|
|
| |
RUMUS
FORMULAE |
| |
|
1.
|
\(x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}\) |
15.
|
\(y=\dfrac{u}{v},\dfrac{dy}{dx}=\dfrac{v\dfrac{du}{dx}-u\dfrac{dv}{dx}}{v^2}\) |
| |
|
|
|
|
2.
|
\(a^m\times a^n=a^{m+n}\) |
16.
|
\(\dfrac{dy}{dx}=\dfrac{dy}{du}\times\dfrac{du}{dx}\)
|
| |
|
|
|
|
3.
|
\(a^m\div a^n=a^{m-n}\) |
17.
|
Luas di bawah lengkung / Area under the curve
\(=\int_a^b y\ dx\) atau (or) \(=\int_a^b x\ dy\)
|
| |
|
|
|
|
4.
|
\((a^m)^n=a^{mn}\) |
18.
|
Isipadu kisaran / Volume of revolution
\(=\int_a^b\pi y^2\ dx\) atau (or) \(=\int_a^b \pi x^2\ dy\)
|
| |
|
|
|
|
5.
|
\(\log_a{mn}=\log_a{m}+\log_a{n}\) |
19.
|
\(I=\dfrac{Q_1}{Q_0}\times 100\) |
| |
|
|
|
|
6.
|
\(\log_a{\dfrac{m}{n}}=\log_a{m}-\log_a{n}\) |
20.
|
\(\bar{I}=\dfrac{\sum W_iI_i}{\sum W_i}\) |
| |
|
|
|
|
7.
|
\(\log_a{m^n}=n\log_a{m}\) |
21.
|
\({}^nP_r=\dfrac{n!}{(n-r)!}\) |
| |
|
|
|
|
8.
|
\(\log_a{b}=\dfrac{\log_c{b}}{\log_c{a}}\) |
22.
|
\({}^nC_r=\dfrac{n!}{(n-r)!r!}\) |
| |
|
|
|
|
9.
|
\(T_n=a+(n+1)d\) |
23.
|
\(P(X=r)={}^nC_rp^rq^{n-r}, p+q=1\) |
| |
|
|
|
|
10.
|
\(S_n=\dfrac{n}{2}[2a+(n-1)d]\) |
24.
|
Min / Mean, \(\mu=np\)
|
| |
|
|
|
|
11.
|
\(T_n=ar^{n-1}\) |
25.
|
\(\sigma=\sqrt{npq}\) |
| |
|
|
|
|
12.
|
\(S_n=\dfrac{a(r^n-1)}{r-1}=\dfrac{a(1-r^n)}{1-r},r\ne 1\) |
26.
|
\(z=\dfrac{X-\mu}{\sigma}\)
|
| |
|
|
|
|
13.
|
\(S_n=\dfrac{a}{r-1},|r|\lt1\) |
27.
|
Panjang lengkok, \(s=j\theta\)
Arc length, \(s=r\theta\)
|
| |
|
|
|
|
14.
|
\(y=uv,\dfrac{dy}{dx}=u\dfrac{dv}{dx}+v\dfrac{du}{dx}\) |
28.
|
Luas sektor, \(L=\dfrac{1}{2}j^2\theta\)
Area of sector, \(A=\dfrac{1}{2}r^2\theta\)
|
|
|
29.
|
\(\sin^2{A}+\text{kos}^2A=1\)
\(\sin^2{A}+\cos^2{A}=1\)
|
41.
|
Titik yang membahagi suatu tembereng garis /
A point dividing a segment of a line
\((x,y)=\left( \dfrac{nx_1+mx_2}{m+n},\dfrac{ny_1+ny_2}{m+n} \right)\)
|
| |
|
|
|
|
30.
|
\(\text{sek}^2A=1+\tan^2{A}\)
\(\sec^2{A}=1+\tan^2{A}\)
|
42.
