Two solid spheres \(A\) and \(B\) of equal volumes but of different densities \(d_A\) and \(d_B\) are connected by a string. They are fully immersed in a fluid of density \(d_F\). They get arranged into an equilibrium state as shown in the figure with a tension in the string. The arrangement is possible only if

A geostationary satellite above the equator is orbiting around the earth at a fixed distance \(r_1\) from the center of the earth. A second satellite is orbiting in the equatorial plane in the opposite direction to the earth's rotation, at a distance \(r_2\) from the center of the earth, such that \(r_1=1.21 r_2\). The time period of the second satellite as measured from the geostationary satellite is \(\frac{24}{p}\) hours. The value of \(p\) is

A point object is placed at distance of \(20 \mathrm{~cm}\) from a thin planoconvex lens of focal length \(15 \mathrm{~cm}\). The plane surface of the lens is now silvered. The image created by the system is at

Statement 1 Two cylinders, one hollow (metal) and the other solid (wood) with the same mass and identical dimensions are simultaneously allowed to roll without slipping down an inclined plane from the same height. The hollow cylinder will reach the bottom of the inclined plane first.
and Statement 2 By the principle of conservation of energy, the total kinetic energies of both the cylinders are identical when they reach the bottom of the incline.

Two beams of red and violet colours are made to pass separately through a prism (angle of the prism is \(60^{\circ}\) ). In the position of minimum deviation, the angle of refraction will be

A conducting square loop of side \(L\), mass \(M\) and resistance \(R\) is moving in the \(X Y\) plane with its edges parallel to the \(X\) and \(Y\) axes. The region \(y \geq 0\) has a uniform magnetic field, \(\vec{B}=B_0 k\). The magnetic field is zero everywhere else. At time \(t=0\), the loop starts to enter the magnetic field with an initial velocity \(v_0 \hat{\jmath} \mathrm{~m} / \mathrm{s}\), as shown in the figure. Considering the quantity \(K=\frac{B_0^2 L^2}{R M}\) in appropriate units, ignoring self-inductance of the loop and gravity, which of the following statements is/are correct:

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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

Column I gives a list of possible set of parameters measured in some experiments. The variations of the parameters in the form of graphs are shown in Column II. Match the set of parameters given in Column I with the graphs given in Column II. Indicate your answer by darkening the appropriate bubble of the \(4 \times 4\) matrix given in the ORS.

The value of
\(\sec ^{-1}\left(\frac{1}{4} \sum_{k=0}^{10} \sec (\frac{7 \pi}{12}+\frac{k \pi}{2}\right) \sec (\frac{7 \pi}{12}\) \(+\frac{(k+1) \pi}{2}))\) in the interval \(\left[-\frac{\pi}{4}, \frac{3 \pi}{4}\right]\) equals

At \(298 \mathrm{~K}\), the limiting molar conductivity of a weak monobasic acid is \(4 \times 10^{2} \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\). At \(298 \mathrm{~K}\), for an aqueous solution of the acid the degree of dissociation is \(\boldsymbol{\alpha}\) and the molar conductivity is \(\mathbf{y} \times 10^{2} \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\). At \(298 \mathrm{~K}\), upon 20 times dilution with water, the molar conductivity of the solution becomes \(3 \mathbf{y} \times 10^{2} \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\)
The value of $\mathrm{y}$ is _________ .

Let \(\mathbb{N}\) denote the set of all natural numbers, and \(\mathbb{Z}\) denote the set of all integers. Consider the functions \(f: \mathbb{N} \rightarrow \mathbb{Z}\) and \(g: \mathbb{Z} \rightarrow \mathbb{N}\) defined by
\(f(n)= \begin{cases}(n+1) / 2 & \text { if } n \text { is odd } \\ (4-n) / 2 & \text { if } n \text { is even }\end{cases}\) And
\(g(n)=\left\{\begin{array}{cc}3+2 n & \text { if } n \geq 0 \\ -2 n & \text { if } n <0\end{array}\right.\)
Define \((g \circ f)(n)=g(f(n))\) for all \(n \in \mathbb{N}\), and \((f \circ g)(n)=f(g(n))\) for all \(n \in \mathbb{Z}\).
Then which of the following statements is (are) TRUE?

A ball is thrown from ground at an angle $\mathrm{\theta }$ with horizontal and with an initial speed ${\mathrm{u}}_0.$ For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is ${\mathrm{V}}_1.$ After hitting the ground, ball rebounds at the same angle $\mathrm{\theta }$ but with a reduced speed of $\frac{{\mathrm{u}}_0}{\mathrm{\alpha }}.$ Its motion continues for a long time as shown in figure. If the magnitude of average velocity of the ball for entire duration of motion is $0.8{\mathrm{V}}_1,$ the value of $\mathrm{\alpha }$ is______

Two point charges \(-Q\) and \(+Q / \sqrt{3}\) are placed in the \(x y\)-plane at the origin \((0,0)\) and a point \((2,0)\), respectively, as shown in the figure. This results in an equipotential circle of radius \(R\) and potential \(V=0\) in the \(x y\)-plane with its center at \((b, 0)\). All lengths are measured in meters.

The value of $R$ is ________ meter.