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 [ • ] denote the greatest integer function. If the domain of the function $f(x)=\sin ^{-1}\left(\frac{x+[x]}{3}\right)$ is $[\alpha, \beta)$, then $\alpha^2+\beta^2$ is equal to:

Let one root of the quadratic equation in $x$ :

$$ \left(k^2-15 k+27\right) x^2+9(k-1) x+18=0 $$

be twice the other. Then the length of the latus rectum of the parabola $y^2=6 k x$ is equal to:

Let $e_1$ and $e_2$ be two distinct roots of the equation $x^2-a x+2=0$. Let the sets $\left\{a \in \mathbb{R}: e_1\right.$ and $e_2$ are the eccentricities of hyperbolas $\}=(\alpha, \beta)$, and $\left\{a \in \mathbb{R}: e_1\right.$ and $e_2$ are the eccentricities of an ellipse and a hyperbola, respectively $\}=(\gamma, \infty)$.

Then $\alpha^2+\beta^2+\gamma^2$ is equal to: