If $f(x)=\left|\begin{array}{ccc}2 \cos ^2 x & \sin 2 x & \sin x \\ \sin 2 x & 2 \sin ^2 x & -\cos x \\ \sin x & -\cos x & 0\end{array}\right|$, then

$$ \left.\int_0^{\pi / 4}|2| f(x) \mid+5 f^{\prime}(x)\right) d x= $$

The number of arbitrary constants that appear in the general solution of the differential equation $\left(\frac{d^4 y}{d x^4}+\frac{d^2 y}{d x^2}\right)^{3 / 2}=5 \frac{d^3 y}{d x^3}$ is

Assertion (A) The degree of the differential equation $y^{\prime \prime}+2 x y^{\prime}+\log _e\left(\frac{d y}{d x}\right)=0$ is 2 .

Reason (R) The degree of a differential equation is the highest degree of the highest order derivative occurring in the equation, after the equation is expressed in the form of a polynomial in differential coefficients. The correct option among the following

Let $S$ be the family of curves given by the general solution of the differential equation $\frac{y^2 e^{-1 / y}}{\sqrt{x}} d x-2 \sec \sqrt{x} d y=0$. Then, the equation of the curve belonging to $S$ and passing through $\left(\pi^2, 1\right)$ is