Problem And Solution SEAMO 2017 Paper D

SEAMO

9. Evaluate

\(\sqrt{2017+2016\sqrt{2017+2016\sqrt{2017+…}}}\)

(A) 2015
(B) 2016
(C) 2017
(D) 2018
(E) 2019


MisalkanΒ 

\(\sqrt{2017+2016\sqrt{2017+2016\sqrt{2017+…}}}=a\)

\(β‡’\sqrt{2017 + 2016π‘Ž} = π‘Ž\)
\(β‡’2017 + 2016π‘Ž = π‘Ž^2\)
\(β‡’π‘Ž^2 βˆ’ 2016π‘Ž βˆ’ 2017 = 0\)
\(β‡’(π‘Ž βˆ’ 2017)(π‘Ž + 1) = 0\)
Diperoleh nilai \(π‘Ž = 2017\) atau \(π‘Ž = βˆ’1\), karena nilainya positif maka yang memenuhi \(π‘Ž =2017\)


10. It is given that \(a+\frac{1}{a}=5\), find \(a^4+\frac{1}{a^4}\)

(A) 523
(B) 527
(C) 631
(D) 635
(E) None of the above


gunakan rumus \((π‘Ž + 𝑏)^2 = π‘Ž^2 + 𝑏^2 + 2π‘Žπ‘\)
\(π‘Ž +\frac{1}{π‘Ž}= 5\)
\(β‡’(π‘Ž +\frac{1}{π‘Ž})^2= 5^2\)
\(β‡’π‘Ž^2 +\frac{1}{π‘Ž^2} + 2π‘Ž (\frac{1}{π‘Ž}) = 25 ⟹ π‘Ž^2 +\frac{1}{π‘Ž^2} = 25 βˆ’ 2 = 23\)
Selanjutnya
\((π‘Ž^2 +\frac{1}{π‘Ž^2})^2= 23^2\)
\(β‡’π‘Ž^4 +\frac{1}{π‘Ž^4 }+ 2π‘Ž^4\frac{1}{π‘Ž^4} = 529\)
\(β‡’π‘Ž^4 +\frac{1}{π‘Ž^4} = 529 βˆ’ 2 = 527\)


11. Evaluate \(\log_{\frac{1}{2}}8+\log_2 64 – \log_5 \frac{1}{125}\)

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6


Belum tersedia


12. ABCD is a rectangle with E the midpoint of AB and DF βŠ₯ CE .
Given that AB = 6 and BC = 4 . Find the length of DF.


(A) 3.6
(B) 4.2
(C) 4.8
(D) 5.4
(E) None of the above


Belum tersedia


13. The least number which, when divided by 52 leaves a remainder 33, when divided by 78 leaves 59 as remainder and when divided by 117 leaves 98 as remainder is
(A) 553
(B) 293
(C) 468
(D) 449
(E) 458


Belum tersedia


14. In the figure shown below, the area of \(Δ𝐴𝐡𝐢\) is \(8\)cmΒ², \(𝐴𝐸 = 𝐷𝐸\) and \(𝐡𝐷 = 2𝐢𝐷\).Find the total area of the shaded regions.

(A) 3.2 cmΒ²
(B) 3.6 cmΒ²
(C) 4.0 cmΒ²
(D) 4.4 cmΒ²
(E) None of the above


Karena \(𝐴𝐸 = 𝐸𝐷\) maka \([𝐴𝐸𝐡] = [𝐡𝐸𝐷]\) dan \([𝐴𝐸𝐹] = [𝐸𝐹𝐷]\), misalkanΒ \([𝐴𝐸𝐡] = 2π‘Ž\) dan\([𝐴𝐸𝐹] = 𝑏\).
Karena \(𝐡𝐷 = 2𝐷𝐢\) maka \([𝐹𝐷𝐢] =\frac{1}{2}[𝐡𝐹𝐷] =\frac{1}{2}(2π‘Ž + π‘₯) = π‘Ž +\frac{1}{2}π‘₯\).
Karena \(𝐡𝐷 = 2𝐷𝐢\) maka \([𝐴𝐷𝐢] =\frac{1}{2}[𝐴𝐡𝐷] =\frac{1}{2}(4π‘Ž) = 2π‘Ž\)
Selanjutnya,
\([𝐴𝐡𝐢] = [𝐴𝐡𝐷] + [𝐴𝐷𝐢] = 4π‘Ž + 2π‘Ž = 6π‘Ž = 8 ⟹ π‘Ž =\frac{4}{3}\)
\([𝐴𝐷𝐢] = 2π‘₯ + π‘Ž +\frac{1}{2}π‘₯ =\frac{4}{3}+\frac{5}{2}π‘₯ = 2 (\frac{4}{3}) ⟹\frac{5}{2}π‘₯ =\frac{8}{3}βˆ’\frac{4}{3}⟹\frac{5}{2}π‘₯ =\frac{4}{3}⟹ π‘₯ =\frac{8}{15}\)
Jadi luas daerah yang diarsir adalah \(π‘₯ + 2π‘Ž =\frac{8}{15}+\frac{8}{3}=\frac{16}{5}= 3,2\)


15. In Ξ”ABC, AN = BM = AB, ∠C = 38Β°. Find ∠APB.


(A) 114Β°
(B) 104Β°
(C) 118Β°
(D) 120Β°
(E) 122Β°


Belum tersedia


16. \(p\) is the difference between a real number and its reciprocal \(q\) is the difference between the square of the same real number and the square of the reciprocal. Then the value of \(p^4 + q^2 + 4p^2\) is
(A) \(2q^2\)
(B) \(3q^2\)
(C) \(\frac{1}{2}q^2\)
(D) \(\frac{3}{4}q^2\)
(E) None of the above


Belum tersedia


17. A motorboat takes 6hrs to travel from port A to port B, which is on the same side of the river. It takes the boat 8 hrs. to return to port A. It is given the speed of the current is 2.5 km/h. Find the speed of the boat in still water.
(A) 10.5 km/h
(B) 13.5 km/h
(C) 16.5 km/h
(D) 17.5 km/h
(E) 18.5 km/h


Belum tersedia


18. Given an equilateral triangle, what is the ratio of area of its inscribed circle to the area of its circumscribed circle?


(A) 1:2
(B) 1:3
(C) 1:4
(D) 1:5
(E) None of the above


Belum tersedia


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