\(\mathbf{a}, \mathbf{b}\) are non-collinear vectors, \(|\mathbf{a}|=2 \sqrt{2},|\mathbf{b}|=3\) and the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(45^{\circ}\). Then, the lengths of the diagonals of the parallelogram whose adjacent sides are represented by the vectors \(5 \mathbf{a}+2 \mathbf{b}\) and \(\mathbf{a}-3 \mathbf{b}\) are

If \(\hat{i}-2 \hat{j}+3 \hat{k}, 2 \hat{i}+3 \hat{j}+\hat{k},-3 \hat{i}-\hat{j}-2 \hat{k}\) are the position vectors of three points A, B, C respectively, then \(\mathrm{A}, \mathrm{B}, \mathrm{C}\)

Let \(\overline{\mathrm{i}}-2 \overline{\mathrm{j}}+\overline{\mathrm{k}}, \overline{\mathrm{i}}+\overline{\mathrm{j}}-2 \overline{\mathrm{k}}, 2 \overline{\mathrm{i}}-\overline{\mathrm{j}}-\overline{\mathrm{k}}\) and \(\overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}}\) be the position vectors of four points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and D respectively. If a point P divides AB in the ratio \(2: 1\) internally and a point \(Q\) divides \(C D\) in the ratio \(1: 2\) externally, then the ratio in which the point with position vector \(5 \overline{\mathrm{i}}-6 \overline{\mathrm{j}}-5 \overline{\mathrm{k}}\) divides PQ is

\((\bar{a}+2 \bar{b}-\bar{c}) \cdot\{(\bar{a}-\bar{b}) \times(\bar{a}-\bar{b}-\bar{c})\}=\)

The number of ways in which $3$ boys and $2$ girls can sit on a bench so that no two boys are adjacent is  ______

If the equations \(2 a x^2-3 b x+4 c=0\) and \(3 x^2-4 x+5=0\) have a common root, then \(\frac{a+b}{b+c}\) is equal to \((a, b, c \in \boldsymbol{R})\)

If the point p represents the comple \(x\) number \(z=x+i y\) in the argand plane and if \(\frac{z+i}{z-1}\) is a purely imaginary number then the locus of \(p\) is

The rms value of the electric field of an electromagnetic wave emitted by a source is \(660 \mathrm{NC}^{-1}\). The average energy density of the electromagnetic wave is

Ferrous ion charges to \(X\) ion, on reacting with acidified hydrogen peroxide.
The number of \(d\)-electrons present in \(X\) and its magnetic moment (in BM) are, respectively :

If a function \(\mathrm{f}: \mathbf{R}-\{l\} \mathbf{R}-\{m\}\) defined by \(\mathrm{f}(\mathrm{x})=\frac{x+3}{x-2}\) is a bijection, then \(3 l-2 m=\)

If \({ }^{(n-1)} C_r=\left(k^2-3\right){ }^n C_{r+1}\), then an interval containing the values of \(k\), is

If \(g(x)=x^2+x-2\) and \(\frac{1}{2}(g \circ f)(x)=2 x^2-5 x+2\), then one such function \(f(x)=\)

Which of the following molecular formula and type of defect are correct for the following statement?
"In \(\mathrm{FeO}\) crystal, some \(\mathrm{Fe}^{2+}\) ions are missing and the loss of positive charge is made up by the presence of \(\mathrm{Fe}^{3+}\) ions".