The question is from CAT Calendars. We need to verify the given statements which is about recurring days. CAT exam is known to test on basics rather than high funda ideas. CAT also tests multiple ideas in the same question, and 2IIMs CAT question bank provides you with CAT questions that can help you gear for CAT Exam CAT 2023. Set Theory (especially constructing venn diagrams) is a frequently tested topic. Make sure you know the basics from this chapter.

Question 5: How many of the following statements have to be true?

i. No year can have 5 Sundays in the month of May and 5 Thursdays in the month of June.

ii. If Feb 14^{th} of a certain year is a Friday, May 14^{th} of the same year cannot be a Thursday

iii. If a year has 53 Sundays, it can have 5 Mondays in the month of May.

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**Statement (I):** No year can have 5 Sundays in the month of May and 5 Thursdays in the month of June.

A year has 5 Sundays in the month of May => it can have 5 each of Sundays, Mondays and Tuesdays, or 5 each of Saturdays, Sundays and Mondays, or 5 each of Fridays, Saturdays and Sundays. Or, the last day of the Month can be Sunday, Monday or Tuesday.

Or, the 1^{st} of June could be Monday, Tuesday or Wednesday. If the first of June were a Wednesday, June would have 5 Wednesdays and 5 Thursdays. So, statement I need not be true.

**Statement (II):** If Feb 14^{th} of a certain year is a Friday, May 14^{th} of the same year cannot be a Thursday

From Feb 14 to Mar 14, there are 28 or 29 days, 0 or 1 odd day

Mar 14 to Apr 14, there are 31 days, or 3 odd days

Apr 14 to May 14 there are 30 days or 2 odd days

So, Feb 14 to May 14, there are either 5 or 6 odd days

So, if Feb 14 is Friday, May 14 can be either Thursday or Wednesday. So, statement II need not be true.

**Statement (III):** If a year has 53 Sundays, it can have 5 Mondays in the month of May

Year has 53 Sundays => It is either a non-leap year that starts on Sunday, or leap year that starts on Sunday or Saturday.

Non-leap year starting on Sunday: Jan 1^{st} = Sunday, Jan 29^{th} = Sunday. Feb 5^{th} is Sunday. Mar 5^{th} is Sunday, Mar 26^{th} is Sunday. Apr 2^{nd} is Sunday. Apr 30^{th} is Sunday, May 1st is Monday. May will have 5 Mondays. So, statement III can be true.

The question is **"How many of the following statements have to be true?"**

Choice B is the correct answer.

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