CAT 2021 Question Paper Slot 2

Using condition 1 :

Ravi had a total of 21 points in timeliness. 3 of the four customers among Atal, Bihari, Chirag, and Deepak gave him the same ratings. Atal gave the highest rating in timeliness in comparison with Bihari and Chirag. Hence he must have given the distinct rating.

Using condition 2 :

The possibilities are A - 9, B - 4, C - 4, D - 4.

A - 6, B - 5, C - 5, D - 5.

Ravi received a total of 29 points in the Packaging and for this, the possibility of the four scores are (5, 7, 8, 9) awarded by the four customers.

In Hygiene the sum of the ratings awarded was 26. This could possibly be awarded by considering the following cases :

A-(4,6,7,9) , B-( 5,6,7,8), C-(4,5,8,9).

Using condition 4: Bihari gave the highest rating in packaging and thus Bihari must have given a 9 rating in the packaging.

Chirag gave the highest rating in Hygiene. In condition 3 it was mentioned that Chirag gave the same points for packaging and hygiene. Since 9 was rated by Bihari packaging it cannot be awarded by Chirag in packaging and hygiene. Since Chirag was awarded the highest in Hygiene. He must award 8 points in Hygiene and Packaging.

Hence of the three possibilities among A, B, and C for Hygiene only B is the possible case with 8 as the maximum score.

In condition 5 it was mentioned that everyone awarded Ravi between 5 and 7 in Behaviour. Unique maximum and minimum ratings in this parameter were given by Atal and Deepak respectively.

Hence Atal must have awarded 7, Deepak 6, Bihari, and Chirag 6 each in Behaviour.

The two possible cases are :

Case 1 :

Case 2 :

The ratings awarded by Atal and Deepak in Packaging are among 5 and 7.

The ratings awarded by Atal, Bihari, Deepak are among 5,6, and 7.

Atal individual ranking in Packaging and Hygiene are the same. The same is true for Deepak.

Since Atal and Deepak can give the ranking among 3 and 4 in Packaging as Bihari is first and Chirag is second in this parameter.

They can rank 3 or 4 in the Hygiene parameter also. Hence Bihari must rate 7 points in Hygiene.

In both the possibilities Bihari and Chirag award a total of 26 points. Hence he wins 40 because the total ratings are greater than 25 received from Bihari and Chirag.

Since he gets a total of 120 in bonuses and tips. He must have 80 from Atal and Deepak.

This is possible if he gets a tip of 30 ad 50 from them respectively.

In case 1 irrespective of Atal standing at rank 3 or rank 4 in Hygiene and Packaging Atal total rating is greater than 25 which implies Ravi gets a tip from Atal but this is not a possible case because Ravi needs a total of Rs 80 from Atal and Deepak. From Atal if he gets Rs 20 as a bonus he cannot get a total of Rs 120 and hence this case fails.

Hence case 1 fails.

In case 2 there are two possibilities :

Atal ranking 3 in both the parameters and Deepak 4^th. Atal ranking 4^th in both the parameters and Deepak 3^rd

In the case where Atal ranks 3^rd in Packaging and Hygiene the total score is 26 and is not a feasible case.

Case - 2A:

Case - 2B :

Case - 2A: fails because Atal's total rating is greater than 25 which should not be the case.

Atal has given a rating of 6 in timeliness

Using condition 1 :

Ravi had a total of 21 points in timeliness. 3 of the four customers among Atal, Bihari, Chirag, and Deepak gave him the same ratings. Atal gave the highest rating in timeliness in comparison with Bihari and Chirag. Hence he must have given the distinct rating.

Using condition 2 :

The possibilities are A - 9, B - 4, C - 4, D - 4.

A - 6, B - 5, C - 5, D - 5.

Ravi received a total of 29 points in the Packaging and for this, the possibility of the four scores are (5, 7, 8, 9) awarded by the four customers.

In Hygiene the sum of the ratings awarded was 26. This could possibly be awarded by considering the following cases :

A-(4,6,7,9) , B-( 5,6,7,8), C-(4,5,8,9).

Using condition 4: Bihari gave the highest rating in packaging and thus Bihari must have given a 9 rating in the packaging.

Chirag gave the highest rating in Hygiene. In condition 3 it was mentioned that Chirag gave the same points for packaging and hygiene. Since 9 was rated by Bihari packaging it cannot be awarded by Chirag in packaging and hygiene. Since Chirag was awarded the highest in Hygiene. He must award 8 points in Hygiene and Packaging.

Hence of the three possibilities among A, B, and C for Hygiene only B is the possible case with 8 as the maximum score.

In condition 5 it was mentioned that everyone awarded Ravi between 5 and 7 in Behaviour. Unique maximum and minimum ratings in this parameter were given by Atal and Deepak respectively.

