CAT 2023 QA Questions and Solutions
Question 1
If \( n \) is the least positive integer such that \( 168 \) is a factor of \( 1134^n \), and \( m \) is the least positive integer such that \( 1134^m \) is a factor of \( 168^n \), then \( m + n \) equals:
A. 24
B. 12
C. 15
D. 9
Answer: C. 15
Detailed Solution:
1. Prime Factorization:
– 168 = \( 2^3 \times 3 \times 7 \)
– 1134 = \( 2 \times 3^3 \times 7 \)
2. To find n:
– For 1134n to have all the prime factors of 168, the power of 2 in 1134n should be at least 3.
– So, n = 3 (because 1134 has 1 power of 2 and we need 3 in total).
3. To find m:
– We need to find the smallest m such that 1134m has all prime factors of 168n.
– For 1683, the power of 2 is 9, so m = 12.
Final Answer: 15
Question 2
If \( \frac{\sqrt{y} + \sqrt{z}}{\sqrt{x} + \sqrt{z}} = \frac{\sqrt{x} + \sqrt{y}}{\sqrt{y} + \sqrt{z}} \), then which of the following must always be true?
A. \( x, y, z \) are in Arithmetic Progression
B. \( y, x, z \) are in Arithmetic Progression
C. \( \sqrt{x}, \sqrt{y}, \sqrt{z} \) are in Arithmetic Progression
D. \( \sqrt{y}, \sqrt{x}, \sqrt{z} \) are in Arithmetic Progression
Answer: B. \( y, x, z \) are in Arithmetic Progression
Detailed Solution:
– Start by cross-multiplying the given equation:
\[ (\sqrt{y} + \sqrt{z})(\sqrt{y} + \sqrt{x}) = (\sqrt{x} + \sqrt{z})(\sqrt{y} + \sqrt{z}) \]
– Expand both sides:
LHS: \( y + \sqrt{yx} + \sqrt{yz} + \sqrt{zx} \)
RHS: \( x + \sqrt{xy} + \sqrt{xz} + \sqrt{yz} \)
– After simplification, we see that the terms satisfy an arithmetic relationship for y, x, and z.
Conclusion: \( y, x, z \) are in Arithmetic Progression.
Final Answer: y, x, z are in Arithmetic Progression
