CAT 2023 Question Paper With Answers & Explanation

Q. 1 If x and y are real numbers such that x² + (x – 2y – 1)² = –4y(x + y), then the value x – 2y is

1) –1

2) 0

3) 1

4) 2

Correct Answer
and
Explanation

Correct Answer

Explanation

Q. 2 If x and y are positive real numbers such that log_x(x² + 12) = 4 and 3 log_y x = 1, then x + y equals

1) 11

2) 10

3) 68

4) 20

Correct Answer
and
Explanation

Correct Answer

Explanation

Q. 3

1) 2√7

2) 3√7

3) 4√5

4) 3√31

Correct Answer
and
Explanation

Correct Answer

Explanation

Q. 4 Let n be the least positive integer such that 168 is a factor of 1134ⁿ. If m is the least positive integer such that 1134ⁿ is a factor of 168^m, then m + n equals

1) 15

2) 9

3) 24

4) 12

Correct Answer
and
Explanation

Correct Answer

Explanation

Q. 5 Let α and β be the two distinct roots of the equation 2x² – 6x + k = 0, such that (α + β) and αβ are the distinct roots of the equation x² + px + p = 0. Then, the value of 8(k – p) is

Correct Answer
and
Explanation

Correct Answer

Explanation

Q. 6 The equation x³ + (2r + 1)x² + (4r – 1)x + 2 = 0 has –2 as one of the roots. If the other two roots are real, then the minimum possible non-negative integer value of r is

Correct Answer
and
Explanation

Correct Answer

Explanation

Q. 7 The number of integer solutions of equation 2|x|(x² + 1) = 5x² is

Correct Answer
and
Explanation

Correct Answer

Explanation

Q. 8 The minor angle between the hours hand and minutes hand of a clock was observed at 8:48 am. The minimum duration, in minutes, after 8:48 am when this angle increases by 50% is

1) 24/11

2) 4

3) 2

4) 36/11

Correct Answer
and
Explanation

Correct Answer

Explanation

The angle moved by hour hand in 8 hours = 30°× 8 = 240°
Angle difference in 1 minute between hour hand and minute hand = (11/2)°
Angle difference in 48 minutes between hour hand and minute hand 48 × (11/2)° = 264°
So minor angle between them = 264° – 240° = 24° 50% of 24° = 12°
Hence, 12° will increase in 12/(11/2) = 24/11 minutes.

Q. 9 The salaries of three friends Sita, Gita and Mita are initially in the ratio 5 : 6 : 7, respectively. In the first year, they get salary hikes of 20%, 25% and 20%, respectively. In the second year, Sita and Mita get salary hikes of 40% and 25%, respectively, and the salary of Gita becomes equal to the mean salary of the three friends. The salary hike of Gita in the second year is

1) 26%

2) 28%

3) 30%

4) 25%

Correct Answer
and
Explanation

Correct Answer

Explanation

Let the initial salaries of Sita, Gita and Mita be 5x, 6x and 7x respectively.
Then, in first year their salaries will be 5x × 1.2, 6x × 1.25 and 7x × 1.2 i.e., 6x, 7.5x and 8.4x respectively.
In second year salaries of Sita and Mita will be 6x × 1.4 and 8.4x × 1.25 i.e., 8.4x and 10.5x respectively.
In second year, salary of Gita = (8.4x + 10.5x)/2 = 9.45x
Hence, in second year the percentage increase in salary of Gita = (9.45x – 7.5x)/7.5x × 100 = 26%.

Q. 10 A mixture P is formed by removing a certain amount of coffee from a coffee jar and replacing the same amount with cocoa powder. The same amount is again removed from mixture P and replaced with same amount of cocoa powder to form a new mixture Q. If the ratio of coffee and cocoa in the mixture Q is 16 : 9, then the ratio of cocoa in mixture P to that in mixture Q is

1) 4 : 9

2) 1 : 3

3) 1 : 2

4) 5 : 9

Correct Answer
and
Explanation

Correct Answer

Explanation

Q. 11 Brishti went on an 8-hour trip in a car. Before the trip, the car had travelled a total of x km till then, where x is a whole number and is palindromic, i.e., x remains unchanged when its digits are reversed. At the end of the trip, the car had travelled a total of 26862 km till then, this number again being palindromic. If Brishti never drove at more than 110 km/h, then the greatest possible average speed at which she drove during the trip, in km/h, was

1) 110

2) 100

3) 80

4) 90

Correct Answer
and
Explanation

Correct Answer

Explanation

From options:
Option (1): 26862 – 8 × 110 = 25982 (Not possible)
Option (2): 26862 – 8 × 100 = 26062 (Possible)
Since we have to find greatest possible average speed, so we can leave option (3) and option (4).
Hence, the required greatest possible average speed is 100 km/h.

