If \(e^{\left(\sinh ^{-1} 2+\cosh ^{-1} \sqrt{6}\right)}=(a+(b+\sqrt{c}) \sqrt{a}+b \sqrt{c})\), then \(a+b+c=\)

Let \(\mathrm{A}\) be a matrix such that \(\mathrm{AB}\) is a scalar matrix where \(B=\left[\begin{array}{ll}1 & 2 \\ 0 & 3\end{array}\right]\) and \(\operatorname{det}(3 A)=27\). Then \(3 A^{-1}+A^2=\)

Let \([x]\) denote the greatest integer not more than \(x\). If \(A\) and \(B\) are the domains of the functions \(f(x)=\frac{x-[x]}{\sqrt{|x|-x}}\) and \(g(x)=\frac{x-[x]}{\sqrt{|x|+x}}\) respectively, then

If \(\alpha\) is a root of the equation \(25 \cos ^2 \theta+5 \cos \theta-12=0\), for \(\frac{\pi}{2} < \alpha < \pi\), then \(\sin 2 \alpha=\)

The set of all real values of \(x\) for which the expansion of \(\left(125 x^2-\frac{27}{x}\right)^{-2 / 3}\) is valid, is

The range of the function \(f(x)=\log _{0.5}\left(x^4-2 x^2+3\right)\) is

The general solution of the differential equation \(x^2 y d x-\left(x^3+y^3\right) d y=0\) is

An alternating emf given by the equation \(\mathrm{E}=200 \sin\) \((50 \pi \mathrm{t})\) (Where \(\mathrm{E}\) is in volts and \(\mathrm{t}\) is in seconds) is applied across a series combination of an inductor and a resistor having inductive reactance \(40 \Omega\) and resistance \(30 \Omega\) respectively. At time \(t=1 \mathrm{~s}\), the power dissipated by the resistor is close to \[ \left(\cos 53^{\circ}=0.6\right) \]

Which of the following elements has the lowest melting point?

If \(1.3 .5+3.5 .7+5.7 .9+\ldots\) to \(n\) terms \(=n(n+1) f(n)\), then \(f(2)=\)

What are the metal ions present in carnallite?

If the order of a differential equation $\frac{d^{2}y}{d{x}^2}-2{\left(\frac{dy}{dx}\right)}^3+\sin \left(\frac{dy}{dx}\right)+y=0$ is $l$ and the degree of the differential equation ${\left(1+\frac{d^{2}y}{d{x}^2}\right)}^{\frac{2}{3}}={\left[2-{\left(\frac{dv}{dx}\right)}^3\right]}^{\frac{3}{2}}$ is $m$, then the differential equation corresponding to the family of curves $y=A{x}^l+B{e}^{mx}$, where $A$ and $B$ are arbitrary constants, is

\[ \begin{aligned} & \text { Given } \mathrm{H}_2(\mathrm{~g})+\frac{1}{2} \mathrm{O}_2(\mathrm{~g}) \rightarrow \mathrm{H}_2 \mathrm{O}(l) ; \Delta \mathrm{H}=-285 \mathrm{~kJ} \ & \mathrm{~N}_2 \mathrm{O}_5(\mathrm{~g})+\mathrm{H}_2 \mathrm{O}(l) \rightarrow 2 \mathrm{HNO}_3(l) ; \Delta \mathrm{H}=-76.6 \mathrm{~kJ} \ & \mathrm{~N}_2(\mathrm{~g})+3 \mathrm{O}_2(\mathrm{~g})+\mathrm{H}_2(\mathrm{~g}) \rightarrow 2 \mathrm{HNO}_3(l) ; \Delta \mathrm{H}=-348.2 \mathrm{~kJ} \ & \text { Calculate the } \Delta \mathrm{H} \text { of } 2 \mathrm{~N}_2(\mathrm{~g})+5 \mathrm{O}_2(\mathrm{~g}) \rightarrow 2 \mathrm{~N}_2 \mathrm{O}_5(\mathrm{~g}) \end{aligned} \] Calculate the \(\Delta \mathrm{H}\) of \(2 \mathrm{~N}_2(\mathrm{~g})+5 \mathrm{O}_2(\mathrm{~g}) \rightarrow 2 \mathrm{~N}_2 \mathrm{O}_5(\mathrm{~g})\).