JEE (Advanced) 2020 Paper-2

A solution of 0.1 M weak base (B) is titrated with 0.1 M of a strong acid (HA). The variation of pH of the solution with the volume of HA added is shown in the figure below. What is the $$p𝐾_b$$ of the base? The neutralization reaction is given by $$B + HA \rightarrow BH^+ + A^-$$.

Liquids A and B form ideal solution for all compositions of A and B at 25 $$^\circ C$$. Two such solutions with 0.25 and 0.50 mole fractions of A have the total vapor pressures of 0.3 and 0.4 bar, respectively. What is the vapor pressure of pure liquid B in bar?

The figure below is the plot of potential energy versus internuclear distance (𝑑) of $$H_2$$ molecule in the electronic ground state. What is the value of the net potential energy 𝐸0 (as indicated in the figure) in kJ mol$$^{−1}$$, for $$𝑑 = 𝑑_0$$ at which the electron-electron repulsion and the nucleus-nucleus repulsion energies are absent? As reference, the potential energy of H atom is taken as zero when its electron and the nucleus are infinitely far apart.
Use Avogadro constant as $$6.023 \times 10^{23}$$ mol$$^{−1}$$.

Consider the reaction sequence from P to Q shown below. The overall yield of the major product Q from P is 75%. What is the amount in grams of Q obtained from 9.3 mL of P? (Use density of P = 1.00 g mL$$^{−1}$$; Molar mass of C = 12.0, H =1.0, O =16.0 and N = 14.0 g mol$$^{−1}$$)

Tin is obtained from cassiterite by reduction with coke. Use the data given below to determine the minimum temperature (in K) at which the reduction of cassiterite by coke would take place.
At $$298 K: \triangle_fH^0(SnO_2(s)) = -581.0$$ KJ mol$$^{−1}$$, $$\triangle_fH^0 (CO_2(g)) = -394.0$$ KJ mol$$^{−1}$$, $$S^0(SnO_2(s)) = 56.0 J K^{-1} mol^{-1}$$, $$S^0(Sn(s)) = 52.0 J K^{-1} mol^{-1}$$ , $$S^0(C(s)) = 6.0 J K^{-1} mol^{-1}$$, $$S^0(CO_2(g)) = 210.0 J K^{-1} mol^{-1}$$.
Assume that the enthalpies and the entropies are temperature independent.

An acidified solution of 0.05 M $$Zn^{2+}$$ is saturated with 0.1 M $$H_2S$$. What is the minimum molar concentration (M) of $$H^+$$ required to prevent the precipitation of ZnS?
Use $$K_{sp}(ZnS) = 1.25 \times 10^{-22}$$ and overall dissociation constant of $$H_2S, K_{NET} = K_1K_2 = 1 \times 10^{-21}$$.

For a complex number z, let Re(z) denote the real part of z. Let 𝑆 be the set of all complex numbers z satisfying $$z^4 − \mid z\mid^4 = 4 𝑖 z^2$$, where $$i = \sqrt{−1}$$. Then the minimum possible value of $$\mid z_1 − z_2\mid^2$$, where $$z_1, z_2 \in S$$ with $$Re(z_1) > 0$$ and $$Re(z_2) < 0$$, is _____

The probability that a missile hits a target successfully is 0.75. In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than 0.95, is _____

Let $$O$$ be the centre of the circle $$𝑥^2 + 𝑦^2 = 𝑟^2$$, where $$𝑟 > \frac{\sqrt{5}}{2}$$. Suppose PQ is a chord of this circle and the equation of the line passing through P and Q is $$2𝑥 + 4𝑦 = 5$$. If the centre of the circumcircle of the triangle $$OPQ$$ lies on the line $$𝑥 + 2𝑦 = 4$$, then the value of 𝑟 is _____

The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a $$2 \times 2$$ matrix such that the trace of A is 3 and the trace of $$A^3$$ is −18, then the value of the determinant of A is _____