JEE Mathematics 2024 Question with Solution

Given: $\vec{a} + 5\vec{b}$ is collinear with $\vec{c}$, $\vec{b} + 6\vec{c}$ is collinear with $\vec{a}$, and $\vec{a} + \alpha\vec{b} + \beta\vec{c} = 0$ where $\vec{b}$ and $\vec{c}$ are non-collinear.

Find: $\alpha + \beta$.

Since $\vec{a} + 5\vec{b}$ is collinear with $\vec{c}$, there exists a scalar $k_1$ such that

So,

Also, $\vec{b} + 6\vec{c}$ is collinear with $\vec{a}$, so there exists a scalar $k_2$ such that

Substituting $\vec{a} = k_1\vec{c} - 5\vec{b}$, we get

Now compare coefficients of the non-collinear vectors $\vec{b}$ and $\vec{c}$:

so

And

therefore

Hence,

Using

we substitute for $\vec{a}$:

Since $\vec{b}$ and $\vec{c}$ are non-collinear, their coefficients must be zero:

So,

Therefore,

The correct option is A.

Given:

Find: $\alpha + \beta$ in

From the first relation,

Substitute into the second:

Comparing coefficients of $\vec{b}$ and $\vec{c}$ gives

Thus,

So,

Comparing this with

we get

Hence,

So the correct option is A.