If the point $\left(a, 8,-2\right)$ divides the line segment joining the points $\left(1,4,6\right)$ and $\left(5,2,10\right)$ in the ratio $m:n$ then $\frac{2m}{n}-\frac{a}{3}=$

The line \(x=\frac{\pi}{4}\) divides the area of the region bounded by \(y=\sin x, y=\cos x\) and \(x\)-axis \(\left(0 \leq x \leq \frac{\pi}{2}\right)\) into two regions of areas \(A_1\) and \(A_2\). Then \(A_1, A_2\) equals

\(\int_0^{\pi / 2} \frac{1}{1+\tan ^{2020}(x)} d x=\)

If \(\int \frac{d x}{x^2+2 x+2}=f(x)+c\), then \(f(x)\) is equal to :

If the distance \(s\) described in time ' \(t\) ' by a particle moving on a straight line is given by \(s=t^5-40 t^3+30 t^2+80 t-250\), then its minimum acceleration is

If \(f(x)=\sqrt{2 x-1}+5 \cos ^{-1}\left(\frac{2 x-1}{3}\right)\) then the domain of the function \(f(x)\) is

The mean deviation about the mean for the following data: $5,6,7,8,6,9,13,12,15$ is

In a colloidal solution, both the dispersed phase and dispersion medium are in liquid phase. What is the type of colloid ?

A oil drop having a mass \(4.8 \times 10^{-10} \mathrm{~g}\) and charge \(2.4 \times 10^{-18} \mathrm{C}\) stands still between two charged horizontal plates separated by a distance of \(1 \mathrm{~cm}\). If now the polarity of the plates is changed, instantaneous acceleration of the drop is : \(\left(g=10 \mathrm{~ms}^{-2}\right)\)

A metal loop of area \(10 \mathrm{~cm}^2\) is placed in a region such that its area vector points along \(\hat{\mathrm{k}}\). The region contains a uniform magnetic field of magnitude \(1.73 \mathrm{~T}\) that points in the direction \(\hat{i}+\hat{j}+\hat{k}\). When the magnetic field is switched off, the field decreases to zero at a steady rate in \(10 \mathrm{~s}\), then the magnitude of emf induced in the loop is

Match the items of List - I with those of the entires of List - II
\(List - I\)
\(\begin{aligned} & \text { (I) } \sin ^2 5^{\circ}+\sin ^2 10^{\circ}+ \\ & \sin ^2 15^{\circ}+\ldots+\sin ^2 90^{\circ}=\end{aligned}\)
\(\begin{aligned} & \text { (II) } \tan ^2 5^{\circ} \cdot \tan ^2 10^{\circ} \text {. } \\ & \tan ^2 15^{\circ} \ldots \tan ^2 85^{\circ}= \\ & \end{aligned}\)
\(\begin{aligned} & \text { (III) } \cos ^2 5^{\circ}+\cos ^2 10^{\circ} \\ & +\cos ^2 15^{\circ}+\ldots+\cos ^2 180^{\circ}=\end{aligned}\)
\(\begin{gathered}\text { (IV) } \cot 5^{\circ}+\cot 10^{\circ}+\cot 15^{\circ} \\ +\ldots .+\cot 175^{\circ}=\end{gathered}\)
\(List - II\)
(A) \(0\)
(B) \(\frac{19}{2}\)
(C) \(18\)
(D) \(1\)
(E) \(-1\)

In a \(\triangle A B C, 2 a c \sin \frac{1}{2}(A-B+C)\) is equal to

If the slope of the tangent drawn at any point \((x, y)\) on a curve is \((x+y)\), then the equation of that curve is