JEE MAIN 2024 ONLINE 29TH JANUARY MORNING SHIFT

An electron is moving under the influence of the electric field of a uniformly charged infinite plane sheet \(\mathrm{S}\) having surface charge density \(+\sigma\). The electron at \(t=0\) is at a distance of \(1 \mathrm{~m}\) from \(S\) and has a speed of \(1 \mathrm{~m} / \mathrm{s}\). The maximum value of \(\sigma\) if the electron strikes \(S\) at \(t=1 \mathrm{~s}\) is \(\alpha\left[\frac{m \epsilon_0}{e}\right] \frac{C}{m^2}\), the value of \(\alpha\) is ___________.

A solution of \(\mathrm{H}_2 \mathrm{SO}_4\) is \(31.4 \% \mathrm{H}_2 \mathrm{SO}_4\) by mass and has a density of \(1.25 \mathrm{~g} / \mathrm{mL}\). The molarity of the \(\mathrm{H}_2 \mathrm{SO}_4\) solution is _________ \(\mathrm{M}\) (nearest integer)

[Given molar mass of \(\mathrm{H}_2 \mathrm{SO}_4=98 \mathrm{~g} \mathrm{~mol}^{-1}\)]

\(\text { If } \frac{{ }^{11} C_1}{2}+\frac{{ }^{11} C_2}{3}+\ldots+\frac{{ }^{11} C_9}{10}=\frac{n}{m} \text { with } \operatorname{gcd}(n, m)=1 \text {, then } n+m \text { is equal to }\) _______.

A ball rolls off the top of a stairway with horizontal velocity \(u\). The steps are \(0.1 \mathrm{~m}\) high and \(0.1 \mathrm{~m}\) wide. The minimum velocity \(u\) with which that ball just hits the step 5 of the stairway will be \(\sqrt{x} \mathrm{~ms}^{-1}\) where \(x=\) __________ [use \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\) ].

The number of species from the following which are paramagnetic and with bond order equal to one is _________.

$$\mathrm{H}_2, \mathrm{He}_2^{+}, \mathrm{O}_2^{+}, \mathrm{N}_2^{2-}, \mathrm{O}_2^{2-}, \mathrm{F}_2, \mathrm{Ne}_2^{+}, \mathrm{B}_2$$

Let \(\alpha, \beta\) be the roots of the equation \(x^2-x+2=0\) with \(\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)\). Then \(\alpha^6+\alpha^4+\beta^4-5 \alpha^2\) is equal to ___________.