Question

For real numbers $\alpha, \beta, \gamma, \delta$ and $\mu$, consider the matrix
$$
M=\left[\begin{array}{ccc}
\alpha & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \
\frac{1}{\sqrt{3}} & \beta & \frac{1}{\sqrt{3}} \
\gamma & \delta & \mu
\end{array}\right]
$$
Suppose that $\mathrm{MM}^{\mathrm{T}}=\mathrm{I}$, where $\mathrm{M}^{\mathrm{T}}$ is the transpose of the matrix M, and I is the $3 \times 3$ identity matrix.
Let $\overrightarrow{\mathrm{u}}=\alpha \hat{\mathrm{i}}+\frac{1}{\sqrt{3}} \hat{\mathrm{j}}+\gamma \hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{v}}=\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}+\delta \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{w}}=-\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{3}} \hat{\mathrm{j}}+\mu \hat{\mathrm{k}}$.
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 $\mathrm{x} \overrightarrow{\mathrm{u}}+\mathrm{y} \overrightarrow{\mathrm{v}}+\mathrm{z} \overrightarrow{\mathrm{w}}=\hat{\mathrm{j}}$ for some real numbers x, y and z, then the value of x is | (2) 1 |
| (R) The value of $|\overrightarrow{\mathrm{u}} .(\overrightarrow{\mathrm{v}} \times \overrightarrow{\mathrm{w}})|$ is | (3) $\frac{1}{\sqrt{2}}$ |
| (S) The value of $|\overrightarrow{\mathrm{u}} \times(\overrightarrow{\mathrm{v}} \times \overrightarrow{\mathrm{w}})|$ is | (4) $\frac{1}{\sqrt{3}}$ |
| | (5) $\frac{5}{6}$ |

Accepted Answer

(A) (P) → (5), (Q) → (4), (R) → (2), (S) → (1)

Answer

(B) (P) → (4), (Q) → (5), (R) → (1), (S) → (2)

Answer

(C) (P) → (5), (Q) → (3), (R) → (2), (S) → (1)

Answer

(D) (P) → (5), (Q) → (4), (R) → (1), (S) → (2)

List-I	List-II
(P) The value of $γ^{2} + δ^{2}$ is	(1) 0
(Q) If $x u + y v + z w = \hat{j}$ for some real numbers x, y and z, then the value of x is	(2) 1
(R) The value of $∣ u . (v \times w) ∣$ is	(3) $\frac{1}{2}$
(S) The value of $∣ u \times (v \times w) ∣$ is	(4) $\frac{1}{3}$
	(5) $\frac{5}{6}$

JEE Advanced · 2026 · 17 May 2026 · Paper 1 · Vector Algebra

Solution

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