|
Luas segi tiga / Area of triangle
\(=\dfrac{1}{2}|(x_1y_2+x_2y_3+x_3y_1)-(x_2y_1+x_3y_2+x_1y_3)|\)
|
| |
|
|
|
|
31.
|
\(\text{kosek}^2A=1+\text{kot}^2A\)
\(\cosec^2A=1+\cot^2{A}\)
|
43.
|
\(|\utilde{r}|=\sqrt{x^2+y^2}\) |
| |
|
|
|
|
32.
|
\(\sin{2A}=2\sin{A}\text{ kos } A\)
\(\sin{2A}=2\sin{A}\cos{A}\)
|
44.
|
\(\hat{r}=\dfrac{x\utilde{i}+y\utilde{j}}{\sqrt{x^2+y^2}}\) |
| |
|
|
|
|
33.
|
\(\begin{aligned} \text{kos }2A&=\text{kos}^2A-\sin^2{A} \\ &=2\text{ kos}^2A-1 \\ &=1-2\sin^2{A} \end{aligned}\)
\(\begin{aligned} \cos{2A}&=\cos^2{A}-\sin^2{A} \\ &=2\cos^2{A}-1 \\ &=1-2\sin^2{A} \end{aligned}\)
|
|
|
| |
|
|
|
|
34.
|
\(\tan{2A}=\dfrac{2\tan{A}}{1-\tan^2{A}}\) |
|
|
| |
|
|
|
|
35.
|
\(\sin{(A\pm B)}=\sin{A}\text{ kos }B\pm \text{kos }A\sin{B}\)
\(\sin{(A\pm B)}=\sin{A}\cos{B}\pm \cos{A}\sin{B}\)
|
|
|
| |
|
|
|
|
36.
|
\(\text{kos }(A\pm B)=\text{kos }A\text{ kos }B\mp \sin{A}\sin{B}\)
\(\cos{(A\pm B)}=\cos{A}\cos{B}\mp \sin{A}\sin{B}\)
|
|
|
| |
|
|
|
|
37.
|
\(\tan{(A\pm B)}=\dfrac{\tan{A}\pm \tan{B}}{1\mp \tan{A}\tan{B}}\) |
|
|
| |
|
|
|
|
38.
|
\(\dfrac{a}{\sin{A}}=\dfrac{b}{\sin{B}}=\dfrac{c}{\sin{C}}\) |
|
|
| |
|
|
|
|
39.
|
\(a^2=b^2+c^2-2bc\text{ kos }A\)
\(a^2=b^2+c^2-2bc\cos{A}\)
|
|
|
| |
|
|
|
|
40.
|
Luas segi tiga / Area of triangle
\(=\dfrac{1}{2}ab\sin{C}\)
|
|
|
|
|
|
|
| |
|
Bahagian A
Section A
[64 markah]
[64 marks]
Jawab semua soalan.
Answer all questions.
|
| |
|
1.
|
|
(a)
|
Permudahkan \(k^{-\frac{1}{2}}\times3k^{-\frac{1}{3}}\)
Simplify \(k^{-\frac{1}{2}}\times3k^{-\frac{1}{3}}\)
|
| |
[ \(2\) markah / \(2\) marks ]
|
|
(b)
|
Tunjukkan bahawa \(5^{x+1}-5^{x-1}+5^x\) boleh dibahagi tepat dengan \(29\) bagi semua integer positif \(x\).
Shows that \(5^{x+1}-5^{x-1}+5^x\) is divisible by \(29\) for all positive integers \(x\).
|
| |
[ \(2\) markah / \(2\) marks ]
|
|
| |
Jawapan / Answer :
|
| |
|
|
| |
|
|
|
|
| |
|
2.
|
Rajah \(1\) menunjukkan sebuah pendulum dengan panjang \(20\) cm yang tergantung pada \(O\) dan berada pada kedudukan awal \(L\). Apabila pendulum tersebut ditarik ke kedudukan \(A\) yang mempunyai perbezaan ketinggian menegak, \(h\) dengan \(4\) cm daripada kedudukan asalnya dan kemudian dilepaskan, pendulum itu berayun.
