JEE Mathematics 2023 Question with Solution

Given: The urns $A, B, C$ are chosen with equal probability. Urn $A$ has $4$ red and $6$ black balls, urn $B$ has $5$ red and $5$ black balls, and urn $C$ has $\lambda$ red and $4$ black balls. Also, $P(C\mid \text{Red}) = 0.4$.

Find: The square of the side length of the largest equilateral triangle inscribed in $y^2 = \lambda x$ with one vertex at the vertex of the parabola.

Using Bayes' theorem,

Since each urn is selected at random,

Also,

Therefore,

From the given working,

Now consider the parabola $y^2 = 6x$.

For the largest equilateral triangle, the upper vertex is taken as

Using the condition shown in the solution,

So,

which gives

Hence the point becomes

Therefore, the square of the side length is

Therefore, the required square of the side length is $432$.

Given: $P(C\mid \text{Red}) = 0.4$ and the parabola is $y^2 = \lambda x$.

Find: $\ell^2$ for the largest inscribed equilateral triangle.

The extracted solution states the final answer as $432$ and shows the intermediate result

Then the parabola becomes

The triangle vertices used are the origin and the symmetric points

From the angle condition,

Thus,

Substituting,

Now the side squared is computed as

So the required answer is $432$.

Note: Another textual approach in the solution mentions a different value of $\lambda$, but it conflicts with the extracted worked solution and the displayed final answer. As instructed, the solution's worked conclusion is treated as the authority.