Answer
$(a)\space 3.368\space km/s$
$(b)\space 7069.8\space s$
Work Step by Step
(a) Here we use the equation $V=\sqrt {\frac{GM_{E}}{r}}$ to find the speed of the Exomars. V - speed of the Exomars, $M_{M}$ - the mass of Mars, G - universal gravitational constant, r - radius of the orbit.
$V=\sqrt {\frac{GM_{M}}{r}}$; Let's plug known values into this equation
$V=\sqrt {\frac{6.67\times10^{-11}\space Nm^{2}/kg^{2}\times0.642\times10^{24}\space kg}{400\times10^{3}m+3.39\times10^{6}m}}$
$V=\sqrt {\frac{4.3\times10^{13}}{3.79\times10^{6}}}\space m/s= 3368.3 \space m/s=3.368\space km/s$
(b) Here we use the equation $T=\sqrt {\frac{4\pi^{2}r^{3}}{GM_{M}}}$ to find the orbital period.
$T=\sqrt {\frac{4\pi^{2}r^{3}}{GM_{M}}}$ ; Let's plug known values into this equation.
$T= 2\pi \sqrt {\frac{r^{3}}{GM_{M}}}$
$T=2\pi \sqrt {\frac{(3.79\times10^{6}m)^{3}}{6.67\times10^{-11}\space Nm^{2}/kg^{2}\times5.97\times10^{24}\space kg}}$
$
T= 2\pi\sqrt {\frac{54.44\times10^{18}m^{3}}{4.3\times10^{13}m^{3}/s^{2}}}$
$T=2\pi\times1125.2\space s=7069.8\space s$
Orbital period = 7069.8 s
