UAS 24-25 FAD1014 Mathematics II

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UNIVERSITI MALAYA UNIVERSITI MALAYA

PEPERIKSAAN ASASI SAINS HAYAT / FIZIKAL EXAMINATION FOR FOUNDATION IN LIFE / PHYSICAL SCIENCES

SESI AKADEMIK 2024/2025 : SEMESTER 2 ACADEMIC SESSION 2024/2025 : SEMESTER 2

FAD1014 : Matematik II 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 two tigers at top, hibiscus flower in center, single tiger at bottom, with banner reading "ILMU PUNCHA KEMAJUAN" above]

(Kertas soalan ini mengandungi 7 soalan dalam 8 halaman yang dicetak) (This question paper consists of 7 questions on 8 printed pages)

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FAD1014

FORMULA

Properties of Integrals: $$\int kf(u),du = k\int f(u),du \qquad\qquad\qquad \int [f(u) \pm g(u)],du = \int f(u),du \pm \int g(u),du$$ $$\int_a^a f(x),dx = 0 \qquad\qquad\qquad\qquad\quad \int_a^b f(x),dx = -\int_b^a f(x),dx$$ $$\int_a^c f(x),dx = \int_a^b f(x),dx + \int_b^c f(x),dx \qquad\quad \int u,dv = uv - \int v,du$$

Integration rules: $$\int du = u + C \qquad\qquad\quad \int \sin u,du = -\cos u + C \qquad\qquad \int \frac{du}{a^2 + u^2} = \frac{1}{a}\tan^{-1}\left(\frac{u}{a}\right) + C$$ $$\int u^n,du = \frac{u^{n+1}}{n+1} + C \qquad\quad \int \cos u,du = \sin u + C \qquad\qquad \int \frac{du}{\sqrt{a^2 - u^2}} = \sin^{-1}\left(\frac{u}{a}\right) + C$$ $$\int \frac{du}{u} = \ln|u| + C \qquad\qquad \int \sec^2 u,du = \tan u + C \qquad\qquad \int \frac{du}{u\sqrt{u^2 - a^2}} = \frac{1}{a}\sec^{-1}\left(\frac{|u|}{a}\right) + C$$ $$\int e^u,du = e^u + C \qquad\qquad \int \csc^2 u,du = -\cot u + C$$ $$\int a^u,du = \frac{1}{\ln a}a^u + C \qquad\quad \int \csc u \cot u,du = -\csc u + C$$ $$\qquad\qquad\qquad\qquad\quad \int \sec u \tan u,du = \sec u + C$$

Powers of Natural Numbers: $$\sum_{k=1}^{n} k = \frac{1}{2}n(n+1)$$ $$\sum_{k=1}^{n} k^2 = \frac{1}{6}n(n+1)(2n+1)$$ $$\sum_{k=1}^{n} k^3 = \frac{1}{4}n^2(n+1)^2$$

Binomial series: $$(a+x)^n = a^n + na^{n-1}x + \frac{n(n-1)}{2!}a^{n-2}x^2 + \frac{n(n-1)(n-2)}{3!}a^{n-3}x^3 + \cdots$$ $$\qquad = a^n + \binom{n}{1}a^{n-1}x + \binom{n}{2}a^{n-2}x^2 + \binom{n}{3}a^{n-3}x^3 + \cdots$$

Taylor and Maclaurin Series: $$f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)(x-a)^2}{2!} + \cdots + \frac{f^{(n-1)}(a)(x-a)^{n-1}}{(n-1)!}$$

Conic sections formulas: $$(x-h)^2 = 4a(y-k) \qquad\qquad \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \qquad\qquad \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$$ $$(y-k)^2 = 4a(x-h) \qquad\qquad \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \qquad\qquad \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$$ $$(x-h)^2 + (y-k)^2 = r^2$$

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FAD1014

BAHAGIAN A / PART A

1. Nyatakan BENAR atau PALSU bagi pernyataan-pernyataan di bawah.

(a) Jika $\int \cos u , du = \sin u + c$, maka $\int \cos 3x , dx = \sin 3x + c$.

(b) Satu-satunya cara untuk menyelesaikan $\int \sin x \cos x , dx$ ialah dengan menggunakan penggantian $u = \cos x$.

(c) Hiperbola adalah lengkungan berbentuk $U$ di mana setiap titiknya mempunyai jarak yang sama antara fokus dan direktriks.

(d) $$\int_1^2 \frac{2x^3}{\sqrt{x^2+1}} , dx = \frac{2(2)^3}{\sqrt{2^2+1}} - \frac{2(1)^3}{\sqrt{1^2+1}}$$

(e) Sifat simetri dalam segitiga Pascal menjelaskan $\displaystyle \binom{n}{r} = \binom{n}{n-r}$ untuk kesemua $r,n \in \mathbb{N}$, di mana $r \leq n$.

(f) $\displaystyle \int \frac{2x-1}{(x-1)^2+4} , dx$ boleh diselesaikan dengan menggunakan penggantian $x - 1 = 2 \tan \theta$.

(g) Pembezaan pada jawapan akhir suatu fungsi pengamiran adalah sama dengan fungsi asalnya.

(h) Bulatan adalah kes bagi elips di mana kedua-dua titik fokus bertindih di pusat bulatannya.

State TRUE or FALSE for the statements below.

(a) If $\int \cos u , du = \sin u + c$, then $\int \cos 3x , dx = \sin 3x + c$.

(b) The only way to solve $\int \sin x \cos x , dx$ is by using substitution $u = \cos x$.

