Let \(A=\left[\begin{array}{ccc}1 & -1 & 2 \\ 0 & 3 & 4\end{array}\right], B=\left[\begin{array}{ccc}4 & 0 & -3 \\ -1 & -2 & -3\end{array}\right]\) and \(C=\left[\begin{array}{cccc}2 & -3 & 0 & 1 \\ 5 & -1 & -4 & 2 \\ -1 & 0 & 0 & 3\end{array}\right]\), what is \(A^T B\) ?

Determinant of skew-symmetric matrix of order "three" is always

If one of the lines represented by $a{x}^2+2hxy+b{y}^2=0$ passes through $(2,3)$ and the other passes through $(4,5)$, then $a+2 h+b$ equals

\(\int_1^5(|x-3|+|1-x|) d x=\)

If \(\frac{3+2 i \sin \theta}{1-2 i \sin \theta}\) is a real number and \(0 < \theta < 2 \pi\), then \(\theta\) is equal to

Let \(\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}\) be two vectors. If \(A_1\) is the area of the quadrilateral having \(\vec{a}, \vec{b}\) as its diagonals and \(A_2\) is the area of the parallelogram having \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}\) as its two adjacent sides, then \(\mathrm{A}_1 \cdot \mathrm{A}_2=\)

If \(O(0,0,0), A(1,2,1), B(2,1,3)\) and \(C(-1,1,2)\) are the vertices of a tetrahedron, then the acute angle between its face \(O A B\) and edge \(B C\) is

\(1 \mathrm{~g}\) of a solute is added to \(18 \mathrm{~g}\) of water. In this aqueous solution, what is the mass per cent of the solute?

Two balls of charge \(q_1\) and \(q_2\) initially have a velocity of the same magnitude and direction. After a uniform electric field has been applied during a certain time, the direction of the velocity of the first ball changes by \(60^{\circ}\), and the velocity magnitude is reduced by half. The direction of the velocity of the second ball changes thereby by \(90^{\circ}\).
In what proportion will the velocity of the second ball change? Determine the magnitude of the charge-to-mass ratio for the second ball if it is equal to \(k_1\) for the first ball. The electrostatic interaction between the balls should be neglected.

The RMS value of emf given by $E=(8\sin \omega t+6\cos \omega t)$ volt is

The time taken for \(10 \%\) completion of a first order reaction is 20 minutes. The time required for the completion of \(19 \%\) of the same reaction in minutes is

A short magnet oscillates with a frequency \(10 \mathrm{~Hz}\) where horizontal component of Earth's magnetic field is \(12 \mu \mathrm{T}\). A downward current of \(15 \mathrm{~A}\) is established in a vertical wire placed \(20 \mathrm{~cm}\) west of the magnet. The new frequency of the magnet is

The values of \(x\) for which the inequality \(\frac{8 x^2+16 x-51}{(2 x-3)(x+4)}>3\) holds, are