Math JEE Adv. 2025 paper 1

 

To solve the problem, we'll follow these steps:

Step 1: Assign Coordinates

Let’s place the triangle $\mathrm{\Delta }PQR$ in the plane for simplicity:

• Let $P=(0,0)$, $Q=(1,0)$, and $R=(0,1)$.

Step 2: Find the Coordinates of $S$

Given the vector equation:

$$\stackrel{<span\; style="background-color:\; white;">‾</span>}{\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">P</span>}}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">5</span>}\stackrel{<span\; style="background-color:\; white;">‾</span>}{\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">Q</span>}}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">6</span>}\stackrel{<span\; style="background-color:\; white;">‾</span>}{\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">R</span>}}<span\; style="background-color:\; white;">=</span>\stackrel{<span\; style="background-color:\; white;">‾</span>}{\mathrm{<span\; style="background-color:\; white;">0</span>}}<span\; style="background-color:\; white;">,</span>$$

we express the vectors in terms of coordinates:

$$\stackrel{<span\; style="background-color:\; white;">‾</span>}{\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">P</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">P</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">S</span>}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">0</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">x</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">0</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">y</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">,</span>$$$$\stackrel{<span\; style="background-color:\; white;">‾</span>}{\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">Q</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">Q</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">S</span>}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">x</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">0</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">y</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">,</span>$$$$\stackrel{<span\; style="background-color:\; white;">‾</span>}{\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">R</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">R</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">S</span>}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">0</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">x</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">y</span>}<span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">.</span>}$$

Substituting into the equation:

$$<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;">x</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;">y</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;">1</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">x</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;">y</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;">0</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">x</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;">y</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</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;">0</span>}<span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">.</span>}$$

This gives the system:

$$<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">x</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">x</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">6</span>}<span\; style="background-color:\; white;">(</span><span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">x</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">0</span>}\text{<span style="background-color: white;">and</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">y</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">(</span><span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">y</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">6</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">y</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">0.</span>}$$

Solving for $x$ and $y$:

$$<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">x</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">5</span>}\mathrm{<span\; style="background-color:\; white;">x</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">6</span>}\mathrm{<span\; style="background-color:\; white;">x</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">0</span>}\text{<span style="background-color: white;">\hspace{0.25em}\hspace{0.05em}</span>}<span\; style="background-color:\; white;">⟹</span>\text{<span style="background-color: white;">\hspace{0.25em}\hspace{0.05em}</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">12</span>}\mathrm{<span\; style="background-color:\; white;">x</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">0</span>}\text{<span style="background-color: white;">\hspace{0.25em}\hspace{0.05em}</span>}<span\; style="background-color:\; white;">⟹</span>\text{<span style="background-color: white;">\hspace{0.25em}\hspace{0.05em}</span>}\mathrm{<span\; style="background-color:\; white;">x</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">5</span>}}{\mathrm{<span\; style="background-color:\; white;">12</span>}}<span\; style="background-color:\; white;">,</span>$$$$<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">y</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">5</span>}\mathrm{<span\; style="background-color:\; white;">y</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">6</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">6</span>}\mathrm{<span\; style="background-color:\; white;">y</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">0</span>}\text{<span style="background-color: white;">\hspace{0.25em}\hspace{0.05em}</span>}<span\; style="background-color:\; white;">⟹</span>\text{<span style="background-color: white;">\hspace{0.25em}\hspace{0.05em}</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">12</span>}\mathrm{<span\; style="background-color:\; white;">y</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">6</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">0</span>}\text{<span style="background-color: white;">\hspace{0.25em}\hspace{0.05em}</span>}<span\; style="background-color:\; white;">⟹</span>\text{<span style="background-color: white;">\hspace{0.25em}\hspace{0.05em}</span>}\mathrm{<span\; style="background-color:\; white;">y</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

Thus, $S=(\frac{5}{12},\frac{1}{2})$.

Step 3: Find the Midpoints $E$ and $F$

• $E$ is the midpoint of $PR$:

  $$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">0</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">0</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">,</span>\frac{\mathrm{<span\; style="background-color:\; white;">0</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">0</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">0.5</span>}<span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">.</span>}$$

]

• $F$ is the midpoint of $QR$:

  $$\mathrm{<span\; style="background-color:\; white;">F</span>}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">0</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">,</span>\frac{\mathrm{<span\; style="background-color:\; white;">0</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">0.5</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">0.5</span>}<span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">.</span>}$$

]

Step 4: Compute the Lengths

• Length of $EF$:

  $$\mathrm{<span\; style="background-color:\; white;">E</span>}\mathrm{<span\; style="background-color:\; white;">F</span>}<span\; style="background-color:\; white;">=</span>\sqrt{<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">0.5</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">0</span>}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">+</span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">0.5</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">0.5</span>}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">0.5.</span>}$$

]

• Length of $ES$:

  $$\mathrm{<span\; style="background-color:\; white;">E</span>}\mathrm{<span\; style="background-color:\; white;">S</span>}<span\; style="background-color:\; white;">=</span>\sqrt{{<span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">5</span>}}{\mathrm{<span\; style="background-color:\; white;">12</span>}}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">0</span>}<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">+</span>{<span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">-</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">5</span>}}{\mathrm{<span\; style="background-color:\; white;">12</span>}}<span\; style="background-color:\; white;">\approx </span>\mathrm{<span\; style="background-color:\; white;">0.4167.</span>}$$

]

Step 5: Calculate the Ratio

$$\frac{\text{<span style="background-color: white;">Length of </span>}\mathrm{<span\; style="background-color:\; white;">E</span>}\mathrm{<span\; style="background-color:\; white;">F</span>}}{\text{<span style="background-color: white;">Length of </span>}\mathrm{<span\; style="background-color:\; white;">E</span>}\mathrm{<span\; style="background-color:\; white;">S</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">0.5</span>}}{\mathrm{<span\; style="background-color:\; white;">0.4167</span>}}<span\; style="background-color:\; white;">\approx </span>\mathrm{<span\; style="background-color:\; white;">1.2.</span>}$$

Final Answer

The value is $\boxed{{\displaystyle 1.2}}$.

Note: The answer falls within the range $[1.15,1.25]$ as specified.