UAS 22-23 FAD1014 Mathematics II
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FAD1014
BAHAGIAN A / PART A
1. (a) Nyatakan BENAR atau PALSU bagi pernyataan-pernyataan di bawah. Berikan justifikasi sekiranya ia PALSU.
(i) $\displaystyle \int x^2 e^x , dx = \frac{x^3}{3} e^x + c$.
(ii) Penggunaan cara kamiran berjadual dalam kes $\displaystyle \int f(x)g(x) , dx$ hanya boleh digunakan sekiranya pembezaan berulang kali $f(x)$ atau $g(x)$ bersamaan dengan $0$.
(iii) $\displaystyle \int \frac{1}{x-7} , dx = \ln x - \ln 7 + c$.
(iv) Pencarian anti-terbitan adalah pembalikan kepada proses mencari terbitan. Oleh itu, hukum terbitan membawa kepada hukum anti-terbitan.
(6 markah)
(b) Lengkapkan pernyataan di bawah:
(i) Dalam Matematik, kita boleh mencari luas (i) ________ lengkung dengan membahagikan kawasan itu kepada sebilangan segi empat tepat yang akan membentuk kawasan yang sama dengan kawasan yang diukur. Lebar segi empat tepat dipanggil (ii) ________. Jumlah luas segi empat tepat ini dipanggil (iii) ________.
(ii) Kamiran tentu boleh ditunjukkan sebagai $\displaystyle \int_a^b f(x) , dx$, di mana $a$ dan $b$ mewakili (iv) ________ atas dan bawah. Hasil kamiran tentu akan menjadi (v) ________ untuk kawasan di atas paksi-$x$ dan (vi) ________ bagi kawasan di bawah paksi-$x$.
(6 markah)
3/8
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FAD1014
(a) State TRUE or FALSE for the statements below. If FALSE, justify.
(i) $\displaystyle \int x^2 e^x , dx = \frac{x^3}{3} e^x + c$.
(ii) Tabular integration method in $\displaystyle \int f(x)g(x) , dx$ can only be used if derivation multiple of times either $f(x)$ or $g(x)$ is equal to $0$.
(iii) $\displaystyle \int \frac{1}{x-7} , dx = \ln x - \ln 7 + c$.
(iv) Finding the antiderivative is the reverse of finding derivative. Therefore, the rule for derivatives leads to a rule of antiderivatives.
(6 marks)
(b) Complete the statements below:
(i) In Mathematics, we can find the area (i) ________ the curve by partitioning the region into rectangles that form a region that is similar to the region being measured. The width of a rectangle is called a (ii) ________. The total area of these rectangles is called the (iii) ________.
(ii) The definite integral can be shown as $\displaystyle \int_a^b f(x) , dx$, where $a$ and $b$ represent the upper and lower (iv) ________. The value of definite integral is (v) ________ for areas above the $x$-axis and (vi) ________ for areas below the $x$-axis.
(6 marks)
2. (a) Tentukan kehomogenan bagi persamaan pembezaan berikut. Beri justifikasi jawapan anda.
$$\frac{dy}{dx} = \frac{y^2 + xy}{x^2 - xy}$$
(6 markah)
(b) Berikan takrif bagi yang berikut dalam aspek fokus.
(i) Elips
(ii) Hiperbola
(6 markah)
4/8
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FAD1014
(a) Identify the homogeneity of the following differential equation. Justify your answer.
$$\frac{dy}{dx} = \frac{y^{2} + xy}{x^{2} - xy}$$
(6 marks)
(b) Give a definition for the following in the foci aspect.
(i) Ellipse
(ii) Hyperbola
(6 marks)
5/8
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FAD1014
BAHAGIAN B / PART B
3. (a) Buktikan $\displaystyle\int x e^{x^{2}},dx = \frac{1}{2}e^{x^{2}} + c.$ Seterusnya, selesaikan $\displaystyle\int x^{3} e^{x^{2}},dx.$
(7 markah)
(b) Rantau $R$ dibatasi oleh lengkung $y = \sqrt{3x}$ , paksi $x$ dan garis $x = 4.$ Cari isipadu apabila $R$ diputarkan pada paksi-$x$ dengan menggunakan
(i) kaedah cakera.
