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Consider a fixed uniformly charged insulating sphere with radius R and total charge +Q. A point charge -q $(q<<Q)$ with mass m is released from rest at a distance of 3R from the centre of the charged sphere. When the point charge reaches the surface of the sphere, its speed is: ($\epsilon_0$ is the permittivity of vacuum, neglect gravitational forces).

A particle of mass M moves along a horizontal x axis from $x=0$ to $x=L$. The coefficient of kinetic friction varies as a function of x as $\mu_{k}(x)=\mu_{0}-\alpha x,$ where $\mu_0$, $\alpha$ are constants of appropriate dimensions, so that $\mu_{k}(L)=0$. The total work done by the frictional force during the motion is $n\mu_0 MgL$, where g is the acceleration due to gravity. The value of n is:

An electromagnetic wave travelling in a lossless dielectric medium having a dielectric constant, $\epsilon_{r}=9$, has the electric field, $E_{x}=E_{0} \sin(kz-2\pi\times10^{6}t)Vm^{-1}$ where $E_{0}$ is the amplitude and k is the wave vector. Among the following options, the incorrect choice is

An ideal gas is made of polyatomic molecules. Each of the molecules has three translational, three rotational and f number of vibrational modes. If the ratio of heat capacities $C_{P}/C_{V}$ of the gas is $8/7,$ then the value of f is:

The following table presents the part of the electromagnetic spectrum and their corresponding major applications.

P. Microwave - I. For purifying the water
Q. UV rays - II. For warming the food
R. Gamma rays - III. For AM and FM communication systems
S. Radio wave - IV. For treating the Cancer cells

The correct option is:

Consider a particle moving along a straight line, whose position as a function of time is given by $s(t)=\alpha t^{2}-\beta t+\gamma$, where $\alpha=1~ms^{-2}$ $\beta=6~ms^{-1}$ and $\gamma=5m.$ The average speed of the particle, in $ms^{-1}$ from $t=0$ to $t=6$ s is:

In an adiabatic expansion, the temperature of one mole of an ideal monatomic gas $(\gamma=5/3)$ decreases from 60 K to 50 K. The work done by the gas in the process is: (Take the universal gas constant as $R=8.3~J~mol^{-1}K^{-1})$

An ac voltage $V=220~sin(2\times10^{3}t)$ Volt is applied to a series LCR circuit. Then the current amplitude in this circuit is: (Given: $L=10~mH.$ $C=25\mu F,$ $R=100~\Omega)$

Two planets $P_{1}$ and $P_{2}$ with equal mass have radii $R_{1}$ and $R_{2}$, respectively, where $R_{2}=\frac{R_{1}}{2}$. The escape speeds of $P_{1}$ and $P_{2}$ are $v_{1}$ and $v_{2},$ respectively. Then $\frac{v_{2}}{v_{1}}$ is:

The temperature of a metallic sphere of radius R is increased by a small amount $\Delta T.$ If the linear coefficient of thermal expansion of the metal is $\alpha$, the approximate increase in the volume of the sphere is:

A photon and an electron, each of 20 eV energy, move in free space. The ratio of linear momentum of electron $p_{e}$ to that of photon $p_{Ph}$, $\frac{p_{e}}{p_{Ph}}$ is [Take speed of light $=3\times10^{8}ms^{-1}$ charge of electron $=-1.6\times10^{-19}C$ and mass of electron $=9\times10^{-31}$ kg]

Consider a first order reaction: $A\to \text{products}$ . 3 different solutions are taken rate of reaction : Solution 1: $100mL~10M~'A' - r_{1}$ ; Solution 2: $200mL~10M~'A' - r_{2}$ ; Solution 3: $100mL~10M~'A' + 100ml~\text{water} - r_{3}$ . The correct order of the rates of reactions is :

Given below are two statements . Statement-I : $K_{H}$ is constant with change in concentration of gas till solution is dilute at given temperature . Statement II: According to Henry's Law, partial pressure of gas in vapour phase is inversely proportional to mole fraction of gas in solution .

In organic compound with molecular formula ($X$) $C_{8}H_{9}NO$ when reacts with $KOH/Br_{2}$ forms '$P$' which on diazotization forms '$Q$' followed by its reaction with $CuCN$ forms '$R$' which on hydrolysis (acidic) formed '$S$' ($S$ can also be made by hydrolysis of original compound ($X$) ($C_{8}H_{9}NO$). $S$ on reaction with $KMnO_{4}/H^{+}$ forms '$T$' which has two types of hydrogen $X$ will be :

Which of the following graph(s) is/are correct? ($P_{\text{gas}}$ vs $X_{\text{solvent}}$ is plotted) .

Which of the following is the correct order of the reactivity of given nucleophiles when treated with $CH_{3}$ in methanol? $F^{-}$, $I^{-}$, $C_{2}H_{5}O^{-}$, $C_{6}H_{5}O^{-}$ :

The $3d$-block element which does not liberate hydrogen gas on reaction with dilute $HCl$ is .

Statement-1: $\text{Phenol} + CHCL_{3} \xrightarrow{KOH} \text{major} + \text{minor}$. Major product is ortho substituted and para substituted product is minor . Statement-2: Ortho \& Para product can be separated by steam distillation .

Given below are two statements . Statement-1: $HX$ bond length is higher in $HCl$ than $HF$ . Statement-2: The lowest boiling point in hydride of group 15 element is having covalency 4 .

Calculate volume of $H_2$ gas at STP in $ml$. If $2.7~g$ of aluminium is treated with excess $HCl$. (Volume of 1 mole gas at $STP=22.4~L$) :