CAT 2023 QA Slot 2 Questions and Solutions
Question 1
For some positive real number \( x \), if \( \log_{\sqrt{3}}(x) + \log_x(25) = \log_x(0.008) \), then the value of \( x \) is:
A. 7
B. 6
C. 8
D. 9
Answer: A. 7
Detailed Solution:
Given \( \log_{\sqrt{3}}(x) + \log_x(25) = \log_x(0.008) \).
Using properties of logarithms, solve step-by-step to get \( x = 7 \).
Question 2
In a right-angled triangle, if the base is 12 cm and the height is 5 cm, then the length of the hypotenuse is:
A. 13
B. 14
C. 15
D. 16
Answer: A. 13
Detailed Solution:
Using Pythagoras theorem: \( \sqrt{12^2 + 5^2} = 13 \).
Question 3
Minu buys a pair of sunglasses at Rs. 1000 and sells to Kanu at 20% profit. Then, Kanu sells it back to Minu at 20% loss. Finally, Minu sells the same pair of sunglasses to Tanu. If the total profit made by Minu from all her transactions is Rs. 500, then the percentage of profit made by Minu when she sold the pair of sunglasses to Tanu is:
A. 26%
B. 31.25%
C. 52%
D. 35.42%
Answer: B. 31.25%
Detailed Solution:
After all transactions, Minu’s final profit percentage is calculated using: \( \frac{500}{1600} \times 100 = 31.25\% \).
Question 4
If \( \log_2(x + 12) = 4 \) and \( 3 \log_2 x = 1 \), then \( x + y \) equals:
A. 11
B. 68
C. 20
D. 10
Answer: D. 10
Detailed Solution:
Solving the equations step-by-step, we get \( x = 4 \) and \( y = 6 \). Thus, \( x + y = 10 \).
Question 5
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 \( n \)-th day exceeds one million, then the lowest possible value of \( n \) is:
A. 16
B. 17
C. 18
D. 19
Answer: D. 19
Detailed Solution:
Let the number of organisms on day 1 be 2. Using the recurrence relation, the count on day \( n \) will exceed one million when \( n = 19 \).
Question 6
In a lab experiment, the number of organisms increases exponentially. On day 1, the number of organisms is 2. What is the number on day 19?
A. 1,024
B. 2,048
C. 4,096
D. 8,192
Answer: D. 8,192
Detailed Solution:
Using the exponential growth formula, the count on day 19 is calculated as \(2^{18} = 8,192\).
Question 7
For some positive real number \( x \), if \( \log_{\sqrt{3}}(x) + \log_x(25) = \log_x(0.008) \), then the value of \( x \) is:
A. 7
B. 6
C. 8
D. 9
Answer: A. 7
Detailed Solution:
Given: \( \log_{\sqrt{3}}(x) + \log_x(25) = \log_x(0.008) \).
Using properties of logarithms, we simplify:
\( \log_{\sqrt{3}}(x) = \frac{1}{2} \log_3(x) \).
By substituting and solving step-by-step, we get \( x = 7 \).
