XAT 2021 QADI questions and solutions

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1. XAT 2021 QADI | Algebra - Logarithms

If log₄ m + log₄ n = log₂ (m + n) where m and n are positive real numbers, then which of the following must be true?

A.
$\frac{1}{m}$ + $\frac{1}{n}$ = 1
B.
m = n
C.
m² + n² = 1
D.
$\frac{1}{m}$ + $\frac{1}{n}$ = 2
E.
No values of m and n can satisfy the given equation

Answer: Option E

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Explanation :

log₄ mn = log₂ (m + n)

$\sqrt{m n}$ = (m + n)

Squaring on both sides

m² + n² + mn = 0

Since m, n are positive real numbers, no value of m and n satisfy the above equations.

2. XAT 2021 QADI | Arithmetic - Ratio, Proportion & Variation

Mr. Jose buys some eggs. After bringing the eggs home, he finds two to be rotten and throws them away. Of the remaining eggs, he puts five-ninth in his fridge, and brings the rest to his mother’s house. She cooks two eggs and puts the rest in her fridge. If her fridge cannot hold more than five eggs, what is the maximum possible number of eggs bought by Mr. Jose?

A.
9
B.
17
C.
11
D.
20
E.
29

Answer: Option C

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Explanation :

Let the number of eggs bought = 9x + 2
number of eggs left after throwing away 2 = 9x
number of eggs kept in fridge = 5x
number of eggs brought to his mothers' house = 4x
number of eggs left after cooking 2 which are kept in fridge = 4x - 2

Given, 4x - 2 <= 5

⇒ x ≤ $\frac{7}{4}$

Hence the max value of x is 1
Max number of eggs bought = 11

3. XAT 2021 QADI | Arithmetic - Simple & Compound Interest

Mohan has some money (₹M) that he divides in the ratio of 1:2. He then deposits the smaller amount in a savings scheme that offers a certain rate of interest, and the larger amount in another savings scheme that offers half of that rate of interest. Both interests compound yearly. At the end of two years, the total interest earned from the two savings schemes is ₹830. It is known that one of the interest rates is 10% and that Mohan deposited more than ₹1000 in each saving scheme at the start. What is the value of M?

A.
7500
B.
6000
C.
To solve this, the other interest rate must also be given.
D.
4500
E.
12000

Answer: Option B

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Explanation :

Let the total amount be 3x
Case 1:
Smaller amount = x, rate of interest = 10
Larger amount = 2x, rate of interest = 5

Total amount received at the end of two years (smaller amount) = ( $1 + \frac{10}{100}$ )² = 1.21 x. CI = 0.21x

Total amount received at the end of two years (larger amount) = 2x ( $1 + \frac{5}{100}$ )² = 2.205x CI = 0.205x

Given, 0.21x + 0.205x = 830
⇒ x = 2000
2x = 4000

Case 2:
Smaller amount = x, rate of interest = 20
Larger amount = 2x, rate of interest = 10
Total amount received at the end of two years (smaller amount) = x( $1 + \frac{20}{100}$ )² = 1.44x. CI = 0.44x

Total amount received at the end of two years (larger amount) = 2x ( $1 + \frac{10}{100}$ )² = 2.42x CI = 0.42x

Given, 0.44x+0.42x = 830
⇒ x = 965.11 which is not valid since it should be greater than 1000

4. XAT 2021 QADI | Modern Math - Probability

A small store has five units of a new phone model in stock: two white, two black, and one red. Three customers arrive at the shop to buy a unit each. Each one has a pre- determined choice of the colour and will not buy a unit of any other colour. All the three customers are equally likely to have chosen any of the three colours. What is the probability that the store will be able to satisfy all the three customers?

A.
$\frac{4}{5}$
B.
$\frac{7}{9}$
C.
$\frac{2}{3}$
D.
$\frac{8}{9}$
E.
$\frac{1}{3}$

5. XAT 2021 QADI | Geometry - Basics

At any point of time, let x be the smaller of the two angles made by the hour hand with the minute hand on an analogue clock (in degrees). During the time interval from 2:30 p.m. to 3:00 p.m., what is the minimum possible value of x?

A.
45
B.
105
C.
90
D.
0
E.
75

6. XAT 2021 QADI | Arithmetic - Ratio, Proportion & Variation

One third of the buses from City A to City B stop at City C, while the rest go non-stop to City B. One third of the passengers, in the buses stopping at City C, continue to City B, while the rest alight at City C. All the buses have equal capacity and always start full from City A. What proportion of the passengers going to City B from City A travel by a bus stopping at City C?

