JEE Advance - Mathematics (2025 - Paper 1 Online - No. 16)
Let $\vec{w} = \hat{i} + \hat{j} - 2\hat{k}$, and $\vec{u}$ and $\vec{v}$ be two vectors, such that $\vec{u} \times \vec{v} = \vec{w}$ and $\vec{v} \times \vec{w} = \vec{u}$. Let $\alpha, \beta, \gamma$, and $t$ be real numbers such that
$\vec{u} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k},\ \ \ - t \alpha + \beta + \gamma = 0,\ \ \ \alpha - t \beta + \gamma = 0,\ \ \ \alpha + \beta - t \gamma = 0.$
Match each entry in List-I to the correct entry in List-II and choose the correct option.
| List – I | List – II |
|---|---|
| (P) $\lvert \vec{v} \rvert^2$ is equal to | (1) 0 |
| (Q) If $\alpha = \sqrt{3}$, then $\gamma^2$ is equal to | (2) 1 |
| (R) If $\alpha = \sqrt{3}$, then $(\beta + \gamma)^2$ is equal to | (3) 2 |
| (S) If $\alpha = \sqrt{2}$, then $t + 3$ is equal to | (4) 3 |
| (5) 5 |
(P) $\to$ (2) (Q) $\to$ (1) (R) $\to$ (4) (S) $\to$ (5)
(P) $\to$ (2) (Q) $\to$ (4) (R) $\to$ (3) (S) $\to$ (5)
(P) $\to$ (2) (Q) $\to$ (1) (R) $\to$ (4) (S) $\to$ (3)
(P) $\to$ (5) (Q) $\to$ (4) (R) $\to$ (1) (S) $\to$ (3)
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