The Diagram \(1\) shows a \(20\) cm length pendulum hanging at point \(O\) which has an initial position at point \(L\). When the pendulum is pulled to position \(A\) which has a verticle height difference, \(h\) of \(4\) cm from its initial position and then released, the pendulum oscillates.
|
| |
|
| |
Rajah \(1\) / Diagram \(1\)
|
| |
Cari jarak yang dilalui oleh pendulum dari titik \(A\) ke titik \(B\).
Find the distance travelled by the pendulum from point \(A\) to point \(B\).
[ Guna / use \(\pi=3.142\) ]
|
| |
[ \(4\) markah / \(4\) marks ]
|
| |
Jawapan / Answer :
|
|
| |
|
|
|
|
| |
|
3.
|
Rajah \(2\) menunjukkan graf garis lurus \(xy\) melawan \(x\).
Diagram \(2\) shows a straight-line graph of \(xy\) against \(x\).
|
| |
|
| |
Rajah \(2\) / Diagram \(2\)
|
| |
Diberi \(y=\dfrac{7}{x}-1\), hitung nilai \(a\) dan \(b\).
Given that \(y=\dfrac{7}{x}-1\), calculate the value of \(a\) and \(b\).
|
| |
[ \(4\) markah / \(4\) marks ]
|
| |
Jawapan / Answer :
|
|
| |
|
|
|
|
| |
|
4.
|
Satu kajian menunjukkan bahawa \(30\%\) daripada murid di sebuah bandar berbasikal ke sekolah. Jika \(10\) orang murid dari bandar itu dipilih secara rawak, hitung kebarangkalian bahawa
A study indicates that \(30\%\) of the students in a city cycle to school. If \(10\) students from the city are chosen at random, calculate the probability that
|
| |
|
(a)
|
tepat \(2\) orang murid tidak berbasikal ke sekolah,
exactly \(2\) students not cycle to school,
|
| |
[ \(1\) markah / \(1\) mark ]
|
|
(b)
|
sekurang-kurangnya \(2\) orang murid berbasikal ke sekolah.
at least \(2\) students cycle to school.
|
| |
[ \(3\) markah / \(3\) marks ]
|
|
| |
Jawapan / Answer :
|
|
| |
|
|
|
|
| |
|
5.
|
Rajah \(3\) menunjukkan segi tiga \(DEF\).
Diagram \(3\) shows a triangle \(DEF\).
|
| |
|
| |
Rajah \(3\) / Diagram \(3\)
|
| |
|
(a)
|
Tunjukkan bahawa luas segi tiga \(DEF\) ialah
Shows that the area of the triangle \(DEF\) is
|
| |
\(\dfrac{1}{2}|(x_1y_2+x_2y_3+x_3y_1)-(x_2y_1+x_3y_2+x_1y_3)|\) |
| |
[ \(3\) markah / \(3\) marks ]
|
|
(b)
|
Seterusnya, cari luas segi tiga \(D(-6,-5)\), \(E(10,5)\) dan \(F(1,8)\).
Hence, find the area of the triangle \(D(-6,-5)\), \(E(10,5)\) and \(F(1,8)\).
|
| |
[ \(2\) markah / \(2\) marks ]
|
|
| |
Jawapan / Answer :
|
|
| |
|
|
|
|
| |
|
6.
|
Rajah \(4\) menunjukkan lengkung \(f(x)=x^2-8x+7\).
Diagram \(4\) shows a curve \(f(x)=x^2-8x+7\).
|
| |
|
| |
Rajah \(4\) / Diagram \(4\)
|
| |
|
(a)
|
Cari
Find
|
| |
|
(i)
|
nilai \(\text{had}_{x\to0} f(x)\).
the value \(\lim_{x\to0}f(x)\).
|
|
(ii)
|
nilai-nilai yang mungkin bagi \(a\) jika \(\text{had}_{x\to a}f(x)=-5\).
the possible values of \(a\) if \(\lim_{x\to a}f(x)=-5\).
|
| |
[ \(3\) markah / \(3\) marks ]
|
|
|
(b)
|
Tentukan fungsi kecerunan, \(\dfrac{dy}{dx}\) dengan menggunakan prinsip pertama.
