Seamo Past Paper F 2019 [PDF]

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QUESTIONS 1 TO 10 ARE WORTH 3 MARKS EACH 1. Find the integer part of



3. What fraction of all the 9-digit numbers formed using the digits 1 to 9, without repetition, is divisible by 36?



1 1 1 1 1 1 2015 + 2016 + 2017 + 2018 + 2019



(A) (B) (C) (D) (E)



400 401 402 403 None of the above



(A)



1 9



(B)



2 9



(C)



1 3



(D)



4 9



(E) None of the above 2. A box contains five cards, numbered 1, 2, 3, 4, and 5.



4. Find the remainder when



Three cards are randomly selected without replacement from the box.



(- − 2)01 + (- − 3)21 is divided by



What is the probability that “4” is the largest value selected?



(A)



1 5



(B)



3 10



(C)



2 5



(D)



1 2



- 2 − 5- + 6 (A) (B) (C) (D) (E)



-+1 1 2 None of the above



5. Find the coefficient of - 3 in (- − 1)(- − 2)(- − 3) … (- − 10)



(E) None of the above



(A) 120 (B) 132 (C) 1200 (D)1320 (E) None of the above



SEAMO 2019 Paper F © TCIMO



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6. Given that in quadrilateral 5678, i. ∠576 = ∠658 = 105° ii. ∠567 = ∠587 = 45°



8. Gary wrote some numbers on each side of 3 cards, and laid them on a table, as shown.



Given that the sum of numbers on each card are equal and the numbers on the hidden sides are prime, find the sum of the three hidden numbers. (A) (B) (C) (D) (E)



Which of the following is true? (A) (B) (C) (D) (E)



56 = 78 56 = 58 56 > 78 56 < 78 None of the above



9. Find the number of positive integers -, such that 4- > + 1 is prime.



7. Suppose



(A) (B) (C) (D) (E)



25 ?+@ = 4



and



A1 + B?CA1 + B@C =



44 43 42 41 None of the above



15 2



0 1 2 3 None of the above



Evaluate ?@. 10. Given that (A) (B) (C) (D) (E)



1 2 3 4 None of the above



0



2- = 20192121 −



1



0 2121 2019



Evaluate G2121



EB1 + - 2 − -F (A) (B) (C) (D) (E)



2



0 1 2019 2020 None of the above



SEAMO 2019 Paper F © TCIMO



13. 5678J is a regular pentagon. A circle can be drawn through the pentagon such that it is tangent to 78 at 8 and 11. H and I are positive integers such to 56 at 5. that H = I2 + 15I + 26 is a perfect square. What is the angle formed by the minor arc 58? Find the value of I. QUESTIONS 11 TO 20 ARE WORTH 4 MARKS EACH



(A) (B) (C) (D) (E)



19 23 29 31 None of the above



12. The right rectangular prism shown below has sides of length 3, 4, and 5.



(A) (B) (C) (D) (E)



The centres of its faces are joined to form an octahedron. Find the volume of this octahedron.



108° 120° 135° 148° None of the above



14. Find the value of I given that K



(1 + 10)(1 + 102 )(1 + 10> ) … A1 + 102 C = 1 + 10 + 102 + 10L + ⋯ + 102NN



(A) (B) (C) (D) (E)



(A) (B) (C) (D) (E)



10 11 12 14 None of the above



SEAMO 2019 Paper F © TCIMO



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5 6 7 8 None of the above



15. 5678 is a square with 56 = O , and 17. Find the least possible value of 5JPQ is a rectangle, such that J lies 8 18 on side 67 and 8 lies on side PQ. S(-) = + 1 + cos - 1 − cos Given that 5J = I, what is the length of side JP?



where - can be any real number for which S(-) is defined. (A) (B) (C) (D) (E)



8 13 18 25 None of the above



18. Let ∆567 be an equilateral triangle inscribed in a circle. H is a point on the arc 67 as shown below. (A)



O2 I



(B)



I2 O



(C)



√2O2 I



(D)



√2I2 O



Given that X ∙ H5 = H6 + H7 , find the value of X.



(E) None of the above



16. Suppose Z, \ and ] are real numbers greater than 1, evaluate 1



1 1 + + ] Z 1 + log `a b EZF 1 + log ba c E F 1 + log c a ` E\F \ ] (A) (B) (C) (D) (E)



1 2 3 4 None of the above



(A)



1 2



(B)



2 3



(C) 1 (D)



3 2



(E) None of the above



4



SEAMO 2019 Paper F © TCIMO



19. Suppose Z, \, ] and numbers such that



d



are



real 22. Three girls 5, 6 and 7, and seven boys are to be lined up in a row. Let I be the number of ways to arrange them such that 6 must lie between 5 and 7 , and 5 and 6 must be separated by exactly 4 boys.



Z+\+]+d =0 and 1 1 1 1 Z\]d e + + + f = 4 Z \ ] d



Then, determine



Find the value of ZL + \ L + ] L + d L . (A) (B) (C) (D) (E)



-4 8 -12 16 None of the above



h



g j. N!



23. Determine the number of 3-element subsets of the set 5 = {1, 2, 3, … , 99} for which the sum of the three elements is a multiple of 3.



24. In ∆567 with sides Z, \ and ] , it is known that \ + ] = 6 and its area 20. Determine the number of fractions in m = Z2 − (\ − ])2 . the pattern below that are expressed in lowest terms. Find the possible largest value of m. 1 2 3 1010 , , ,…, 2019 2018 2017 1010 25. Let - be a positive real number. (A) (B) (C) (D) (E)



400 500 600 700 None of the above



Suppose O = ⌊log01 -⌋ and I = plog01



⌊r⌋ denotes the largest integer less than or equal to r.



QUESTIONS 21 TO 25 ARE WORTH 6 MARKS EACH 21. Simplify



Find the least 4O2 − 3I2 .



3 4 + +⋯ 1! + 2! + 3! 2! + 3! + 4! +



100 q -



possible



2019 2017! + 2018! + 2019!



End of Paper SEAMO 2019 Paper F © TCIMO



5



value



of



SEAMO 2019 Paper F – Answers Multiple-Choice Questions Questions 1 to 10 carry 3 marks each. Q1 D



Q2 B



Q3 B



Q4 C



Q5 D



Q6 A



Q7 E



Q8 A



Q9 B



Q10 C



Questions 11 to 20 carry 4 marks each. Q11 B



Q12 A



Q13 E



Q14 C



Q15 A



Q16 C



Q17 D



Q18 C



Q19 C



Q20 A



Free-Response Questions Questions 21 to 25 carry 6 marks each. 21



22



23



24



25



1 1 − 2 2019!



840



52305



36 17



−48



© SEAMO 2019



Paper F



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