JEE Main 2026 (Online) 6th April Morning Shift

A three coulomb charge moves from the point $(0,-2,-5)$ to the point $(5,1,2)$ in an electric field expressed as $\vec{E}=2 x \hat{\mathrm{i}}+3 \mathrm{y}^2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}} \mathrm{N} / \mathrm{C}$. The work done in moving the charge is $\_\_\_\_$ J.

4.7 g of phenol is heated with Zn to give product X . If this reaction goes to $60 \%$ completion then the number of moles of compound X formed will be

$\_\_\_\_$ $\times 10^{-2}$.(Nearest Integer)

(Given molar mass in $\mathrm{g} \mathrm{mol}^{-1}: \mathrm{H}: 1, \mathrm{C}: 12, \mathrm{O}: 16$ )

For the functions $f(\theta)=\alpha \tan ^2 \theta+\beta \cot ^2 \theta$, and $g(\theta)=\alpha \sin ^2 \theta+\beta \cos ^2 \theta, \alpha>\beta>0$, let $\min\limits_{0<\theta<\frac{\pi}{2}} f(\theta)=\max\limits_{0<\theta<\pi} g(\theta)$. If the first term of a G.P. is $\left(\frac{\alpha}{2 \beta}\right)$, its common ratio is $\left(\frac{2 \beta}{\alpha}\right)$ and the sum of its first 10 terms is $\frac{m}{n}, \operatorname{gcd}(m, n)=1$, then $m+n$ is equal to $\_\_\_\_$ .

A certain gas is isothermally compressed to $\left(\frac{1}{3}\right)^{\mathrm{rd}}$ of its initial volume $\left(V_{\mathrm{o}}=3\right.$ litre) by applying required pressure. If the bulk modulus of the gas is $3 \times 10^5 \mathrm{~N} / \mathrm{m}^2$, the magnitude of work done on the gas is $\_\_\_\_$ J.

Consider the reaction $\mathrm{X} \rightleftharpoons \mathrm{Y}$ at 300 K . If $\Delta \mathrm{H}^\theta$ and K are $28.40 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and $1.8 \times 10^{-7}$ at the same temperature, then the magnitude of $\Delta \mathrm{S}^\theta$ for the reaction in $\mathrm{JK}^{-1} \mathrm{~mol}^{-1}$ is $\_\_\_\_$ . (Nearest integer)

(Given : $\mathrm{R}=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}, \ln 10=2.3, \log 3=0.48, \log 2=0.30$ )

Let $y=y(x)$ be the solution of the differential equation $\left(x^2-x \sqrt{x^2-1}\right) d y+\left(y\left(x-\sqrt{x^2-1}\right)-x\right) d x=0, x \geq 1$. If $y(1)=1$, then the greatest integer less than $y(\sqrt{5})$ is $\_\_\_\_$ .