JEE Main 2026 (Online) 5th April Evening Shift

When an external resistance of $5 \Omega$ is connected across terminals of a cell, a current of 0.25 A flows through it. When the $5 \Omega$ resistor is replaced by a $2 \Omega$ resistor, a current of 0.5 A flows through it. The internal resistance of the cell is $\_\_\_\_$ $\Omega$.

At 298 K , the molar conductivity of $x \%(\mathrm{w} / \mathrm{w}) \mathrm{MX}$ solution (aqueous) is $123.5 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}$. The conductance of same solution is $1.9 \times 10^{-3} \mathrm{~S}$. The value of $x$ is $\_\_\_\_$ $\times 10^{-2}$.

(Given: cell constant $=1.3 \mathrm{~cm}^{-1}$; molar mass of MX is $75 \mathrm{~g} \mathrm{~mol}^{-1}$, density of aqueous solution of MX at 298 K is $1.0 \mathrm{~g} \mathrm{~mL}^{-1}$ )

From the point $(-1,-1)$, two rays are sent making angles of $45^{\circ}$ with the line $x+y=0$. These rays get reflected from the mirror $x+2 y=1$. If the equations of the reflected rays are $\mathrm{a} x+\mathrm{b} y=9$ and $c x+d y=7, a, b, c, d \in \mathbf{Z}$, then the value of $a d+b c$ is $\_\_\_\_$ .

The heat extracted out of $x$ gram of water initially at $50^{\circ} \mathrm{C}$ to $\operatorname{cool}$ it down to $0^{\circ} \mathrm{C}$ is sufficient to evaporate $(1000-x)$ gram of water also initially at $50^{\circ} \mathrm{C}$. The value of $x$ (closest integer) is $\_\_\_\_$ .

(Take latent heat of water $2256 \mathrm{~kJ} / \mathrm{kg} . \mathrm{K}$, specific heat capacity of water $4200 \mathrm{~J} / \mathrm{kg} . \mathrm{K}$ )

The total number of unpaired electrons present in the $d^3, d^4$ (low spin) $d^5$ (high spin), $\mathrm{d}^6$ (high spin) and $\mathrm{d}^7$ (low spin) octahedral complex systems is $\_\_\_\_$ .

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function such that $f(x)+3 f\left(\frac{\pi}{2}-x\right)=\sin x, x \in \mathbf{R}$. Let the maximum value of $f$ on $\mathbf{R}$ be $\alpha$. If the area of the region bounded by the curves $g(x)=x^2$ and $h(x)=\beta x^3, \beta>0$, is $\alpha^2$, then $30 \beta^3$ is equal to $\_\_\_\_$ .