CAT 2023 QA Slot 3 Questions and Solutions
Question 1
Let \( n \) and \( m \) be two positive integers such that there are exactly 41 integers greater than \( 8^m \) and less than \( 8^n \), which can be expressed as powers of 2. Then, the smallest possible value of \( n + m \) is:
A. 44
B. 14
C. 16
D. 42
Answer: C. 16
Detailed Solution:
Given that there are 41 integers between \( 8^m \) and \( 8^n \), which are powers of 2. We need to find the values of \( n \) and \( m \).
Smallest value for \( n + m = 16 \).
Question 2
Suppose \( f(x, y) \) is a real-valued function such that \( f(3x + 2y, 2x – 5y) = 19x \) for all real numbers \( x \) and \( y \). The value of \( x \) for which \( f(x, 2x) = 27 \), is:
A. 3
B. 4
C. 5
D. 6
Answer: A. 3
Detailed Solution:
Given \( f(3x + 2y, 2x – 5y) = 19x \). Setting \( y = 2x \), we get:
\( f(x, 2x) = 27 \Rightarrow x = 3 \).
