Paper D 2022 [PDF]

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QUESTIONS 1 TO 10 ARE WORTH 3 MARKS EACH



3. If 7.03³ = 347.428927 then 0.703³ is equal to



1. Which of the following statements has a larger value?



(A) 0.034728927 (B) 0.34728927



2 × 2 × 2 × … × 2, ⏟ ⏟ 5 × 5 × 5 × …× 5 329 2's



141 5's



(𝑖)



(𝑖𝑖)



(C) 3.4728927 (D) 0.0034728927 (E) 0.00034728927



(A) (𝑖) > (𝑖𝑖) (B) (𝑖) = (𝑖𝑖) (C) (𝑖) < (𝑖𝑖) (D) Both (A) and (B)



4. Let 𝑚 and 𝑛 be 2 integers such that 𝑚 > 𝑛 . Suppose 𝑚 + 𝑛 = 20, 𝑚2 + 𝑛 2 = 328, find 𝑚2 − 𝑛 2.



(E) Both (B) and (C)



(A) 280 (B) 292 2. Each time the hands of a standard 12-hour clock form a 180° angle, a bell chimes once. From midnight today until midnight tomorrow, how many chimes will be heard?



(C) 300 (D) 320 (E) 340



(A) 20 (B) 21 (C) 22 (D) 23 (E) 24



SEAMO 2022 Paper D



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5. Find the sum



7. Let 𝑎𝑏𝑐𝑑𝑒𝑓 be a 6-digit integer such that 𝑓𝑎𝑏𝑐𝑑𝑒 is 5 times the value of 𝑎𝑏𝑐𝑑𝑒𝑓. Find (𝑎 + 𝑏 + 𝑐 + 𝑑 + 𝑒 + 𝑓).



2022 2022 2022 2022 + + +⋯+ 1×2 2×3 3×4 2021 × 2022 (A) 2020



(A) 25



(B) 2021



(B) 26



(C) 2022



(C) 27



(D) 2023



(D) 28



(E) 2024



(E) 30



8. If the radius of a sphere increases by 5%, what is the corresponding percentage increase in its volume corrected to 1 decimal place?



6. Find the smallest positive integer k such that (291 + 𝑘) is divisible by 127. (A) 122



[Volume of Sphere =



(B) 123 (C) 124



(A) 11.5%



(D) 125



(B) 12.8%



(E) 126



(C) 15%



4 × 𝜋𝑟 3 ] 3



(D) 15.8% (E) 4%



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SEAMO 2022 Paper D



9. A football match can be won, drawn or lost with equal probability. A school plays 2 matches each week. The probability that at least one match is drawn that week, is



𝑚 𝑛



QUESTIONS 11 TO 20 ARE WORTH 4 MARKS EACH 11. Solve the equation log 𝑥 (2𝑥 2 − 7𝑥 + 12) = 2



.



Find (𝑚 + 𝑛).



(A) 0, 1



(A) 13



(B) 1



(B) 14



(C) −2, 3



(C) 15



(D) 2, 3



(D) 16



(E) 3, 4



(E) 17



12. Solve the equation 10. Find the positive integral value of 𝑥, 𝑦, for 𝑥 < 10, 𝑦 < 10, so that the following statement is satisfied. Find (𝑥 + 𝑦).



1 + sin θ − 2 cos 2 θ = 0 for 0° ≤ θ ≤ 90°. [Hint: sin2 θ + cos 2 θ = 1]



2𝑥 9𝑦 = 2𝑥9𝑦



(A) 10°



(A) 6



(B) 20°



(B) 7



(C) 30°



(C) 8



(D) 45°



(D) 9



(E) 60°



(E) 10



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13. The last 2-digits of 92022 is



15. If 𝑥 2 − 10𝑥 + 1 = 0, find the value of 𝑥4 +



(A) 21



𝑥4



.



