If \(\int \frac{x^4+1}{x^2+1} d x=A x^3+B x^2+C x+D \operatorname{Tan}^{-1} x+E\), then \(A+B+C+D=\)

If \(f(x)=\sqrt{x+2 \sqrt{2 x-4}}+\sqrt{x-2 \sqrt{2 x-4}}\), then the value of \(10 \times f^{\prime}(102)=\)

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) \(-1\)

\(\lim _{n \rightarrow \infty}\left(\frac{1^2}{n^3+1^3}+\frac{2^2}{n^3+2^3}+\ldots+\frac{1}{2 n}\right)=\)

Let \(f\) be defined on \(D=[R-\{-1,1\}\) by \(f(x)=\frac{|x|}{1-|x|}\), then

If \(\int \frac{\sqrt[4]{x}}{\sqrt{x}+\sqrt[4]{x}} d x=\frac{2}{3}\left[A \sqrt[4]{x^3}+B \sqrt[4]{x^2}+C \sqrt[4]{x}+D \log (1+\sqrt[4]{x})\right]+K\) then \(\frac{2}{3}(A+B+C+D)=\)

In a \(\triangle A B C\), if \(r_1=12, r_2=18\) and \(r_3=36\), then \(b=\)

A rod is found to be \(0.05 \mathrm{~cm}\) longer at \(40^{\circ} \mathrm{C}\) than it is at \(10^{\circ} \mathrm{C}\). The length of the rod at \(0^{\circ} \mathrm{C}\) is
(coefficient of linear expansion of the material of the rod \(\left.=1.5 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}\right)\)

A stationary hydrogen atom emits a photon corresponding to first line of Lyman series. Recoil velocity of the atom is nearly

A block of mass 2 kg is placed on a rough horizontal surface. If a horizontal force of 20 N acting on the block produces an acceleration of \(7 \mathrm{~m} \mathrm{~s}^{-2}\) in it, then the coefficient of kinetic friction between the block and the suface is (Aceleration due to gravity \(=10 \mathrm{~m} \mathrm{~s}^{-2}\) )

Suppose that the \(x\)-coordinates of the points \(A\) and \(B\) satisfy \(x^2+2 x-a^2=0\) and their \(y\)-coordinates satisfy \(y^2+4 y-b^2=0\). Then, the equation of the circle with \(A B\) as its diameter is

Two bodies each of mass \(m\) are hung from a balance whose scale pans differ in a vertical height by \(h\). If the mean density of the earth is \(\rho\), the error in weighing is

The photoelectric threshold for a certain metal surface is $360 \mathrm{Å}$. If the metal surface is irradiated by a wavelength of $1100 \mathrm{Å}$, the kinetic energy of the emitted photoelectrons is