(ii) kaedah cangkerang.
(7 markah)
(a) Show that $\displaystyle\int x e^{x^{2}},dx = \frac{1}{2}e^{x^{2}} + c.$ Hence, solve $\displaystyle\int x^{3} e^{x^{2}},dx.$
(7 marks)
(b) The region $R$ is bounded by the curve $y = \sqrt{3x}$, $x$-axis and the line $x = 4$. Find the volume obtained when $R$ is rotated about the $x$-axis by using
(i) washer method.
(ii) shell method.
(7 marks)
4. (a) Cari hasil tambah siri berikut,
$$1^{2} + 3^{2} + 5^{2} + \cdots + \text{sebutan ke-}n.$$
(5 markah)
(b) Tunjukkan bahawa $\displaystyle\frac{1}{(r+4)(r+5)} = \frac{1}{r+4} - \frac{1}{r+5}.$ Seterusnya, dapatkan hasil tambah $n$ sebutan pertama bagi siri tersebut dengan menggunakan kaedah beza,
$$\frac{1}{5 \cdot 6} + \frac{1}{6 \cdot 7} + \frac{1}{7 \cdot 8} + \cdots + \frac{1}{(n+4)(n+5)}.$$
(9 markah)
6/8
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FAD1014
(a) Find the summation of the following series, $$1^{2} + 3^{2} + 5^{2} + \cdots + n\text{-th terms.}$$ (5 marks)
(b) Show that $\displaystyle\frac{1}{(r+4)(r+5)} = \frac{1}{r+4} - \frac{1}{r+5}$ Hence, find the summation of the first $n$ terms of the following series by using method of differences, $$\frac{1}{5 \cdot 6} + \frac{1}{6 \cdot 7} + \frac{1}{7 \cdot 8} + \cdots + \frac{1}{(n+4)(n+5)}.$$ (9 marks)
5. (a) Tentukan sebutan tanpa $x$ dalam pengembangan $\displaystyle\left(x^{2} + \frac{1}{3x^{2}}\right)^{10}$. (7 markah)
(b) Dapatkan kembangan Maclaurin bagi $\displaystyle\ln\left[\sqrt[3]{\frac{1+2x}{1-2x}}\right]$ dalam kuasa menaik $x$, sehingga sebutan $x^{3}$. (7 markah)
(a) Determine the term which is independent of $x$ in the expansion $\displaystyle\left(x^{2} + \frac{1}{3x^{2}}\right)^{10}$ (7 marks)
(b) Find the Maclaurin's expansion of $\displaystyle\ln\left[\sqrt[3]{\frac{1+2x}{1-2x}}\right]$ in ascending powers of $x$, up to $x^{3}$ term. (7 marks)
7/8
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FAD1014
6. (a) Tentukan persamaan elips sekiranya koordinat bagi fokus dan bucu adalah masing-masing $(-1, 4), (-1, 2),$ dan $(-1, 3 \pm \sqrt{3})$. (5 markah)
(b) Diberi bahawa panjang menegak paksi konjugat bagi hiperbola adalah 8 dan persamaan-persamaan asimptot adalah $y = \pm \dfrac{2}{3}x$.
(i) Cari persamaan hiperbola berkenaan, dan koordinat bagi fokus-fokus dan bucu-bucu.
(ii) Lakarkan graf bagi hiperbola tersebut. Labelkan garis asimptot dan titik-titik yang berkaitan. (9 markah)
(a) Determine the equation of the ellipse if the coordinates of foci and vertices are given as $(-1, 4), (-1, 2),$ and $(-1, 3 \pm \sqrt{3})$, respectively. (5 marks)
(b) Given that the vertical length of the conjugate axis of a hyperbola is 8 and the equations of the asymptotes are $y = \pm \dfrac{2}{3}x$.
(i) Find the equation of the hyperbola, and the coordinates for its foci and vertices.
(ii) Sketch the graph of the hyperbola. Label the asymptotes and the relevant points. (9 marks)
TAMAT/END
8/8
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