CAT 2023 QA Slot 3 Quiz
7. Algebra – Surds & Indices
Let \( n \) be any natural number such that \( 5^{n-1} < 3^{n+1} \).
The least integer value of \( m \) that satisfies \( 3^{n+1} < 2^{n+m} \) for all such \( n \) is?
Solution: The least value of \( m \) is 5.
For \( n = 5 \), we need \( 3^6 < 2^{5 + m} \). This gives \( 729 < 2^{10} \), so \( m = 5 \).
14. Arithmetic – Percentage
The population of a town in 2020 was 100,000. The population decreased by \( y\% \) from 2020 to 2021, and increased by \( x\% \) from 2021 to 2022.
If the population in 2022 was greater than in 2020 and \( x – y = 10 \), what was the lowest possible population in 2021?
Solution: The lowest population in 2021 is 73,000.
Using the formula for successive percentage change, the population reduces by 27%, giving \( 73,000 \) as the minimum.
15. Arithmetic – Ratio, Proportion & Variation
Two collectors A and B collect coins weekly in the ratio 3:4. If A collects a multiple of 7 in 5 weeks and B collects a multiple of 24 in 3 weeks, what is the minimum number of coins collected by A in one week?
Solution: The minimum number of coins collected by A in one week is 42.
The least common multiple of the constraints gives \( x = 14 \), making \( 3x = 42 \).
20. Algebra – Progressions
Let \( a_n = 46 + 8n \) and \( b_n = 98 + 4n \) be two sequences. Find the sum of all terms common to both sequences.
Solution: The sum of all common terms is 14,900.
The common terms form an arithmetic sequence with 50 terms, summing to \( 14,900 \).
