JEE Advanced – Page 7

Radioactive decay of Potassium

December 30th, 2021jeefirst0 Comments

JEE Advanced 2019 Paper 1, Question 4

In a radioactive sample, ${}^{40}_{19}{\rm K}$ nuclei decay into stable ${}^{40}_{20}{\rm Ca}$ nuclei with decay constant $4.5 \times 10^{-10}$ per year or into stable ${}^{40}_{18}{\rm Ar}$ nuclei with decay constant $0.5 \times 10^{-10}$ per year. In this sample all the stable ${}^{40}_{20}{\rm Ca}$ and ${}^{40}_{18}{\rm Ar}$ nuclei are produced by the ${}^{40}_{19}{\rm K}$ nuclei only. In time $t_1 \times 10^9$ years, if the ratio of the sum of stable ${}^{40}_{20}{\rm Ca}$ and ${}^{40}_{18}{\rm Ar}$ nuclei to the radioactive ${}^{40}_{19}{\rm K}$ nuclei is 99, the value of $t_1$ will be [Given $\ln 10 = 2.3$ ],

1.15
9.2
2.3
4.6

Solution

Let us first consider the situation where a sample of $X$ nuclei decays into a single type of nuclei $Y$ . The number $N$ of $X$ nuclei changes with …

Rod heated by wire

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JEE Advanced 2019 Paper 1, Question 3

A current carrying wire heats a metal rod. The wire provides a constant power $P$ to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature $T$ in the metal rod changes with time $t$ as

(1) $\begin{equation*} T(t) = T_0 (1 + \beta t^{\frac{1}{4}}) \end{equation*}$

where $\beta$ is a constant with appropriate dimension while $T_0$ is a constant with dimension of temperature. The heat capacity of the metal is,

$\frac{4 P [T(t) - T_0]^3}{\beta^4 T_0^4}$
$\frac{4 P [T(t) - T_0]^4}{\beta^4 T_0^5}$
$\frac{4 P [T(t) - T_0]^2}{\beta^4 T_0^3}$
$\frac{4 P [T(t) - T_0]}{\beta^4 T_0^2}$

Solution

The heat capacity of an object is defined by the relation

(2) $\begin{equation*} \Delta Q = C \Delta T \end{equation*}$

where $\Delta Q$ is the heat that the object absorbs and $\Delta T$ is the resulting temperature change of …

Insulating spherical shell with a hole

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JEE Advanced 2019 Paper 1, Question 2.

A thin spherical insulating shell of radius $R$ carries a uniformly distributed charge such that the potential at its surface is $V_0$ . A hole with a small area $\alpha 4 \pi R^2 \ (\alpha \ll 1)$ is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?

The potential at the center of the shell is reduced by $2 \alpha V_0$
The magnitude of the eletric field at the center of the shell is reduced by $\frac{\alpha V_0}{2 R}$
The ratio of the potential at the center of the shell to that of the point at $R/2$ from center towards

…

Spherical gas cloud

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JEE Advanced 2019 Paper 1, Question 1

Consider a spherical gas cloud of mass density $\rho(r)$ in free space where $r$ is the radial distance from its center. The gaseous cloud is made of particles of equal mass $m$ moving in circular orbits about the common center with the same kinetic energy $K$ . The force acting on the particles is their mutual gravitational force. If $\rho(r)$ is constant in time, the particle number density $n(r) = \rho(r)/m$ is [ $G$ is the universal gravitational constant]