In a radioactive decay chain reaction, ${}_{90}^{230}\mathrm{Th}$ nucleus decays into ${}_{84}{}^{214}\mathrm{Po}$ nucleus. The ratio of the number of $\alpha$ to number of ${\beta }^{-}$ particles emitted in this process is _______.

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A uniform thin cylindrical disk of mass \(M\) and radius \(R\) is attached to two identical massless springs of spring constant \(k\) which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance \(d\) from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is \(L\). The disk is initially at its equilibrium position with its centre of mass \((C M)\) at a distance Lfrom the wall. The disk rolls without slipping with velocity \(\mathbf{v}_0=v_0 \hat{\mathbf{i}}\) The coefficient of friction is \(\mu\).
Question :
The net external force acting on the disk when its centre of mass is at displacement \(x\) with respect to its equilibrium position is

In a radioactive sample, ${\mathrm{}}_{19}^{40}\mathrm{K}$ nuclei either decay into stable ${\mathrm{}}_{20}^{40}\mathrm{C}\mathrm{a}$ nuclei with decay constant $4.5\times {10}^{-10}$ per year or into stable ${\mathrm{}}_{18}^{40}\mathrm{A}\mathrm{r}$ nuclei with decay constant $0.5\times {10}^{-10}$ per year. Given that in this sample all the stable ${\mathrm{}}_{20}^{40}\mathrm{C}\mathrm{a}$ and ${\mathrm{}}_{18}^{40}\mathrm{A}\mathrm{r}$ nuclei are produced by the ${\mathrm{}}_{19}^{40}\mathrm{K}$ nuclei only. In time $\mathrm{t}\times {10}^9$ years, if the ratio of the sum of stable ${\mathrm{}}_{20}^{40}\mathrm{C}\mathrm{a}$ and ${\mathrm{}}_{18}^{40}\mathrm{A}\mathrm{r}$ nuclei to the radioactive ${\mathrm{}}_{19}^{40}\mathrm{K}$ nuclei is $99,$ the value of t will be : [Given $\mathrm{l}\mathrm{n}\mathrm{}10=2.3$ ]

A conducting solid sphere of radius \(R\) and mass \(M\) carries a charge \(Q\). The sphere is rotating about an axis passing through its center with a uniform angular speed \(\omega\). The ratio of the magnitudes of the magnetic dipole moment to the angular momentum about the same axis is given as \(\alpha \frac{Q}{2 M}\). The value of \(\alpha\) is _______ .

Students I, II and III perform an experiment for measuring the acceleration due to gravity \((g)\) using a simple pendulum. They use different lengths of the pendulum and/or record time for different number of oscillations. The observations are shown in the table.
Least count for length \(=0.1 \mathrm{~cm}\)
Least count for time \(=0.1 \mathrm{~s}\)

If \(E_{\mathrm{I}}, E_{\mathrm{II}}\) and \(E_{\mathrm{III}}\) are the percentage errors in \(g\), i.e., \(\left(\frac{\Delta g}{g} \times 100\right)\) for students I, II and III, respectively.

A particle is moved along a path $\mathrm{A}\mathrm{B}-\mathrm{B}\mathrm{C}-\mathrm{C}\mathrm{D}-\mathrm{D}\mathrm{E}-\mathrm{E}\mathrm{F}-\mathrm{F}\mathrm{A},$ as shown in figure, in presence of a force $\overrightarrow{\mathrm{F}}=\left(\mathrm{\alpha }\mathrm{y}\widehat{\mathrm{i}}+2\mathrm{\alpha }\mathrm{x}\widehat{\mathrm{j}}\right)\mathrm{N},$ where $x$ and $\mathrm{y}$ are in meter and $\mathrm{\alpha }=-1\mathrm{N}/{\mathrm{m}}^{-1}.$ The work done on the particle by this force $\overrightarrow{\mathrm{F}}$ will be ____ Joule.

An ideal gas undergoes a four step cycle as shown in the $P-V$ diagram below. During this cycle, heat is absorbed by the gas in:

Match the column.
Normals at \(P, Q, R\) are drawn to \(y^2=4 x\) which intersect at \((3,0)\). Then,

Let \(y(x)\) be the solution of the differential equation
\(x^2 \frac{d y}{d x}+x y=x^2+y^2, x>\frac{1}{e}\)
satisfying \(y(1)=0\). Then the value of \(2 \frac{(y(e))^2}{y\left(e^2\right)}\) is _____ .

A block of mass \(5 \mathrm{~kg}\) moves along the \(x\)-direction subject to the force \(F=(-20 x+10) \mathrm{N}\), with the value of \(x\) in metre. At time \(t=0 \mathrm{~s}\), it is at rest at position \(x=1 \mathrm{~m}\). The position and momentum of the block at \(t=(\pi / 4) \mathrm{s}\) are

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column 1	Column 2	Column 3
(I) $x^2+y^2=a^2$	(i) $my=m^{2}x+a$	(P) $\left(\frac{a}{m^2},\frac{2a}{m}\right)$
(II) $x^2+a^2{y}^2=a^2$	(ii) $y=mx+a \sqrt{m^2+1}$	(Q) $\left(\frac{-ma}{\sqrt{m^2+1}},\frac{a}{\sqrt{m^2+1}}\right)$
(III) $y^2=4ax$	(iii) $y=mx+ \sqrt{a^2{m}^2-1}$	(R) $\left(\frac{-a^{2}m}{\sqrt{a^2{m}^2+1}},\frac{1}{\sqrt{a^2{m}^2+1}}\right)$
(IV) $x^2-a^2{y}^2=a^2$	(iv) $y=mx+\sqrt{a^2{m}^2+1}$	(S) $\left(\frac{-a^{2}m}{\sqrt{a^2{m}^2-1}},\frac{-1}{\sqrt{a^2{m}^2-1}}\right)$

If a tangent to a suitable conic (Column 1) is found to be $y=x+8$ and its point of contact is $(8,16)$, then which of the following options is the only Correct combination?

The stoichiometric reaction of $516 \mathrm{g}$ of dimethyldichlorosilane with water results in a tetrameric cyclic product $\mathbf{X}$ in $75\%$ yield. The weight (in $\mathrm{g}$) of $\mathbf{X}$ obtained is ____________
\(\left[\right.\)Use, molar mass\(.\left(\text{gmol} ^{-1}\right):\text H =1,\text C =12,\)\(\text O =16, \text{Si} =28, \text{Cl} =35.5]\)

Two solid cylinders \(P\) and \(Q\) of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder \(P\) has most of its mass concentrated near its surface, while \(Q\) has most of its mass concentrated near the axis. Which statement(s) is(are) correct?