Math JEE Adv. 2025 paper 1

 

To solve the problem, we'll analyze the given frequency distribution step-by-step and match the entries in List-I to List-II.

Given Data:

• Values (x): 4, 5, 8, 9, 6, 12, 11

• Frequencies (f): 5, $f_1$, $f_2$, 2, 1, 1, 3

• Total frequency: 19

• Median: 6

Step 1: Determine $f_1$ and $f_2$

The total frequency is:

$$\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">+</span>{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">+</span>{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">19</span>}\text{<span style="background-color: white;">\hspace{0.25em}\hspace{0.05em}</span>}<span\; style="background-color:\; white;">⟹</span>\text{<span style="background-color: white;">\hspace{0.25em}\hspace{0.05em}</span>}{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">+</span>{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">7.</span>}$$

Since the median is 6, the cumulative frequency up to the median must cover the 10th observation (as $\frac{19+1}{2}=10$).

Cumulative Frequencies:

• Up to 4: 5

• Up to 5: $5+f_1$

• Up to 6: $5+f_1+1=6+f_1$

For the median to be 6, the 10th observation must fall in the 6-value category:

$$\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">+</span>{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">\le </span>\mathrm{<span\; style="background-color:\; white;">10</span>}<span\; style="background-color:\; white;">\le </span>\mathrm{<span\; style="background-color:\; white;">6</span>}<span\; style="background-color:\; white;">+</span>{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}\text{<span style="background-color: white;">\hspace{0.25em}\hspace{0.05em}</span>}<span\; style="background-color:\; white;">⟹</span>\text{<span style="background-color: white;">\hspace{0.25em}\hspace{0.05em}</span>}{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">4.</span>}$$

Thus, $f_2=7-f_1=3$.

Verification of $7f_1+9f_2$:

$$\mathrm{<span\; style="background-color:\; white;">7</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">9</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">28</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">27</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">55.</span>}$$

So, $(P)\to 55$.

Step 2: Compute the Mean ($\mu$)

First, calculate the sum of $x\times f$:

$$\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">8</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">9</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">6</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">12</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">11</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">20</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">20</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">24</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">18</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">6</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">12</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">33</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">133.</span>}$$

The mean is:

$$\mathrm{<span\; style="background-color:\; white;">\mu </span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">133</span>}}{\mathrm{<span\; style="background-color:\; white;">19</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">7.</span>}$$

Step 3: Compute Mean Deviation about the Mean ($\alpha$)

Calculate $\mathrm{\mid }x-\mu \mathrm{\mid }\times f$ for each value:

$$\mathrm{<span\; style="background-color:\; white;">\mid </span>}\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}\mathrm{<span\; style="background-color:\; white;">\mid </span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">15</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">\mid </span>}\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}\mathrm{<span\; style="background-color:\; white;">\mid </span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">8</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">\mid </span>}\mathrm{<span\; style="background-color:\; white;">8</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}\mathrm{<span\; style="background-color:\; white;">\mid </span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">\mid </span>}\mathrm{<span\; style="background-color:\; white;">9</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}\mathrm{<span\; style="background-color:\; white;">\mid </span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">\mid </span>}\mathrm{<span\; style="background-color:\; white;">6</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}\mathrm{<span\; style="background-color:\; white;">\mid </span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">\mid </span>}\mathrm{<span\; style="background-color:\; white;">12</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}\mathrm{<span\; style="background-color:\; white;">\mid </span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">\mid </span>}\mathrm{<span\; style="background-color:\; white;">11</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}\mathrm{<span\; style="background-color:\; white;">\mid </span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">12.</span>}$$

Total deviation:

$$\mathrm{<span\; style="background-color:\; white;">15</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">8</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">12</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">48.</span>}$$

Thus, $19\alpha =48$, so $(Q)\to 48$.

Step 4: Compute Mean Deviation about the Median ($\beta$)

The median is 6. Calculate $\mathrm{\mid }x-6\mathrm{\mid }\times f$:

$$\mathrm{<span\; style="background-color:\; white;">\mid </span>}\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">6</span>}\mathrm{<span\; style="background-color:\; white;">\mid </span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">10</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">\mid </span>}\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">6</span>}\mathrm{<span\; style="background-color:\; white;">\mid </span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">\mid </span>}\mathrm{<span\; style="background-color:\; white;">8</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">6</span>}\mathrm{<span\; style="background-color:\; white;">\mid </span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">6</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">\mid </span>}\mathrm{<span\; style="background-color:\; white;">9</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">6</span>}\mathrm{<span\; style="background-color:\; white;">\mid </span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">6</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">\mid </span>}\mathrm{<span\; style="background-color:\; white;">6</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">6</span>}\mathrm{<span\; style="background-color:\; white;">\mid </span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">0</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">\mid </span>}\mathrm{<span\; style="background-color:\; white;">12</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">6</span>}\mathrm{<span\; style="background-color:\; white;">\mid </span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">6</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">\mid </span>}\mathrm{<span\; style="background-color:\; white;">11</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">6</span>}\mathrm{<span\; style="background-color:\; white;">\mid </span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">15.</span>}$$

Total deviation:

$$\mathrm{<span\; style="background-color:\; white;">10</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">6</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">6</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">0</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">6</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">15</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">47.</span>}$$

Thus, $19\beta =47$, so $(R)\to 47$.

Step 5: Compute Variance (${\sigma }^2$)

Calculate $(x-\mu {)}^2\times f$:

$$<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">45</span>}<span\; style="background-color:\; white;">,</span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">4</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">16</span>}<span\; style="background-color:\; white;">,</span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">8</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">,</span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">9</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">8</span>}<span\; style="background-color:\; white;">,</span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">6</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">,</span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">12</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">25</span>}<span\; style="background-color:\; white;">,</span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">11</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">48.</span>}$$

Total squared deviation:

$$\mathrm{<span\; style="background-color:\; white;">45</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">16</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">8</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">25</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">48</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">146.</span>}$$

Thus, $19{\sigma }^2=146$, so $(S)\to 146$.

Final Matching:

• $(P)\to 55$

• $(Q)\to 48$ (3)

• $(R)\to 47$ (2)

• $(S)\to 146$ (1)

Correct Option: C

$$\boxed{{\displaystyle \mathrm{<span\; style="color:\; black;"><span\; style="background-color:\; white;">C</span></span>}}}$$