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JEE Mains | PYQ

Let \( \hat{u} \) and \( \hat{v} \) be unit vectors inclined at an acute angle such that \( \left|\hat{u}\times\hat{v}\right|=\frac{\sqrt{3}}{2} \). If \( \vec{A}=\lambda \hat{u}+\hat{v}+(\hat{u}\times\hat{v}) \), then \( \lambda \) is equal to:

Two adjacent sides of a parallelogram \( PQRS \) are given by \( \overrightarrow{PQ}=\hat{j}+\hat{k} \) and \( \overrightarrow{PS}=\hat{i}-\hat{j} \). If the side \( PS \) is rotated about the point \( P \) by an acute angle \( \alpha \) in the plane of the parallelogram so that it becomes perpendicular to the side \( PQ \), then \( \sin^{2}\!\left(\dfrac{5\alpha}{2}\right)-\sin^{2}\!\left(\dfrac{\alpha}{2}\right) \) is equal to:

Let the vectors \( \vec{a}=-\hat{i}+\hat{j}+3\hat{k} \) and \( \vec{b}=\hat{i}+3\hat{j}+\hat{k} \). For some \( \lambda,\mu\in \mathbb{R} \), let \( \vec{c}=\lambda \vec{a}+\mu \vec{b} \). If \( \vec{c}\cdot(3\hat{i}-6\hat{j}+2\hat{k})=10 \) and \( \vec{c}\cdot(\hat{i}+\hat{j}+\hat{k})=-2 \), then \( |\vec{c}|^2 \) is equal to:

The square of the distance of the point of intersection of the lines \( \vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(a\hat{i} - \hat{j}),\, a \ne 0 \) and \( \vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + a\hat{k}) \) from the origin is:

For three unit vectors \( \vec{a}, \vec{b}, \vec{c} \) satisfying \( \lvert \vec{a}-\vec{b} \rvert^{2} + \lvert \vec{b}-\vec{c} \rvert^{2} + \lvert \vec{c}-\vec{a} \rvert^{2} = 9 \) and \( \lvert 2\vec{a} + k\vec{b} + k\vec{c} \rvert = 3 \), the positive value of \( k \) is:

Let \( \vec{a}, \vec{b}, \vec{c} \) be three vectors such that \( \vec{a}\times\vec{b} = 2(\vec{a}\times\vec{c}) \). If \( |\vec{a}| = 1 \), \( |\vec{b}| = 4 \), \( |\vec{c}| = 2 \), and the angle between \( \vec{b} \) and \( \vec{c} \) is \( 60^\circ \), then \( |\vec{a}\cdot\vec{c}| \) is equal to:

Let \( \vec a=4\hat i-\hat j+3\hat k \), \( \vec b=10\hat i+2\hat j-\hat k \) and a vector \( \vec c \) be such that \( 2(\vec a\times \vec b)+3(\vec b\times \vec c)=\vec 0 \). If \( \vec a\cdot \vec c=15 \), then \( \vec c\cdot(\hat i+\hat j-3\hat k) \) is equal to:

Let \( \vec{a} = -\hat{i} + 2\hat{j} + 2\hat{k} \), \( \vec{b} = 8\hat{i} + 7\hat{j} - 3\hat{k} \) and \( \vec{c} \) be a vector such that \( \vec{a} \times \vec{c} = \vec{b} \). If \( \vec{c} \cdot (\hat{i} + \hat{j} + \hat{k}) = 4 \), then \( \left|\vec{a} + \vec{c}\right|^2 \) is equal to:

Let \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \) and \( \vec{b} = \lambda \hat{j} + 2\hat{k}, \lambda \in \mathbb{Z} \) be two vectors. Let \( \vec{c} = \vec{a} \times \vec{b} \) and \( \vec{d} \) be a vector of magnitude \( 2 \) in \( yz \)-plane. If \( |\vec{c}| = \sqrt{53} \), then the maximum possible value of \( (\vec{c}\cdot \vec{d})^2 \) is equal to:

Let \( \vec{a} = -\vec{i} + \vec{j} + 2\vec{k},\ \vec{b} = \vec{i} - \vec{j} - 3\vec{k},\ \vec{c} = \vec{a} \times \vec{b} \) and \( \vec{d} = \vec{c} \times \vec{a}. \) Then \( (\vec{a} - \vec{b}) \cdot \vec{d} \) is equal to:

Let \( \vec{c} \) and \( \vec{d} \) be vectors such that \( |\vec{c} + \vec{d}| = \sqrt{29} \) and \( \vec{c} \times (2\hat{i} + 3\hat{j} + 4\hat{k}) = (2\hat{i} + 3\hat{j} + 4\hat{k}) \times \vec{d} \). If \( \lambda_1, \lambda_2 \) \( (\lambda_1 > \lambda_2) \) are the possible values of \( (\vec{c} + \vec{d}) \cdot (-7\hat{i} + 2\hat{j} + 3\hat{k}) \), then the equation \( K^2 x^2 + (K^2 - 5K + \lambda_1)xy + \left(3K + \frac{\lambda_2}{2}\right) y^2 - 8x + 12y + \lambda_2 = 0 \) represents a circle, for \( K \) equal to:

Let \( PQR \) be a triangle such that \( \overrightarrow{PQ} = -2\hat{i} - \hat{j} + 2\hat{k} \) and \( \overrightarrow{PR} = a\hat{i} + b\hat{j} - 4\hat{k},\ a, b \in \mathbb{Z} \). Let \( S \) be the point on \( QR \), which is equidistant from the lines \( PQ \) and \( PR \). If \( |\overrightarrow{PR}| = 9 \) and \( \overrightarrow{PS} = \hat{i} - 7\hat{j} + 2\hat{k} \), then the value of \( 3a - 4b \) is ________.

Let \( \vec{a} = \sqrt{7}\,\hat{i} + \hat{j} - \hat{k} \) and \( \vec{b} = \hat{j} + 2\hat{k} \). If \( \vec{r} \) is a vector such that \( \vec{r} \times \vec{a} + \vec{a} \times \vec{b} = \vec{0} \) and \( \vec{r} \cdot \vec{a} = 0 \), then \( |3\vec{r}|^{2} \) is equal to:

Let $\overrightarrow{AB} = 2\hat{i} + 4\hat{j} - 5\hat{k}$ and $\overrightarrow{AD} = \hat{i} + 2\hat{j} + \lambda\hat{k}, \lambda \in \mathbb{R}$. Let the projection of the vector $\vec{v} = \hat{i} + \hat{j} + \hat{k}$ on the diagonal $\overrightarrow{AC}$ of the parallelogram $ABCD$ be of length one unit. If $\alpha, \beta$, where $\alpha > \beta$, be the roots of the equation $\lambda^2x^2 - 6\lambda x + 5 = 0$, then $2\alpha - \beta$ is equal to:

Let \( \vec{a} = 2\hat{i} + \hat{j} - 2\hat{k} \), \( \vec{b} = \hat{i} + \hat{j} \) and \( \vec{c} = \vec{a} \times \vec{b} \). Let \( \vec{d} \) be a vector such that \( \lvert \vec{d} - \vec{a} \rvert = \sqrt{11} \), \( \lvert \vec{c} \times \vec{d} \rvert = 3 \) and the angle between \( \vec{c} \) and \( \vec{d} \) is \( \frac{\pi}{4} \). Then \( \vec{a} \cdot \vec{d} \) is equal to:

If \( \vec{a} \) and \( \vec{b} \) are two vectors such that \( |\vec{a}|=2 \) and \( |\vec{b}|=3 \), then the maximum value of \( 3\big|\!(3\vec{a}+2\vec{b})\!\big|+4\big|\!(3\vec{a}-2\vec{b})\!\big| \) is:

Let \( \vec{a}=2\hat{i}-\hat{j}-\hat{k},\ \vec{b}=\hat{i}+3\hat{j}-\hat{k} \) and \( \vec{c}=2\hat{i}+\hat{j}+3\hat{k} \). Let \( \vec{v} \) be the vector in the plane of the vectors \( \vec{a} \) and \( \vec{b} \), such that the length of its projection on the vector \( \vec{c} \) is \( \frac{1}{\sqrt{14}} \). Then \( |\vec{v}| \) is equal to:

The shortest distance between the lines \( \vec{r}=\left(\frac{1}{3}\hat{i}+2\hat{j}+\frac{8}{3}\hat{k}\right)+\lambda(2\hat{i}-5\hat{j}+6\hat{k}) \) and \( \vec{r}=\left(-\frac{2}{3}\hat{i}-\frac{1}{3}\hat{k}\right)+\mu(\hat{j}-\hat{k}), \lambda,\mu \in \mathbb{R}, \) is:

If \( \vec{a}=\hat{i}+\hat{j}+\hat{k} \), \( \vec{b}=\hat{j}-\hat{k} \) and \( \vec{c} \) be three vectors such that \( \vec{a}\times\vec{c}=\vec{b} \) and \( \vec{a}\cdot\vec{c}=3 \), then \( \vec{c}\cdot(\vec{a}-2\vec{b}) \) is equal to ______.

The square of the distance of the point \( P(5, 6, 7) \) from the line \( \dfrac{x-2}{2} = \dfrac{y-5}{3} = \dfrac{z-2}{4} \) is equal to:

Let a vector \( \vec{a} = \sqrt{2}\,\hat{i} -\hat{j} + \lambda \hat{k};\; \lambda > 0 \), make an obtuse angle with the vector \( \vec{b} = -\lambda^{2}\hat{i} + 4\sqrt{2}\,\hat{j} + 4\sqrt{2}\,\hat{k} \) and an angle \( \theta,\; \frac{\pi}{6} < \theta < \frac{\pi}{2} \) with the positive z-axis. If the set of all possible values of \( \lambda \) is \( (\alpha,\beta) - \{\gamma\} \) then \( \alpha + \beta + \gamma \) is equal to ______ .