Hence Atal must have awarded 7, Deepak 6, Bihari, and Chirag 6 each in Behaviour.

The two possible cases are :

Case 1 :

Case 2 :

The ratings awarded by Atal and Deepak in Packaging are among 5 and 7.

The ratings awarded by Atal, Bihari, Deepak are among 5,6, and 7.

Atal individual ranking in Packaging and Hygiene are the same. The same is true for Deepak.

Since Atal and Deepak can give the ranking among 3 and 4 in Packaging as Bihari is first and Chirag is second in this parameter.

They can rank 3 or 4 in the Hygiene parameter also. Hence Bihari must rate 7 points in Hygiene.

In both the possibilities Bihari and Chirag award a total of 26 points. Hence he wins 40 because the total ratings are greater than 25 received from Bihari and Chirag.

Since he gets a total of 120 in bonuses and tips. He must have 80 from Atal and Deepak.

This is possible if he gets a tip of 30 ad 50 from them respectively.

In case 1 irrespective of Atal standing at rank 3 or rank 4 in Hygiene and Packaging Atal total rating is greater than 25 which implies Ravi gets a tip from Atal but this is not a possible case because Ravi needs a total of Rs 80 from Atal and Deepak. From Atal if he gets Rs 20 as a bonus he cannot get a total of Rs 120 and hence this case fails.

Hence case 1 fails.

In case 2 there are two possibilities :

Atal ranking 3 in both the parameters and Deepak 4^th. Atal ranking 4^th in both the parameters and Deepak 3^rd

In the case where Atal ranks 3^rd in Packaging and Hygiene the total score is 26 and is not a feasible case.

Case - 2A: 

Case - 2B :

Case - 2A: fails because Atal's total rating is greater than 25 which should not be the case.

Among Atal and Deepak, one of them gives a tip of 30 and the other gives a tip of 50. Hence 30 or 50 any case is possible

Using condition 1 :

Ravi had a total of 21 points in timeliness. 3 of the four customers among Atal, Bihari, Chirag, and Deepak gave him the same ratings. Atal gave the highest rating in timeliness in comparison with Bihari and Chirag. Hence he must have given the distinct rating.

Using condition 2 :

The possibilities are A - 9, B - 4, C - 4, D - 4.

A - 6, B - 5, C - 5, D - 5.

Ravi received a total of 29 points in the Packaging and for this, the possibility of the four scores are (5, 7, 8, 9) awarded by the four customers.

In Hygiene the sum of the ratings awarded was 26. This could possibly be awarded by considering the following cases :

A-(4,6,7,9) , B-( 5,6,7,8), C-(4,5,8,9).

Using condition 4: Bihari gave the highest rating in packaging and thus Bihari must have given a 9 rating in the packaging.

Chirag gave the highest rating in Hygiene. In condition 3 it was mentioned that Chirag gave the same points for packaging and hygiene. Since 9 was rated by Bihari packaging it cannot be awarded by Chirag in packaging and hygiene. Since Chirag was awarded the highest in Hygiene. He must award 8 points in Hygiene and Packaging.

Hence of the three possibilities among A, B, and C for Hygiene only B is the possible case with 8 as the maximum score.

In condition 5 it was mentioned that everyone awarded Ravi between 5 and 7 in Behaviour. Unique maximum and minimum ratings in this parameter were given by Atal and Deepak respectively.

Hence Atal must have awarded 7, Deepak 6, Bihari, and Chirag 6 each in Behaviour.

The two possible cases are :

Case 1 :

Case 2 :

The ratings awarded by Atal and Deepak in Packaging are among 5 and 7.

The ratings awarded by Atal, Bihari, Deepak are among 5,6, and 7.

Atal individual ranking in Packaging and Hygiene are the same. The same is true for Deepak.

Since Atal and Deepak can give the ranking among 3 and 4 in Packaging as Bihari is first and Chirag is second in this parameter.

They can rank 3 or 4 in the Hygiene parameter also. Hence Bihari must rate 7 points in Hygiene.

In both the possibilities Bihari and Chirag award a total of 26 points. Hence he wins 40 because the total ratings are greater than 25 received from Bihari and Chirag.

Since he gets a total of 120 in bonuses and tips. He must have 80 from Atal and Deepak.

This is possible if he gets a tip of 30 ad 50 from them respectively.

In case 1 irrespective of Atal standing at rank 3 or rank 4 in Hygiene and Packaging Atal total rating is greater than 25 which implies Ravi gets a tip from Atal but this is not a possible case because Ravi needs a total of Rs 80 from Atal and Deepak. From Atal if he gets Rs 20 as a bonus he cannot get a total of Rs 120 and hence this case fails.

Hence case 1 fails.

