The tangent to the circle $x^2 + y^2 = 10$ at the point $(3,1)$ touches the circle $x^2 + y^2 - 2\sqrt{10}\,x - 20y + k = 0$, then the value of $k$ is...

If the eccentricity of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1, (a > b)$ is $\dfrac{2}{3}$ and its focal chord is $3x + 2y - 6 = 0$, then the value of $a^2 + b^2$ is...

If $\lim\limits_{x \to \infty}\left[\dfrac{x^2+x+1}{x+1} - ax - b\right] = 3$, then $a - b = $...

The correct logical equivalence from the following is /are ___
(I) $p \to (q \to r) \equiv (p \wedge q) \to r$
(II) $(p \to q) \to r \equiv p \to (q \vee r)$
(III) $(p \to q) \to r \equiv (p \to r) \wedge (\sim q \to r)$
(IV) $p \to (q \to r) \equiv q \to (p \to r)$