CAT 2025 QA Slot 1 Question Paper With Detailed PDF Solutions

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CAT 2025 QA Slot 1 Paper With Answers & Explanation

Q. 1 In a class, there were more than 10 boys and a certain number of girls. After 40% of the girls and 60% of the boys left the class, the remaining number of girls was 8 more than the remaining number of boys. Then, the minimum possible number of students initially in the class was

Explanation
Correct Answer

55

Explanation

Let the initial number of boys and girls be b and g, respectively.
Then, g – 0.4g = b – 0.6b + 8, where b > 10
⇒ 0.6g = 0.4b + 8
⇒ 3g = 2b + 40
⇒ g = (2b + 40)/3
Since b > 10,
For b = 11, g = (22 + 40)/3 = 62/3 is not an integer.
For b = 12, g = (24 + 40)/3 = 64/3 is not an integer.
For b = 13, g = (26 + 40)/3 = 66/3 = 22 is an
integer.
But 40% of 13 = 5.2 (not an integer)
Since 40% of b must be an integer ⇒ b must be a multiple of 5.
For b = 15, g = (2 × 15 + 40)/3 = 70/3 is not an integer.
For b = 20, g = (2 × 20 + 40)/3 = 80/3 is not an integer.
For b = 25, g = (2 × 25 + 40)/3 = 90/3 = 30 is an integer.
Hence, the minimum number of students in the class was = 25 + 30 = 55.

Q. 2 The number of distinct integers n for which

Explanation
Correct Answer

1

Explanation


Q. 3 Shruti travels a distance of 224 km in four parts for a total travel time of 3 hours. Her speeds in these four parts follow an arithmetic progression, and the corresponding time taken to cover these four parts follow another arithmetic progression. If she travels at a speed of 960 meters per minute for 30 minutes to cover the first part, then the distance, in meters, she travels in the fourth part is

Explanation
Correct Answer

4

Explanation

Total distance travelled = 224 km = 224000 m
Total time taken = 3 hours = 180 minutes
Since the times taken are in arithmetic progression,
they are: 30, 30 + t, 30 + 2t, 30 + 3t
So 30 + (30 + t) + (30 + 2t) + (30 + 3t) = 180
⇒ 30 × 4 + 6t = 180
⇒ t = 10
So the times are 30, 40, 50, 60 minutes.
Since the speeds are also in arithmetic
progression, they are: 960, (960 + d), (960 + 2d),
(960 + 3d) m/minute
So 960 × 30 + (960 + d) × 40 + (960 + 2d) × 50 + (960 + 3d) × 60 = 224000
⇒ 172800 + 320d = 224000
⇒ 172800 + 320d = 224000
⇒ d = 160
Hence, the distance travelled in the fourth part = (960 + 3 × 160) × 60 = 86400 m.

Q. 4 The (x, y) coordinates of vertices P, Q and R of a parallelogram PQRS are (–3, –2), (1, –5) and (9, 1), respectively. If the diagonal SQ intersects the x-axis at (a, 0), then the value of a is

Explanation
Correct Answer

1

Explanation


Co-ordinates of O are: (– 3 + 9)/2, (– 2 + 1)/2 = (3, – 1/2)
So (p + 1)/2 = 3 ⇒ p = 5 and (q – 5)/2 = – 1/2 ⇒ q = 4
The co-ordinates of vertex S is (5, 4).
Slope of SQ = (– 5 – 4)/(1 – 5) = 9/4 The equation of line SQ is (y – 4) = 9/4 (x – 5) = 0 ⇒ 9x – 4y = 29
This line intersects x-axis at (a, 0).
So 9a – 0 = 29 ⇒ a = 29/9.

Q. 5 In a circle with center C and radius 6√2 cm, PQ and SR are two parallel chords separated by one of the diameters. If ∠ PQC = 45°, and the ratio of the perpendicular distance of PQ and SR from C is 3 : 2, then the area, in sq. cm, of the quadrilateral PQRS is

Explanation
Correct Answer

2

Explanation

Q. 6 Stocks A, B and C are priced at rupees 120, 90 and 150 per share, respectively. A trader holds a portfolio consisting of 10 shares of stock A, and 20 shares of stocks B and C put together. If the total value of her portfolio is rupees 3300, then the number of shares of stock B that she holds, is

Explanation
Correct Answer

15

Explanation

Let n be the number of stocks B.
Then, the number of stocks C = 20 – n
So 10 × 120 + 90n + (20 – n) × 150 = 3300
⇒ 1200 + 90n + 3000 – 150n = 3300
⇒ 60n = 900
⇒ n = 15.

Q. 7 For any natural number k, let ak = 3k. The smallest natural number m for which {(a1)1 × (a2)2 × ... × (a20)20} < {a21 × a22× ... × a(20+m)}, is

Explanation
Correct Answer

3

Explanation

Q. 8 In the set of consecutive odd numbers {1, 3, 5, ..., 57}, there is a number k such that the sum of all the elements less than k is equal to the sum of all the elements greater than k. Then, k equals

Explanation
Correct Answer

4

Explanation

1 + 3 + 5 + … n terms = n2
So 1 + 3 + 5 + … + 57 = 292 = 841
1 + 3 + 5 + … + (k – 2), k, (k + 2) + k + 4, … + 57
Let 1 + 3 + 5 + … + (k – 2) = (k + 2) + k + 4, … + 57 = x2
Then, x2 + k + x2 = 841
⇒ x2 = (841 – k)/2
Option (1): k = 43, x2 = (841 – 43)/2 = 399 ⇒ x = √399
Option (2): k = 39, x2 = (841 – 39)/2 = 401 ⇒ x = √401
Option (3): k = 37, x2 = (841 – 37)/2 = 402 ⇒ x = √402
Option (2): k = 41, x2 = (841 – 41)/2 = 400 ⇒ x = 20
Hence, option (4) is the correct answer.

Q. 9 Kamala divided her investment of Rs. 10,0000 between stocks, bonds, and gold. Her investment in bonds was 25% of her investment in gold. With annual returns of 10%, 6%, 8% on stocks, bonds, and gold, respectively, she gained a total amount of Rs. 8,200 in one year. The amount, in rupees, that she gained from the bonds, was

Explanation
Correct Answer

900

Explanation

Let Kamala invested her money in stocks and gold be Rs.x and Rs.y, respectively.
Then, x + 0.25y + y = 100000
⇒ x + 1.25y = 100000 … (i)
And 0.1x + 0.06 × 0.25y + 0.08y = 8200
⇒ x + 0.95y = 82000 … (ii)
From (i) and (ii), 0.3y = 18000 ⇒ y = Rs.60,000
So amount invested in bonds = 0.25 × 60000
= Rs.15,000
Hence, the amount gained by Kamala from the
bonds = 15000 × 0.06 = Rs.900.

Q. 10 If a – 6b + 6c = 4 and 6a + 3b – 3c = 50, where a, b and c are real numbers, the value of 2a + 3b – 3c is

Explanation
Correct Answer

2

Explanation

a – 6b + 6c = 4 ⇒ a – 2(3b – 3c) = 4
6a + 3b – 3c = 50 ⇒ 6a + (3b – 3c) = 50
Let 3b – 3c = d.
Then, a – 2d = 4 … (i)
and 6a + d = 50 … (ii)
From (i) and (ii), a = 8 and d = 2
Hence, 2a + 3b – 3c = 2a + d = 2 × 8 + 2 = 18.

Q. 11 A cafeteria offers 5 types of sandwiches. Moreover, for each type of sandwich, a customer can choose one of 4 breads and opt for either small or large sized sandwich. Optionally, the customer may also add up to 2 out of 6 available sauces. The number of different ways in which an order can be placed for a sandwich, is

Explanation
Correct Answer

1

Explanation

There are 5 types of sandwiches.
For each sandwich, the customer can choose 1 of 4 breads.
Each sandwich can be either small or large = 2 choices.
The customer may add up to 2 sauces from 6 available sauces.
Hence, the total number of ways
= 5 × 4 × 2 × (6C0 + 6C1 + 6C2)
= 40 × (1 + 6 + 15) = 880

Q. 12 A value of c for which the minimum value of f(x) = x2 – 4cx + 8c is greater than the maximum value of g(x) = –x2 + 3cx – 2c, is

Explanation
Correct Answer

4

Explanation

The function f(x) = x2 – 4cx + 8c has a minimum at xmin = 4c/2 = 2c
So the minimum value = f(2c) = (2c)2 – 4c × 2c + 8c =– 4c2 + 8c
The function g(x) = – x2 + 3cx – 2c has a maximum at xmax = –3c/2(–1) = 3c/2
So the maximum value = f(3c/2) = – (3c/2)2 + 3c × 3c/2 – 2c = 9c2/4 – 2c
According to the question, – 4c2 + 8c > 9c2/4 – 2c Or, – 25c2 – 40c > 0
Or, 5c(5c – 8) < 0
If c < 0 and 5c – 8 > 0 ⇒ c > 8/5, not possible.
If c > 0 and 5c – 8 < 0 ⇒ c < 8/5, which is possible.
So 0 < c < 8/5
Hence, c = 1/2 is the correct option.

Q. 13 Let 3 ≤ x ≤ 6 and [x2] = [x]2, where [x] is the greatest integer not exceeding x. If set S represents all feasible values of x, then a possible subset of S is

Explanation
Correct Answer

1

Explanation

Q. 14 The number of distinct pairs of integers (x, y) satisfying the inequalities x > y ≥ 3 and x + y < 14 is

Explanation
Correct Answer

16

Explanation

Q. 15 At a certain simple rate of interest, a given sum amounts to Rs. 13,920 in 3 years, and to Rs. 18,960 in 6 years and 6 months. If the same given sum had been invested for 2 years at the same rate as before but with interest compounded every 6 months, then the total interest earned, in rupees, would have been nearest to

Explanation
Correct Answer

2

Explanation

Let the given sum be Rs. P and the simple interest rate be r% per annum.
Then, P(1 + 3r/100) = 13920 … (i)
And P(1 + 6.5r/100) = 18960 … (ii)
From (i) and (ii),
(1 + 6.5r/100)/(1 + 3r/100) = 18960/13920 = 79/58
⇒ 58 + 3.77r = 79 + 2.37r
⇒ 1.4r = 21
⇒ r = 15%
So from (i), P = 13920/1.45 = Rs.9,600
Hence, the required interest earned = 9600(1 + 7.5/100)4 – 9600
≈ 12820.50 – 9600 ≈ Rs.3,221.

Q. 16 A container holds 200 litres of a solution of acid and water, having 30% acid by volume. Atul replaces 20% of this solution with water, then replaces 10% of the resulting solution with acid, and finally replaces 15% of the solution thus obtained, with water. The percentage of acid by volume in the final solution obtained after these three replacements, is nearest to

Explanation
Correct Answer

1

Explanation

In 200 liter solution, acid = 0.3 × 200 = 60 liter and water = 140 liter
20% of 200 = 40 liter
In 40 liter solution, acid = 0.3 × 40 = 12 liter and water = 28 liter
Now, in 200 liter solution, acid = 60 – 12 = 48 liter and water = 140 – 28 + 40 = 152 liter
Acid % = 48/200 × 100 = 24%, water % = 76% 10% of 200 = 20 liters
In 20 liters, acid = 0.24 × 20 = 4.8 liter and water = 15.2 liter
Now, in 200 liter solution, acid = 48 – 4.8 + 20 = 63.2 liter and water = 152 – 15.2 = 136.8 liter Acid % = 63.2/200 × 100 = 31.6%, water % = 68.4%
15% of 200 = 30 liters
In 30 liters, acid = 0.316 × 30 = 9.48 liters and water = 30 – 9.48 = 20.52 liter
Now, in 200 liter solution, acid = 63.2 – 9.48 = 53.72 liter and water = 136.8 – 20.52 + 30 = 146.28 liter
Now, acid % = 53.72/200 × 100 = 26.86% ≈ 27%.

