JEE Mathematics 2024 Question with Solution

Given: $y = x$ is the bisector of angle $B$, side $AC$ has equation $2x - y = 2$, $A = (4,6)$ and $B = (\alpha,\beta)$. Also, $2AB = BC$.

Find: $\alpha + 2\beta$.

From the solution, the key conclusion used is that point $B$ lies on the bisector $y=x$, so

$$\alpha = \beta$$

Hence,

$$\alpha + 2\beta = 3\alpha$$

The extracted working on the page further states that simplifying the condition $2AB = BC$ gives

$$\alpha = 12\text{ and/or related values}$$

and then concludes

$$3\alpha = 42$$

Therefore,

$$\alpha + 2\beta = 42$$

So, the correct option is A.

Note: The provided the solution is internally inconsistent in places, but it explicitly concludes that $\alpha + 2\beta = 42$ and marks option A as correct.