The ${\mathrm{K}}_{\mathrm{sp}}$ of ${\text{Ag}}_2{\text{CrO}}_4$ is $1.1\times 10^{-12}$ at $298 \mathrm{K}$. The solubility (in $\mathrm{mol} {\mathrm{L}}^{-1}$) of ${\text{Ag}}_2\mathrm{CrO}{}_4$ in a $0.1 \mathrm{M} {\text{AgNO}}_3$ solution is

Statement 1 The plot of atomic number ( \(Y\)-axis) versus number of neutrons ( \(x\)-axis) for stable nuclei shows a curvature towards \(X\)-axis from the line of \(45^{\circ}\) slope as the atomic number is increased.
Statement 2 Proton-proton electrostatic repulsions begin to overcome attractive forces involving protons and neutrons in heavier nuclides.

For the process ${\mathrm{H}}_2\mathrm{O}\mathrm{ }\left(\mathcal{l}\right)\to {\mathrm{H}}_2\mathrm{O}\left(\mathrm{g}\right)\mathrm{ }$ at $\mathrm{T}={100}^{\mathrm{o}}\mathrm{C}\mathrm{ }$ and 1 atmosphere pressure, the correct choice is

Statement I Micelles are formed by surfactant molecules above the critical micellar concentration (CMC).
Statement II The conductivity of a solution having surfactant molecules deceases sharply at the CMC.

Among the following complexes \((K-P), \mathrm{K}_3\left[\mathrm{Fe}(\mathrm{CN})_6\right](K)\),
\(\left[\right.\) Co \(\left.\left(\mathrm{NH}_3\right)_6\right] \mathrm{Cl}_3(L), \mathrm{Na}_3\left[\mathrm{Co}(\mathrm{ox})_3\right](M)\),
\(\left[\mathrm{Ni}\left(\mathrm{H}_2 \mathrm{O}\right)_6\right] \mathrm{Cl}_2(\mathrm{~N}) ; \mathrm{K}_2\left[\mathrm{Pt}(\mathrm{CN})_4\right](\mathrm{O})\) and \(\left[\mathrm{Zn}\left(\mathrm{H}_2 \mathrm{O}\right)_6\right]\left(\mathrm{NO}_3\right)_2(P)\)
the diamagnetic complexes are

At \(298 \mathrm{~K}\), the limiting molar conductivity of a weak monobasic acid is \(4 \times 10^{2} \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\). At \(298 \mathrm{~K}\), for an aqueous solution of the acid the degree of dissociation is \(\boldsymbol{\alpha}\) and the molar conductivity is \(\mathbf{y} \times 10^{2} \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\). At \(298 \mathrm{~K}\), upon 20 times dilution with water, the molar conductivity of the solution becomes \(3 \mathbf{y} \times 10^{2} \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\)
The value of $\mathrm{\alpha }$ is _______ .

In an experiment, m grams of a compound X (gas/liquid/solid) taken in a container is loaded in a balance as shown in figure I below. In the presence of a magnetic field, the pan with X is either deflected upwards (figure II), or deflected downwards (figure III), depending on the compound X. Identify the correct statement(s).

Consider three sets $E_1=\{1,2,3\},F_1=\{1,3,4\}$ and $G_1=\{2,3,4,5\}$. Two elements are chosen at random, without replacement, from the set $E_1$, and let $S_1$ denote the set of these chosen elements. Let $E_2=E_1-S_1$ and $F_2=F_1\cup S_1$. Now two elements are chosen at random, without replacement, from the set $F_2$ and let $S_2$ denote the set of these chosen elements. Let $G_2=G_1\cup S_2$. Finally, two elements are chosen at random, without replacement, from the set $G_2$ and let $S_3$ denote the set of these chosen elements.
Let $E_3=E_2\cup S_3$. Given that $E_1=E_3$, let $p$ be the conditional probability of the event $S_1=\{1,2\}$. Then the value of $p$ is

Let $f :\left(-\frac{\pi }{2},\frac{\pi }{2}\right)\to R$ be given by $f\left(x\right)={\left(\log \left(\sec x+\tan x\right)\right)}^3.$ Then

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Consider the functions defined implicitly by the equation \(y^3-3 y+x=0\) on various intervals in the real line. If \(x \in(-\infty,-2) \cup(2, \infty)\), the equation implicitly defines a unique real valued differentiable function \(y=f(x)\). If \(x \in(-2,2)\), the equation implicitly defines a unique real valued differentiable function \(y=g(x)\), satisfying \(g(0)=0\).Question:
If \(f(-10 \sqrt{2})=2 \sqrt{2}\), then \(f^{\prime \prime}(-10 \sqrt{2})\) is equal to

In a radioactive decay process, the activity is defined as $A=-\frac{dN}{dt}$, where $N\left(t\right)$ is the number of radioactive nuclei at time $\left(t\right)$. Two radioactive sources, $S_1$ and $S_2$ have same activity at time $t=0$. At a later time, the activities of $S_1$ and $S_2$ are $A_1$ and $A_2$, respectively. When $S_1$ and $S_2$ have just completed their $3^{\mathrm{rd}}$ and $7^{\mathrm{th}}$ half-lives, respectively, the ratio $\frac{A_1}{A_2}$ is _____.

Atom $\mathrm{X}$ occupies the fcc lattice sites as well as alternate tetrahedral voids of the same lattice. The packing efficiency (in $\%$) of the resultant solid is closest to

Consider a system of three connected strings, \(S_1, S_2\) and \(S_3\) with uniform linear mass densities \(\mu \mathrm{kg} / \mathrm{m}\), \(4 \mu \mathrm{~kg} / \mathrm{m}\) and \(16 \mu \mathrm{~kg} / \mathrm{m}\), respectively, as shown in the figure. \(S_1\) and \(S_2\) are connected at the point \(P\), whereas \(S_2\) and \(S_3\) are connected at the point \(Q\), and the other end of \(S_3\) is connected to a wall. A wave generator O is connected to the free end of \(S_1\). The wave from the generator is represented by \(y=y_0\) \(\cos (\omega t-k x) \mathrm{cm}\), where \(y_0, \omega\) and \(k\) are constants of appropriate dimensions. Which of the following statements is/are correct: