The inverse of the matrix \(\left[\begin{array}{ccc}7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1\end{array}\right]\) is

If the roots of the equation \(x^3+a x^2+b x+c=0\) are in arithmetic progression, then

If \(z_1\) and \(z_2\) are two of the \(n^{\text {th }}\) roots of unity such that the line segment joining them subtends a right angle at the origin then for a positive integer \(\mathrm{k}, \mathrm{n}\) takes the form

The domain of the derivative of the function \(f(x)=\frac{x}{1+|x|}\) is

If \(y=\frac{\log x}{x}\), then \(\frac{d^2 y}{d x^2}\) at \(x=1\)

\(\lim _{x \rightarrow \infty}\left(\frac{2+\sin x}{x^2+3}\right)\) is equal to

The area (in sq. units) of the triangle formed by the lines \(x^2-3 x y+y^2=0\) and \(x+y+1=0\), is

At \(\mathrm{T}(\mathrm{K})\), equal weights of \(\mathrm{H}_2, \mathrm{D}_2\), and \(\mathrm{T}_2\) are present in closed vessel. The pressure exerted by this gaseous mixture is \(\mathrm{P}\) atm. The ratio of partial pressures of \(\mathrm{T}_2, \mathrm{D}_2\) and \(\mathrm{H}_2\) is approximately
(H, D and \(T\) are isotopes of hydrogen).

Identify the \(P\) and \(Q\) of the following reaction
\(P+Q \longrightarrow\left[\mathrm{B}(\mathrm{OH})_4\right]^{-}+\mathrm{H}_3 \mathrm{O}^{+}\)

If \(a b c \neq 0\) and the system of equations \(x+7 a y+2 a z=0, x+6 b y+2 b z=0\), \(x+5 c y+2 c z=0\) has a non-trivial solution, then \(a, b, c\) are in

Twelve cells, each having emf \(E\) volts are connected in series and are kept in a closed box. Some of these cells are wrongly connected with positive and negative terminals reversed. This 12 cell battery is connected in series with an ammeter, an external resistance \(R\) ohms and a two-cell battery (two cells of the same type used earlier, connected perfectly in series). The current in the circuit when the 12-cell battery and 2-cell battery aid each other is 3A and is \(2 \mathrm{~A}\) when they oppose each other. Then, the number of cells in 12-cell battery that are connected wrongly is :

Choose the correct option about the matrices given below
\(\begin{aligned} & A=\left[\begin{array}{ccc}\cos \frac{\pi}{4} & \sin \frac{\pi}{4} & 0 \\ -\sin \frac{\pi}{4} & \cos \frac{\pi}{4} & 0 \\ 0 & 0 & 1\end{array}\right] \\ & B=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos \frac{\pi}{3} & \sin \frac{\pi}{3} \\ 0 & -\sin \frac{\pi}{3} & \cos \frac{\pi}{3}\end{array}\right] \\ & C=\left[\begin{array}{ccc}\cos \frac{\pi}{6} & 0 & \sin \frac{\pi}{6} \\ 0 & 1 & 0 \\ -\sin \frac{\pi}{6} & \cos \frac{\pi}{6} & 0\end{array}\right]\end{aligned}\)
\(D=\left[\begin{array}{ccc}
\cos \frac{\pi}{2} & \sin \frac{\pi}{2} & 0 \\
-\sin \frac{\pi}{2} & \cos \frac{\pi}{2} & 0 \\
0 & 0 & 1
\end{array}\right]\)

The variance of the following continuous frequency distribution is
\(\begin{array}{lllll} \hline \text { Class Interval } & 0-10 & 10-20 & 20-30 & 30-40 \\ \hline \text { Frequency } & 2 & 3 & 4 & 1 \\ \hline \end{array}\)