JEE Main 2025 (22 Jan Shift 1)

The position vectors of two 1 kg particles, (A) and (B), are given by $\vec r_A=(\alpha_1t^2\hat i+\alpha_2t\hat j+\alpha_3t\hat k)$ m and $\vec r_B=(\beta_1t\hat i+\beta_2t^2\hat j+\beta_3t\hat k)$ m, respectively; ($\alpha_1=1$ m/s², $\alpha_2=3n$ m/s, $\alpha_3=2$ m/s, $\beta_1=2$ m/s, $\beta_2=-1$ m/s², $\beta_3=4p$ m/s), where $t$ is time, $n$ and $p$ are constants. At $t=1$ s, $|\vec V_A|=|\vec V_B|$ and velocities $\vec V_A$ and $\vec V_B$ of the particles are orthogonal to each other. At $t=1$ s, the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is $\sqrt L$ kg m² s⁻¹. The value of L is _______.

A particle is projected at an angle of $30^\circ$ from horizontal at a speed of 60 m/s. The height traversed by the particle in the first second is $h_0$ and height traversed in the last second, before it reaches the maximum height, is $h_1$. The ratio $h_0:h_1$ is _______ [Take, $g=10$ m/s²]

The number of molecules/ions that show linear geometry among the following is ________ SO₂, BeCl₂, CO₂, N₃⁻, NO₂, F₂O, XeF₂, NO₂⁺, I₃⁻, O₃

A $\to$ B. The molecule A changes into its isomeric form B by following a first order kinetics at a temperature of 1000 K. If the energy barrier with respect to reactant energy for such isomeric transformation is 191.48 kJ mol⁻¹ and the frequency factor is $10^{20}$, the time required for 50% molecules of A to become B is _______ picoseconds (nearest integer). [R = 8.314 J K⁻¹ mol⁻¹]

Consider the following sequence of reactions : C₆H₅NO₂ —(i) Sn + HCl; (ii) NaNO₂, HCl, 0°C; (iii) Cu₂Cl₂; (iv) Na, Ether→ A (Product) Molar mass of the product formed (A) is _______ g mol⁻¹.

Some CO₂ gas was kept in a sealed container at a pressure of 1 atm and at 273 K. This entire amount of CO₂ gas was later passed through an aqueous solution of Ca(OH)₂. The excess unreacted Ca(OH)₂ was later neutralized with 0.1 M of 40 mL HCl. If the volume of the sealed container of CO₂ was x, then x is _______ cm³ (nearest integer). [Given : The entire amount of CO₂(g) reacted with exactly half of the initial amount of Ca(OH)₂ present in the aqueous solution.]

In Carius method for estimation of halogens, 180 mg of an organic compound produced 143.5 mg of AgCl. The percentage composition of chlorine in the compound is _______ %. (Given : molar mass in g mol⁻¹ of Ag : 108, Cl : 35.5)