\(\frac{d}{d x}\left(\cos ^{-1}\left(\frac{4 x^3}{27}-x\right)\right)=\)

\(\int \sqrt{\mathrm{x}^2+\mathrm{x}+1} \mathrm{dx}\)

The order and degree of the differential equation \(3 x^2 \frac{d^2 y}{d x^2}-\sin \left(\frac{d^3 y}{d x^3}\right)+\cos (x y)=0\) are

For how many distinct values of $x$, the equation $\sin \left[2{\cos }^{-1}\cot \left(2{\tan }^{-1}x\right)\right]=0$ holds?

If \(\mathbf{a}\) makes an acute angle with \(\mathbf{b}, \mathbf{r} \cdot \mathbf{a}=0\) and \(\mathbf{r} \times \mathbf{b}=\mathbf{c} \times \mathbf{b}\), then \(\mathbf{r}=\)

Solution of \(\frac{d y}{d x}=\frac{x \log x^2+x}{\sin y+y \cos y}\) is

In a \(\triangle A B C, \Sigma(b+c) \tan \frac{A}{2} \tan \left(\frac{B-C}{2}\right)\) is equal to

If \((2,-1,2)\) and \((K, 3,5)\) are the triads of direction ratios of two lines and the angle between them is \(45^{\circ}\), then the value of \(K\) is

The gravitational potential energy of a body on the surface of the earth is E. If the body is taken from the surface of the earth to a height equal to \(150 \%\) of the radius of the earth, its gravitational potential energy is

If the wavelength of a photon is \(4000 Å\), then its energy will be

An ideal gas is subjected to cyclic process involving four thermodynamic states, the amounts of heat \((Q)\) and work \((W)\) involved in each of these states are
\(\begin{aligned}
& Q_1=6000 \mathrm{~J}, \quad Q_2=-5500 \mathrm{~J} ; Q_3=-3000 \mathrm{~J} ; \\
& Q_4=3500 \mathrm{~J} \\
& W_1=2500 \mathrm{~J} ; \quad W_2=-1000 \mathrm{~J} ; \quad W_3=-1200 \mathrm{~J} ; \\
& W_4=x \mathrm{~J} .
\end{aligned}\)
The ratio of the net work done by the gas to the total heat absorbed by the gas is \(\eta\). The values of \(x\) and \(\eta\) respectively are

The colour code for a resistance of $22 \mathrm{\Omega }\pm 5\%$ is _____

The initial rates of decrease of \(\mathrm{I}_2\) in acetone - iodine reaction catalysed by \(\mathrm{H}^{+}\) are given in the table.
\(\begin{array}{|c|c|c|c|c|}
\hline \begin{array}{c}
\text { Experiment } \\
\end{array} & \begin{array}{c}
\text { Initial } \\
{\left[\mathrm{I}_2\right]} \\
\left(\mathrm{mol} \mathrm{L}^{-1}\right)
\end{array} & \begin{array}{c}
\text { Initial } \\
{\left[\mathrm{H}^{+}\right]} \\
\left(\mathrm{mol} \mathrm{L}^{-1}\right)
\end{array} & \begin{array}{c}
\text { Initial } \\
{\left[\mathrm{CH}_3 \mathrm{COCH}_3\right]} \\
\left(\mathrm{mol} \mathrm{L}^{-1}\right)
\end{array} & \begin{array}{c}
\text { Initial rate of decrease of } \mathrm{I}_2 \\
\left(\mathrm{mol} \mathrm{L}^{-1} \mathrm{~s}^{-1}\right)
\end{array} \\
\hline 1 & 0.01 & 0.1 & 0.1 & 0.096 \\
2 & 0.01 & 0.2 & 0.1 & 0.192 \\
3 & 0.02 & 0.2 & 0.1 & 0.192 \\
4 & 0.01 & 0.2 & 0.2 & 0.384 \\
\hline
\end{array}\)
The order with respect to \(\mathrm{I}_2, \mathrm{H}^{+}\), acetone and total order of the reaction respectively are: