The solution of \(\mathrm{e}^{y-x} \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{y(\sin x+\cos x)}{(1+y \log y)}\) is

The joint equation of two lines passing through the origin and perpendicular to the lines given by \(2 x^2+5 x y+3 y^2=0\) is

The argument of \(\frac{1+i \sqrt{3}}{\sqrt{3}+i}, i=\sqrt{-1}\) is

Five persons \(A, B, C, D\) and \(E\) are seated in a circular arangement, if each of them is given a hat of one of the three colours red, blue and green, then the number of ways, of distributing the hats such that the person seated in adjacent seats get different coloured hats, is

Let \(S\) be the set of all real values of \(\lambda\) such that a plane passing through the points \(\left(-\lambda^2, 1,1\right),\left(1,-\lambda^2, 1\right)\) and \(\left(1,1,-\lambda^2\right)\) also passes through the point \((-1,-1,1)\). Then \(S\) is equal to

If planes $\overrightarrow{r}.\left(p\widehat{i}-\widehat{j}+2\widehat{k}\right)+3=0$ and $\overrightarrow{r}.\left(2\widehat{i}-p\widehat{j}-\widehat{k}\right)-5=0$ include angle $\frac{\pi }{3}$ then the value of $p$ is

A random variable \(\mathrm{X}\) has the following probability distribution

Then \(F(4)=\)

Time is a crucial factor for ________.

The quantity \(\frac{\mathrm{PV}}{\mathrm{KT}}\) represents \((\mathrm{K}=\) Boltzmann constant)

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

From a disc of mass 'M' and radius 'R' a circular hole of diameter \(\mathrm{R}\) is cut whose rim passes through the centre. The moment of inertia of the remaining part of the disc about perpendicular axis passing through the centre is

In \(\triangle \mathrm{ABC}\), with usual notations, if \(\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}\), then the value of \(\cos \mathrm{A}+\cos \mathrm{B}+\cos \mathrm{C}\) is

If \(p_1, p_2, p_3\) are altitudes of a triangle \(A B C\) from the vertices \(A, B, C\) respectively and if \(\Delta\) is the area of the triangle, \(S\) is the semi perimeter of the triangle, then
\(\frac{\operatorname{Cos} \mathrm{A}}{\mathrm{p}_1}+\frac{\operatorname{Cos} \mathrm{B}}{\mathrm{p}_2}+\frac{\operatorname{Cos} \mathrm{C}}{\mathrm{p}_3}\)