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JEE Advance | PYQ

Let \( \vec{a}, \vec{b} \) be two vectors, and let \( P, Q \) and \( R \) be the points with position vectors \( \vec{a}, \vec{b} \) and \( \vec{a} + \vec{b} \), respectively, with respect to the origin \( O \). If \( |\vec{a} + \vec{b}| = \sqrt{21} \), \( |\vec{a} - \vec{b}| = 3 \), and \( \vec{a} \) and \( (\vec{a} - \vec{b}) \) are perpendicular to each other, then the area of the triangle \( OPR \) is:

Let \( O \) be the origin and \( \overrightarrow{OA}=2\hat{i}+2\hat{j}+\hat{k} \), \( \overrightarrow{OB}=\hat{i}-2\hat{j}+2\hat{k} \) and \( \overrightarrow{OC}=\dfrac{1}{2}\left(\overrightarrow{OB}-\lambda \overrightarrow{OA}\right) \) for some \( \lambda>0 \). If \( \left|\overrightarrow{OB}\times\overrightarrow{OC}\right|=\dfrac{9}{2} \), then which of the following statements is (are) TRUE?

Let \( \vec{u}, \vec{v} \) and \( \vec{\omega} \) be vectors in three-dimensional space, where \( \vec{u} \) and \( \vec{v} \) are unit vectors which are not perpendicular to each other and \( \vec{u}\!\cdot\!\vec{w} = 1 \), \( \vec{v}\!\cdot\!\vec{w} = 1 \), \( \vec{w}\!\cdot\!\vec{w} = 4 \). If the volume of the parallelepiped, whose adjacent sides are represented by the vectors \( \vec{u}, \vec{v} \) and \( \vec{w} \) is \( \sqrt{2} \), then the value of \( \lvert 3\vec{u} + 5\vec{v} \rvert \) is ________.

Let \( \vec{p} = 2\hat{i} + \hat{j} + 3\hat{k} \) and \( \vec{q} = \hat{i} - \hat{j} + \hat{k} \). If for some real number \( \alpha, \beta \) and \( \gamma \), we have
\( 15\hat{i} + 10\hat{j} + 6\hat{k} = \alpha(2\vec{p} + \vec{q}) + \beta(\vec{p} - 2\vec{q}) + \gamma(\vec{p} \times \vec{q}) \), then the value of \( \gamma \) is ________________.

Let \( \overrightarrow{OP} = \dfrac{\alpha - 1}{\alpha}\, \hat{i} + \hat{j} + \hat{k} \), \( \overrightarrow{OQ} = \hat{i} + \dfrac{\beta - 1}{\beta}\, \hat{j} + \hat{k} \) and \( \overrightarrow{OR} = \hat{i} + \hat{j} + \dfrac{1}{2}\hat{k} \), be three vectors, where \( \alpha, \beta \in \mathbb{R} - \{0\} \) and \( O \) denotes the origin. If \( (\,\overrightarrow{OP} \times \overrightarrow{OQ}\,) \cdot \overrightarrow{OR} = 0 \) and the point \( (\alpha, \beta, 2) \) lies on the plane \( 3x + 3y - z + l = 0 \), then the value of \( l \) is ______________.

For real numbers \( \alpha, \beta, \gamma, \delta \) and \( \mu \), consider the matrix
.
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 $\vec{u} = \alpha\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \gamma\hat{k}$, $\vec{v} = \frac{1}{\sqrt{2}}\hat{i} + \beta\hat{j} + \delta\hat{k}$ and $\vec{w} = -\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \mu\hat{k}$.
Match each entry in List-I to the correct entry in List-II and choose the correct option.

Let the position vectors of the points $P, Q, R$ and $S$ be $\vec{a} = \hat{i} + 2\hat{j} - 5\hat{k}$, $\vec{b} = 3\hat{i} + 6\hat{j} + 3\hat{k}$, $\vec{c} = \frac{17}{5}\hat{i} + \frac{16}{5}\hat{j} + 7\hat{k}$ and $\vec{d} = 2\hat{i} + \hat{j} + \hat{k}$, respectively. Then which of the following statements is true?