(d) $$\int_1^2 \frac{2x^3}{\sqrt{x^2+1}} , dx = \frac{2(2)^3}{\sqrt{2^2+1}} - \frac{2(1)^3}{\sqrt{1^2+1}}$$

(e) The symmetry of Pascal's Triangle suggests that $\displaystyle \binom{n}{r} = \binom{n}{n-r}$ for all $r,n \in \mathbb{N}$, where $r \leq n$.

(f) $\displaystyle \int \frac{2x-1}{(x-1)^2+4} , dx$ can be solved by using the substitution of $x - 1 = 2 \tan \theta$.

[Handwritten annotation in right margin: $u = a \tan \theta$, $u^2 + a^2$]

(g) The derivative of a resulting integrated function is the same as the original function.

(h) Circle is a case of an ellipse where both foci coincide at the centre of the circle.

(8 markah/marks)

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FAD1014

2. (a) Isi tempat kosong dengan jawapan yang tepat.

(i) Penyelesaian kepada persamaan pembeza yang memenuhi syarat awal dikenali sebagai penyelesaian ________________________________________.

(ii) Pengamiran dengan kaedah penggantian adalah songsangan daripada pembezaan ________________________________________.

(iii) Suatu jujukan adalah janjang aritmetik dan juga geometrik jika setiap sebutan di dalam jujukan itu adalah ________________________________________.

(b) Pertimbangkan siri berikut: $8x - 4x^2 + 2x^3 - x^4 + \cdots$

(i) Nyatakan jenis siri dan berikan justifikasi.

(ii) Tentukan sama ada ia adalah siri terhingga atau tidak terhingga.

(ii) Apakah sebutan kedua dalam siri itu?

(iii) Tuliskan sebutan kelima bagi siri itu.

(a) Fill in the blank with the correct answer.

(i) The solution to a differential equation that satisfies initial condition is known as ________________________________________ solution.

(ii) The integration by substitution method is the reverse of the ________________________________________ in differentiation.

(iii) A sequence is both an arithmetic and geometric progression if all the terms in the sequence are ________________________________________.

(b) Consider the series: $8x - 4x^2 + 2x^3 - x^4 + \cdots$

(i) State the type of the series and justify.

(ii) Determine whether it is a finite or an infinite series.

(ii) What is the second term in the series?

(iii) Write the fifth term of the series.

(8 markah/marks)

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FAD1014

5. (a) Diberi $\frac{dy}{dx} = e^{-2y}$ dan $y(5) = 0$. Cari nilai $p$ jika $y(p) = 3$.

(b) Diberi persamaan pembeza adalah seperti berikut, $$\frac{dy}{dx} = \frac{y^2 + xy}{x^2 - xy}.$$

Dengan menggunakan penggantian $y = vx$, tunjukkan penyelesaian umum persamaan pembeza di atas boleh diungkapkan dalam bentuk $\ln(xy) + \frac{x}{y} = A$.

(a) Given $\frac{dy}{dx} = e^{-2y}$ and $y(5) = 0$. Find the value of $p$ such that $y(p) = 3$.

(b) Given the differential equation as follows, $$\frac{dy}{dx} = \frac{y^2 + xy}{x^2 - xy}.$$

By using the substitution $y = vx$, show that the general solution of the above differential equation can be expressed in the form of $\ln(xy) + \frac{x}{y} = A$.

(14 markah/marks)

6. (a) Ungkapkan $\frac{3-x}{(1-2x)(2+x)}$ sebagai pecahan separa. Kemudian, dapatkan kembangan Binomial untuk $\frac{3-x}{(1-2x)(2+x)}$ sehingga ke sebutan $x^2$. Nyatakan julat bagi $x$ di mana kembangan di atas adalah sah.

(b) Jika empat sebutan pertama bagi siri Maclaurin $f(x) = e^x$ ialah $1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots$, dapatkan (i) kembangan untuk $g(x) = e^{3x}$. (ii) anggaran untuk $\int_0^1 \frac{e^{3x}-1}{x} , dx$.

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FAD1014

(a) Express $\frac{3-x}{(1-2x)(2+x)}$ as partial fractions. Then, obtain the Binomial expansion of $\frac{3-x}{(1-2x)(2+x)}$ until the term of $x^2$. State the range of values of $x$ where the expansion is valid. [handwritten: $-\frac{1}{2} < x < \frac{1}{2}$]

(b) If the first four terms in the Maclaurin series of $f(x) = e^x$ is $1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots$, find the (i) expansion of $g(x) = e^{3x}$. (ii) approximation of $\int_0^1 \frac{e^{3x}-1}{x} , dx$.

(14 markah/marks)

7. (a) Carikan pusat dan jejari untuk bulatan $x^2 + y^2 + 5x - 6y - 5 = 0$.

(b) Tunjukkan persamaan $16x^2 + 4y^2 - 64x - 40y + 100 = 0$ adalah sebuah elips dan cari pusat, fokus, dan bucu-bucunya.

(c) Tunjukkan bahawa garisan $x + y = 1$ adalah tangen kepada hiperbola $2x^2 - 3y^2 = 6$ dan cari koordinat titik sentuhannya.

(a) Find the centre and radius of the circle $x^2 + y^2 + 5x - 6y - 5 = 0$.

(b) Show that equation $16x^2 + 4y^2 - 64x - 40y + 100 = 0$ represent an ellipse and find the centre, foci, and vertices.

(c) Show that the line $x + y = 1$ is a tangent to the hyperbola $2x^2 - 3y^2 = 6$ and find the coordinate of the point of contact.

(14 markah/marks)

TAMAT END

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Verbatim transcription via Kimi K2.6 vision subagents.