1. Evaluate the definite integral: \(\int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx\)
2. If \(f(x) = \begin{cases} x^2 \sin(1/x), & x \neq 0 \\ 0, & x = 0 \end{cases}\), then at \(x = 0\), \(f(x)\) is:
3. Let \(\vec{a}, \vec{b}, \vec{c}\) be three non-coplanar unit vectors such that the angle between any two is \(\frac{\pi}{3}\). If \(\vec{a} \times \vec{b} + \vec{b} \times \vec{c} = p\vec{a} + q\vec{b} + r\vec{c}\), find \(p^2 + q^2 + r^2\).
4. The value of \(\lim_{n \to \infty} \sum_{k=1}^n \frac{n}{n^2 + k^2}\) is:
5. If \(\omega\) is a complex cube root of unity, then the value of \(\det \begin{pmatrix} 1 & \omega & \omega^2 \\ \omega & \omega^2 & 1 \\ \omega^2 & 1 & \omega \end{pmatrix}\) is: