat, dengan setiap pelajar teknologi maklumat membayar RM \(8\ 000\) dan setiap pelajar kejuruteraan membayar RM \(12\ 000\) sebagai yuran tahunan.
the minimum total tuition fees collected per academic year if there are \(125\) information technologies student, with each information technologies student paying RM \(8\ 000\) and each engineering student paying RM \(12\ 000\) as annual tuition fees.
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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 the 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.
Rajah \(7\) menunjukkan dua zarah, \(A\) dan \(B\), bergerak di sepanjang suatu garis lurus masing-masing melalui dua titik tetap, \(K\) dan \(L\).
Diagram \(7\) shows two particles, \(A\) and \(B\), move along a straight line passing two fixed points, \(K\) and \(L\) respectively.
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Rajah \(7\) / Diagram \(7\)
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Zarah \(A\) melalui titik tetap \(K\) dan zarah \(B\) melalui titik tetap \(L\) secara serentak. Jarak \(KL\) ialah \(20\) m. Halaju \(A\), \(v\) ms\(^{-1}\), diberi oleh \(v_A=3t-2t^2+2\) dengan \(t\) ialah masa, dalam saat, selepas meninggalkan \(K\) manakala zarah \(B\) bergerak dengan halaju seragam \(-2\) ms\(^{-1}\). Zarah \(A\) berhenti seketika di titik \(M\).
[ Anggap gerakan ke kanan ialah positif ]
Particle \(A\) passes the fixed point, \(K\) and particle \(B\) passes the fixed point, \(L\) simultaneously. The distance of \(KL\) is \(20\) m. The velocity of \(A\), \(v\) ms\(^{-1}\), is given by \(v_A=3t-2t^2+2\), where \(t\) is time, is seconds, after leaving \(K\) while \(B\) travels with a constant velocity of \(-2\) ms\(^{-1}\). Particle \(A\) stops instantaneously at the point \(M\).
[ Assume motion to the right is positive ]
Cari
Find
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(a)
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cari halaju maksimum, dalam ms\(^{-1}\), bagi zarah \(A\),
the maximum velocity, in ms\(^{-1}\), of the particle \(A\),
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[ \(3\) markah / \(3\) marks ]
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(b)
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jarak, dalam m, \(M\) daripada \(K\).
the distance, in m, \(M\) from \(K\).
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[ \(4\) markah / \(4\) marks ]
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(c)
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jarak, dalam m, antara zarah \(A\) dan zarah \(B\) apabila \(A\) berada di titik \(M\).
the distance, in m, between \(A\) and \(B\) when \(A\) is at the point \(M\).
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[ \(3\) markah / \(3\) marks ]
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Jawapan / Answer :
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KERTAS SOALAN TAMAT
END OF QUESTION PAPER |
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