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\begin{document}







\problem
Find the Laplace transform of $g(t):= \left[ \begin{array}{ll} 3t & 0\leq t\leq 1 \\ 3 & 1\leq t \end{array} \right. $ 

\correct
$(3-3e^{-s})s^{-2}$
\wrong
$\frac{3}{s^{2}}+\frac{3}{s}$
\wrong
$e^{-3s}s^{-2}$
\wrong
$(1-e^{-3s})s^{-2}$
\wrong
$\frac{3+s}{s^{2}}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \problem Which of the following numbers is a root of $z^{6}+1=0$. \correct $\frac{1}{2}(\sqrt{3}-i)$
\wrong $-1$
\wrong $e^{\smallfrac{2\pi i}{6}}$
\wrong $1+i$
\wrong $\frac{\sqrt{2}}{2}(1-i)$







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \problem
Let $y(x) = u_1(x)\cos(x) + u_2(x)\sin(x)$ be a particular solution of $$y'' + y = \tan(x),\qquad 0 < x < {\pi\over2}$$ given by the method of variation of parameters. Find $u_2$. \correct
{$u_2 = -\cos(x)$}
\wrong
{$u_2 = \sin(x)$}
\wrong
{$u_2 = \sec^2(x)$}
\wrong
{$u_2 = -\sec(x)$}
\wrong
{$u_2 = \cot(x)$}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \problem
Find the Laplace transform of $f(t)=te^{t}.$ 

\correct $F(s)=\frac{1}{(s-1)^{2}}$
\wrong $F(s)=\frac{1-s}{s^2}$
\wrong $F(s)=\frac{1}{s^{2}}$
\wrong $F(s)=s+e^{-s}$
\wrong $F(s)={s^{-2}}+e^{-s}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 



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