JEE Mathematics 2025 Question with Solution

Given: Two parallel lines are at distance $5$ units. Take them as

$$y=0$$

and

$$y=5$$

Point $P$ lies between them at unit distance from one line, so choose

$$P(0,1)$$

with $Q$ on one line and $R$ on the other.

Find: The value of $(QR)^2$.

For an equilateral triangle, all sides are equal. The extracted solution states that using coordinate geometry with rotational symmetry, the side length of the equilateral triangle is

$$4\sqrt{3}$$

Therefore,

$$QR=4\sqrt{3}$$

Now square the side length:

$$(QR)^2=(4\sqrt{3})^2=16\cdot 3=48$$

Therefore, the required value is $48$.

Given: The parallel lines are taken as

$$y=0 \quad \text{and} \quad y=5$$

and $P$ is chosen as

$$P(0,1)$$

Find: The square of the side $QR$ of the equilateral triangle.

The first extracted approach explicitly concludes that the side length is

$$4\sqrt{3}$$

and then identifies

$$QR=4\sqrt{3}$$

Since the triangle is equilateral, this directly gives the required side.

Hence,

$$(QR)^2=\left(4\sqrt{3}\right)^2=48$$

So the numerical answer is $48$.

The second extracted approach contains inconsistent intermediate reasoning, but its final conclusion also states $48$. Therefore the accepted answer is $48$.