Given that $\vec{F}(x, y, z)=\sin (y) \hat{x}+\cos (x) \hat{y}+5 \hat{z}$, the integral $\iint_S \vec{F}(x, y, z) \cdot \overrightarrow{d s}$ over the unit sphere $S$ centered at the origin evaluates to $\_\_\_\_$ . (Round off to one decimal place)
A is an $m \times m$ skew-symmetric matrix with real-valued entries, and $x$ is an $m$-dimensional column vector with real-valued entries such that $x^T x=1$. The quantity $x^T A x$ evaluates to $\_\_\_\_$ . (Answer in integer)
Which one of the following statements is ALWAYS correct about a collection of $p$ column vectors, each having $n$ real-valued entries?
Consider the second-order differential equation
$$ \frac{d^2 y}{d x^2}+\frac{d y}{d x}+y=0 $$
with initial conditions
$$ y(0)=1,\left.\frac{d y}{d x}\right|_{x=0}=1 $$
The solution is given by
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