CAT 2022 QA Slot 1 questions and solutions

1. CAT 2022 QA Slot 1 | Arithmetic - Profit & Loss

Ankita buys 4 kg cashews, 14 kg peanuts and 6 kg almonds when the cost of 7 kg cashews is the same as that of 30 kg peanuts or 9 kg almonds. She mixes all the three nuts and marks a price for the mixture in order to make a profit of ₹1752. She sells 4 kg of the mixture at this marked price and the remaining at a 20% discount on the marked price, thus making a total profit of ₹744. Then the amount, in rupees, that she had spent in buying almonds is

A.
1176
B.
1680
C.
2520
D.
1440

2. CAT 2022 QA Slot 1 | Arithmetic - Ratio, Proportion & Variation

In a village, the ratio of number of males to females is 5 : 4. The ratio of number of literate males to literate females is 2 : 3. The ratio of the number of illiterate males to illiterate females is 4 : 3. If 3600 males in the village are literate, then the total number of females in the village is

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Answer: 43200

Video Explanation

Explanation :

Number of male literates = 3600

$\frac{Number of literate males}{Number of literate females}$ = $\frac{2}{3}$

⇒ $\frac{3600}{Number of literate females}$ = $\frac{2}{3}$

⇒ Number of literate females = 5400

Let number of illiterate males and females be 4x and 3x respectively.

⇒ $\frac{Total males}{Total females}$ = $\frac{5}{4}$

⇒ $\frac{3600 + 4 x}{5400 + 3 x}$ = $\frac{5}{4}$

⇒ 14400 + 16x = 27000 + 15x

⇒ x = 12,600

⇒ Total females in the village = 5400 + 3x = 5400 + 37,800 = 43,200

Hence, 43200.

3. CAT 2022 QA Slot 1 | Arithmetic - Ratio, Proportion & Variation

Pinky is standing in a queue at a ticket counter. Suppose the ratio of the number of persons standing ahead of Pinky to the number of persons standing behind her in the queue is 3 : 5. If the total number of persons in the queue is less than 300, then the maximum possible number of persons standing ahead of Pinky is

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4. CAT 2022 QA Slot 1 | Geometry - Quadrilaterals & Polygons

A trapezium ABCD has side AD parallel to BC. ∠BAD = 90°, BC = 3 cm and AD = 8 cm. If the perimeter of this trapezium is 36 cm, then its area, in sq. cm, is

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Answer: 66

Video Explanation

Explanation :

Let AB = x

In right ∆CED,CE = AB = x
⇒ CD = $\sqrt{x^{2} + 25}$

In the given figure,
⇒ 3 + 8 + x + $\sqrt{x^{2} + 25}$ = 36
⇒ $\sqrt{x^{2} + 25}$ = 25 – x
⇒ x² + 25 = 625 + x² – 50x
⇒ 50x = 600
⇒ x = 12

Now, area of a trapezium = 1/2 × (sum of parallel sides) × height
= 1/2 × 11 × 12 = 66

Hence, 66.

5. CAT 2022 QA Slot 1 | Arithmetic - Profit & Loss

Amal buys 110 kg of syrup and 120 kg of juice, syrup being 20% less costly than juice, per kg. He sells 10 kg of syrup at 10% profit and 20 kg of juice at 20% profit. Mixing the remaining juice and syrup, Amal sells the mixture at ₹ 308.32 per kg and makes an overall profit of 64%. Then, Amal’s cost price for syrup, in Rupees per kg, is

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6. CAT 2022 QA Slot 1 | Algebra - Number Theory

Let a and b be natural numbers. If a² + ab + a = 14 and b² + ab + b = 28, then (2a + b) equals

A.
9
B.
10
C.
8
D.
7

7. CAT 2022 QA Slot 1 | Modern Math - Permutation & Combination

The number of ways of distributing 20 identical balloons among 4 children such that each child gets some balloons but no child gets an odd number of balloons, is

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8. CAT 2022 QA Slot 1 | Geometry - Coordinate Geometry

Let ABCD be a parallelogram such that the coordinates of its three vertices A, B, C are (1, 1), (3, 4) and (−2, 8), respectively. Then, the coordinates of the vertex D are

A.
(4, 5)
B.
(0, 11)
C.
(-4, 5)
D.
(-3, 4)

9. CAT 2022 QA Slot 1 | Algebra - Number Theory

Let A be the largest positive integer that divides all the numbers of the form 3^k + 4^k + 5^k and B be the largest positive integer that divides all the numbesr of the form 4^k + 3(4^k) + 4^k+2, where k is any positive integer. Then (A + B) equals

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10. CAT 2022 QA Slot 1 | Algebra - Quadratic Equations

Let a, b and c be non-zero real numbers such that b² < 4ac, and f(x) = ax² + bx + c. If the set S consists of all integers m such that f(m) < 0, then the set S must necessarily be

A.
the empty set
B.
the set of all integers
C.
either the empty set or the set of all integers
D.
the set of all positive integers

11. CAT 2022 QA Slot 1 | Arithmetic - Mixture, Alligation, Removal & Replacement

A mixture contains lemon juice and sugar syrup in equal proportion. If a new mixture is created by adding this mixture and sugar syrup in the ratio 1 : 3, then the ratio of lemon juice and sugar syrup in the new mixture is

A.
1 : 4
B.
1 : 6
C.
1 : 5
D.
1 : 7

12. CAT 2022 QA Slot 1 | Algebra - Number Theory

For natural numbers x, y and z, if xy + yz = 19 and yz + xz = 51, then the minimum possible value of xyz is

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13. CAT 2022 QA Slot 1 | Algebra - Progressions

For any natural number n, suppose the sum of the first n terms of an arithmetic progression is (n+ 2n²). If the n^th term of the progression is divisible by 9, then the smallest possible value of n is

A.
9
B.
8
C.
4
D.
7

14. CAT 2022 QA Slot 1 | Arithmetic - Average

The average weight of students in a class increases by 600 gm when some new students join the class. If the average weight of the new students is 3 kg more than the average weight of the original students, then the ratio of number of original students to the number of new students is

A.
3 : 1
B.
4 : 1
C.
1 : 2
D.
1 : 4

15. CAT 2022 QA Slot 1 | Algebra - Inequalities & Modulus

The largest real value of a for which the equation |x + a| + |x - 1| = 2 has an infinite number of solutions for x is

A.
-1
B.
2
C.
0
D.
1

16. CAT 2022 QA Slot 1 | Geometry - Circles

All the vertices of a rectangle lie on a circle of radius R. If the perimeter of the rectangle is P, then the area of the rectangle is

A.
$\frac{P^{2}}{2}$ - 2PR
B.
$\frac{P^{2}}{8}$ - 2R²
C.
$\frac{P^{2}}{8}$ - $\frac{R^{2}}{2}$
D.
$\frac{P^{2}}{16}$ - R²

Answer: Option B

Video Explanation

Explanation :

Let the longer side of the rectangle be a and shorter side by b.

In the figure given, (2R)² = a² + b²

⇒ P = 2(a + b)

⇒ $\frac{P}{2}$ = a + b

Squaring both sides.

⇒ $\frac{P^{2}}{4}$ = a² + b² + 2ab

⇒ $\frac{P^{2}}{4}$ - 4R² = 2ab

⇒ $\frac{P^{2}}{8}$ - 2R² = ab

Area of the rectangle = a × b = ab = $(\frac{P^{2}}{8} - {2R}^{2})$

Hence, option (b).

17. CAT 2022 QA Slot 1 | Algebra - Functions & Graphs

For any real number x, let [x] be the largest integer less than or equal to x. If $\sum_{n = 1}^{N} [\frac{1}{5} + \frac{n}{25}]$ = 25, then N is

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Answer: 44

Video Explanation

Explanation :

$[\frac{1}{5} + \frac{n}{25}]$ = 0 when $\frac{1}{5} + \frac{n}{25}$ < 1 ⇒ n < 20

$[\frac{1}{5} + \frac{n}{25}]$ = 1 when 1 ≤ $\frac{1}{5} + \frac{n}{25}$ < 2 ⇒ 20 ≤ n < 45

$\sum_{n = 1}^{44} [\frac{1}{5} + \frac{n}{25}]$ = $\sum_{n = 1}^{19} [\frac{1}{5} + \frac{n}{25}]$ + $\sum_{n = 20}^{44} [\frac{1}{5} + \frac{n}{25}]$ = 0 + (1 + 1 + 1 + … + 1 (25 times)) = 25

∴ N = 44

Hence, 44.

18. CAT 2022 QA Slot 1 | Arithmetic - Time, Speed & Distance

Trains A and B start traveling at the same time towards each other with constant speeds from stations X and Y, respectively. Train A reaches station Y in 10 minutes while train B takes 9 minutes to reach station X after meeting train A. Then the total time taken, in minutes, by train B to travel from station Y to station X is

A.
12
B.
15
C.
10
D.
6

19. CAT 2022 QA Slot 1 | Arithmetic - Average

The average of three integers is 13. When a natural number n is included, the average of these four integers remains an odd integer. The minimum possible value of n is:

A.
5
B.
1
C.
3
D.
4

Answer: Option A

Video Explanation

Explanation :

Sum of the original 3 numbers = 3 × 13 = 39.

Now, $\frac{39 + n}{4}$ = odd = 2k - 1

⇒ 39 + n = 8k – 4

⇒ 43 + n = 8k

For k to be an integer least possible natural value of n = 5

Hence, option (a).

20. CAT 2022 QA Slot 1 | Venn Diagram

In a class of 100 students, 73 like coffee, 80 like tea and 52 like lemonade. It may be possible that some students do not like any of these three drinks. Then the difference between the maximum and minimum possible number of students who like all the three drinks is

A.
52
B.
47
C.
48
D.
53

21. CAT 2022 QA Slot 1 | Arithmetic - Simple & Compound Interest

Alex invested his savings in two parts. The simple interest earned on the first part at 15% per annum for 4 years is the same as the simple interest earned on the second part at 12% per annum for 3 years. Then, the percentage of his savings invested in the first part is

A.
40%
B.
62.5%
C.
60%
D.
37.5%

Answer: Option D

Video Explanation

Explanation :

Let the amount invested in first and second parts is F and S respectively.

⇒ $\frac{F \times 15 \times 4}{100}$ = $\frac{S \times 12 \times 3}{100}$

⇒ $\frac{F}{S}$ = $\frac{3}{5}$

⇒ % amount invested in first part = $\frac{3}{3 + 5}$ × 100% = 37.5%

Hence, option (d).

22. CAT 2022 QA Slot 1 | Algebra - Inequalities & Modulus

Let 0 ≤ a ≤ x ≤ 100 and f(x) = |x - a| + |x - 100| + |x - a - 50|. Then the maximum value of f(x) becomes 100 when a is equal to

A.
0
B.
25
C.
50
D.
100

CAT 2022 QA Slot 1 | Previous Year CAT Paper

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