JEE MAIN 2026 ONLINE 24TH JANUARY EVENING SHIFT

Let S be a set of 5 elements and $\mathrm{P}(\mathrm{S})$ denote the power set of S . Let E be an event of choosing an ordered pair (A, B) from the set $\mathrm{P}(\mathrm{S}) \times \mathrm{P}(\mathrm{S})$ such that $\mathrm{A} \cap \mathrm{B}=\emptyset$. If the probability of the event $E$ is $\frac{3^p}{2^q}$, where $p, q \in N$, then $p+q$ is equal to

A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_1$. A small co-centric cylinder of length $L / 2$ and radius $R / 3$ carved from this cylinder has moment of inertia about its axis equals to $I_2$. The ratio $I_1 / I_2$ is $\_\_\_\_$ .

The unsaturated ether on acidic hydrolysis produces carbonyl compounds as shown below :-

Based on this, predict the solution/reagent that will help to distinguish "P" and "Q" obtained in the following reaction :-

Let $z=(1+i)(1+2 i)(1+3 i) \ldots .(1+n i)$, where $i=\sqrt{-1}$. If $|z|^2=44200$, then $n$ is equal to $\_\_\_\_$

When 300 J of heat given to an ideal gas with $C_p=\frac{7}{2} R$ its temperature raises from $20^{\circ} \mathrm{C}$ to $50^{\circ} \mathrm{C}$ keeping its volume constant. The mass of the gas is (approximately) $\_\_\_\_$ g. $(\mathrm{R}=8.314 \mathrm{~J} / \mathrm{mol} . \mathrm{K})$

The half-life of ${ }^{65} \mathrm{Zn}$ is 245 days. After $x$ days, $75 \%$ of original activity remained. The value of $x$ in days is $\_\_\_\_$ . (Nearest integer)

(Given: $\log 3=0.4771$ and $\log 2=0.3010$ )