UPSC NDA 2 2017 Mathematics Solved Paper

Concept:The number of notes counted per minute follows an arithmetic progression (AP) after the first 10 minutes.Explanation:In the first 10 minutes, $10 \times 150 = 1500$ notes are counted.Remaining notes: $4500 - 1500 = 3000$.From $a_{10}$ onward, the counts form an AP with $a_{10}=150$ and $d = -2$.However, the counting of the remaining notes starts from $a_{11}$.So, the AP for the remaining part has first term $a = a_{11} = 150 - 2 = 148$, and $d = -2$.Let $n$ be the number of minutes after the 10th minute.Sum of these $n$ terms must equal 3000:$S_n = \frac{n}{2}[2 \times 148 + (n-1)(-2)] = 3000$.Simplify: $\frac{n}{2}[296 - 2(n-1)] = 3000$ → $\frac{n}{2}[296 - 2n + 2] = 3000$ → $\frac{n}{2}[298 - 2n] = 3000$.Multiply by 2: $n(298 - 2n) = 6000$.Divide by 2: $n(149 - n) = 3000$.Rearrange: $149n - n^2 = 3000$ → $n^2 - 149n + 3000 = 0$.Factorize: $(n-24)(n-125) = 0$.So, $n = 24$ or $n = 125$.After 24 minutes, the term $a_{10+24} = a_{34} = 150 + (34-10)(-2) = 150 - 48 = 102$ (still positive).$n=125$ would eventually give negative counts, so $n=24$ is correct.Total time = first 10 minutes + 24 minutes = 34 minutes.Answer:B. 34 minutes