\(\displaystyle\int \dfrac{x + 1}{x(1 + x e^x)} dx =\)

\(S \equiv x^2+y^2-2 x-4 y-4=0\) and \(S^{\prime} \equiv x^2+y^2-4 x-2 y-16=0\) are two circles the point \((-2,-1)\) lies

The domain of the function \(\sin ^{-1}\left(x^2-4\right)\) is

Solution of the differential equation \(y \dfrac{d y}{d x}+x=0\) represents a family of

\(\text { If A }(\operatorname{adj} A)=5 I \text {, where I is the identity matrix of order } 3 \text {, then }|\operatorname{adj} A|=\)

Codes used for vehicle identification consists of two distinct English alphabets followed by two distinct digits from 1 to 9 . How many of them end with an even number.

If \(\omega=\frac{-1+\sqrt{3 i}}{2} \operatorname{then}\left(3+\omega+3 \omega^{2}\right)^{4}\) is

Carnot heat engine takes \(300 \mathrm{~J}\) of heat from a source at \(627^{\circ} \mathrm{C}\) and gives some part of it to \(\operatorname{sink}\) at \(27^{\circ} \mathrm{C}\). Work done by engine in one cycle is

An inorganic compound \(\mathrm{W}\) undergoes the following reactions:
\(
\begin{gathered}
W+\dfrac{\mathrm{Na}_2 \mathrm{CO}_3}{\mathrm{O}_2 / \text { heat }} \rightarrow X+\dfrac{\mathrm{H}^{+}}{} \rightarrow Y_{(s)} \\
Y(a q)+\mathrm{KCl}(\mathrm{aq}) \rightarrow Z_{(S)}
\end{gathered}
\)
Z appears in the form of orange crystals and is used as an oxidising agent in acid medium. Identify the compound W.

The radius of the circle passes through the foci of a conic \(\dfrac{x^2}{16}+\dfrac{y^2}{9}=1\) and has its centre at \((0,3)\), then the diameter of the circle is ---

In which one of the case the chemical reaction given in Column I does not match with the name of the reaction given in Column II?
\(\begin{array}{|l|l|l|} \hline \text{S.No.} & \text{Column I - Reactions} & \text{Column II - Name of reaction.} \\ \hline \text{A} & \mathrm{C}_6 \mathrm{H}_5 \mathrm{CHO}+ \text{Conc. NaOH, heat} \rightarrow \mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}_2 \mathrm{OH}+\mathrm{C}_6 \mathrm{H}_5 \mathrm{COONa} & \text{Cannizzaro's reaction} \\ \hline \text{B} & \mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}_3+ \text{ (i) } \mathrm{CrO}_2 \mathrm{Cl}_2 / \mathrm{CS}_2 \text{ (ii) } \mathrm{H}_3 \mathrm{O}^{+} \rightarrow \mathrm{C}_6 \mathrm{H}_5 \mathrm{CHO} & \text{Etard reaction.} \\ \hline \text{C} & \mathrm{C}_6 \mathrm{H}_6+\mathrm{CO} / \mathrm{HCl} \text{ (anh. } \mathrm{AlCl}_3 \rightarrow \mathrm{C}_6 \mathrm{H}_5 \mathrm{CHO} & \text{Kolbe's reaction} \\ \hline \text{D} & \mathrm{RCN}+ \text{ (i) } \mathrm{SnCl}_2+\mathrm{HCl} \text{ (ii) } \rightarrow \mathrm{RCHO} & \text{Stephen's reaction.} \\ \hline \end{array}\)

The wavelength of the radiation emitted by a body depends upon

Which of the following will form a biodegradable polymer?