UAS 23-24 FAC1004 Advanced Mathematics II
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FAC 1004
(d) Fungsi $e^x$ boleh dinyatakan dalam bentuk gabungan 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.
(i) $\tanh(x)$ boleh dinyatakan sebagai $\frac{e^x - e^{-x}}{e^x + e^{-x}}$ dalam bentuk eksponen.
(ii) Kegunaan utama fungsi hiperbolik adalah untuk menentukan sudut.
(iii) $\cosh^2(x) - 1 = \sinh^2(x)$.
(iv) Semua fungsi hiperbola boleh disongsangkan tanpa sebarang sekatan domain.
(a) Let $i^n = -1$, where $n$ is positive integers. Give two possible values for $n$.
(b) Determine the principal argument for the following radian angles:
(i) $\frac{9\pi}{5}$
(ii) $-\pi$
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FAC 1004
(c) [Diagram: Right-angled triangle ABC with right angle at vertex B. Side AB is labeled $y$ along the horizontal base. Side BC is labeled $2x$ along the vertical. Angle at vertex A is labeled $\alpha$. Angle at vertex C is labeled $\beta$. Hypotenuse is AC.]
Figure 1
Figure 1 shows a right angle triangle ABC. Supposed AB = $y$ and $\alpha = \tan^{-1}\left(\frac{2x}{y}\right)$. Find
(i) BC.
(ii) AC.
(iii) $\cot^{-1}\left(\frac{2x}{y}\right)$.
(iv) $\sec(\beta)$.
(d) The function $e^x$ can be expressed in the following sum of exponential function
$$e^x = \frac{e^x + e^{-x}}{2} + \frac{e^x - e^{-x}}{2},$$
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.
(i) $\tanh(x)$ can be expressed as $\frac{e^x - e^{-x}}{e^x + e^{-x}}$ in exponential form.
(ii) Hyperbolic functions are mainly used to determine angles.
(iii) $\cosh^2(x) - 1 = \sinh^2(x)$.
(iv) All hyperbolic functions are invertible without any domain restriction.
(12 markah/marks)
4 / 10
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FAC 1004
- (a) Pertimbangkan suatu linear bersandar persamaan pembeza bukan homogen dengan pekali-pekali linear
$$(a_1 x + b_1 y + c_1),dx + (a_2 x + b_2 y + c_2),dy = 0.$$
Senaraikan langkah-langkah yang perlu dilakukan untuk mendapatkan penyelesaian am.
(b) Katakan suatu persamaan pembeza am
$$K(x,y),dx + S(x,y),dy = 0.$$
Apakah kriteria utama supaya persamaan pembeza tersebut ialah jenis tepat?
(c) Sebuah tangki dengan isipadu $V\ \text{m}^3$ mengandungi $V_0\ \text{m}^3$ air suling yang telah dicampur dengan $P\ \text{g}$ gula. Larutan $H$ dengan $M_1\ \text{g/m}^3$ kepekatan gula telah dipam masuk ke dalam tangki pada kadar $R_1\ \text{m}^3/\text{s}$. Larutan tersebut juga akan dipam keluar pada kadar aliran yang $R_2\ \text{m}^3/\text{s}$.
[Andaikan larutan itu dilarutkan secara seragam]
Tuliskan persamaan atau ketaksamaan pemboleh ubah yang sesuai Ketika
(i) tangki itu akan melimpah.
(ii) tangki itu akan kosong.
(iii) isipadu larutan di dalam tangki tidak berubah.
(a) Consider a linear dependent of non-homogeneous differential equation with linear coefficients
$$(a_1 x + b_1 y + c_1),dx + (a_2 x + b_2 y + c_2),dy = 0.$$
List the steps needed in order to obtain the general solution.
(b) Let a general differential equation
$$K(x,y),dx + S(x,y),dy = 0.$$
What is the main criterion so that the differential equation is exact?