In case 2 there are two possibilities :

Atal ranking 3 in both the parameters and Deepak 4^th. Atal ranking 4^th in both the parameters and Deepak 3^rd

In the case where Atal ranks 3^rd in Packaging and Hygiene the total score is 26 and is not a feasible case.

Case - 2A: 

Case - 2B :

Case - 2A: fails because Atal's total rating is greater than 25 which should not be the case.

Atal has given the maximum rating in Behaviour.

Using condition 1 :

Ravi had a total of 21 points in timeliness. 3 of the four customers among Atal, Bihari, Chirag, and Deepak gave him the same ratings. Atal gave the highest rating in timeliness in comparison with Bihari and Chirag. Hence he must have given the distinct rating.

Using condition 2 :

The possibilities are A - 9, B - 4, C - 4, D - 4.

A - 6, B - 5, C - 5, D - 5.

Ravi received a total of 29 points in the Packaging and for this, the possibility of the four scores are (5, 7, 8, 9) awarded by the four customers.

In Hygiene the sum of the ratings awarded was 26. This could possibly be awarded by considering the following cases :

A-(4,6,7,9) , B-( 5,6,7,8), C-(4,5,8,9).

Using condition 4: Bihari gave the highest rating in packaging and thus Bihari must have given a 9 rating in the packaging.

Chirag gave the highest rating in Hygiene. In condition 3 it was mentioned that Chirag gave the same points for packaging and hygiene. Since 9 was rated by Bihari packaging it cannot be awarded by Chirag in packaging and hygiene. Since Chirag was awarded the highest in Hygiene. He must award 8 points in Hygiene and Packaging.

Hence of the three possibilities among A, B, and C for Hygiene only B is the possible case with 8 as the maximum score.

In condition 5 it was mentioned that everyone awarded Ravi between 5 and 7 in Behaviour. Unique maximum and minimum ratings in this parameter were given by Atal and Deepak respectively.

Hence Atal must have awarded 7, Deepak 6, Bihari, and Chirag 6 each in Behaviour.

The two possible cases are :

Case 1 :

Case 2 :

The ratings awarded by Atal and Deepak in Packaging are among 5 and 7.

The ratings awarded by Atal, Bihari, Deepak are among 5,6, and 7.

Atal individual ranking in Packaging and Hygiene are the same. The same is true for Deepak.

Since Atal and Deepak can give the ranking among 3 and 4 in Packaging as Bihari is first and Chirag is second in this parameter.

They can rank 3 or 4 in the Hygiene parameter also. Hence Bihari must rate 7 points in Hygiene.

In both the possibilities Bihari and Chirag award a total of 26 points. Hence he wins 40 because the total ratings are greater than 25 received from Bihari and Chirag.

Since he gets a total of 120 in bonuses and tips. He must have 80 from Atal and Deepak.

This is possible if he gets a tip of 30 ad 50 from them respectively.

In case 1 irrespective of Atal standing at rank 3 or rank 4 in Hygiene and Packaging Atal total rating is greater than 25 which implies Ravi gets a tip from Atal but this is not a possible case because Ravi needs a total of Rs 80 from Atal and Deepak. From Atal if he gets Rs 20 as a bonus he cannot get a total of Rs 120 and hence this case fails.

Hence case 1 fails.

In case 2 there are two possibilities :

Atal ranking 3 in both the parameters and Deepak 4^th. Atal ranking 4^th in both the parameters and Deepak 3^rd

In the case where Atal ranks 3^rd in Packaging and Hygiene the total score is 26 and is not a feasible case.

Case - 2A: 

Case - 2B :

Case - 2A : fails because Atal's total rating is greater than 25 which should not be the case.

Deepak gives a rating of 7 in Packaging.

$$2.25\leq2+2^{n+2}\leq202$$

$$2.25-2\le2+2^{n+2}-2\le202-2$$

$$0.25\le2^{n+2}\le200$$

$$\log_20.25\le n+2\le\log_2200$$

$$-2\le n+2\le7.xx$$

$$-4\le n\le7.xx-2$$

$$-4\le n\le5.xx$$

Possible integers = -4, -3, -2, -1, 0, 1, 2, 3, 4, 5

If we see the second expression that is provided, i.e 

$$3+3^{n+1}$$, it can be implied that n should be at least -1 for this expression to be an integer.

So, n = -1, 0, 1, 2, 3, 4, 5.

Hence, there are a total of 7 values.

Given x, y, z are three terms in an arithmetic progression.

Considering x = a, y = a+d, z = a+2*d.

Using the given equation x*y*z = 5*(x+y+z)

 a*(a+d)*(a+2*d) = 5*(a+a+d+a+2*d)

=a*(a+d)*(a+2*d)  = 5*(3*a+3*d) = 15*(a+d).

= a*(a+2*d) = 15.

