Problems And Solutions SEAMO PAPER D 2021

SEAMO SMP

11. The integer 𝑁 is formed by writing all integers from 1001 to 1021
consecutively.

i.e. 𝑁 = 100110021003 … 1021.

Find the greatest power of 3 that is a factor of 𝑁.
(A) 9
(B) 27
(C) 81
(D) 243
(E) None of the above



12. Find the value of π‘₯ given that

\(\frac{9}{2} = 2 + \frac{1} {0 + \frac{1}{2 +\frac{1}{π‘₯}}}\)

(A) 3
(B) 2
(C) 1
(D) 0
(E) None of the above


\(\frac{9}{2} = 2 + \frac{5}{2} = 2 + \frac{1}{\frac{2}{5}} = 2 + \frac{1}{0 +\frac{1}{\frac{5}{2}}}= 2 +\frac{1}{0 +\frac{1}{2 +\frac{1}{2}}}\)
Jadi nilai \(π‘₯\) nya adalah 2


13. Find the constant term in the expansion of

\((2π‘₯^2 +\frac{1}{π‘₯^3})^{10}\)

(A) 12400
(B) 12440
(C) 13400
(D) 13440
(E) None of the above


Gunakan rumus binomial newton
Konstanta \((2π‘₯^2 +\frac{1}{π‘₯^3})^{10}\)

diperoleh dari \({10\choose 6}(2π‘₯^2)^6(\frac{1}{π‘₯^3})^4\)

Konstantanya adalah \({10\choose 6}2^6 =\frac{10!}{6!.4!}Γ— 64 = 13440\)


14. Suppose \(𝐴𝐡𝐢𝐷\) is a square of side length of 1. \(𝑀\) is the midpoint of \(𝐴𝐡\).


Find the area of the shaded region.
(A) \(\frac{1}{8}\)
(B) \(\frac{1}{10}\)
(C) \(\frac{1}{12}\)
(D) \(\frac{2}{25}\)
(E) None of the above


\([ABCD]=1\)

Perhatikan \(Δ𝑀𝑂𝑁 β‰ˆ Δ𝐡𝐢𝑁\), diperoleh perbandinganΒ 
\(\frac{𝑂𝑀}{𝐡𝐢}=\frac{𝑂𝑁}{𝐡𝑁}=\frac{1}{2}\)

\([𝑂𝑀𝐡] =\frac{1}{8}[𝐴𝐡𝐢𝐷] =\frac{1}{8}\)

\([𝑂𝑁𝑀] =\frac{1}{3}[𝑂𝑀𝐡] =\frac{1}{3}Γ—\frac{1}{8}=\frac{1}{24}\)

Karena \([ONM]=[OPM]\), maka luas arsiran adalah \(2 Γ—\frac{1}{24}=\frac{1}{12}\)


15. A fair die is numbered 1 to 6 on each face. \(π‘Ž, 𝑏\) and \(𝑐\) are the results of the first, second and third throws, respectively.
What is the probability that \(π‘Ž > 𝑏 > 𝑐\)?
(A) \(\frac{2}{21}\)
(B) \(\frac{3}{32}\)
(C) \(\frac{4}{43}\)
(D) \(\frac{5}{24}\)
(E) None of the above


Ruang sampelnya sebanyak \(6 Γ— 6 Γ— 6 = 216\) cara
Banyak susunan \(π‘Ž > 𝑏 > 𝑐\) adalah \({6\choose 3}=\frac{6!}{3!.3!}=20\) cara
Jadi peluangnya \(\frac{20}{216}=\frac{5}{54}\)


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