Numbers & Algebra - Numbers and Algebra MCQ

Option: D

Explanation :

Number of students who answered 1 or more questions wrongly = 2^N-1.
Number of students who answered 2 or more questions wrongly = 2^N-2
Hence number of students who answered 1 question wrongly = 2^N-1- 2^N-2 = 2^N-2
Similarly it can be shown that number of students who answered 2 questions wrongly = 2^N-2 - 2^N-3 = 2^N-3
Similarly we can find number of students who answered K questions wrongly where K ≥ 3
Hence total number of questions attempted wrongly
s = 2^N-2+2(2^N-3)+3(2^N-4 )+...+(N-1)(2⁰) + N(1) ........(1)
∴ s/2 = 2^N-3+ 2(2^n-4) +.......+ (N-2)(2⁰) + (N-1)/2 + N/2.......... (2)
Substracting equations (1) from (2)
s/2 = 2^N-2 + 2N-3 +...+2⁰+1/2
=> s = 2^N + 2^N-2+2^N-3+...+1 = 2^N -1
= 8191
=> N = 13

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Option: A

Explanation :

Worst scenario is when other four get equal number of votes.
Let the winning candidate get x votes.
∴ x > (261-x) / 4
=> x > 52 
x = 53

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Option: B

Explanation :

Consider a set of three numbers a, b and c.

 Thus, a + b + c = 3AM.

But  

which cannot be found out. Thus, I is false,

 Sum of the numbers = a + b + c = 3AM.

Thus, II is true.

.

But    

 which cannot be found out. Thus, III is false.

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