Since all x, y, z are positive integers and y-x > 2. a, a+d, a+2*d are integers.

The common difference is positive and greater than 2.

Among the different possibilities are : (a=1, a+2d = 5), (a, =3, a+2d = 5), (a = 5, a+2d = 3), (a=15, a+2d = 1)

Hence the only possible case satisfying the condition is :

a = 1, a+2*d = 15.

x = 1, z = 15.

z-x = 14.

Given the 4 digit number :

Considering the number in thousands digit is a number in the hundredth digit is b, number in tens digit is c, number in the units digit is d.

Let the number be abcd.

Given that a+b+c = 14. (1)

                 b+c+d = 15. (2)

                 c = d+4. (3).

In order to find the maximum number which satisfies the condition, we need to have abcd such that a is maximum which is the digit in thousands place in order to maximize the value of the number. b, c, and d are less than 9 each as they are single-digit numbers.

Substituting (3) in (2) we have b+d+4+d = 15, b+2*d = 11.  (4)

Subtracting (2) and (1) : (2) - (1) = d = a+1.   (5)

Since c cannot be greater than 9 considering c to be the maximum value 9 the value of d is 5.

If d = 5, using d = a+1, a = 4.

Hence the maximum value of a = 4 when c = 9, d = 5.

Substituting b+2*d = 11. b = 1.

The highest four-digit number satisfying the condition is 4195

Raj invested Rs 10000 in the first year. Assuming the loss he faced was x%.

The amount after 1 year is 10,000*(1 - x/100). = 10000 - 100*x.

Given the balance was greater than Rs 5000 and hence x < 50 percent.

When Raj invested this amount in the second year he earned a profit which is five times that of the first-year percentage.

Hence the amount after the second year is : (10000 - 100x)(1+$$\frac{\left(5\cdot x\right)}{100}$$).

Raj gained a total of 35 percent over the period of two years and hence the 35 percent is Rs 3500.

Hence the final amount is Rs 13,500.

(10000 - 100x)(1+$$\frac{\left(5\cdot x\right)}{100}$$) = 13,500

$$\left(100+5\cdot x\right)\cdot\left(100\ -\ x\right)\ =\ 13500$$

10000 - 100*x +500*x - 5*$$x^2$$ = 13500.

$$5x^2-400x+3500\ =\ 0$$

Solving the equation the roots are :

x = 10, x = 70.

Since x < 50, x = 10 percent.

This question is an application of the product rule in probability and combinatorics.

In the product rule, if two events A and B can occur in x and y ways, and for an event E, both events A and B need to take place, the number of ways that E can occur is xy. This can be expanded to 3 or more events as well.

Event 1: Distribution of balloons

Since each child gets at least 4 balloons, we will initially allocate these 4 balloons to each of them. 

So we are left with 15 - 4 x 3 = 15 - 12 = 3 balloons and 3 children. 

Now we need to distribute 3 identical balloons to 3 children. 

This can be done in $$^{n+r-1}C_{r-1}$$ ways, where n = 3 and r = 3. 

So, number of ways = $$^{3+3-1}C_{3-1}=^5C_2=\frac{5\times\ 4}{2\times\ 1}=10$$

Event 2: Distribution of pencils

Since each child gets at least one pencil, we will allocate 1 pencil to each child. We are now left with 6 - 3 = 3 pencils.

We now need to distribute 3 identical pencils to 3 children.

This can be done in $$^{n+r-1}C_{r-1}$$ ways, where n = 3 and r = 3.

So, number of ways = $$^{3+3-1}C_{3-1}=^5C_2=\frac{5\times\ 4}{2\times\ 1}=10$$

Event 3: Distribution of erasers

We need to distribute 3 identical erasers to 3 children.

This can be done in $$^{n+r-1}C_{r-1}$$ ways, where n = 3 and r = 3.

So, number of ways = $$^{3+3-1}C_{3-1}=^5C_2=\frac{5\times\ 4}{2\times\ 1}=10$$

Applying the product rule, we get the total number of ways = 10 x 10 x 10 = 1000.

Considering the length of train A = La, length of train B = Lb.

The speed of train A be 5*x, speed of train B be 3*x.

From the information provided :

The front end of A crossed the rear end of B 46 seconds after the front ends of the trains had crossed each other.

In this case, train A traveled a distance equivalent to the length of train B which is Lb at a speed of 5*x+3*x = 8*x because both the trains are traveling in the opposite direction.

Hence (8*x)*(46) = Lb.

In the information provided :

It took another 69 seconds for the rear ends of the trains to cross each other.

In the next 69 seconds 

The train B traveled a distance equivalent to the length of train A in this 69 seconds.

Hence (8*x)*(69) = La.

La/Lb = 69/46 = 3/2 = 3 : 2