In a \(\triangle A B C, 2 A+C=300^{\circ}\). If the circumradius of the \(\triangle A B C\) is eight times its inradius then \(\sin \frac{C}{2}=\)

If \(\bar{a}, \bar{b}, \bar{c}\) are three unit vectors such that \(|\bar{a}-\bar{b}|^2+|\bar{b}-\bar{c}|^2+|\bar{c}-\bar{a}|^2=15\), then \(|\bar{a}-\bar{b}-\bar{c}|^2-4(\bar{b} \cdot \bar{c})=\)

\(\mathrm{ABC}\) is a triangle and the radical centre of the circles with \(\mathrm{AB}, \mathrm{BC}, \mathrm{CA}\) as the diameters is \((-6,5)\). Now if \(A=(3,2), B=(2,1)\) then \(C=\)

Seven balls are drawn simultaneously from bag containing 5 white and 6 green balls. The probability of drawing 3 white and 4 green balls is :

If the equations
\[
\begin{aligned}
& k x+(k+1) y+(k-1) z=0, \\
& (k-1) x+(k+2) y+k z=0 \text { and } \\
& (k+1) x+k y+(k+2) z=0
\end{aligned}
\]
have a non trivial solution, then the sum of all the possible values of \(k\) is

The difference of the order and degree of the differential equation \(\left(\frac{d^2 y}{d x^2}\right)^{-\frac{7}{2}}\left(\frac{d^3 y}{d x^3}\right)^2-\left(\frac{d^2 y}{d x^2}\right)^{-\frac{5}{2}}\left(\frac{d^4 y}{d x^4}\right)=0\) is

\(\int \frac{d x}{(1+\sqrt{x})^{2022}}=\)

Which of the following reactions are not feasible?
I. \(\mathrm{CH}_3 \mathrm{CH}=\mathrm{CH}_2 \frac{\mathrm{HCl}}{\left(\mathrm{C}_6 \mathrm{H}_5 \mathrm{CO}\right)_2 \mathrm{O}_2} \longrightarrow \mathrm{CH}_3 \mathrm{CH}_2 \mathrm{CH}_2 \mathrm{Cl}\)
II. \(\mathrm{C}_6 \mathrm{H}_6+\mathrm{CH}_3 \mathrm{CH}_2 \mathrm{CH}_2 \mathrm{Cl} \stackrel{\mathrm{AlCl}_3}{\longrightarrow} \mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}\left(\mathrm{CH}_3\right)_2\)
III. \(\mathrm{CH}_3 \mathrm{CHClCH}_2 \mathrm{Cl} \underset{\Delta}{\stackrel{\mathrm{KOH}}{\longrightarrow}} \mathrm{CH}_3 \mathrm{C} \equiv \stackrel{+}{\mathrm{CH}}\)
IV. \(\mathrm{CH}_3 \mathrm{CH}=\mathrm{CHCH}_3 \stackrel{\mathrm{KMnO}_4 / \mathrm{H}^{+}}{\longrightarrow} \mathrm{CH}_3 \mathrm{COOH}\)

If the area of the circle \(x^2+y^2=2\) is divided into two parts by the parabola \(y=x^2\), then the area (in sq units) of the larger part is

If the length of the sub-tangent at any point \(P\) on a curve is proportional to the abscissa of the point P. then the equation of that curve is ( C is an arbitrary constant)

The electric field intensity (E) at a distance of 3 m from a uniform long straight wire of linear charge density 0.2 m \(\mathrm{Cm}^{-1}\) is

For a reaction \(2 \mathrm{CO}(\mathrm{g})+\mathrm{O}_2(\mathrm{~g}) \rightleftharpoons 2 \mathrm{CO}_2(\mathrm{~g})\), \(\Delta_{\mathrm{r}} \mathrm{G}^0=-128 \mathrm{~kJ}\) at \(300 \mathrm{~K}\). If \(\Delta_{\mathrm{r}} \mathrm{S}^0\) of the reaction is \(-40 \mathrm{~J} \mathrm{~K}^{-1}\), calculate \(\Delta_{\mathrm{r}} \mathrm{U}\) of the reaction.

The period of \(\sin ^4 x+\cos ^4 x\) is