CAT Model Paper 8 Questions and Answers with Explanation Part 11

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Answer the questions based on the data given below:

Several lecturers from four disciplines, viz. Physics, Chemistry, Mathematics and Biology were selected. They were to attend a recently organized faculty conclave. All applicants were classified into five age groups, namely:

Q_59-To-60_Table of All Applicants Were Classified into Five Age Groups
Age RangeAge Group
Age less than 50Young
Middle aged
Senior
Stalwart
Age 80 and AboveRetired

For the conclave, not more than two lecturers of age group were short-listed for representing a discipline. Initially, the maximum possible lecturers were short -listed. However, only five more than half of them got selected and they finally sat for the conclave. The following observations about the selected lecturers were recorded:

If two lecturers were excluded from the retired lecturers, then the number of seniors and stalwarts, each is one less than the number in each of the other three age groups.

Total number of lecturers in Physics, Chemistry and Biology put together is the square of an integer.

There is no Biology lecturer who is middle aged or stalwart and there is no Physics lecturer who is a senior. Every other discipline is represented by at least one lecturer from each age group.

Rocky and Platy are young Physics lecturers

Q: 59. If the number of Physics lecturers is greater than the number of Chemistry lecturers, then which of the following is NOT possible to determine?

(A) Number of middle-aged Physics lecturers

(B) Number of retired Physics lecturers

(C) Number of middle-aged Chemistry lecturers

(D) Number of retired Chemistry lecturers

Ans: C

Solution:

The maximum possible lecturers were short-listed. Since there are 5 age groups and 4 different disciplines; and there can be at most two lecturers of a particular age group representing a particular discipline, the number of lecturers short-listed

Number of lecturers selected from them .

Now,

(I) Tells that out of these 25, the number of lecturers from various age groups is:

Q_59_1_Table of the Number of Lecturers from Various Age Groups
Young5
Middle-aged5
Senior4
Stalwarts4
Retired7
Total25

Now coming directly to (III) ,

Q_59_2_Table of the Number of Lecturers from Various Age Groups
PhysicsChemistryMathematicsBiology
YoungYoungYoungYoung
Middle-agedMiddle-agedMiddle-aged
SeniorSeniorSenior
StalwartStalwartStalwart
RetiredRetiredRetiredRetired

Each of them is having at least one lecturer. So, assigning one to each of them, the table looks like:

Q_59_3_Table of Each of Them is Having at Least One Lecturer
PhysicsChemistryMathematicsBiology
Young (1)Young (1)Young (1)Young (1)
Middle-aged (1)Middle-aged (1)Middle-aged (1)Middle-aged (1)
Senior (1)Senior (1)Senior (1)
Stalwart (1)Stalwart (1)Stalwart (1)Stalwart (1)
Retired (1)Retired (1)Retired (1)Retired (1)

The balance lecturers are:

Q_59_4_Table of Balance Lecturers
Young1
Middle-aged2
Senior1
Stalwarts1
Retired3
Total8

Going back to (II) , total number of lecturers for physics, chemistry and biology can be counted to lie between 12 and 20. Moreover, the total is a square,

Total in these three is 16 and there are 9 lecturers from mathematics stream.

Proceeding, (IV) tells us that Rocky and Platy are Young physics lecturers.

So, the balance Young lecturer goes in Physics. And since none of the cells can have more than 2 and total for Mathematics is 9, we get a better version of the table as:

Q_59_5_Table of Balance Young Lecturer Goes in Physics
PhysicsChemistryMathematicsBiology
Young (2)Young (1)Young (1)Young (1)
Middle-aged (1)Middle-aged (1)Middle-aged (1)
Senior (1)Senior (2)Senior (1)
Stalwart (1)Stalwart (1)Stalwart (2)
Retired (1)Retired (1)Retired (1)Retired (1)

And the balance lecturers now:

Q_59_6_Table of Balance Lecturers
Middle Aged1
Retired2
Total3

Final Arrangement:

Q_59_7_Table of Final Arrangement
PhysicsChemistryBiologyMathematicsTotal
Young21115
Middle-aged1 or 21 or 2X25
SeniorX1124
Stalwart11X24
Retired1 or 21 or 21 or 227
Total5 or 6 or 75 or 6 or 73 or 4925
16925

Note that the column for Mathematics is complete. And in Biology, count for Retired can be 2 at max. And there is no provision for middle aged in Biology. Thus, out of the two balance retired lecturers, at most one can go in Biology and in that case, the remaining retired and middle-aged ones must go into Physics or Chemistry or both.

(4) ; If now, minimum lecturers for Physics as well as that of Chemistry are 5. But since the number of Physics lecturers is greater than the number of Chemistry lecturers, at least one of the balances three must go into Physics.

Case I: Middle-aged lecturer goes into Physics.

Now, out of the remaining two retired lecturers one has to go into Physics and the remaining one can go either in Chemistry or Biology.

Case II: Retired lecturer goes into Physics.

Here also, the count for retired in Physics is full (2) . Middle Aged cannot go in Biology and if he goes into Chemistry, the initial condition that “the number of Physics lecturers is greater than the number of Chemistry lecturers” can never get satisfied. So, the middle aged goes into Physics and we get the same configuration as in case I.

Q_59_7_Table of Retired Lecturer Goes into Physics
Physics (7)Chemistry (5 or 6)Mathematics (9)Biology (3 or 4)
Young (2)Young (1)Young (1)Young (1)
Middle-aged (2)Middle-aged (1)Middle-aged (2)
Senior (1)Senior (2)Senior (2)
Stalwart (1)Stalwart (1)Stalwart (2)
Retired (2)Retired (1)Retired (2)Retired (1)

The one remaining retired lecturer goes into either Chemistry or Biology.

Hence the number of retired Chemistry lecturers cannot be determined. Hence Answer option (C)

Q: 60. What can be said about the total number of Physics and Chemistry lecturers put together?

(I) It is less than or equal to 12.

(II) It is more than or equal to 12.

(III) It is less than or equal to 13.

(A) Only I

(B) Both II and III

(C) Both I and II

(D) Only III

Ans: B

Solution:

In the previous problem it is already explained that the middle-aged lecturer and at least one retired lecturer goes into Physics or Chemistry or both, the total is at least 12 and at most 13. Only It is more than or equal to 12 and It is less than or equal to 13 is correct.