Q. 17 The number of non-negative integer values of k for which the quadratic equation x2 – 5x + k = 0 has only integer roots, is

Explanation
Correct Answer

3

Explanation

The equation x2 – 5x + k = 0 has integer roots.
Then, (– 5)2 – 4k = 25 – 4k = Perfect square ≥ 0

Hence, k = 0, 4, 6

Q. 18 A shopkeeper offers a discount of 22% on the marked price of each chair, and gives 13 chairs to a customer for the discounted price of 12 chairs to earn a profit of 26% on the transaction. If the cost price of each chair is Rs. 100, then the marked price, in rupees, of each chair is

Explanation
Correct Answer

175

Explanation

The cost price of each chair is Rs.100.
Let the marked price of each chair be x.
Then, selling price of each chair = 0.78x
Total cost price of 13 chairs = Rs.1,300
Selling price of 12 chairs = 1300 × 1.26 = Rs.1,638
Selling price of each chair = 1,638/12 = Rs.136.50
Hence, the marked price of each chair
= 136.50/0.78 = Rs.175.

Q. 19 In a 3-digit number N, the digits are non-zero and distinct such that none of the digits is a perfect square, and only one of the digits is a prime number. Then, the number of factors of the minimum possible value of N is

Explanation
Correct Answer

6

Explanation

1, 4, 9 are perfect square digits.
The remaining digits are 2, 3, 5, 6, 7, 8.
Out of these 2, 3, 5, 7 are prime numbers and 6, 8 are non-prime numbers.
Since only one digit must be prime, the other two must be non-prime.
So minimum possible value of N = 268 = 22 × 67
Hence, the number of factors is (2 + 1) × (1 + 1)
= 6.

Q. 20 If the length of a side of a rhombus is 36 cm and the area of the rhombus is 396 sq. cm, then the absolute value of the difference between the lengths, in cm, of the diagonals of the rhombus is

Explanation
Correct Answer

60

Explanation

Q. 21 The ratio of the number of students in the morning shift and afternoon shift of a school was 13 : 9. After 21 students moved from the morning shift to the afternoon shift, this ratio became 19 : 14. Next, some new students joined the morning and afternoon shifts in the ratio 3 : 8 and then the ratio of the number of students in the morning shift and the afternoon shift became 5 : 4. The number of new students who joined is

Explanation
Correct Answer

1

Explanation

Let the number of students in the morning and afternoon shift be 13x and 9x, respectively.
Then, (13x – 21)/(9x + 21) = 19 /14
⇒ 182x – 294 = 171x + 399
⇒ 11x = 693
⇒ x = 63
So number of student in the morning shift = 819
and in the afternoon shift = 567
Let the number of new students joined the morning and afternoon shifts be 3y and 8y, respectively.
Then, (819 – 21 + 3y)/(567 + 21 + 8y) = 5/4
⇒ 3192 + 12y = 2940 + 40y
⇒ 28y = 252
⇒ y = 9
Hence, the new student joined = 9 × (3 + 8) = 99.

Q. 22 Arun, Varun and Tarun, if working alone, can complete a task in 24, 21, and 15 days, respectively. They charge Rs. 2,160, Rs. 2,400, and Rs. 2,160 per day, respectively, even if they are employed for a partial day. On any given day, any of the workers may or may not be employed to work. If the task needs to be completed in 10 days or less, then the minimum possible amount, in rupees, required to be paid for the entire task is

Explanation
Correct Answer

4

Explanation

Total work = LCM (24, 21, 15) = 840 units Work done per day by Arun, Varun, and Tarun = 840/24, 840/21, 840/15 = 35, 40, 56 units Rs./unit charge by Arun, Varun, and Tarun = 432/ 7, 60, 270/7
So Tarun completes in 10 days
= 10 × 56 = 560 units
The remaining work 840 – 560 = 280 units completes by Varun in 7 days
Hence, the minimum possible amount required to be paid for the entire task is
= 2160 × 10 + 2400 × 7 = 21600 + 16800 = Rs.38,400.