Under the influence of the coulomb field of charge \(+Q\), a charge \(-q\) is moving around it in an elliptical orbit. Find out the correct statement(s).

Airplanes A and B are flying with constant velocity in the same vertical plane at angles ${30}^{\mathrm{o}}$ and ${60}^{\mathrm{o}}$ with respect to the horizontal respectively as shown in the figure. The speed of A is $100\sqrt{3}\mathrm{}\mathrm{m}{\mathrm{s}}^{-1}$ . At time t = 0 s, an observer in A finds B at a distance of 500 m. This observer sees B moving with a constant velocity perpendicular to the line of motion of A. If at $\mathrm{t}={\mathrm{t}}_0$ , A just escapes being hit by B, t₀ in seconds is

A perfectly reflecting mirror of mass $\mathrm{M}$ mounted on a spring constitutes a spring-mass system of angular frequency $\mathrm{\Omega }$ such that $\frac{4\mathrm{\pi }\mathrm{M}\mathrm{\Omega }}{\mathrm{h}}={10}^{24}{\mathrm{m}}^{-2}$ with $\mathrm{h}$ as Planck's constant. $\mathrm{N}$ photons of wavelength $\mathrm{\lambda }=8\mathrm{\pi }\times {10}^{-6}\mathrm{m}$ strike the mirror simultaneously at normal incidence such that the mirror gets displaced by $1\mathrm{\mu }\mathrm{m}.$ If the value of $\mathrm{N}$ is $\mathrm{x}\times {10}^{12},$ then the value of $\mathrm{x}$ is________.
[Consider the spring as massless]

A particle of mass $\text{0.2 kg}$ is moving in one dimension under a force that delivers a constant power $\text{0.5 W}$ to the particle. If the initial speed $\left({in\; ms}^{-1}\right)$ of the particle is zero, the speed $\left({in\; ms}^{-1}\right)$ after $\text{5 s}$ is

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 $m^{-3}$ )

A double star system consists of two stars \(A\) and \(B\) which have time period \(T_A\) and \(T_B\). Radius \(R_A\) and \(R_B\) and mass \(M_A\) and \(M_B\). Choose the correct option.

A student is performing an experiment using a resonance column and a tuning fork of frequency \(244 s^{-1}\). He is told that the air in the tube has been replaced by another gas (assume that the column remains filled with the gas). If the minimum height at which resonance occurs is \((0.350 \pm 0.005) m\), the gas in the tube is (Useful information: \(\sqrt{167\text{ RT}}=640\text{ J}^{\frac{1}{2}}\text{ mol}^{-\frac{1}{2}} ;\) \(\sqrt{140\text{ RT}}=590\text{ J}^{\frac{1}{2}}\text{ mole}^{-\frac{1}{2}}\) . The molar masses M in grams are given in the options. Take the values of \(\sqrt{\frac{10}{\text M }}\) for each gas as given there.)

The value of  $I={\displaystyle \sum }_{k=1}^{98}\underset{k}{\overset{k+1}{{\displaystyle \int }}}\frac{k+1}{x(x+1)}dx,$ then

A disc of mass \(M\) and radius \(R\) is free to rotate about its vertical axis as shown in the figure. A battery operated motor of negligible mass is fixed to this disc at a point on its circumference. Another disc of the same mass \(M\) and radius \(R / 2\) is fixed to the motor's thin shaft. Initially, both the discs are at rest. The motor is switched on so that the smaller disc rotates at a uniform angular speed \(\omega\). If the angular speed at which the large disc rotates is \(\omega / n\), then the value of \(n\) is ______ .

Benzene and naphthalene form an ideal solution at room temperature. For this process, the true statement(s) is (are)

Let $f\left(x\right)=\frac{\sin \pi x}{x^2}, x>0.$
Let $x_1<x_2<x_3<\dots <x_n<\dots$ be all the points of local maximum of $f$ and $y_1<y_2<y_3<\dots <y_n<\dots$ be all the points of local minimum of $f.$
Then which of the following options is/are correct?

If \(f(x)\) is twice differentiable function such that \(f(a)=0, f(b)=2, f(c)=1, f(d)=2\), \(f(e)=0\), where \(a < b < c < d < e\), then the minimum number of zeros of \(g(x)=\left\{f^{\prime}(x)\right\}^2+f^{\prime \prime}(x) \cdot f(x)\) in the interval \([a, e]\) is

The potential energy of a particle of mass m at a distance r from a fixed point O is given by $V\left(r\right)=k{r}^2/2$, where k is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius R about the point O. If v is the speed of the particle and L is the magnitude of its angular momentum about O, which of the following statements is (are) true?