Radioactive decay kinetics

A sample initially contains only U-238 isotope of uranium. With time, some of the U-238 radioactively decays into Pb-206 while the rest of it remains undisintegrated.

When the age of the sample is $\mathbf{P}\times10^8$ years, the ratio of mass of Pb-206 to that of U-238 in the sample is found to be 7. The value of P is _____.

[Given: Half-life of U-238 is $4.5\times10^9$ years; $\log_e 2 = 0.693$]

A radioactive element has a half life of 200 days. The percentage of original activity remaining after 83 days is ______. (Nearest integer)

(Given : antilog 0.125 = 1.333, antilog 0.693 = 4.93)

Assuming 1 μg of trace radioactive element X with a half life of 30 years is absorbed by a growing tree. The amount of X remaining in the tree after 100 years is _______ $\times10^{-1}$ μg.

[Given: ln 10 = 2.303; log 2 = 0.30]

$\ce{^{238}_{92}U}$ is known to undergo radioactive decay to form $\ce{^{206}_{82}Pb}$ by emitting alpha and beta particles. A rock initially contained $68\times10^{-6}$ g of $\ce{^{238}_{92}U}$. If the number of alpha particles that it would emit during its radioactive decay of $\ce{^{238}_{92}U}$ to $\ce{^{206}_{82}Pb}$ in three half-lives is $Z\times10^{18}$, then what is the value of $Z$?

In the decay sequence,

$$\ce{^{238}_{92}U ->[-x_1] ^{234}_{90}Th ->[-x_2] ^{234}_{91}Pa ->[-x_3] ^{234}\mathbf{Z} ->[-x_4] ^{230}_{90}Th}$$

$x_1$, $x_2$, $x_3$ and $x_4$ are particles/radiation emitted by the respective isotopes. The correct option(s) is(are)

A plot of the number of neutrons $(N)$ against the number of protons $(P)$ of stable nuclei exhibits upward deviation from linearity for atomic number, $Z>20$. For an unstable nucleus having $N/P$ ratio less than 1, the possible mode(s) of decay is(are)

A closed vessel with rigid walls contains 1 mol of $^{238}_{92}\mathrm U$ and 1 mol of air at 298 K. Considering complete decay of $^{238}_{92}\mathrm U$ to $^{206}_{82}\mathrm{Pb}$, the ratio of the final pressure to the initial pressure of the system at 298 K is

In the nuclear transmutation
$$\ce{^9_4Be + X -> ^8_4Be + Y}$$
(X, Y) is(are)

Bombardment of aluminum by $\alpha$-particle leads to its artificial disintegration in two ways, (i) and (ii) as shown. Products X, Y and Z respectively are,

The number of neutrons emitted when $^{235}_{92}\mathrm U$ undergoes controlled nuclear fission to $^{142}_{54}\mathrm{Xe}$ and $^{90}_{38}\mathrm{Sr}$ is

The total number of $\alpha$ and $\beta$ particles emitted in the nuclear reaction $\ce{^{238}_{92}U -> ^{214}_{82}Pb}$ is

STATEMENT-1: The plot of atomic number (y-axis) versus number of neutrons (x-axis) for stable nuclei shows a curvature towards x-axis from the line of 45° slope as the atomic number is increased.

and

STATEMENT-2: Proton-proton electrostatic repulsions begin to overcome attractive forces involving protons and neutrons in heavier nuclides.

A positron is emitted from $\ce{^{23}_{11}Na}$. The ratio of the atomic mass and atomic number of the resulting nuclide is