Q. 12 In an examination, the average marks of 4 girls and 6 boys is 24. Each of the girls has the same marks while each of the boys has the same marks. If the marks of any girl is at most double the marks of any boy, but not less than the marks of any boy, then the number of possible distinct integer values of the total marks of 2 girls and 6 boys is

1) 22

2) 19

3) 21

4) 20

Correct Answer
and
Explanation

Correct Answer

Explanation

Q. 13 Gita sells two objects A and B at the same price such that she makes a profit of 20% on object A and a loss of 10% on object B. If she increases the selling price such that objects A and B are still sold at an equal price and a profit of 10% is made on object B, then the profit made on object A will be nearest to

1) 45%

2) 42%

3) 47%

4) 49%

Correct Answer
and
Explanation

Correct Answer

Explanation

Q. 14 The amount of job that Amal, Sunil and Kamal can individually do in a day, are in harmonic progression. Kamal takes twice as much time as Amal to do the same amount of job. If Amal and Sunil work for 4 days and 9 days, respectively, Kamal needs to work for 16 days to finish the remaining job. Then the number of days Sunil will take to finish the job working alone, is

Correct Answer
and
Explanation

Correct Answer

Explanation

Work done by Amal, Sunil and Kamal is in HP.
So time taken by Amal, Sunil and Kamal will be in AP.
Let Amal complete the work in n days.
Then, Kamal will complete the work in 2n days and Sunil will complete the work in (n + 2n)/2 = 1.5n days.
Ratio of time taken = n : 1.5n : 2n = 2 : 3 : 4
Ratio of work done = 1/2 : 1/3 : 1/4 = 6 : 4 : 3
Total units of work = 6 × 4 + 4 × 9 + 3 × 16 = 108
Hence, time taken by Sunil alone to complete the work is 108/4 = 27 days.

Q. 15 Anil invests Rs. 22000 for 6 years in a certain scheme with 4% interest per annum, compounded half-yearly. Sunil invests in the same scheme for 5 years, and then reinvests the entire amount received at the end of 5 years for one year at 10% simple interest. If the amounts received by both at the end of 6 years are same, then the initial investment made by Sunil, in rupees, is

Correct Answer
and
Explanation

Correct Answer

20808

Explanation

Q. 16 Arvind travels from town A to town B, and Surbhi from town B to town A, both starting at the same time along the same route. After meeting each other, Arvind takes 6 hours to reach town B while Surbhi takes 24 hours to reach town A. If Arvind travelled at a speed of 54 km/h, then the distance, in km, between town A and town B is

Correct Answer
and
Explanation

Correct Answer

972

Explanation

Q. 17 Let C be the circle x² + y² + 4x – 6y – 3 = 0 and L be the locus of the point of intersection of a pair of tangents to C with the angle between the two tangents equal to 60°. Then, the point at which L touches the line x = 6 is

1) (6, 8)

2) (6, 4)

3) (6, 3)

4) (6, 6)

Correct Answer
and
Explanation

Correct Answer

Explanation

Q. 18 A quadrilateral ABCD is inscribed in a circle such that AB : CD = 2 : 1 and BC : AD = 5 : 4. If AC and BD intersect at the point E, then AE : CE equals

1) 8 : 5

2) 5 : 8

3) 1 : 2

4) 2 : 1

Correct Answer
and
Explanation

Correct Answer

Explanation

Q. 19 In a right-angled ΔABC, the altitude AB is 5 cm, and the base BC is 12 cm. P and Q are two points on BC such that the areas of ΔABP, ΔABQ and ΔABC are in arithmetic progression. If the area of ΔABC is 1.5 times the area of ΔABP, the length of PQ, in cm, is

Correct Answer
and
Explanation

Correct Answer

Explanation

Q. 20 The number of all natural numbers up to 1000 with non-repeating digits is

1) 738

2) 648

3) 504

4) 585

Correct Answer
and
Explanation

Correct Answer

Explanation

Number of non-repeating one digit numbers
= 1 × 9 = 9
Number of non-repeating two digit numbers
= 9 × 9 = 81
Number of non-repeating three digit numbers
= 9 × 9 × 8 = 648
Hence, total numbers = 9 + 81 + 648 = 738.

Q. 21

1) √x, √z and √y are in arithmetic progression

2) x, y and z are in arithmetic progression

3) √x, √y and √z are in arithmetic progression

4) y, x and z are in arithmetic progression

Correct Answer
and
Explanation

Correct Answer

Explanation

Q. 22 A lab experiment measures the number of organisms at 8 am every day. Starting with 2 organisms on the first day, the number of organisms on any day is equal to 3 more than twice the number on the previous day. If the number of organisms on the nth day exceeds one million, then the lowest possible value of n is

Correct Answer
and
Explanation

CAT 2023 QA - Slot 1