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 water cooler of storage capacity 120 litres can cool water at constant rate of P watts. In a closed circulation system (as shown schematically in the figure), the water from the cooler is used to cool an external device that generates constantly 3 kW of heat (thermal load). The temperature of water fed into the device cannot exceed ${30}^{o}C$ and the entire stored 120 litres of water is initially cooled to ${10}^{o}C$ . The entire system is thermally insulated. The minimum value of P (in watts) for which the device can be operated for 3 hours is:

(Specific heat of water is 4.2 kJ $k{g}^{-1}{K}^{-1}$ and the density of water is 1000 kg $m^{-3}$ )

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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 \(\mu\).
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Question :
The centre of mass of the disk undergoes simple harmonic motion with angular frequency \(\omega\) equal to

Two coherent monochromatic point sources $S_1$ and $S_2$ of wavelength $\lambda =600nm$ are placed symmetrically on either side of the center of the circle as shown. The sources are separated by a distance $d=1.8mm.$ This arrangement produces interference fringes visible as alternate bright and dark spots on the circumference of the circle. The angular separation between two consecutive bright spots is $\mathrm{\Delta }\theta$ . Which of the following options is/are correct?

Two balls, having linear momenta \(\overrightarrow{\mathbf{p}}_1=p \hat{\mathbf{i}}\) and \(\overrightarrow{\mathbf{p}}_2=-p \hat{\mathbf{i}}\), undergo a collision in free space. There is no external force acting on the balls. Let \(\overrightarrow{\mathbf{p}}_1^{\prime}\) and \(\overrightarrow{\mathbf{p}}_2^{\prime}\) be their final momenta. The following option(s) is/are NOT ALLOWED for any non-zero value of \(p, a_1, a_2, b_1, b_2, c_1\) and \(c_2\)

A glass beaker has a solid, plano-convex base of refractive index 1.60, as shown in the figure. The radius of curvature of the convex surface (SPU) is \(9 \mathrm{~cm}\), while the planar surface (STU) acts as a mirror. This beaker is filled with a liquid of refractive index \(n\) up to the level QPR. If the image of a point object \(\mathrm{O}\) at a height of \(h\) (OT in the figure) is formed onto itself, then, which of the following option(s) is(are) correct?

Match the following for the given process

The total number of cyclic structural as well as stereoisomers possible for a compound with the molecular formula \(=\mathrm{C}_5 \mathrm{H}_{10}\) is

Two adjacent sides of a parallelogram \(A B C D\) are given by \(\overrightarrow{\mathbf{A B}}=2 \hat{\mathbf{i}}+10 \hat{\mathbf{j}}+11 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{A D}}=-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\). The side \(A D\) is rotated by an acute angle \(\alpha\) in the plane of the parallelogram so that \(A D\) becomes \(A D^{\prime}\). If \(A D^{\prime}\) makes a right angle with the side \(A B\), then the cosine of the angle \(\alpha\) is given by

Let \(k \in \mathbb{R}\). If \(\lim _{x \rightarrow 0^+}(\sin (\sin k x)+\cos x+x)^{\frac{2}{x}}=e^6\), then the value of \(k\) is

The ellipse \(E_{1}: \frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) is inscribed in a rectangle \(R\) whose sides are parallel to the coordinate axes. Another ellipse \(E_{2}\) passing through the point \((0,4)\) circumscribes the rectangle \(R\). The eccentricity of the ellipse \(E_{2}\) is

Let \(S_1=\{(i, j, k): i, j, k \in\{1,2, \ldots, 10\}\}\)
\(S_2=\{(i, j): 1 \leq i < j+2 \leq 10, i, j\) \(\in\{1,2, \ldots, 10\}\} \)
\(S_3=\{(i, j, k, l): 1 \leq i < j < k < l, i,\) \(j, k, l \in\{1,2, \ldots . ., 10\}\}\)
and \(S_4=\{(i, j, k, l): i, j, k\) and \(l\) are distint elements in \(\{1,2, \ldots, 10\}\}\) If the total number of elements in the set \(S_r\) is \(n_r, r=1,2,3,4\) then which of the following statements is (are) TRUE?

A ball is projected from the ground at an angle of ${45}^o$ with the horizontal surface. It reaches a maximum height of $120\mathrm{ m}$ and returns to the ground. Upon hitting the ground for the first time, it loses half of its kinetic energy. Immediately after the bounce, the velocity of the ball makes an angle of ${30}^o$ with the horizontal surface. The maximum height it reaches after the bounce (in meters) is _____________.