NIMCET 2024 — Mathematics PYQ

1. Core Property of Median

The median is the middlemost value of a sorted dataset. Unlike the mean, the median of a combined set depends heavily on the distribution of individual values around their respective medians, rather than just their sums.

However, a fundamental rule governs the combination of two datasets:

If two sets have medians $m_1$ and $m_2$ where $m_1\le m_2$, the median of the combined set must fall within the range bounded by the two individual medians.

$$\min (m_1,m_2)\le \text{Combined Median}\le \max (m_1,m_2)$$

2. Applying the Data

• Median of Set $A$ ($m_1$) = $2$

• Median of Set $B$ ($m_2$) = $4$

Therefore, the median of the combined set must satisfy:

$$2\le \text{Combined Median}\le 4$$

3. Evaluating the Options

• "At least $2$": This condition states that the median $\ge 2$. Since our combined median lies in the range $[2,4]$, it is guaranteed to be at least $2$. This matches perfectly.

• Why can't it be smaller? Since $m_1=2$, at least half of the numbers in set $A$ are $\ge 2$. Similarly, since $m_2=4$, at least half of the numbers in set $B$ are $\ge 4$ (which also means they are $\ge 2$). When you pool all these numbers together, more than half of the elements in the combined set will be $\ge 2$, pushing the center position to a value that is at least $2$.

Correct Answer: (D) at least $2$