(A) 1112



(B) 81



(B) 3345



(C) 09



(C) 7963



(D) 61



(D) 9602



(E) 01



14. In the cryptarithm below, each letter represents a unique digit. What is the digit represented by B?



1



(E) 9935



16. Evaluate 16 3456782 − 345674 × 345682 (A) 1 (B) 2 (C) 3 (D) 16



(A) 0



(E) 345678



(B) 1 (C) 2 (D) 3 (E) 4



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SEAMO 2022 Paper D



17. In 𝐴𝐵𝐶, 𝐴𝐵 = 𝐴𝐶, 𝐷 and 𝐸 are points 19. How many integers 𝑛 between 1 to on 𝐴𝐶 and 𝐴𝐵, respectively, such that 2022 are there such that 1𝑛 + 2𝑛 + 𝐴𝐷 = 𝐷𝐸 = 𝐵𝐸 and 𝐵𝐷 = 𝐵𝐶. Find ∠𝐴. 3𝑛 + 4𝑛 + 5𝑛 is divisible by 5? (A) 1503 (B) 1507 (C) 1508 (D) 1510 (E) 1517 (A) 30° (B) 34° (C) 37° (D) 41°



20. Which of the following expressions have the greatest value? (A) √8 + √8



(E) 45°



(B) √9 + √7 (C) √10 + √6 18. Given that 𝑎 = √6 − 2 and 𝑏 = 2√2 − √6, then



(D) √11 + √5 (E) √12 + √4



(A) 𝑎 > 𝑏 (B) 𝑎 = 𝑏 (C) 𝑎 < 𝑏 (D) 𝑏 = √2𝑎 (E) 𝑎 = √2𝑏



SEAMO 2022 Paper D



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QUESTIONS 21 TO 25 ARE WORTH 6 MARKS EACH 21. Find the largest n such that there is only one whole number 𝑘 that satisfies 8 𝑛 7 < < 15 𝑛 + 𝑘 13



22. In 𝐴𝐵𝐶 shown below, ∠𝐴 = 60° and the length of 𝐴𝐵 and 𝐴𝐶 are roots of



23. Given that 𝑎 satisfies 𝑎2 − 𝑎 − 1 = 0 , find the value of 𝑎8 + 7 𝑎−4.



24. Let 𝑛 = 20212 + 20222 . Find the value of √2𝑛 − 1.



25. A circle is inscribed in an equilateral triangle and a square is inscribed in the circle. The ratio of the area of 𝑎



𝑥 2 − 5𝑥 + 3 = 0. 𝐷 is a point on 𝐵𝐶



square to that of the triangle is



such that 𝐴𝐷 is the bisector of ∠𝐴.



It is known 𝑎, 𝑏 and 𝑐 are positive integers such that 𝑎 and 𝑏 have no common factor and 𝑐 has no square factors. Find 𝑎 + 𝑏 + 𝑐.



𝐴𝐵 and 𝐴𝐶 are tangential to a circle drawn using 𝐷 as its centre. If the radius of the circle is



𝑎√𝑏 𝑐



𝑏√𝑐



.



, where 𝑎



and 𝑐 have no common factor and 𝑏 has no square factor, find 𝑎 + 𝑏 + 𝑐.



End of Paper



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SEAMO 2022 Paper D



SEAMO 2022 Paper D – Answers Multiple-Choice Questions Questions 1 to 10 carry 3 marks each. Q1 (A)



Q2 (C)



Q3 (B)



Q4 (D)



Q5 (B)



Q6 (E)



Q7 (C)



Q8 (D)



Q9 (B)



Q10 (B)



Questions 11 to 20 carry 4 marks each. Q11 (E)



Q12 (C)



Q13 (B)



Q14 (A)



Q15 (D)



Q16 (A)



Q17 (E)



Q18 (A)



Q19 (E)



Q20 (A)



23 48



24 4043



25 8



Free-Response Questions Questions 21 to 25 carry 6 marks each. 21 112



© SEAMO 2022



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Paper D



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