Consider the function $f : (0, \infty) \to (-\infty, \infty)$ given by

$f(x) = \sqrt{x} \log_e(x) - x + 1$.

Then which one of the following statements is TRUE?

Let $P$ be the point on the parabola $y = x^2$ such that the slope of the tangent to the parabola at the point $P$ is $4$. Let $Q$ be the point in the first quadrant lying on the circle $x^2 + y^2 = 2$ such that the slope of the tangent to the circle at the point $Q$ is $-1$. Let $R$ be the point in the first quadrant lying on the ellipse $x^2 + 4y^2 = 8$ such that the slope of the tangent to the ellipse at the point $R$ is $-\frac{1}{2}$. Then the radius of the circle passing through the points $P, Q$ and $R$ is

Which one of the following matrices can be obtained by performing elementary row transformations on the $3 \times 3$ identity matrix?

Considering only the principal values of the inverse trigonometric functions, the value of

$$\cot^{-1}(\cot(-11)) + 10 \sin\left(2 \cos^{-1}\left(\frac{1}{\sqrt{2}}\right)\right) + 10\sin(2 \tan^{-1}(2))$$

is