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The half-life of a radioactive isotope $X$ is 20 years. It decays to another element $Y$ which is stable. The two elements $X$ and $Y$ were found to be in the ratio 1:7 in a sample of a given rock. The age of the rock is estimated to be:
Two simple harmonic motions of angular frequency $100\text{ rad s}^{-1}$ and $1000\text{ rad s}^{-1}$ have the same displacement amplitude. The ratio of their maximum acceleration will be:
The decay constant of a radio isotope is $\lambda$. If $A_1$ and $A_2$ are its activities at times $t_1$ and $t_2$ respectively, the number of nuclei which have decayed during the time $(t_1 - t_2)$ is:
The activity of a radioactive sample is measured as $N_0$ counts per minute at $t=0$ and $N_0/e$ counts per minute at $t=5$ min. The time (in minute) at which the activity reduces to half its value is:
A radioisotope X with a half-life 1.4 × 10^9 yr decays into Y which is stable. A sample of the rock from a cave was found to contain X and Y in the ratio 1:7. The age of the rock is:
The mass of a ${}_{3}^{7}\mathrm{Li}$ nucleus is 0.042 u less than the sum of the masses of its nucleons. The binding energy per nucleon of ${}_{3}^{7}\mathrm{Li}$ nucleus is nearly:
Which one of the following equations of motion represents simple harmonic motion where $k$, $k_0$, $k_1$, and $a$ are all positive?
A particle is executing SHM along a straight line. Its velocities at distances $x_1$ and $x_2$ from the mean position are $v_1$ and $v_2$, respectively. Its time period is:
A person can see objects clearly only when they lie between 50 cm and 400 cm from his eyes. In order to increase the maximum distance of distinct vision to infinity, the type and power of the correcting lens, the person has to use, will be:
A radioactive nucleus ${}_{Z}^{A}\mathrm{X}$ undergoes spontaneous decay in the sequence ${}_{Z}\mathrm{X} \rightarrow {}_{Z-1}\mathrm{B} \rightarrow {}_{Z-3}\mathrm{C} \rightarrow {}_{Z-2}\mathrm{D}$, where $Z$ is the atomic number of element $\mathrm{X}$. The possible decay particles in the sequence are:
In the given figure, a diode D is connected to an external resistance $R = 100 \, \Omega$ and an e.m.f of $3.5\text{ V}$. If the barrier potential developed across the diode is $0.5\text{ V}$, the current in the circuit will be:
The unit of permittivity of free space $\varepsilon_0$ is
If $T_1, T_2, T_3, T_4$ and $T_5$ represent the tension in the string of a simple pendulum when the bob is at the left extreme, right extreme, mean, any intermediate left and any intermediate right positions, respectively. Then, which of the following relations are correct? (A) $T_1 = T_2$ (B) $T_3 > T_2$ (C) $T_4 > T_3$ (D) $T_3 = T_4$ (E) $T_5 > T_2$ Choose the most appropriate answer from the options given below:
The magnifying power of a telescope is 9. When it is adjusted for parallel rays, the distance between the objective and eyepiece is 20 cm. The focal lengths of lenses are:
A nucleus of uranium decays at rest into nuclei of thorium and helium. Then:
A nucleus with Z = 92 emits the following in a sequence: $\alpha, \beta^-, \beta^-, \alpha, \alpha, \alpha, \alpha, \alpha, \beta^-, \beta^-, \alpha, \beta^+, \beta^+, \alpha$. The Z of the resulting nucleus is:
An automobile engine develops $100\text{ kW}$ when rotating at a speed of $1800\text{ rev/min}$. What torque does it deliver?
Given below are two statements: Statement-I: The law of radioactive decay states that the number of nuclei undergoing the decay per unit time is inversely proportional to the total number of nuclei in the sample. Statement-II: The half life of a radionuclide is the sum of the life time of all nuclei, divided by the initial concentration of the nuclei at time t=0. In the light of the above statements, choose the most appropriate answer from the options given below:
A light bulb and an inductor coil are connected to an AC source through a key as shown in the figure below. The key is closed and after some time an iron rod is inserted into the interior of the inductor. The glow of the light bulb:
To get output $1$ for the following circuit, the correct choice for the input is: