The smallest ketone and its next homologoue are reacted with \(\mathrm{NH}_2 \mathrm{OH}\) to form oxime

The ground state energy of hydrogen atom is $-13.6\mathrm{ }\mathrm{e}\mathrm{V}$ . Consider an electronic state $\mathrm{\Psi }$ of ${\mathrm{H}\mathrm{e}}^{+}$ whose energy, azimuthal quantum number and magnetic quantum number are $-3.4\mathrm{ }\mathrm{e}\mathrm{V},\mathrm{ }2$ and $0$ respectively. Which of the following statement(s) is(are) true for the state $\mathrm{\Psi }$ ?

Paragraph:
The reactions of \(\mathrm{Cl}_{2}\) gas with cold-dilute and hot-concentrated \(\mathrm{NaOH}\) in water give sodium salts of two (different) oxoacids of chlorine, \(\boldsymbol{P}\) and \(\boldsymbol{Q}\), respectively. The \(\mathrm{Cl}_{2}\) gas reacts with \(\mathrm{SO}_{2}\) gas, in presence of charcoal, to give a product \(\boldsymbol{R}\). \(\boldsymbol{R}\) reacts with white phosphorus to give a compound \(\boldsymbol{S}\). On hydrolysis, \(\boldsymbol{S}\) gives an oxoacid of phosphorus, \(\boldsymbol{T}\).
Question:
\(P\) and \(Q\), respectively, are the sodium salts of

For the electrochemical cell, $\text{Mg(s)}\left|{\text{Mg}}^{\text{2+}}\left(\text{aq,1}\text{M}\right)\right|\left|{\text{Cu}}^{\text{2+}}\left(\text{aq,1}\text{M}\right)\right|\text{Cu }\left(\text{s}\right)$ the standard emf of the cell is $\text{2}\text{.70 V}$ at $\text{300 K}\text{.}$ When the concentration of ${\text{Mg}}^{\text{2+}}$ is changed to $\text{x M,}$ the cell potential changes to $\text{2}\text{.67 V}$ at $\text{300 K}\text{.}$ The value of $\text{x}$ is ________.
(given, $\frac{\text{F}}{\text{R}}{\text{=11500 K V}}^{-\text{1}}\text{,}$ where $\text{F}$ is the Faraday constant and $\text{R}$ is the gas constants, $\text{ln}\left(\text{10}\right)\text{=}\text{2}\text{.30}$ )

Match the compounds in LIST-$\mathrm{I}$ with the observations in LIST-$\mathrm{II}$, and choose the correct option.

	LIST-$\mathrm{I}$		LIST-$\mathrm{II}$
$\left(\mathrm{I}\right)$	Aniline	$\left(\mathrm{P}\right)$	Sodium fusion extract of the compound on boiling with ${\mathrm{FeSO}}_4$, followed by acidification with conc. ${\mathrm{H}}_2{\mathrm{SO}}_4$, gives Prussian blue color.
$\left(\mathrm{II}\right)$	$\mathrm{o}-$Cresol	$\left(\mathrm{Q}\right)$	Sodium fusion extract of the compound on treatment with sodium nitroprusside gives blood red color.
$\left(\mathrm{III}\right)$	Cysteine	$\left(\mathrm{R}\right)$	Addition of the compound to a saturated solution of ${\mathrm{NaHCO}}_3$ results in effervescence.
$\left(\mathrm{IV}\right)$	Caprolactam	$\left(\mathrm{S}\right)$	The compound reacts with bromine water to give a white precipitate.
		$\left(\mathrm{T}\right)$	Treating the compound with neutral ${\mathrm{FeCl}}_3$ solution produces violet color.

The number of resonance structures for N is

A tin chloride $\mathrm{Q}$ undergoes the following reactions (not balanced)
$\mathrm{Q}+\mathrm{C}{\mathrm{l}}^{-}\to \mathrm{X}$
$\mathrm{Q}+\mathrm{M}{\mathrm{e}}_3\mathrm{N}\to \mathrm{Y}$
$\mathrm{Q}+\mathrm{C}\mathrm{u}\mathrm{C}{\mathrm{l}}_2\to \mathrm{Z}+\mathrm{C}\mathrm{u}\mathrm{C}\mathrm{l}$
$\mathrm{X}$ is a monoanion having pyramidal geometry. Both $\mathrm{Y}$ and $\mathrm{Z}$ are neutral compounds. Choose the correct option(s).

The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of mass about an axis, and (ii) its motion about an instantaneous axis passing through the centre of mass.
These axes need not be stationary. Consider, for example, a thin uniform disc welded (rigidly fixed) horizontally at its rim to a massless, stick, as shown in the figure. When the disc-stick system is rotated about the origin on a horizontal frictionless plane with angular speed \(\omega\), the motion at any instant can be taken as a combination of (i) a rotation of the centre of mass of the disc about the \(z\)-axis and (ii) a rotation of the disc through an instantaneous vertical axis passing through its centre of mass (as is seen from the changed orientation of points \(P\) and \(Q\) ). Both these motions have the same angular speed \(\omega\) in this case


Now consider two similar systems as shown in the figure: Case
(a) the disc with its face vertical and parallel to \(x-z\) plane; Case
(b) the disc with its face making an angle of
\(45^{\circ}\) with \(x-y\) plane and its horizontal diameter parallel to \(x\)-axis. In both the cascs, the disc is welded at point \(\mathrm{P}\), and the systems are rotated with constant angular speed \(\omega\) about the \(z\) axis.

Question : Which of the following statements regarding the angular speed about the instantaneous axis (passing through the centre of mass) is correct?

The tangent \(P T\) and the normal \(P N\) to the parabola \(y^2=4 a x\) at a point \(P\) on

Consider two straight lines, each of which is tangent to both the circle $x^2+y^2=\frac{1}{2}$ and the parabola $y^2=4x$ . Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O(0, 0)$ and whose semi-major axis is $OQ$. If the length of the minor axis of this ellipse is $\sqrt{2}$, then the which of the following statement(s) is (are) TRUE?

If $3^{\text{x}}=4^{\text{x}-1}$, then x =

Let $f :\left(0, \infty \right)\to R$ be given by $f\left(x\right)= \underset{\frac{1}{x}}{\overset{x}{\int }}{e}^{-\left(t+\frac{1}{t}\right)}\frac{dt}{t},$ then

A planet of mass M, has two natural satellites with masses $m_1\mathrm{a}\mathrm{n}\mathrm{d}{m}_2$ . The radii of their circular orbits are $R_1 \mathrm{a}\mathrm{n}\mathrm{d} R_2$ respectively. Ignore the gravitational force between the satellites. Define $v_1,L_1,K_1 \mathrm{a}\mathrm{n}\mathrm{d} T_1$ to be, respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite $1; \mathrm{a}\mathrm{n}\mathrm{d} v_2,L_2,K_2\mathrm{a}\mathrm{n}\mathrm{d} T_2$ and to be the corresponding quantities of satellite 2. Given $m_1/m_2=2 \mathrm{a}\mathrm{n}\mathrm{d} R_1/R_2=\frac{1}{4}$ , match the ratios in List-I to the numbers in List-II.

	LIST -I		LIST-II
A)	$\frac{v_1}{v_2}$	P)	$\frac{1}{8}$
B)	$\frac{L_1}{L_2}$	Q)	1
C)	$\frac{K_1}{K_2}$	R)	2
D)	$\frac{T_1}{T_2}$	S)	8