JEE MAIN - Mathematics (2024 - 29th January Morning Shift - No. 2)
Let $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be three non-zero vectors such that $$\vec{b}$$ and $$\vec{c}$$ are non-collinear. If $$\vec{a}+5 \vec{b}$$ is collinear with $$\vec{c}, \vec{b}+6 \vec{c}$$ is collinear with $$\vec{a}$$ and $$\vec{a}+\alpha \vec{b}+\beta \vec{c}=\overrightarrow{0}$$, then $$\alpha+\beta$$ is equal to
30
$$-$$30
$$-$$25
35
Explanation
$$\begin{aligned} & \vec{a}+5 \vec{b}=\lambda \vec{c} \\ & \vec{b}+6 \vec{c}=\mu \vec{a} \end{aligned}$$
Eliminating $$\vec{a}$$
$$\begin{aligned} & \lambda \overrightarrow{\mathrm{c}}-5 \overrightarrow{\mathrm{b}}=\frac{6}{\mu} \overrightarrow{\mathrm{c}}+\frac{1}{\mu} \overrightarrow{\mathrm{b}} \\ & \therefore \mu=\frac{-1}{5}, \lambda=-30 \\ & \alpha=5, \beta=30 \end{aligned}$$
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