Let $f:R\to R$ be given by $f\left(x\right)$ $=\left\{\begin{array}{cc}{\mathrm{x}}^5+5{\mathrm{x}}^4+10{\mathrm{x}}^3+10{\mathrm{x}}^2+3\mathrm{x}+1,& \mathrm{x}<0;\\ {\mathrm{x}}^2-\mathrm{x}+1,& 0\le \mathrm{x}<1;\\ \frac{2}{3}{\mathrm{x}}^3-4{\mathrm{x}}^2+7\mathrm{x}-\frac{8}{3},& 1\le \mathrm{x}<3;\\ \left(\mathrm{x}-2\right){\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{e}}\left(\mathrm{x}-2\right)-\mathrm{x}+\frac{10}{3},& \mathrm{x}\ge 3\end{array}\right.$
Then which of the following options is/are correct?

A hyperbola, having the transverse axis of length \(2 \sin \theta\), is confocal with the ellipse \(3 x^2+4 y^2=12\). Then, its equation is

Consider all rectangles lying in the region $\left\{\left(x, y\right)\in R\times R:0\le x\le \frac{\pi }{2}and0\le y\le 2\sin \left(2x\right)\right\}$ and having one side on the $x$-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is

For \(x>0, \lim _{x \rightarrow 0}\left((\sin x)^{1 / x}+\left(\frac{1}{x}\right)^{\sin x}\right)\) is

Consider a triangle \(A B C\) and let \(a, b\) and \(c\) denote the lengths of the sides opposite to vertices \(A, B\) and \(C\) respectively. Suppose \(a=6, b=10\) and the area of the triangle is \(15 \sqrt{3}\). If \(\angle A C B\) is obtuse and if \(r\) denotes the radius of the incircle of the triangle, then \(r^2\) is equal to

Consider three planes
\[
P_1: x-y+z=1, P_2: x+y-z=-1
\]
and \(P_3: x-3 y+3 z=2\)
Let \(L_1, L_2\) and \(L_3\) be the lines of intersection of the planes \(P_2\) and \(P_3, P_3\) and \(P_1, P_1\) and \(P_2\), respectively.
Statement 1 Atleast two of the lines \(L_1, L_2\) and \(L_3\) are non-parallel.
Statement 2 The three planes do not have a common point.

The total number of distinct $x\in \left[0, 1\right]$ for which, ${\int }_0^x\frac{t^2}{1+t^4}dt=2x-1$ is

KI in acetone, undergoes S_N2 reaction with each of P, Q, R and S. The rates of the reaction vary as

The graph between \(1 / \lambda\) and stopping potential \((V)\) of three metals having work function \(\phi_1, \phi_2\) and \(\phi_3\) in an experiment of photoelectric effect is plotted as shown in the figure. Which of the following statement(s) is/are correct? [Here \(\lambda\) is the wavelength of the incident ray].

A parallel plate capacitor has a dielectric slab of dielectric constant K between its plates that covers 1/3 of the area of its plates, as shown in the figure. The total capacitance of the capacitor is C while that of the portion with dielectric in between is ${\mathrm{C}}_1$ . When the capacitor is charged, the plate area covered by the dielectric gets charge ${\mathrm{Q}}_1$ and the rest of the area gets charge ${\mathrm{Q}}_2$ . The electric field in the dielectric is ${\mathrm{E}}_1$ and that in the other portion is ${\mathrm{E}}_2$ . Choose the correct option/options, ignoring edge effects.

Let \((x, y)\) be such that \(\sin ^{-1}(a x)+\cos ^{-1}(y)+\cos ^{-1}(b x y)=\frac{\pi}{2}\). Match the statements in Column I with the values in Column II.

Paragraph:
The reaction of \(\mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\) with freshly prepared \(\mathrm{FeSO}_{4}\) solution produces a dark blue precipitate called Turnbull's blue. Reaction of \(\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\) with the \(\mathrm{FeSO}_{4}\) solution in complete absence of air produces a white precipitate \(\mathbf{X}\), which turns blue in air. Mixing the \(\mathrm{FeSO}_{4}\) solution with \(\mathrm{NaNO}_{3}\), followed by a slow addition of concentrated \(\mathrm{H}_{2} \mathrm{SO}_{4}\) through the side of the test tube produces a brown ring.
Question:
Precipitate \(\mathbf{X}\) is

In terms of potential difference $V$ , electic current $I$ , permittivity ${\epsilon }_0$ , permeability ${\mu }_0$ and speed of light $c$ , the dimensionally correct equation(s) is (are)