Determine gradient function, \(\dfrac{dy}{dx}\) using the first principle.
|
| |
[ \(3\) markah / \(3\) marks ]
|
|
| |
Jawapan / Answer :
|
|
| |
|
|
|
|
| |
|
7.
|
Selesaikan sistem persamaan linear yang berikut dengan menggunakan kaedah penghapusan:
Solve the following system of linear equations using the elimination method:
|
| |
\(\begin{aligned} 2x+3y-z&=20\\ 3x+2y+z&=20\\ x+4y+2z&=15 \end{aligned}\) |
| |
[ \(5\) markah / \(5\) marks ]
|
| |
Jawapan / Answer :
|
|
| |
|
|
|
|
| |
|
8.
|
|
(a)
|
Sesaran, \(S_m\), bagi komuter yang bergerak di atas landasan selepas \(t\) saat diberi oleh \(s(t)=t^2+2t\), dengan keadaan \(t\ge0\). Dengan menggunakan prinsip pertama, cari halaju komuter itu apabila \(t=7\).
The displacement, \(S_m\), for a commuter which moves on the rail after \(t\) seconds is given as \(s(t)=t^2+2t\), where \(t\ge0\). By using the first derivative, find the velocity of the commuter when \(t=7\).
|
| |
[ \(4\) markah / \(4\) marks ]
|
|
(b)
|
Diberi \(f(x)=\dfrac{(x^2-1)^3}{x^2+1}\), cari \(f'(x)\).
Given \(f(x)=\dfrac{(x^2-1)^3}{x^2+1}\), find \(f'(x)\).
|
| |
[ \(3\) markah / \(3\) marks ]
|
|
| |
Jawapan / Answer :
|
|
| |
|
|
|
|
| |
|
9.
|
|
| |
Rajah \(5\) / Diagram \(5\)
|
| |
Diberi sebuah silinder dengan lilitan bulatan \(\dfrac{2\pi}{\sqrt{2}-1}\) cm dan tinggi \((\sqrt{2}+1)\) cm diisi dengan air minuma. Tunjukkan bahawa isipadu air minuman di dalam silinder itu ialah \((7+5\sqrt{2})\pi\) cm\(^3\).
Given a cylinder with a circumference of a circle \(\dfrac{2\pi}{\sqrt{2}-1}\) cm and the height \((\sqrt{2}+1)\) cm, is filled with drink water. Show that the volume of the drinking water in the cylinder is \((7+5\sqrt{2})\pi\) cm\(^3\).
|
| |
[ \(4\) markah / \(4\) marks ]
|
| |
Jawapan / Answer :
|
|
| |
|
|
|
|
| |
|
10.
|
|
(a)
|
Cari punca-punca bagi persamaan kuadratik \(x^2-kx+4=0\) dalam sebutan \(k\).
Find the roots of quadratic equations \(x^2-kx+4=0\) in terms of \(k\).
|
| |
[ \(2\) markah / \(2\) marks ]
|
|
(b)
|
Rajah \(6\) menunjukkan dua graf fungsi kuadratik \(f(x)=ax^2+bx+c\) dan \(g(x)=p(x-3)^2+1\).
Diagram \(6\) shows the graph of two quadratic functions \(f(x)=ax^2+bx+c\) and \(g(x)=p(x-3)^2+1\).
|
| |
|
| |
Rajah \(6\) / Diagram \(6\)
|
| |
|
(i)
|
Nyatakan julat \(a\) dalam sebutan \(p\).
State the range of \(a\) in terms of \(p\).
|
| |
[ \(1\) markah / \(1\) mark ]
|
| |
|
|
(ii)
|
Tulis fungsi kuadratik \(g(x)\) yang baharu, jika graf fungsi kuadratik \(g(x)\) bergerak \(5\) unit ke kiri.
