CAT 2019 Question Paper Slot 2 with Solutions, Download PDF

Instructions

A large store has only three departments, Clothing, Produce, and Electronics. The following figure shows the percentages of revenue and cost from the three departments for the years 2016, 2017 and 2018. The dotted lines depict percentage levels. So for example, in 2016, 50% of store's revenue came from its Electronics department while 40% of its costs were incurred in the Produce department.

In this setup, Profit is computed as (Revenue - Cost) and Percentage Profit as Profit/Cost × 100%.
It is known that
1. The percentage profit for the store in 2016 was 100%.
2. The store’s revenue doubled from 2016 to 2017, and its cost doubled from 2016 to 2018.
3. There was no profit from the Electronics department in 2017.
4. In 2018, the revenue from the Clothing department was the same as the cost incurred in the Produce department.

Question 61

What percentage of the total profits for the store in 2016 was from the Electronics department?

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Video Solution

Solution

We can make the following table from the web chart given in the question:

If we consider the total cost in the year

2016 as 100, then according to Statement 1, the total revenue in 2016 must be 200.

The store’s revenue doubled from 2016 to 2017, thus the total revenue in the year 2017 = 400.

We can find the revenue for the individual department in the year 2017, from the table.

There was no profit from the Electronics department in 2017, thus, we can find the total cost in 2017= 300

Considering the statement 4, we can find the total revenue in 2018 and tabulate the following table.

Profit in 2016 = 200-100= 100
Profit in the electronic department in 2016= 100-30=70
The total profits= for the store in 2016 were from the Electronics department 70%

Question 62

What was the approximate difference in profit percentages of the store in 2017 and 2018?

Video Solution

Solution

We can make the following table from the web chart given in the question:

If we consider the total cost in the year

2016 as 100, then according to Statement 1, the total revenue in 2016 must be 200.

The store’s revenue doubled from 2016 to 2017, thus the total revenue in the year 2017 = 400.

We can find the revenue for the individual department in the year 2017, from the table.

There was no profit from the Electronics department in 2017, thus, we can find the total cost in 2017= 300

Considering the statement 4, we can find the total revenue in 2018 and tabulate the following table.

Profit percentage in 2017= (400-300)/300 %= 33.33%

Profit percentage in 2018= (250-200)/200 %= 25%

The approximate difference in profit percentages of the store in 2017 and 2018= (33.33-25)%= 8.33%

Option C.

Instructions

Students in a college are discussing two proposals --
A: a proposal by the authorities to introduce dress code on campus, and
B: a proposal by the students to allow multinational food franchises to set up outlets on college campus.

A student does not necessarily support either of the two proposals.

In an upcoming election for student union president, there are two candidates in fray:
Sunita and Ragini. Every student prefers one of the two candidates.

A survey was conducted among the students by picking a sample of 500 students. The following information was noted from this survey.

1. 250 students supported proposal A and 250 students supported proposal B.
2. Among the 200 students who preferred Sunita as student union president, 80% supported proposal A.
3. Among those who preferred Ragini, 30% supported proposal A.
4. 20% of those who supported proposal B preferred Sunita.
5. 40% of those who did not support proposal B preferred Ragini.
6. Every student who preferred Sunita and supported proposal B also supported proposal A.
7. Among those who preferred Ragini, 20% did not support any of the proposals.

Question 63

Among the students surveyed who supported proposal A, what percentage preferred Sunita for student union president?

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Solution

Total number of students surveyed= 500

Every student prefers one of the two candidates. Ragini(R) and Sunita(S).

Thus, R+S=500.

According to statement 2, "Among the 200 students who preferred Sunita as student union president, 80% supported proposal A."

The number of students who support Sunita(S)=200

The number of students who supported Ragini(R)=300

According to statements 2 and 3, 160 students who supported Sunita also supported the proposal A & 90 students who supported Ragini also supported proposal A.

According to statements 4 and 6, we can make the following Venn diagram for Sunita.

According to statement 5 and 7, we can make the following Venn diagram.

The number of students who preferred Sunita and the proposal A=160

=160/250= 64%

Question 64

What percentage of the students surveyed who did not support proposal A preferred Ragini as student union president?

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Solution

Total number of students surveyed= 500

Every student prefers one of the two candidates. Ragini(R) and Sunita(S).

Thus, R+S=500.

According to statement 2, "Among the 200 students who preferred Sunita as student union president, 80% supported proposal A."

The number of students who support Sunita(S)=200

The number of students who supported Ragini(R)=300

According to statements 2 and 3, 160 students who supported Sunita also supported the proposal A & 90 students who supported Ragini also supported proposal A.

According to statements 4 and 6, we can make the following Venn diagram for Sunita.

According to statement 5 and 7, we can make the following Venn diagram.

The percentage of the students surveyed who did not support proposal A preferred Ragini as student union president = 210/250=84%

Answer 84

Question 65

What percentage of the students surveyed who supported both proposals A and B preferred Sunita as student union president?

Video Solution

Solution

Total number of students surveyed= 500

Every student prefers one of the two candidates. Ragini(R) and Sunita(S).

Thus, R+S=500.

According to statement 2, "Among the 200 students who preferred Sunita as student union president, 80% supported proposal A."

The number of students who support Sunita(S)=200

The number of students who supported Ragini(R)=300

According to statements 2 and 3, 160 students who supported Sunita also supported the proposal A & 90 students who supported Ragini also supported proposal A.

According to statements 4 and 6, we can make the following Venn diagram for Sunita.

According to statement 5 and 7, we can make the following Venn diagram.