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FAC 1004
(c) A $V\ \text{m}^3$ tank contains $V_0\ \text{m}^3$ distilled water mixed with $P\ \text{g}$ sugar. Solution $H$ with $M_1\ \text{g/m}^3$ sugar concentration is pumped into the tank at the rate of $R_1\ \text{m}^3/\text{s}$. The solution is also pumped out at the rate of $R_2\ \text{m}^3/\text{s}$.
[Assume the solution is uniformly mixed]
Determine the suitable equation or inequality of variables when
(i) the tank will overflow.
(ii) the tank will empty.
(iii) the volume of solution in the tank does not change.
(12 markah/marks)
6 / 10
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BAHAGIAN B / PART B
3. Pertimbangkan nombor kompleks yang memenuhi kesemua ketaksamaan kompleks berikut. $$-\frac{\pi}{3} \leq \arg(z+2) \leq \frac{\pi}{2}$$ $$|z - i| < |z - 3i|$$ $$|z + 3 - 2i| \leq 3$$
(a) Carikan persamaan-persamaan Cartes bagi sempadan-sempadan yang terlibat.
(b) Lakar dan lorek rantau berkenaan. Labelkan kesemua sempadan.
(c) Tentukan sama ada $z = 2$ berada di dalam rantau. Berikan justifikasi anda.
Consider the complex numbers satisfying all of the following complex inequalities. $$-\frac{\pi}{3} \leq \arg(z+2) \leq \frac{\pi}{2}$$ $$|z - i| < |z - 3i|$$ $$|z + 3 - 2i| \leq 3$$
(a) Find the Cartesian equations of the involved boundaries.
(b) Sketch and shade the region. Label its boundaries.
(c) Determine whether $z = 2$ lies in the region. Justify your answer.
(14 markah/marks)
4. (a) Katakan $y = \tan^{-1}(x)$. Oleh itu, $\tan(y) = x$ . Buktikan $$\frac{dy}{dx} = \frac{1}{1+x^2}.$$
Seterusnya, carikan $\frac{dy}{dx}$ bagi $y = \tan^{-1}(e^{3x})$.
(b) Buktikan bahawa $\sinh(x+y) = \sinh x \cosh y + \sinh y \cosh x$ dengan menggunakan definisi fungsi hiperbolik $$\sinh(x) = \frac{e^x - e^{-x}}{2} \text{ dan } \cosh(x) = \frac{e^x + e^{-x}}{2}.$$
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(c) Carikan semua nilai yang mungkin bagi $\sinh x$ yang memenuhi persamaan $$\cosh^2 x - \sinh x = 3.$$
[Petunjuk : $\cosh^2 x - \sinh^2 x = 1$]
(d) Selesaikan $$\int \frac{\cosh x}{\sinh x} , dx.$$
(a) Suppose $y = \tan^{-1} x$. Therefore $\tan(y) = x$. Prove $$\frac{dy}{dx} = \frac{1}{1+x^2}.$$
Hence, find $\frac{dy}{dx}$ for $y = \tan^{-1}(e^{3x})$.