Write the new quadratic function of \(g(x)\), if the graph of \(g(x)\) moves \(5\) unit to the left.
|
| |
[ \(1\) markah / \(1\) mark ]
|
|
|
(c)
|
Diberi bahawa suatu garis lurus \(y=mx-3\) menyilang lengkung \(y=x^2-3x+m\) pada dua titik yang berlainan. Cari julat bagi nilai \(m\).
Given that a straight line \(y=mx-3\) intersects the curve \(y=x^2-3x+m\) at two different points. Find the range of values of \(m\).
|
| |
[ \(3\) markah / \(3\) marks ]
|
|
| |
Jawapan / Answer :
|
|
| |
|
|
|
|
| |
|
11.
|
Rajah \(7\) menunjukkan keratan rentas tiga bekas silinder, masing-masing berjejari \(3\) cm, diikat dengan getah yang diregangkan. [Gunakan \(\pi=3.142\)]
Diagram \(7\) shows the cross-section of three cylindrical containers, each of radius \(3\) cm, held together by a stretched elastic band. [Use \(\pi=3.142\)]
|
| |
|
| |
Rajah \(7\) / Diagram \(7\)
|
| |
Cari
Find
|
| |
|
(a)
|
luas kawasan berlorek,
the shaded area,
|
| |
[ \(3\) markah / \(3\) marks ]
|
|
(b)
|
panjang getah yang diregangkan.
the stretched length of the band.
|
| |
[ \(4\) markah / \(4\) marks ]
|
|
| |
Jawapan / Answer :
|
|
| |
|
|
|
|
| |
|
12.
|
|
(a)
|
Selesaikan persamaan
Solve the equation
|
| |
\(\dfrac{{}^{2n}P_n}{(2n-2)!}=\dfrac{10n}{n!}\) |
| |
[ \(3\) markah / \(3\) marks ]
|
|
(b)
|
Puan Sashitta menyusun \(8\) biji manik untuk menjadi satu gelang tangan. Berapakah gelang tangan yang berbeza yang boleh dibentuk jika \(3\) daripada manik itu adalah sama warna?
Puan Sashitta arranges \(8\) beads to form a bracelet. How many different bracelets can she form if \(3\) of the beads are of the same colour?
|
| |
[ \(2\) markah / \(2\) marks ]
|
|
(c)
|
Haris mempunyai \(5\) helai baju, \(7\) pasang seluar panjang dan \(4\) pasang kasut. Berapakah cara Haris boleh memilih dua daripada setiap item untuk dibawa bersama semasa bercuti?
Haris owns \(5\) shirts, \(7\) pairs of pants and \(4\) pairs of shoes. In how many ways can Haris choose two of each item to pack for a vacation?
|
| |
[ \(2\) markah / \(2\) marks ]
|
|
| |
Jawapan / Answer :
|
|
| |
|
|
|
|
| |
|
Bahagian B
Section B
[16 markah]
[16 marks]
Bahagian ini mengandungi tiga soalan. Jawab dua soalan.
This section contains three questions. Answer two questions.
|
| |
|
13.
|
|
(a)
|
Diberi bahawa garis lurus \(y=0\) merupakan garis tangen kepada lengkung \(f(x)=2x^2+(h-1)x+2k^2\), dengan keadaan \(h\) dan \(k\) adalah pemalar.
Given that the straight line \(y=0\) is the tangent line to the curve \(f(x)=2x^2+(h-1)x+2k^2\), where \(h\) and \(k\) are constants.
Ungkapkan \(h\) dalam sebutan \(k\).
Express \(h\) in terms of \(k\).
|
| |
[ \(2\) markah / \(2\) marks ]
|
|
(b)
|
Diberi bahawa fungsi kuadratik \(f(x)=-2x^2+4x+30\) ditakrifkan dalam domain \(-3\le x\le 7\).
