Problems And Solutions SEAMO PAPER E 2021

11. Two real numbers between 0 and 1 are randomly chosen. What is the probability that the difference between the two numbers is greater than \(\frac{1}{4}\) ?
(A) \(\frac{1}{2}\)
(B) \(\frac{9}{16}\)
(C) \(\frac{5}{8}\)
(D) \(\frac{11}{16}\)
(E) None of the above

12. Find the minimum value of \(\Sigma_{k=1}^{20} |𝑛 − 𝑘|\) ,
where \(𝑛\) ranges over all positive integers.
(A) 50
(B) 100
(C) 120
(D) 150
(E) None of the above

13. A jury of 12 people must decide if a defendant is guilty. To come to a decision, an absolute majority of votes is needed. It is known that four
will vote ‘YES’ and three will vote ‘NO’.

Among the rest, four will each toss a fair coin and voted based on their toss. The last person will vote with majority. What is the probability that the defendant is found guilty?
(A) \(\frac{1}{3}\)
(B) \(\frac{1}{4}\)
(C) \(\frac{5}{8}\)
(D) \(\frac{5}{9}\)
(E) None of the above

14. Given the array of numbers below,

Define \(𝑆_𝑖\) the sum of numbers that are either in the \(𝑖^{th}\) row or \(𝑖^{th}\) column in the \(𝑖 × 𝑖\) square as shown in the figure above.
For example, \(𝑆_1 = 1 , 𝑆_2 = 8\) and \(𝑆_3 = 27\).
Evaluate \(𝑆_{21}\).
(A) 9259
(B) 9260
(C) 9261
(D) 9262
(E) None of the above

15. Find the value of \(𝑥\) given that

\(2^{𝑥+1} + 2^𝑥 + 2^{𝑥−1} = 56\sqrt 2\)

(A) \(\frac{7}{2}\)
(B) \(4\)
(C) \(\frac{9}{2}\)
(D) \(5\)
(E) None of the above