CAT 2023 QA Slot 2 Questions and Solutions
Question 1
The number of positive integers less than 50, having exactly two distinct factors other than 1 and itself, is:
A. 13
B. 15
C. 16
D. 17
Answer: B. 15
Detailed Solution:
Positive integers with exactly two distinct factors other than 1 and itself are of the form \( a \times b \) or \( a^3 \), where \( a \) and \( b \) are prime numbers.
Possible combinations include:
If one prime is 2, the other can be 3, 5, 7, etc.
∴ Total count = 15.
Question 2
A container has 40 liters of milk. 4 liters are removed and replaced with 4 liters of water. This process is repeated. The smallest number of times required for milk to be less than water is:
A. 5
B. 6
C. 7
D. 8
Answer: C. 7
Detailed Solution:
After each replacement, \( \frac{9}{10} \) of the milk remains.
After 7 replacements, quantity of milk = \( 40 \times \left( \frac{9}{10} \right)^7 < 20 \).
Thus, 7 replacements are required.
Question 3
Let both series \( a_1, a_2, a_3, \ldots \) and \( b_1, b_2, b_3, \ldots \) be in arithmetic progression with common differences as prime numbers. If a5=b9, a19=b19 and b2=0, , then a11 equals, find \( a_{19} \):
A. 86
B. 84
C. 79
D. 83
Answer: C. 79
Detailed Solution:
Given \( a_5 = 49 \), common differences are 5 and 7 for series \( a_n \) and \( b_n \).
\( a_{19} = 49 + 6 \times 5 = 79 \).
Question 4
Let \( a_n = 13 + 6(n – 1) \) and \( b_n = 15 + 7(n – 1) \). Find the largest three-digit integer common to both sequences:
A. 967
B. 978
C. 986
D. 999
Answer: A. 967
Detailed Solution:
First common term = 43; common difference = 42.
Largest three-digit term = 967.
