Jee Advanced
Vector Algebra
JEE Advanced 2026 Paper 1 Online
For real numbers $\alpha$, $\beta$, $\gamma$, $\delta$ and $\mu$, consider the matrix
$$ M = \begin{bmatrix} \alpha & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \beta & \frac{1}{\sqrt{3}} \\ \gamma & \delta & \mu \end{bmatrix}. $$
Suppose that $MM^{T} = I$, where $M^{T}$ is the transpose of the matrix $M$, and $I$ is the $3 \times 3$ identity matrix. Let
\(\vec{u} = \alpha\,\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \gamma\,\hat{k}, \qquad \vec{v} = \frac{1}{\sqrt{2}}\hat{i} + \beta\hat{j} + \delta\hat{k} \qquad \text{and} \qquad \vec{w} = -\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \mu\hat{k}.\)
Match each entry in List-I to the correct entry in List-II and choose the correct option.
| List-I | List-II |
|---|---|
(P) The value of \(\gamma^2 + \delta^2\) is | (1) 0 |
(Q) If \(x\vec{u} + y\vec{v} + z\vec{w} = \hat{j}\) for some real numbers $x$, $y$ and $z$, then the value of $x$ is | (2) 1 |
(R) The value of \(\left|\vec{u} \cdot (\vec{v} \times \vec{w})\right|\) is | (3) \(\frac{1}{\sqrt{2}}\) |
(S) The value of \(\left|\vec{u} \times (\vec{v} \times \vec{w})\right|\) is | (4) \(\frac{1}{\sqrt{3}}\) |
| (5) \(\frac{5}{6}\) |