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4. Find the sum of all possible positive integers π such that the expression below is an integer.
QUESTIONS 1 TO 10 ARE WORTH 3 MARKS EACH 1. Find the last two digits of 112020. (A) (B) (C) (D) (E)
4π 3 β 16π2 + 29π + 60 2π β 3
01 41 71 91 None of the above
(A) (B) (C) (D) (E)
2. The quadratic equation
42 69 75 81 None of the above
π₯ 2 β 52π₯ + π = 0 5. Evaluate the sum
has roots that are prime numbers.
π=
Find the maximum value of π. (A) (B) (C) (D) (E)
520 576 640 667 None of the above
+β―+
(A)
1 2
(B)
210 421
(C)
105 211
3. Let π(π₯) = π₯ 2 + 2020π₯ + 20. How many ordered pairs of positive integers (π, π) are there such that π(π + π) = π(π) + π(π)? (A) (B) (C) (D) (E)
20 1 + 202 + 204
(D) 1 (E) None of the above
1 2 3 4 None of the above
SEAMO 2020 Paper F Β© SEAMO PTE LTD
1 2 3 + + 1 + 12 + 14 1 + 22 + 24 1 + 32 + 34
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6. 2 white, 3 black and 4 grey marbles are shared equally among 9 students.
9. How many positive integers less than 2020 with the property that the sum of its digits equals 9?
Find the number of ways the marbles can be distributed so that Bran and Sansa gets the same colour and Arya gets a grey marble. (A) (B) (C) (D) (E)
(A) (B) (C) (D) (E)
120 130 140 150 None of the above
50 100 102 202 None of the above
10. The sequence {ππ } is defined by ππ+2 =
7. Given that π is a real number such that π4 + π3 + π2 + 1 = 0.
with π1 = 1 and π2 = 2.
Evaluate π2020 + 2π2010 + 3π2000.
Evaluate π2020.
(A) (B) (C) (D) (E)
(A) (B) (C) (D) (E)
2 4 6 8 None of the above
11. Find the smallest prime factor of
1 1 1 =π+ =π+ π π π
1000 β¦ 01 β 2020 π§ππππ
What is the largest possible value of πππ? (A)
(A) (B) (C) (D) (E)
1 2
(B) 2 (C)
1 2 3 4 None of the above
QUESTIONS 11 TO 20 ARE WORTH 4 MARKS EACH
8. Given that π. π and π are three distinct real numbers such that π+
1 + ππ+1 ππ
3 5 7 11 None of the above
5 2
(D) 3 (E) None of the above
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SEAMO 2020 Paper F Β© SEAMO PTE LTD
12. In the expansion of
15. Given that π and π are real numbers satisfying
π(π₯) = (1 + ππ₯)4 (1 + ππ₯)5 2 2 { 6 β 5π + 4π β 3π + 2ππ β π = 0 πβπ=1
where π and π are positive integers, the coefficient of π₯ 2 is 66.
Find the sum of all possible values of 30π
Evaluate π + π. (A) (B) (C) (D) (E)
π
2 3 4 5 None of the above
(A) (B) (C) (D) (E)
13. The equation π₯ 3 β ππ₯ 2 + ππ₯ β 2020 has three positive integer roots.
. β15 β10 15 30 None of the above
16. The figure below shows a 5 Γ 6 rectangular board with a missing 1 Γ 2 rectangle in the center.
Find the least possible value of π. (A) (B) (C) (D) (E)
101 110 202 220 None of the above How many squares are there in the board?
14. Evaluate the sum π = sin2 0Β° + sin2 2Β° + sin2 4Β° + β― + sin2 180Β° (A) (B) (C) (D) (E)
(A) (B) (C) (D) (E)
80 81 88 90 None of the above
SEAMO 2020 Paper F Β© SEAMO PTE LTD
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14 30 54 56 None of the above
17. In βπ΄π΅πΆ,
20. π΄, π΅, πΆ and π· are four distinct points lying on the circumference of a circle such that chords π΄π΅ and πΆπ· are perpendicular at point πΈ.
(sin π΄ + sin π΅) βΆ (sin π΅ + sin πΆ) βΆ (sin πΆ + sin π΄) = 19 βΆ 20 βΆ 21
Given that πΈπ΄ = 4, πΈπ΅ = 2 and πΈπΆ = 6 , find the radius of the circle.
Find the value of 99cos π΄. (A) (B) (C) (D) (E)
39 41 51 60 None of the above
18. Find the least positive integer π such that
the
equation
10π
β
π₯
β = 98 has
integer solution π₯. βπβ is the largest integer smaller than or equal to π. (A) (B) (C) (D) (E)
(A) (B) (C) (D) (E)
3 4 5 6 None of the above
β270 18 None of the above
QUESTIONS 21 TO 25 ARE WORTH 6 MARKS EACH 21. You need to tile a 10 Γ 1 hallway with a supply of 1 Γ 1 red, 2 Γ 1 red tiles and 2 Γ 1 blue tiles. Find the number of ways you can tile the 10 Γ 1 hallway.
19. How many positive integers π < 100 such that 2(56π ) + π(23π+2 ) β 1 is divisible by 7 for any positive integer π? (A) (B) (C) (D) (E)
β221 15
12 14 18 19 None of the above
22. π₯, π¦ and π§ are real numbers such that π₯+π¦+π§ = 7 {π₯ + π¦ 2 + π§ 2 = 19 π₯ 3 + π¦ 3 + π§ 3 = 64 2
Evaluate π₯ 4 + π¦ 4 + π§ 4.
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SEAMO 2020 Paper F Β© SEAMO PTE LTD
23. In βπ΄π΅πΆ shown below, π΄π·, π΅πΈ and πΆπΉ intersect at π . Suppose π΄π = π, π΅π = π, πΆπ = π and π·π = πΈπ = πΉπ = π₯. Given that π₯ = 3 and π + π + π = 20 , find πππ.
24. Positive integers π, π and π randomly selected from the {1,2,3, β¦ ,2020} with replacement.
are set
Find the probability that πππ + ππ + 2π is divisible by 5.
25. π΄π΅πΆπ· is a convex quadrilateral such that π΄πΆ intersects π΅π· at πΈ . π» is a point lying in the segment π·πΈ such that π΄π» is perpendicular to π·πΈ. Suppose π΅πΈ = πΈπ·, πΆπΈ = 9, πΈπ» = 12, π΄π» = 32 and β π΅πΆπ΄ = 90Β° . Evaluate the length of πΆπ·.
End of Paper
SEAMO 2020 Paper F Β© SEAMO PTE LTD
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SEAMO 2020 Paper F β Answers Multiple-Choice Questions Questions 1 to 10 carry 3 marks each. Q1 A
Q2 D
Q3 D
Q4 B
Q5 B
Q6 C
Q7 C
Q8 E
Q9 C
Q10 A
Questions 11 to 20 carry 4 marks each. Q11 D
Q12 B
Q13 B
Q14 E
Q15 A
Q16 D
Q17 C
Q18 B
Q19 B
Q20 A
Free-Response Questions Questions 21 to 25 carry 6 marks each.
Β© SEAMO 2020
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