Given that the quadratic function \(f(x)=-2x^2+4x+30\) is defined in the domain \(-3\le x\le 7\).
|
| |
|
(i)
|
Dengan menggunakan kaedah penyempurnaan kuasa dua, ungkapkan \(f(x)\) dalam bentuk verteks, dan nyatakan koordinat titik pusingan bagi \(f(x)\).
By using completing the square method, express \(f(x)\) in the vertex form and state the turning point of \(f(x)\).
|
|
(ii)
|
Ungkapkan \(f(x)\) dalam bentuk pintasan, dan seterusnya, lakarkan graf bagi \(f(x)\).
Express \(f(x)\) in the intercept form, and hence, sketch the graph of \(f(x)\).
|
| |
[ \(6\) markah / \(6\) marks ]
|
|
|
| |
Jawapan / Answer :
|
|
| |
|
|
|
|
| |
|
14.
|
Rajah \(8\) menunjukkan garis lurus \(y=-2x+3\) menyilang garis \(x=\beta\) di \(A\) dan menyilang paksi-\(y\) di \(B\).
Diagram \(8\) shows a stright line \(y=-2x+3\) intersects line \(x=\beta\) at \(A\) and intersects \(y\)-axis at \(B\).
|
| |
|
| |
Rajah \(8\) / Diagram \(8\)
|
| |
Diberi bahawa koordinat \(C\) ialah \((\beta, 0)\) dan \(\angle{ABC}=90^\circ\), cari
Given that coordinate of \(C\) is \((\beta, 0)\) and \(\angle{ABC}=90^\circ\), find
|
| |
|
(a)
|
nilai \(\beta\),
the value of \(\beta\),
|
| |
[ \(2\) markah / \(2\) marks ]
|
|
(b)
|
koordinat \(A\),
the coordinate of \(A\),
|
| |
[ \(1\) markah / \(1\) mark ]
|
|
(c)
|
luas segi tiga \(ABC\),
the area of triangle \(ABC\),
|
| |
[ \(2\) markah / \(2\) marks ]
|
|
(d)
|
persamaan lokus bagi \(S\) jika titik \(S\) bergerak dengan keadaan jaraknya dari titik \(B\) sentiasa sama dengan jarak antara titik \(B\) dan titik \(C\).
the equation of the locus \(S\) if the point \(S\) moves with such that its distance from point \(B\) is always the same as the distance between point \(B\) and point \(C\).
|
| |
[ \(3\) markah / \(3\) marks ]
|
|
| |
Jawapan / Answer :
|
|
| |
|
|
|
|
| |
|
15.
|
|
(a)
|
Diberi \(\log_{12}{3}=k\), ungkapkan \(\log_{\sqrt{3}}8\) dalam sebutan \(k\).
Given that \(\log_{12}{3}=k\), express \(\log_{\sqrt{3}}8\) in terms of \(k\).
|
| |
[ \(4\) markah / \(4\) marks ]
|
|
(b)
|
Sekumpulan saintis di Bandar Hawkin menjumpai satu organisma ganjil yang membiak mengikut persamaan \(\ln{\left(\dfrac{x}{e^2}\right)}=t-3\ln{x}\) dengan keadaan \(x\) ialah bilangan sel terhasil dan \(t\) ialah masa pembiakan dalam jam. Setelah \(10\) jam, ungkapkan bilangan sel yang terhasil dalam sebutan \(e\).
A group of scientists in Hawkin City discovered an odd organism that reporduces following the equation \(\ln{\left(\dfrac{x}{e^2}\right)}=t-3\ln{x}\) such that \(x\) is the number of resulting cells and \(t\) is the reproductive time in hours. After \(10\) hours, express the number of resulting cells in terms of \(e\).
|
| |
[ \(4\) markah / \(4\) marks ]
|
|
| |
Jawapan / Answer :
|
|
| |
KERTAS SOALAN TAMAT
END OF QUESTION PAPER |
| |
|
|
|
|
| |
 |
| |
|
|
