Consider two solid spheres P and Q each of density 8 gm $c{m}^{-3}$ and diameters 1 cm and 0.5 cm, respectively. Sphere P is dropped into a liquid of density 0.8 gm $c{m}^{-3}$ and viscosity $\eta =3$ poiseulles. Sphere Q is dropped into a liquid of density 1.6 gm $c{m}^{-\mathrm{3 }}$and viscosity $\eta =2$ poiseulles. the ratio of the terminal velocities of P and Q is.

A large spherical mass $\mathrm{M}$ is fixed at one position and two identical point masses $\mathrm{m}$ are kept on a line passing through the centre of $\mathrm{M}$ (see figure). The point masses are connected by a rigid massless rod of length $\mathrm{l}$ and this assembly is free to move along the line connecting them. All three masses interact only through their mutual gravitational interaction. When the point mass nearer to $\mathrm{M}$ is at a distance $\mathrm{r}=3\mathrm{l}$ from $\mathrm{M}$ , the tension in the rod is zero for $\mathrm{m}=\mathrm{k}\left(\frac{\mathrm{M}}{288}\right)$ . The value of $\mathrm{k}$ 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

Six point charges, each of the same magnitude \(q\), are arranged in different manners as shown in Column-II. In each case, a point \(M\) and a line \(P Q\) passing through \(M\) are shown. Let \(E\) be the electric field and \(V\) be the electric potential at \(M\) (potential at infinity is zero) due to the given charge distribution when it is at rest. Now, the whole system is set into rotation with a constant angular velocity about the line \(P Q\). Let \(B\) the magnetic field at \(M\) and \(\mu\) be the magnetic moment of the system in this condition. Assume each rotating charge to be equivalent to a steady current.

A large square container with thin transparent vertical walls and filled with water $\left(\text{refractive index}\frac{4}{3}\right)$ is kept on a horizontal table. A student holds a thin straight wire vertically inside the water $12\mathrm{cm}$ from one of its corners, as shown schematically in the figure. Looking at the wire from this corner, another student sees two images of the wire, located symmetrically on each side of the line of sight as shown. The separation (in $\mathrm{cm}$) between these images is ________.

A student skates up a ramp that makes an angle $30°$ with the horizontal. He/she starts (as shown in the figure) at the bottom of the ramp with speed $v_0$ and wants to turn around over a semicircular path $xyz$ of radius $R$ during which he/she reaches a maximum height $h$ (at point $y$) from the ground as shown in the figure. Assume that the energy loss is negligible and the force required for this turn at the highest point is provided by his/her weight only. Then ($g$ is the acceleration due to gravity)

Four solid spheres each of diameter \(\sqrt{5} \mathrm{~cm}\) and mass \(0.5 \mathrm{~kg}\) ar placed with their centres at the corners of a square of side \(4 \mathrm{~cm}\). The moment of inertia of the system about the diagonal of the square is \(N \times 10^{-4} \mathrm{~kg}-\mathrm{m}^2\), then \(N\) is

An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is $n$. The internal energy of one mole of the gas is $U_n$ and the speed of sound in the gas is $v_n$. At a fixed temperature and pressure, which of the following is the correct option?

Let \(f(x)\) be a continuously differentiable function on the interval \((0, \infty)\) such that \(f(1)=2\) and \(\underset{t \rightarrow x}{\lim} \frac{t^{10} f(x)-x^{10} f(t)}{t^9-x^9}=1\) for each \(x>0\). Then, for all \(x>0, f(x)\) is equal to

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Let \(\psi_{1}:[0, \infty) \rightarrow \mathbb{R}, \psi_{2}:[0, \infty) \rightarrow \mathbb{R}, f:[0, \infty) \rightarrow \mathbb{R}\) and \(g:[0, \infty) \rightarrow \mathbb{R}\) be functions such that \(f(0)=g(0)=0\),

\(\psi_{1}(x)=e^{-x}+x, \quad x \geq 0\),

\(\psi_{2}(x)=x^{2}-2 x-2 e^{-x}+2, \quad x \geq 0\),

\(f(x)=\int_{-x}^{x}\left(|t|-t^{2}\right) e^{-t^{2}} d t, \quad x>0\)

and \(g(x)=\int_{0}^{x^{2}} \sqrt{t} e^{-t} d t, \quad x>0\)


Question:

Which of the following statements is TRUE ?

Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ be a differentiable function such that its derivative $f^{\text{'}}$ is continuous and $f\left(\pi \right)=-6$. If $F:\left[0,\pi \right]\to \mathrm{ℝ}$ is defined by $F\left(x\right)={\int }_0^{x}f\left(t\right)dt,$ and if ${\int }_0^{\pi }\left(f^{\text{'}}\left(x\right)+F\left(x\right)\right)\cos x dx=2$, then the value of $f\left(0\right)$ is_______

A double slit setup is shown in the figure. One of the slits is in medium $2$ of refractive index $n_2$. The other slit is at the interface of this medium with another medium $1$ of refractive index $n_1\left(\ne n_2\right)$. The line joining the slits is perpendicular to the interface and the distance between the slits is $d$. The slit widths are much smaller than $d$. A monochromatic parallel beam of light is incident on the slits from medium $1$. A detector is placed in medium $2$ at a large distance from the slits, and at an angle $\theta$ from the line joining them, so that $\theta$ equals the angle of refraction of the beam. Consider two approximately parallel rays from the slits received by the detector.
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Which of the following statement(s) is(are) correct?

Let the circles $C_1:x^2+y^2=9$ and $C_2:{\left(x-3\right)}^2+{\left(y-4\right)}^2=16,$ intersect at the points $X$ and $Y.$ Suppose that another circle $C_3:{\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$ satisfies the following conditions:
$\left(i\right)$ centre of $C_3$ is collinear with the centres of $C_1$ and $C_2$
$\left(ii\right) C_1$ and $C_2$ both lie inside $C_3,$ and
$\left(iii\right) C_3$ touches $C_1$ at $M$ and $C_2$ at $N.$
Let the line through $X$ and $Y$ intersect $C_3$ at $Z$ and $W,$ and let a common tangent of $C_1$ and $C_3$ be a tangent to the parabola $x^2=8\alpha y.$
There are some expression given in the List- $\mathrm{I}$ whose values are given in List- $\mathrm{I}\mathrm{I}$ below:
 

	List-  I		List-  II
$\left(I\right)$	$2h+k$	$\left(P\right)$	$6$
$\left(II\right)$	$\frac{\mathrm{L}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }ZW}{\mathrm{L}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }XY}$	$\left(Q\right)$	$\sqrt{6}$
$\left(III\right)$	$\frac{\mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}\mathrm{ }MZN}{\mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}\mathrm{ }ZMW}$	$\left(R\right)$	$\frac{5}{4}$
$\left(IV\right)$	$\alpha$	$\left(S\right)$	$\frac{21}{5}$
		$\left(T\right)$	$2\sqrt{6}$
		$\left(U\right)$	$\frac{10}{3}$

Which of the following is the only INCORRECT combination?