1. Given a matrix \(M= \left[ \begin{array}{cc} 2 & 1\\ 1 & 2 \end{array} \right]\), find \(\cos \left(\frac{\pi M}{6}\right)\)?

2. A cone-shaped surface lamina \(S\) is given by \[z=4-2\sqrt{x^2+y^2},\quad 0\leq z\geq 4\] At each point on S, the density is proportional to the distance between the point and the z-axis. Find the mass \(m\) of the lamina?

1. A satellite is moving in a circular orbit around the Earth. If \(T,V \text{ and } E\) are its average kinetic, average potential and total energies, respectively, then how is \(V\) and \(E\) related to \(T\)?
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1. Consider a single heat source located at the origin with temperature \[T(x,y,z)=\frac{25}{\sqrt{x^2+y^2+z^2}}\] calculate the heat flux across the surface \[S=\left\{(x,y,z):z=\sqrt{1-x^2-y^2},x^2+y^2\leq 1\right\}\]
2. For an enthalpy change of \(2257.03 \;kJ/kg\), calculate the entropy change along a \(T = 100^{\circ} C\) isotherm for water going from saturated liquid to saturated vapor.

1. A 1-D SHO with hamiltonian \(H_o=\frac{p^2}{2m}+\frac{kx^2}{2}\) is subjected to a small perturbation, \(H_1=\alpha x+\beta x^3+\gamma x^4\). The first order correction to the ground state energy is dependent on.
(A) only \(\beta\) (B) \(\alpha\) and \(\gamma\) (C) \(\alpha\) and \(\beta\) (D) only \(\gamma\)

2. A particle is confined to a 1-D potential box with the potential \[\begin{aligned}V(x)&=0,\quad 0\leq x \leq a\\&=\infty, \quad \text{otherwise} \end{aligned}\] For a small perturbation, \(W=\beta x\), within the box, find the first order correction to the ground state energy?

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