UAS 24-25 FAC1004 Advanced Mathematics II
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UNIVERSITI MALAYA
UNIVERSITI MALAYA
PEPERIKSAAN ASASI SAINS FIZIKAL
EXAMINATION FOR FOUNDATION IN PHYSICAL SCIENCES
SESI AKADEMIK 2024/2025 : SEMESTER 2
ACADEMIC SESSION 2024/2025 : SEMESTER 2
FAC1004 : Matematik Lanjutan II
Advanced Mathematics II
Mei/Jun 2025 Masa : 2 jam
May/June 2025 Time : 2 hours
ARAHAN KEPADA CALON :
INSTRUCTIONS TO CANDIDATES :
Calon dikehendaki menjawab semua soalan.
Candidate is required to answer all questions.
[University of Malaya coat of arms: shield with tigers, hibiscus, and banner reading "ILMU PUNCA KEMAJUAN"]
(Kertas soalan ini mengandungi 6 soalan dalam 12 halaman yang dicetak)
(This question paper consists of 6 questions on 12 printed pages)
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FAC1004
BAHAGIAN A / PART A
- (a) "$\pi$ adalah nombor kompleks."
Adakah pernyataan di atas benar? Berikan justifikasi ringkas.
(b) Pertimbangkan $z = x + yi$, jika $\text{Im}(z) = 0$ dan $\text{Re}(z) < 0$, apakah nilai $\arg(z)$?
(c) [Graph of $y = \cos x$ for $0 \le x \le \pi$. The curve starts at $(0,1)$, crosses the $x$-axis at $(\pi/2, 0)$, and ends at $(\pi, -1)$. The axes are labeled $x$ (horizontal) and $y$ (vertical).]
$$y = \cos x, \quad 0 \le x \le \pi$$
Rajah 1
Rajah 1 menunjukkan graf $\cos x$ dengan domain yang dihadkan sebelum disongsangkan kepada $\cos^{-1}(x)$. Nyatakan domain dan julat bagi $\cos^{-1}(x)$.
(d) Fungsi $e^x$ boleh dinyatakan dalam bentuk gabungan fungsi-fungsi eksponen berikut
$$e^x = \frac{e^x + e^{-x}}{2} + \frac{e^x - e^{-x}}{2},$$
di mana
$$\frac{e^x + e^{-x}}{2} = \cosh (x)$$
dan
$$\frac{e^x - e^{-x}}{2} = \sinh(x).$$
Tentukan sama ada pernyataan-pernyataan berikut BENAR atau
PALSU.
2/12
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FAC1004
(i) $\text{sech}(x) = \dfrac{e^{-x}}{e^{x} + e^{-x}}$
(ii) Fungsi hiperbolik adalah untuk menentukan sudut dalam sebuah segitiga.
(iii) Semua fungsi hiperbolik boleh disongsangkan tanpa sebarang sekatan domain.
(iv) $\sinh^{-1}(x) = \ln\left(x + \sqrt{x^{2} + 1}\right)$
(a) “$\pi$ is a complex number.”
Is the statement above true? Justify briefly.
(b) Consider $z = x + yi$, if $\text{Im}(z) = 0$ and $\text{Re}(z) < 0$, what is the value of $\arg(z)$?
(c) [Diagram: Cartesian axes labelled $x$ and $y$. A smooth decreasing curve starts at the point $(0,1)$ on the $y$-axis, crosses the $x$-axis at $(\pi/2,0)$, and ends at $(\pi,-1)$. A dashed horizontal line runs from the $y$-axis at $y=-1$ to the endpoint $(\pi,-1)$. The $x$-axis is marked with $0$, $\pi/2$, and $\pi$; the $y$-axis is marked with $1$, $0$, and $-1$. Below the graph: $y = \cos x,\quad 0 \leq x \leq \pi$. Caption: Figure 1.]
Figure 1 shows a graph of $\cos x$ with a restricted domain before it is inverted to $\cos^{-1}(x)$. State the domain and range of $\cos^{-1}(x)$.
(d) The function $e^{x}$ can be expressed in the following sum of exponential functions $$e^{x} = \frac{e^{x} + e^{-x}}{2} + \frac{e^{x} - e^{-x}}{2},$$ where $$\frac{e^{x} + e^{-x}}{2} = \cosh (x)$$ and $$\frac{e^{x} - e^{-x}}{2} = \sinh(x).$$ Determine whether the following statements are TRUE or FALSE.
3/12
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FAC1004
(i) $\text{sech}(x) = \dfrac{e^{-x}}{e^{x} + e^{-x}}$
(ii) Hyperbolic function is used to determine an angle in a triangle.
(iii) All hyperbolic functions are invertible without any domain restriction.
(iv) $\sinh^{-1}(x) = \ln\left(x + \sqrt{x^{2} + 1}\right)$
(12 markah/marks)
2. (a) Pertimbangkan suatu persamaan pembeza umum jenis Bernoulli, $$\frac{dy}{dx} + P(x),y = Q(x),y^{n}.$$
(i) Apakah penggantian yang dilakukan untuk menukar persamaan pembeza di atas kepada jenis linear?
(ii) Tuliskan persamaan pembeza yang telah ditukar kepada jenis linear tersebut.
(iii) Tuliskan formula untuk mencari faktor integrasi untuk menyelesaikan persamaan pembeza linear dalam (ii).
(iv) Katakan faktor integrasi adalah $\mu(x)$. Pilih satu formula yang betul daripada pilihan-pilihan di bawah dalam mencari penyelesaian umum bagi persamaan Bernoulli.
[Diagram: Three rounded rectangular boxes arranged in a triangular layout, each containing a formula. Top-left box: $\mu(x),y^{1-n} = \int \mu(x)(1-n),Q(x),dx$ Middle-right box: $y^{1-n} = \int \mu(x)(1-n),Q(x),dx$ Bottom-left box: $\mu(x),y^{1-n} = \int \mu(x),Q(x),dx$]
4/12
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FAC1004
(c) Suppose the linearly independent case of nonhomogeneous differential equation
$$(ax + c)dx + (by + d)dy = (ex + f)dy + (gy + h)dx,$$
where $a, b, c, d, e, f, g$ and $h$ are arbitrary constants.
(i) Write formula in terms of the constants that satisfies the equation above.
(ii) The following flowchart is the general process of solving the differential equation. Write the suitable type of differential equation in the blank space.
[Flowchart diagram with three boxes connected by arrows. Top-left box labeled: "Nonhomogenous differential equation of linearly independent case". Arrow points right to a blank rectangular box. Arrow from the blank box points downward, then leftward to a bottom-left box labeled: "Separable differential equation".]
(12 markah/marks)
7/12
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FAC1004
BAHAGIAN B / PART B
3. (a) Andaikan $z$ sebagai nombor kompleks dalam satah Argand memenuhi persamaan kompleks
$$|2z + 6 - 2i| = |2z - 2 + 2i|.$$
(i) Cari persamaan Cartesnya.
(ii) Tentukan nombor kompleks yang memenuhi $|z - 5 - 2i|_{\min}.$
(b) Katakan ketaksamaan-ketaksamaan kompleks berikut
$$|z| > 1, |z - 3| \leq 4 \text{ dan } \text{Re}(z) < \text{Im}(\bar{z}).$$
Lakar dan lorekkan rantau yang memenuhi kesemua ketaksamaan tersebut.
(a) Suppose $z$ be any complex number in the Argand plane satisfying
$$|2z + 6 - 2i| = |2z - 2 + 2i|.$$
(i) Find the Cartesian equation.
(ii) Determine complex number that satisfies $|z - 5 - 2i|_{\min}.$
(b) Let complex inequalities as follows
$$|z| > 1, |z - 3| \leq 4 \text{ and } \text{Re}(z) < \text{Im}(\bar{z}).$$
Sketch and shade the region satisfies all inequalities.
(14 markah/marks)
8/12
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FAC1004
(a) Differentiate the following function with respect to $x$ $$f(x)=\operatorname{sech}^{-1}(4x).$$ $$\Bigl[\text{ Hint}:\frac{d}{dx}\bigl(\operatorname{sech}^{-1}u\bigr)=-\frac{1}{u\sqrt{1-u^{2}}}\frac{du}{dx}\Bigr]$$
(b) Consider the following equations $$\int\frac{dx}{\sqrt{x^{2}-a^{2}}}=\cosh^{-1}!\left(\frac{x}{a}\right)+D$$ and $$\cosh^{-1}x=\ln!\left(x+\sqrt{x^{2}-1}\right).$$ Show that $$\int\frac{dx}{\sqrt{x^{2}-a^{2}}}=\ln!\left(x+\sqrt{x^{2}-a^{2}}\right)+C.$$
(c) Consider an exact differential equation $$\bigl(4x^{3}-8xy^{2}-y\sin xy\bigr),dx+\bigl(16y^{3}-8yx^{2}-x\sin xy\bigr),dy=0.$$
Find $A(x,y)$, if the general solution is $$\bigl(A(x,y)\bigr)^{2}+\cos(xy)=C,$$ where $C$ is a constant.
(14 markah/marks)
6. Sebuah tangki dengan isipadu $222\ \mathrm{m}^{3}$ mengandungi $22\ \mathrm{m}^{3}$ air suling yang telah dilarutkan dengan $20\ \mathrm{kg}$ gula. Larutan gula dengan kepekatan $2\ \mathrm{kg/m^{3}}$ telah dipam masuk ke dalam tangki dengan kadar $5\ \mathrm{m^{3}/min}$. Larutan tersebut juga akan dipam keluar dengan kadar $3\ \mathrm{m^{3}/min}$.
[Andaikan larutan di dalam tangki dilarutkan secara seragam sentiasa]
(a) Katakan $A$ dan $t$ masing-masing mewakili jisim gula dalam unit $\mathrm{kg}$ dan masa dalam unit $\mathrm{min}$.
Tunjukkan bahawa hubungan antara $A$ dan $t$ ialah $$A=4t+44-\frac{24\sqrt{22^{3}}}{(2t+22)^{\frac{3}{2}}}.$$
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Verbatim transcription via Kimi K2.6 vision subagents.