SEAMO PAPER D 2019 [PROBLEM And SOLUTION] - BorneoMath

7. \(AB\) and \(BC\) are diameters of semicircles \(ADB\) and \(CPB\) respectively. The line \(AD\) is tangent to semicircle \(CPB\) at point \(P\). Given that \(∠PBC = 28°\), find \(∠PAC\).

(A) 32°
(B) 33°
(C) 34°
(D) 35°
(E) None of the above

\(AD\) merupakan garis singgung setengah
lingkaran kecil maka \(∠𝑂𝑃𝐴 = 90°\).
Karena \(OP = OB\) maka \(∠𝑂𝐵𝑃 = 𝑂𝑃𝐵 = 28°\)
\(∠𝑃𝑂𝐴 = 28° + 28° = 56°\)
Dengan menggunakan jumlah sudut segitiga
diperoleh \(∠𝑃𝐴𝐶 = 180° − 90° − 56° = 34°\)

8. Line AB intersects a clockface at 10 and 4. How many minutes after 4:00 PM does the angle between the hour and minute hands, in relation to Line AB, equal each other?
(A) \(17\frac{6}{11}\)
(B) \(17\frac{6}{13}\)
(C) \(18\frac{6}{11}\)
(D) \(18\frac{6}{13}\)
(E) None of the above

Misalkan Jarum menunjukkan \(n\) menit
\(∠𝐵𝑂𝑄 = ∠𝐵𝑂𝑃\)
\(⇒4 × 30° −\frac{𝑛}{60}× 360° =\frac{𝑛}{60}× 30°\)
\(⇒120° − 6𝑛 =\frac{1}{2}𝑛\)
\(⇒\frac{13}{2}𝑛 = 120\)
\(⇒𝑛 =\frac{240}{13}= 18\frac{6}{13}\)

9. The sum

\(\frac{1}{1\times 2\times 3}+\frac{1}{2\times 3\times 4}+…+\frac{1}{17\times 18\times 19}+\frac{1}{18\times19\times 20}\)

is \(\frac {m}{n}\) in its lowest terms. Find \((m+n)\).
(A) 603
(B) 749
(C) 787
(D) 938
(E) 949

\(\frac{1}{1\times 2\times 3}=\frac{1}{2}(\frac{1}{1\times 2}-\frac {1}{2\times 3})\)

\(\frac{1}{2\times 3\times 4}=\frac{1}{2}(\frac{1}{2\times 3}-\frac {1}{3\times 4})\)

\(\frac{1}{3\times 4\times 5}=\frac{1}{2}(\frac{1}{3\times 4}-\frac {1}{4\times 5})\)

…

\(\frac{18}{1\times 19\times 20}=\frac{1}{2}(\frac{1}{18\times 19}-\frac {1}{19\times 20})\)

Jumlahkan

\(=\frac{1}{2}(\frac{1}{1\times 2}-\frac {1}{19\times 20})=\frac{1}{2}(\frac{190-1}{19\times 20})=\frac{1}{2}(\frac{189}{380})=\frac{189}{760}\)

diperoleh \(\frac {m}{n}=\frac{189}{760}\), jadi nilai dari
\(m+n=189+760=949\)

10. Evaluate

Log\(\frac{25}{16}\) – 2 log\(\frac{5}{7}\) + log\(\frac{32}{49}\)

(A) log 2
(B) log 3
(C) log 4
(D) log 5
(E) None of the above

11. Simplify

\(\frac{2^{n+4} – 2(2^n)}{2(2^{n+2})}\)

(A) \(2^{n+1}-\frac{1}{8}\)
(B) \(-2^{n+1}\)
(C) \(\frac{7}{8}\)
(D) \(\frac{7}{4}\)
(E) None of the above

\(\frac{2^{𝑛+4} − 2(2^𝑛)}{2(2^{𝑛+2})}\)

\(=\frac{2^42^𝑛 − 2(2^𝑛)}{2(2^22^𝑛)}\)

\(=\frac{16(2^𝑛) − 2(2^𝑛)}{8(2^𝑛)}\)

\(=\frac{14(2^𝑛)}{8(2^𝑛)}\)

\(=\frac{14}{8}=\frac{7}{4}\)

12. Given that

\(\frac{c}{a+b}<\frac{a}{b+c}<\frac{b}{c+a}\)

Which of the following is true?

(A) \(c< a< b\)
(B) \(b< c< a\)
(C) \(a< b< c\)
(D) \(c< b< a\)
(E) None of the above

\(\frac{𝑐}{𝑎 + 𝑏}<\frac{𝑎}{𝑏 + 𝑐}<\frac{𝑏}{𝑐 + 𝑎}\)

\(\frac{𝑐}{𝑎 + 𝑏}+ 1 <\frac{𝑎}{𝑏 + 𝑐}+ 1 <\frac{𝑏}{𝑐 + 𝑎}+ 1\)

\(\frac{𝑐}{𝑎 + 𝑏}+\frac{𝑎 + 𝑏}{𝑎 + 𝑏}<\frac{𝑎}{𝑏 + 𝑐}+\frac{𝑏 + 𝑐}{𝑏 + 𝑐}<\frac{𝑏}{𝑐 + 𝑎}+\frac{𝑐 + 𝑎}{𝑐 + 𝑎}\)

\(\frac{𝑎 + 𝑏 + 𝑐}{𝑎 + 𝑏}<\frac{𝑎 + 𝑏 + 𝑐}{𝑏 + 𝑐}<\frac{𝑎 + 𝑏 + 𝑐}{𝑐 + 𝑎}\)

\(\frac{1}{𝑎 + 𝑏}<\frac{1}{𝑏 + 𝑐}<\frac{1}{𝑐 + 𝑎}\)

Selanjutnya tinjau pertidaksamaan pertama
\(\frac{1}{𝑎+𝑏}<\frac{1}{𝑏+𝑐}⟹ 𝑎 + 𝑏 > 𝑏 + 𝑐 ⟹ 𝑎 > 𝑐\)

tinjau pertidaksamaan kedua
\(\frac{1}{𝑏+𝑐}<\frac{1}{𝑐+𝑎}⟹ 𝑏 + 𝑐 > 𝑐 + 𝑎 ⟹ 𝑏 > 𝑎\)

karena \(𝑎 > 𝑐\) dan \(b > a\), dapat disimpulkan \(𝑐 < 𝑎 < 𝑏\)

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