(b) Prove that $\sinh(x+y) = \sinh x \cosh y + \sinh y \cosh x$ by using the definition of hyperbolic function $$\sinh(x) = \frac{e^x - e^{-x}}{2} \text{ and } \cosh(x) = \frac{e^x + e^{-x}}{2}.$$
(c) Find all possible values of $\sinh x$ that satisfy the equation $$\cosh^2 x - \sinh x = 3.$$
[Hint : $\cosh^2 x - \sinh^2 x = 1$]
(d) Solve $$\int \frac{\cosh x}{\sinh x} , dx.$$
(14 markah/marks)
5. (a) Bezakan fungsi berikut terhadap $x$ $$f(x) = \cosh^{-1}\left(\frac{x}{5}\right).$$
[ Petunjuk : $\frac{d}{dx}(\cosh^{-1} u) = \frac{1}{\sqrt{u^2-1}}\frac{du}{dx}$ ]
(b) Pertimbangkan persamaan berikut $$\tanh^{-1}(x) = \frac{1}{2}\ln\left|\frac{1+x}{1-x}\right|.$$
Tunjukkan bahawa $$\tanh^{-1}\left(\frac{x}{a}\right) = \frac{1}{2}\ln\left|\frac{a+x}{a-x}\right|.$$
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FAC 1004
(c) Pertimbangkan persamaan-persamaan berikut
$$\int \frac{dx}{\sqrt{x^2 - a^2}} = \cosh^{-1} \left( \frac{x}{a} \right) + C,$$
$$\cosh^{-1} x = \ln \left( x + \sqrt{x^2 - 1} \right).$$
Carikan nilai tepat bagi
$$\int_{10}^{20} \frac{\cosh^{-1} \left( \dfrac{x}{5} \right)}{\sqrt{x^2 - 25}} dx , .$$
(d) Katakan $\quad (x + \alpha y + 2)^5 = C(5x - 5y + 2) \quad$ adalah penyelesaian am bagi persamaan pembeza bukan homogen dengan pekali-pekali linear berikut
$$(x - 2y)dx + (4x - 3y + 2)dy = 0,$$
yang mana $\alpha$ ialah nombor nyata dan $C$ ialah pemalar arbitrari.
Kirakan nilai tepat bagi $\alpha$.
(a) Differentiate the following function with respect to x
$$f(x) = \cosh^{-1} \left( \frac{x}{5} \right).$$
$$\left[ , \textit{Hint} : \frac{d}{dx} (\cosh^{-1} u) = \frac{1}{\sqrt{u^2 - 1}} \frac{du}{dx} , \right]$$
(b) Consider the following equation
$$\tanh^{-1} (x) = \frac{1}{2} \ln \left| \frac{1 + x}{1 - x} \right|.$$
Show that
$$\tanh^{-1} \left( \frac{x}{a} \right) = \frac{1}{2} \ln \left| \frac{a + x}{a - x} \right|.$$
(c) Consider the following equations
$$\int \frac{dx}{\sqrt{x^2 - a^2}} = \cosh^{-1} \left( \frac{x}{a} \right) + C,$$
$$\cosh^{-1} x = \ln \left( x + \sqrt{x^2 - 1} \right).$$
Find the exact value of
$$\int_{10}^{20} \frac{\cosh^{-1} \left( \dfrac{x}{5} \right)}{\sqrt{x^2 - 25}} dx , .$$
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FAC 1004
(d) Let $(x + \alpha y + 2)^5 = C(5x - 5y + 2)$ be the general solution of the following non-homogeneous differential equation with linear coefficients
$$(x - 2y)dx + (4x - 3y + 2)dy = 0 , ,$$
where $\alpha$ is a real number and $C$ is an arbitrary constant.
Calculate the exact value of $\alpha$.
(14 markah/marks)
- Katakan suatu persamaan pembeza Bernoulli
$$2x \frac{dy}{dx} + y = \frac{2x}{y} \tan^{-1} x , .$$
(a) Selesaikan persamaan pembeza di atas dan tunjukkan bahawa penyelesaian am dapat ditulis sebagai
$$y^2 = \frac{(x^2 + 1) \tan^{-1} x + C}{x} - 1 , ,$$
yang mana $C$ ialah pemalar arbitrari.
(b) Seterusnya, carikan penyelesaian khusus bagi persamaan pembeza Bernoulli jika $y(1) = \sqrt{\dfrac{\pi}{2}}$.
Suppose a Bernoulli differential equation
$$2x \frac{dy}{dx} + y = \frac{2x}{y} \tan^{-1} x , .$$
(a) Solve the differential equation above and show that the general solution can be written as
$$y^2 = \frac{(x^2 + 1) \tan^{-1} x + C}{x} - 1 , ,$$
where $C$ is an arbitrary constant.
(b) Hence, find the particular solution of the Bernoulli differential equation if $y(1) = \sqrt{\dfrac{\pi}{2}}$.
(14 markah/marks)
TAMAT END
10 / 10
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