According to the Venn diagram, the students surveyed who supported both proposals A and B preferred Sunita as student union president $$\frac{50}{50+50}$$ % =50%

Question 66

How many of the students surveyed supported proposal B, did not support proposal A and preferred Ragini as student union president?

Video Solution

Solution

Total number of students surveyed= 500

Every student prefers one of the two candidates. Ragini(R) and Sunita(S).

Thus, R+S=500.

According to statement 2, "Among the 200 students who preferred Sunita as student union president, 80% supported proposal A."

The number of students who support Sunita(S)=200

The number of students who supported Ragini(R)=300

According to statements 2 and 3, 160 students who supported Sunita also supported the proposal A & 90 students who supported Ragini also supported proposal A.

According to statements 4 and 6, we can make the following Venn diagram for Sunita.

According to statement 5 and 7, we can make the following Venn diagram.

From the diagram, we can understand that option A is correct.

Instructions

For the following questions answer them individually

Question 67

The average of 30 integers is 5. Among these 30 integers, there are exactly 20 which do not exceed 5. What is the highest possible value of the average of these 20 integers?

Video Solution

Solution

It is given that the average of the 30 integers = 5

Sum of the 30 integers = 30*5=150

There are exactly 20 integers whose value is less than 5.

To maximise the average of the 20 integers, we have to assign minimum value to each of the remaining 10 integers

So the sum of 10 integers = 10*6=60

The sum of the 20 integers = 150-60= 90

Average of 20 integers = $$\ \frac{\ 90}{20}$$ = 4.5

Question 68

Amal invests Rs 12000 at 8% interest, compounded annually, and Rs 10000 at 6% interest, compounded semi-annually, both investments being for one year. Bimal invests his money at 7.5% simple interest for one year. If Amal and Bimal get the same amount of interest, then the amount, in Rupees, invested by Bimal is

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Solution

The amount with Amal at the end of 1 year = 12000*1.08+10000*1.03*1.03=23569

Interest received by Amal = 23569-22000=1569

Let the amount invested by Bimal = 100b

Interest received by Bimal = 100b*7.5*1/100=7.5b

It is given that the amount of interest received by both of them is the same

7.5b=1569

b=209.2

Amount invested by Bimal = 100b = 20920

Question 69

What is the largest positive integer n such that $$\frac{n^2 + 7n + 12}{n^2 - n - 12}$$ is also a positive integer?

Video Solution

Solution

$$\ \frac{\ n^2+3n+4n+12}{n^2-4n+3n-12}$$

=$$\ \frac{\ n^{ }\left(n+3\right)+4\left(n+3\right)}{n^{ }\left(n-4\right)+3\left(n-4\right)}$$

=$$\ \frac{\left(\ n+4\right)\left(n+3\right)}{\left(n-4\right)\left(n+3\right)}$$

=$$\ \frac{\left(\ n+4\right)}{\left(n-4\right)}$$

=$$\ \frac{\left(\ n-4\right)+8}{\left(n-4\right)}$$

=$$\ 1+\frac{8}{\left(n-4\right)}$$ which will be maximum when n-4 =8

n=12

D is the correct answer.

Question 70

How many pairs (m, n) of positive integers satisfy the equation $$m^2 + 105 = n^2$$?

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Solution

$$n^2-m^2=105$$

(n-m)(n+m) = 1*105, 3*35, 5*21, 7*15, 15*7, 21*5, 35*3, 105*1.

n-m=1, n+m=105 ==> n=53, m=52

n-m=3, n+m=35 ==> n=19, m=16

n-m=5, n+m=21 ==> n=13, m=8

n-m=7, n+m=15 ==> n=11, m=4

n-m=15, n+m=7 ==> n=11, m=-4

n-m=21, n+m=5 ==> n=13, m=-8

n-m=35, n+m=3 ==> n=19, m=-16

n-m=105, n+m=1 ==> n=53, m=-52

Since only positive integer values of m and n are required. There are 4 possible solutions.

CAT Content

CAT 2019 Question Paper Slot 2

What percentage of the total profits for the store in 2016 was from the Electronics department?

What was the approximate difference in profit percentages of the store in 2017 and 2018?

Among the students surveyed who supported proposal A, what percentage preferred Sunita for student union president?

What percentage of the students surveyed who did not support proposal A preferred Ragini as student union president?

What percentage of the students surveyed who supported both proposals A and B preferred Sunita as student union president?

How many of the students surveyed supported proposal B, did not support proposal A and preferred Ragini as student union president?

The average of 30 integers is 5. Among these 30 integers, there are exactly 20 which do not exceed 5. What is the highest possible value of the average of these 20 integers?

What is the largest positive integer n such that $$\frac{n^2 + 7n + 12}{n^2 - n - 12}$$ is also a positive integer?

How many pairs (m, n) of positive integers satisfy the equation $$m^2 + 105 = n^2$$?

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CAT Content

CAT 2019 Question Paper Slot 2

What percentage of the total profits for the store in 2016 was from the Electronics department?

What was the approximate difference in profit percentages of the store in 2017 and 2018?

Among the students surveyed who supported proposal A, what percentage preferred Sunita for student union president?

What percentage of the students surveyed who did not support proposal A preferred Ragini as student union president?

What percentage of the students surveyed who supported both proposals A and B preferred Sunita as student union president?

How many of the students surveyed supported proposal B, did not support proposal A and preferred Ragini as student union president?

The average of 30 integers is 5. Among these 30 integers, there are exactly 20 which do not exceed 5. What is the highest possible value of the average of these 20 integers?

What is the largest positive integer n such that $$\frac{n^2 + 7n + 12}{n^2 - n - 12}$$ is also a positive integer?

How many pairs (m, n) of positive integers satisfy the equation $$m^2 + 105 = n^2$$?

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