JEE MAIN 2025 ONLINE 28TH JANUARY MORNING SHIFT

Two iron solid discs of negligible thickness have radii $R_1$ and $R_2$ and moment of intertia $I_1$ and $I_2$, respectively. For $R_2=2 R_1$, the ratio of $I_1$ and $I_2$ would be $1 / x$, where $\mathrm{x}=$ _______ .

Consider the following elements $\mathrm{In}, \mathrm{Tl}, \mathrm{Al}, \mathrm{Pb}, \mathrm{Sn}$ and Ge . The most stable oxidation states of elements with highest and lowest first ionisation enthalpies, respectively, are

If $\alpha=1+\sum\limits_{r=1}^6(-3)^{r-1} \quad{ }^{12} \mathrm{C}_{2 r-1}$, then the distance of the point $(12, \sqrt{3})$ from the line $\alpha x-\sqrt{3} y+1=0$ is ________.

The moment of inertia of a solid disc rotating along its diameter is 2.5 times higher than the moment of inertia of a ring rotating in similar way. The moment of inertia of a solid sphere which has same radius as the disc and rotating in similar way, is $n$ times higher than the moment of inertia of the given ring. Here, $\mathrm{n}=$ ________ Consider all the bodies have equal masses.

Consider the following sequence of reactions:

11.25 mg of chlorobenzene will produce ___________$\times 10^{-1} \mathrm{mg}$ of product B.

(Consider the reactions result in complete conversion.)

[Given molar mass of $\mathrm{C}, \mathrm{H}, \mathrm{O}, \mathrm{N}$ and Cl as $12,1,16,14$ and $35.5 \mathrm{~g} \mathrm{~mol}^{-1}$ respectively]

Let $\mathrm{E}_1: \frac{x^2}{9}+\frac{y^2}{4}=1$ be an ellipse. Ellipses $\mathrm{E}_{\mathrm{i}}$ 's are constructed such that their centres and eccentricities are same as that of $\mathrm{E}_1$, and the length of minor axis of $\mathrm{E}_{\mathrm{i}}$ is the length of major axis of $E_{i+1}(i \geq 1)$. If $A_i$ is the area of the ellipse $E_i$, then $\frac{5}{\pi}\left(\sum\limits_{i=1}^{\infty} A_i\right)$, is equal to _______.