Instructions

Faculty members in a management school can belong to one of four departments - Finance and Accounting (F&A), Marketing and Strategy (M&S), Operations and Quants (O&Q) and Behaviour and Human Resources (B&H). The numbers of faculty members in F&A, M&S, O&Q and B&H departments are 9, 7, 5 and 3 respectively.

Prof. Pakrasi, Prof. Qureshi, Prof. Ramaswamy and Prof. Samuel are four members of theĀ school's faculty who were candidates for the post of the Dean of the school. Only one of theĀ candidates was from O&Q.

Every faculty member, including the four candidates, voted for the post. In each department,Ā all the faculty members who were not candidates voted for the same candidate. The rules forĀ the election are listed below.

1. There cannot be more than two candidates from a single department.

2. A candidate cannot vote for himself/herself.

3. Faculty members cannot vote for a candidate from their own department.

After the election, it was observed that Prof. Pakrasi received 3 votes, Prof. Qureshi receivedĀ 14 votes, Prof. Ramaswamy received 6 votes and Prof. Samuel received 1 vote. Prof. PakrasiĀ voted for Prof. Ramaswamy, Prof. Qureshi for Prof. Samuel, Prof. Ramaswamy for Prof.Ā Qureshi and Prof. Samuel for Prof. Pakrasi.

Question 41

Which of the following can be the number of votes that Prof. Qureshi received from aĀ single department?

Question 42

If Prof. Samuel belongs to B&H, which of the following statements is/are true?

Statement A: Prof. Pakrasi belongs to M&S.

Statement B: Prof. Ramaswamy belongs to O&Q

Question 44

Which of the following statements is/are true?

Statement A: Non-candidates from M&S voted for Prof. Qureshi.

Statement B: Non-candidates from F&A voted for Prof. Qureshi.

Instructions

For the following questions answer them individually

Question 45

Let n be the least positive integer such that 168 is a factor of $$1134^{n}$$. If m is the least positive integer such that $$1134^{n}$$ is a factor of $$168^{m}$$, then m + n equals

Question 46

If $$x$$ and $$y$$ are positive real numbers such that $$\log_{x}(x^2 + 12) = 4$$ and $$3 \log_{y} x = 1$$, then $$x + y $$ equals

Question 48

If $$x$$ and $$y$$ are real numbers such that $$x^{2} + (x - 2y - 1)^{2} = -4y(x + y)$$, then the value $$x - 2y$$ is

Question 49

The number of integer solutions of equation $$2|x|(x^{2}+1) = 5x^{2}$$ is

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Question 50

The equation $$x^{3} + (2r + 1)x^{2} + (4r - 1)x + 2 =0$$ has -2 as one of the roots. If the other two roots are real, then the minimum possible non-negative integer value of r is

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