In the figure below, the switches $S_1\mathrm{a}\mathrm{n}\mathrm{d}{S}_2$ are closed simultaneously at $t=0$ and a current starts to flow in the circuit. Both the batteries have the same magnitude of the electromotive force (emf) and the polarities are as indicated in the figure. Ignore mutual inductance between the inductors. The current I in the middle wire reaches its maximum magnitude $I_{max}$ time $t=\tau$. Which of the following statements is (are) true?

A rectangular conducting loop of length $4 \mathrm{cm}$ and width $2 \mathrm{cm}$ is in the $xy$-plane, as shown in the figure. It is being moved away from a thin and long conducting wire along the direction $\frac{\sqrt{3}}{2}\stackrel{\wedge }{x}+\frac{{\displaystyle 1}}{{\displaystyle 2}}\stackrel{\wedge }{y}$ with a constant speed $v$. The wire is carrying a steady current $I=10 \mathrm{A}$ in the positive $x$-direction. A current of $10 \mu \mathrm{A}$ flows through the loop when it is at a distance $d=4 \mathrm{cm}$ from the wire. If the resistance of the loop is $0.1 \Omega$, then the value of $v$ is _____ $\mathrm{m} {\mathrm{s}}^{-1}$.
[Given: The permeability of free space ${\mu }_0=4\pi \times {10}^{-7} \mathrm{N} {\mathrm{A}}^{-2}$]

A particle of mass $m$ is initially at rest at the origin. It is subjected to a force and starts moving along the x - axis. Its kinetic energy changes with time as $dK/dt=\gamma t,$ where $\gamma$ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?

A parallel beam of light is incident from air at an angle $\alpha$ on the side PQ of a right angled triangular prism of refractive index $n=\sqrt{2}$ . Light undergoes total internal reflection in the prism at the face PR when $\alpha$ has a minimum value of ${45}^o.$ The angle $\theta$ of the prism is:

Two identical concave mirrors each of focal length \(f\) are facing each other as shown in the schematic diagram. The focal length \(f\) is much larger than the size of the mirrors. A glass slab of thickness \(t\) and refractive index \(n_0\) is kept equidistant from the mirrors and perpendicular to their common principal axis. A monochromatic point light source \(S\) is embedded at the center of the slab on the principal axis, as shown in the schematic diagram. For the image to be formed on \(S\) itself, which of the following distances between the two mirrors is/are correct:

Two parallel wires in the plane of the paper are distance ${\mathrm{X}}_0$ apart. A point charge is moving with speed u between the wires in the same plane at a distance ${\mathrm{X}}_1$ from one of the wires. When the wires carry current of magnitude $\mathrm{ }\mathrm{I}$  in the same direction, the radius of curvature of the path of the point charge is ${\mathrm{R}}_1$ . In contrast, if the currents $\mathrm{ }\mathrm{I}$ in the two wires have directions opposite to each other, the radius of curvature of the path is ${\mathrm{R}}_2$ . If $\frac{{\mathrm{X}}_0}{{\mathrm{X}}_1}=3$ , value of $\frac{{\mathrm{R}}_1}{{\mathrm{R}}_2}$ is

A string of length $1 \mathrm{m}$ and mass $2\times {10}^{-5} \mathrm{kg}$ is under tension $T$. When the string vibrates, two successive harmonics are found to occur at frequencies $750 \mathrm{Hz}$ and $1000 \mathrm{Hz}$. The value of tension $T$ is _____ newton.

Considering only the principal values of the inverse trigonometric functions, the value of \(\tan \left(\sin ^{-1}\left(\frac{3}{5}\right)-2 \cos ^{-1}\left(\frac{2}{\sqrt{5}}\right)\right)\) is

The positon vector \(\vec{r}\) of a particle of mass m is given by the following equation \(\vec{r}(t)=\alpha t^3 \hat{i} +\beta t^2 \hat{j}\), where \(\alpha=\frac{10}{3} m s^{-3}, \beta=5 m s^{-2}\) and \(m=0.1 kg\). At \(t=1 s\), which of the following statements (s) is (are) true about the particle?

Paragraph:
p-amino-N, N-dimethylaniline is added to a strongly acidic solution of \(X\). The resulting solution is treated with a few drops of aqueous solution of \(Y\) to yield blue colouration due to the formation of methylene blue. Treatment of the aqueous solution of \(Y\) with the reagent potassium hexacyanoferrate(II) leads to the formation of an intense blue precipitate. The precipitate dissolves an excess addition of the reagent. Similarly, treatment of the solution of \(Y\) with the solution of potassium hexacyanoferrate(III) leads to a brown colouration due to the formation of \(Z\).
Question:
The compound \(Z\), is

Fusion of $\mathrm{M}\mathrm{n}{\mathrm{O}}_2$ with $\mathrm{K}\mathrm{O}\mathrm{H}$ in presence of ${\mathrm{O}}_2$ produces a salt $\mathrm{W}.$ Alkaline solution of $\mathrm{W}$ upon electrolytic oxidation yields another salt $\mathrm{X}.$ The manganese containing ions present in $\mathrm{W}$ and $\mathrm{X},$ respectively, are $\mathrm{Y}$ and $\mathrm{Z}.$ Correct statement(s) is (are)

Let $f^{\prime }\left(x\right)= \frac{192x^3}{2+{\sin }^4\pi x}$ for all $x\in \mathbb{ }\mathbb{R}$ with $f\left(\frac{1}{2}\right)=0.$ If $m \le \underset{\frac{1}{2}}{\overset{1}{\int }}f\left(x\right)dx \le M,$ then the possible values of m and M are

For the reaction sequence given below, the correct statement(s) is(are)