With usual notations, in \(\triangle \mathrm{ABC}\), if \(\mathrm{a}=2, \mathrm{~b}=3, \mathrm{c}=5\) and \(\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=\frac{k+7}{30}\),
then \(\mathrm{k}=\)

Suppose that the points \((\mathrm{h}, \mathrm{k}),(1,2)\) and \((-3,4)\) lie on the line \(l_1\). If a line \(l_2\) passing through the points \((h, k)\) and \((4,3)\) is perpendicular to \(l_1\), then \(\left(\frac{k}{h}\right)\) equals

\(f(x) \begin{cases}=\frac{\sqrt{1+p x}-\sqrt{1-p x}}{x} & \text {, if } 1 \leq x < 0 \ =\frac{2 x+1}{x-2} \end{cases}\) \(\text {, if } 0 \leq x \leq 1\)
is continuous in the interval \([-1,1]\), then \(\mathrm{p}=\)

The unit vectors perpendicular to the plane determined by the points \(\mathrm{A}(1,-1,2) \mathrm{~B}(2,0,-1) \mathrm{~C}(0,2,1)\) is

Let \(\mathrm{f}: \mathbb{R}-\{2\} \rightarrow \mathbb{R}-\{1\}\) defined by \(\mathrm{f}(x)=\frac{x-3}{x-2}\) and \(\mathrm{g}: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(\mathrm{g}(x)=3 \mathrm{x}-2\), then sum of all values of \(x\) for which \(\mathrm{f}^{-1}(x)+\mathrm{g}^{-1}(x)=\frac{19}{6}\) is

The line through the points \((1,4),(-5,1)\) intersects the line \(4 x+3 y-5=0\) in the point

The number of values of \(x\) in the interval \([0,3 \pi]\) satisfying the equation \(2 \sin ^2 x+5 \sin x-3=0\) is

Given, for \(\mathrm{Sn}^{4+} / \mathrm{Sn}^{2+}\), standard reduction potential is \(0.15 \mathrm{~V}\) and for \(\mathrm{Au}^{3+} / \mathrm{Au}\), standard reduction potential is \(1.5 \mathrm{~V}\). For the reaction, \(3 \mathrm{Sn}^{2+}+2 \mathrm{Au}^{3+} \longrightarrow 3 \mathrm{Sn}^{4+}+2 \mathrm{Au}\)
the value of \(E_{\text {cell }}^{\circ}\) is,

The value of \(\int \frac{d x}{x\left(x^{n}+1\right)}\) is

How many moles of iodomethane are consumed in the following conversion?
\(\mathrm{CH}_3 \mathrm{NH}_2 \xrightarrow[\Delta]{\mathrm{CH}_3 \mathrm{I}}\left(\mathrm{CH}_3\right)_4 \cdot \mathrm{~N}^{+} \mathrm{I}^{-}\)

If \(\int \frac{\sin x}{\sin (x-\alpha)} d x=A x+B \log \sin (x-\alpha)+c\), then the value of \(\mathrm{A}\) and \(\mathrm{B}\) are respectively (where \(\mathrm{c}\) is a constant of integration)

Let the plane passing through point \((2,1,-1)\) containing line joining the points \((1,3,2)\) and \((1,2,1)\) makes intercepts p,q,r on co-ordinate axes, then \(\mathrm{p}+\mathrm{q}+\mathrm{r}=\)

Calculate the standard enthalpy change for reaction,
\(\begin{aligned}
& \mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}_{(\ell)}+3 \mathrm{O}_{2(\mathrm{~g})} \rightarrow 2 \mathrm{CO}_{2(\mathrm{~g})}+3 \mathrm{H}_2 \mathrm{O}_{(\ell)} \\
& \text {If,}\\
& \text {} \begin{aligned}
\Delta_{\mathrm{f}} \mathrm{H}^{\circ}\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right) & =-280 \mathrm{~kJ} \mathrm{~mol}^{-1} \\
\Delta_{\mathrm{f}} \mathrm{H}^{\circ}\left(\mathrm{CO}_2\right) & =-390 \mathrm{~kJ} \mathrm{~mol}^{-1} \\
\Delta_{\mathrm{f}} \mathrm{H}^{\circ}\left(\mathrm{H}_2 \mathrm{O}\right) & =-285 \mathrm{~kJ} \mathrm{~mol}^{-1}
\end{aligned}
\end{aligned}\)