A.
$\frac{1}{7}$
B.
$\frac{1}{9}$
C.
$\frac{1}{3}$
D.
$\frac{7}{9}$
E.
$\frac{4}{9}$

7. XAT 2021 QADI | Arithmetic - Time, Speed & Distance

Rajesh, a courier delivery agent, starts at point A and makes a delivery each at points B, C and D, in that order. He travels in a straight line between any two consecutive points. The following are known:

(i) AB and CD intersect at a right angle at E, and

(ii) BC, CE and ED are respectively 1.3 km, 0.5 km and 2.5 km long.

If AD is parallel to BC, then what is the total distance (in km) that Rajesh covers in travelling from A to D?

A.
10.2
B.
12
C.
11.5
D.
5.5
E.
18

8. XAT 2021 QADI | Algebra - Functions & Graphs

Let f(x) = $\frac{x^{2} + 1}{x^{2} - 1}$ if x ≠ 1, -1, and 1 if x = 1, -1. Let g(x) = $\frac{x + 1}{x - 1}$ if x ≠ 1, and 3 if x = 1. What is the minimum possible values of $\frac{f (x)}{g (x)}$ ?

A.
$\frac{1}{2}$
B.
-1
C.
$\frac{1}{4}$
D.
$\frac{1}{3}$
E.
1

Answer: Option D

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Explanation :

$\frac{f (x)}{g (x)}$ = $\frac{(x^{2} + 1)}{x^{2} - 1}$ . $\frac{(x - 1)}{x + 1}$ = $\frac{(x^{2} + 1)}{{(x + 1)}^{2}}$

This function is definitely greater than 0

let y = $\frac{(x^{2} + 1)}{{(x + 1)}^{2}}$

⇒ x² (y − 1) + 2yx + (y − 1) = 0 which is quadratic in x

Disctiminant should be greater than 0

4y² - 4(y - 1)² ≥ 0

⇒ y >= 1/2

When x =1, f(x)/g(x) = 1/3
Hence either the value should be greater than 1/2 or should be equal to 1/3

9. XAT 2021 QADI | Arithmetic - Time, Speed & Distance

Swati can row a boat on still water at a speed of 5 km/hr. However, on a given river, it takes her 1 hour more to row the boat 12 km upstream than downstream. One day, Swati rows the boat on this river from X to Y, which is N km upstream from X. Then she rows back to X immediately. If she takes at least 2 hours to complete this round trip, what is the minimum possible value of N?

A.
3
B.
4.8
C.
2
D.
3.6
E.
2.1

Answer: Option B

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Explanation :

Let the speed of the stream be x

Let the speed of the stream be

$\frac{12}{5 - X}$ = $\frac{12}{5 + X}$ + 1

The value of x satisfying the above equation is 1
Now,

$\frac{N}{5 + 1}$ + $\frac{N}{5 - 1}$ ≥ 2

$\frac{2 N + 3 N}{12}$ ≥ 2

⇒ N ≥ 4.8

10. XAT 2021 QADI | Modern Math - Probability

Rahul has just made a 3 × 3 magic square, in which, the sum of the cells along any row, column or diagonal, is the same number N. The enries in the cells are given as expressions in x, y and Z. Find N

A.
12
B.
36
C.
21
D.
40
E.
24

11. XAT 2021 QADI | Arithmetic - Time, Speed & Distance

On the bank of the pristine Tunga river, a deer and a tiger are joyfully playing with each other. The deer notices that it is 40 steps away from the tiger and starts running towards it. At the same time, the tiger starts running away from the deer. Both run on the same straight line. For every five steps the deer takes, the tiger takes six. However, the deer takes only two steps to cover the distance that the tiger covers in three. In how many steps can the deer catch the tiger?

A.
200
B.
To solve this, the length of a deer’s step must also be given
C.
120
D.
360
E.
320

12. XAT 2021 QADI | DI - Tables & Graphs

What BEST is known about the team compositions for the projects rated C?

A.
The three teams have two, three and six members respectively.
B.
All are either two-member or three-member teams.
C.
All are three-member teams.
D.
One is the six-member team, the rest are two-member teams.
E.
All are two-member teams.

Answer: Option E

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Explanation :

Total percentage incentive when number of team members = 1 = 100%
Total percentage incentive when the number of team members = 2 =160%
Total percentage incentive when the number of team members = 3=180%
Total percentage incentive when the number of team members = 4= 190%
Total percentage incentive when the number of team members >4 = 200%
From 1, Number of people in 8 different projects = 6, 3, 3, 3, 3, 2, 2, 2 respectively
From 2, Given, exactly three projects are rated C and 4.8 lakh is paid in total
A minimum of 3 lakhs has to be paid for rating C => 3 *1.6 = 4.8lakhs ⇒ All 2 member teams have been rated C
From 3, one project has been rated A*. Let that project be handled by the team of 3 members ⇒ Incentives = 180% of 6 = 10.8 lakh
Now remaining 6, 3, 3, 3 should be either rated A or B and the total incentives should be equal to 45 - 10.8 - 4.8 = 29.4 lakhs
Let us assume 6 has been rated B ⇒ Incentives = 200% of 3 = 6 lakhs
The remaining 23.4 lakhs should come from 180% $\frac{23.4}{1.8}$ = 13 lakhs

Hence the remaining 3,3,3 can be rated as A, A, B
Hence final ratings are and total payouts are
6 - B - 6lakhs
3- A - 9 lakhs
3-A - 9 lakhs
3-B - 5.4 lakhs
3-A* - 10.8lakhs
2-C - 1.6 lakhs
2-C - 1.6 lakhs
2-C - 1.6 lakhs

13. XAT 2021 QADI | Arithmetic - Percentage

What BEST is known about the team composition for the project rated A*?

A.
A three-member team
B.
Either a three-member team or the six-member team
C.
A two-member team
D.
Either a two-member team or a three-member team
E.
The six-member team

Answer: Option A

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Explanation :

Total percentage incentive when number of team members = 1 = 100%
Total percentage incentive when the number of team members = 2 =160%
Total percentage incentive when the number of team members = 3=180%
Total percentage incentive when the number of team members = 4= 190%
Total percentage incentive when the number of team members >4 = 200%
From 1, Number of people in 8 different projects = 6, 3, 3, 3, 3, 2, 2, 2 respectively
From 2, Given, exactly three projects are rated C and 4.8 lakh is paid in total
A minimum of 3 lakhs has to be paid for rating C => 3 *1.6 = 4.8lakhs ⇒ All 2 member teams have been rated C
From 3, one project has been rated A*. Let that project be handled by the team of 3 members ⇒ Incentives = 180% of 6 = 10.8 lakh
Now remaining 6, 3, 3, 3 should be either rated A or B and the total incentives should be equal to 45 - 10.8 - 4.8 = 29.4 lakhs
Let us assume 6 has been rated B ⇒ Incentives = 200% of 3 = 6 lakhs
The remaining 23.4 lakhs should come from 180% $\frac{23.4}{1.8}$ = 13 lakhs

Hence the remaining 3,3,3 can be rated as A, A, B
Hence final ratings are and total payouts are
6 - B - 6lakhs
3- A - 9 lakhs
3-A - 9 lakhs
3-B - 5.4 lakhs
3-A* - 10.8lakhs
2-C - 1.6 lakhs
2-C - 1.6 lakhs
2-C - 1.6 lakhs

14. XAT 2021 QADI | Algebra - Simple Equations

Total amount of money paid for projects rated A (in lakhs of Rupees) is:

A.
19
B.
15
C.
16
D.
17
E.
18

Answer: Option E

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Explanation :

Total percentage incentive when number of team members = 1 = 100%
Total percentage incentive when the number of team members = 2 =160%
Total percentage incentive when the number of team members = 3=180%
Total percentage incentive when the number of team members = 4= 190%
Total percentage incentive when the number of team members >4 = 200%
From 1, Number of people in 8 different projects = 6, 3, 3, 3, 3, 2, 2, 2 respectively
From 2, Given, exactly three projects are rated C and 4.8 lakh is paid in total
A minimum of 3 lakhs has to be paid for rating C => 3 *1.6 = 4.8lakhs ⇒ All 2 member teams have been rated C
From 3, one project has been rated A*. Let that project be handled by the team of 3 members ⇒ Incentives = 180% of 6 = 10.8 lakh
Now remaining 6, 3, 3, 3 should be either rated A or B and the total incentives should be equal to 45 - 10.8 - 4.8 = 29.4 lakhs
Let us assume 6 has been rated B ⇒ Incentives = 200% of 3 = 6 lakhs
The remaining 23.4 lakhs should come from 180% $\frac{23.4}{1.8}$ = 13 lakhs

Hence the remaining 3,3,3 can be rated as A, A, B
Hence final ratings are and total payouts are
6 - B - 6lakhs
3- A - 9 lakhs
3-A - 9 lakhs
3-B - 5.4 lakhs
3-A* - 10.8lakhs
2-C - 1.6 lakhs
2-C - 1.6 lakhs
2-C - 1.6 lakhs

15. XAT 2021 QADI | Algebra - Simple Equations

What is the number of happy male shoppers?

A.
10
B.
15
C.
5
D.
20
E.
40

16. XAT 2021 QADI | Algebra - Simple Equations

Which among the following is the lowest?

A.
Number of neutral first-time female shoppers
B.
Number of unhappy first-time female shoppers
C.
Number of unhappy first-time male shoppers
D.
Number of neutral first-time male shoppers
E.
Number of happy returning male shoppers

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