If the position vectors of the vertices \(A, B, C\) of a triangle are \(3 \bar{i}+4 \bar{j}-\bar{k}\), \(\overline{\mathrm{i}}+3 \overline{\mathrm{j}}+\overline{\mathrm{k}}, 5(\overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}})\) respectively, then the magnitude of the altitude drawn from \(A\) on to the side \(B C\) is

The vectors \(\mathbf{A B}=3 \mathbf{i}-2 \mathbf{j}+2 \mathbf{k}\) and \(\mathbf{B C}=\mathbf{i}-2 \mathbf{k}\) are the adjacent sides of a parallelogram. The angle between its diagonals is

If $\overrightarrow{\mathbf{a}}=\widehat{\dot{\mathbf{i}}}+\widehat{\dot{\mathbf{j}}}+\widehat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=\widehat{\dot{\mathbf{i}}}+3\widehat{\dot{\mathbf{j}}}+5\widehat{\mathbf{k}}$ and $\overrightarrow{\mathbf{c}}=7\widehat{\dot{\mathbf{i}}}+9\widehat{\dot{\mathbf{j}}}+11\widehat{\mathbf{k}}$ then the area of parallelogram having diagonals $\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}$ is

If \(s=60 t-5 t^2\) denotes the distance covered by a particle in time \(t\), then the distance it covers before coming to rest is.... units.

Let $x, y$ are real numbers. If $\overrightarrow{a}=\left(\sin x\right)\hat{i}+\left(\sin y\right)\hat{j}$ and $\overrightarrow{b}=\left(\cos x\right)\hat{i}+\left(\cos y\right)\hat{j}$, then $|\overrightarrow{a}\times \overrightarrow{b}|$ is

The equation of the plane in normal form which passes through the points $(-2,1,3),(1,1,1)$ and $(2,3,4)$ is

If a team of 4 persons is to be selected out of 4 married couples to play mixed doubles tennis game, then the number of ways of forming a team in which no married couple appears, is

If \(\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}\) is defined by \(\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x})+\mathrm{f}(\mathrm{y}) \forall \mathrm{x}, \mathrm{y} \in\) \(\mathbb{R}\) and \(f(1)=7\), then \(\sum_{r=1}^n f(r)=\)

If \(\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \mathbf{b}=-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\), \(\mathbf{c}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}\), \(\mathbf{n}\) is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\) and \(\theta\) is the angle between \(\mathbf{c}\) and \(\mathbf{n}\) then \(\sin \theta=\)

Identify the order in which the spin only magnetic moment (in BM) increases for the following four ions

The transformed equation of \(x^2-y^2+2 x+4 y=0\) when the origin is shifted to the point \((-1,2)\) is

If \(f: R \rightarrow R\) is defined by
\(f(x)= \begin{cases}|[x-5]|, & \text { for } x < 5 \\ {[|x-5|],} & \text { for } x \geq 5\end{cases}\)
Then, \((f \circ f)\left(-\frac{7}{2}\right)=\)
(here, \([x]\) is the greatest integer not exceeding \(x\))

Match the item given in Column-I (method used for determining colligative property) and Column-II (for the corresponding colligative property) and find the correct order.
\(\begin{array}{|c|c|c|}
\hline & \text { Column I } & \text { Column II } \\
\hline \text { A. } & \text { Beckmann method } & \text { 1. Osmotic pressure } \\
\hline \text { B. } & \begin{array}{l}
\text { Ostwald-Walker method }
\end{array} & \text { 2. Elevation in B.P } \\
\hline \text { C. } & \begin{array}{l}
\text { Berkeley-Hartley method }
\end{array} & \text { 3. Depression in F.P. } \\
\hline \text { D. } & \text { Landsberger method } & \begin{array}{l}
\text { 4. Relative lowering of vapour pressure }
\end{array} \\
\hline
\end{array}\)