$1.$ A solid rocket booster used for a low earth orbit mission has the following characteristics:

A throat diameter of $75$ mm and a combustion chamber volume of $0\mathrm{.}08$ m$3.$ Estimate the characteristic length of the rocket in meters.

$11.$ A rocket uses a conical nozzle with a semi$-$divergence angle of $15.$ Estimate the fractional loss of thrust due to the conical nozzle in percentage.

$2.$ Consider a Rotor$-$stator stage of an axial compressor given below:

Populate the following blanks with the appropriate relations i$.$e across stations

$1,2,3.$ Assume the flow to be adiabatic and non$-$isentropic$.$

$$ Absolute tangential velocity, c$\backslash$theta across stator: c$\backslash$theta $2\_\_\_$ c$\backslash$theta $3$

$$ Relative total temperature across rotor, T$0$R: T$0$R$1\_\_\_\_$T$0$R$2$

$$ Relative velocity, w$,$ across rotor: w$1\_\_\_$ w$2$

$$ Absolute total pressure across the rotor and stator: P$01\_\_\_$ P$02\_\_\_\_$ P$03$

$3.$ State True $/$ False: Characteristic velocity, C$^(*)=\backslash$sqrt$((1)/($r$)((\backslash$gamma $+1)/(2))^(($r$+1)/(\backslash$gamma $-1))((()/($bar$)($R$))$T$\_(02))/($M$)),$ strongly depends on the Nozzle characteristics

$4.$ For a given speed of the rotor of an axial compressor, a decrease in the axial velocity from the design value leads to flow separation on the $\_\_\_\_\_\_\_\_($suction$/$pressure side$)$ of the blade due to $\_\_\_\_\_\_\_\_($positive$/$negative$)$ blade incidence.

$5.$ Fill in the blanks using either of the following phrases: marginally fuel$-$rich$/$stoichiometric$/$marginally fuel$-$lean:

Chamber temperature, Tc$,$ is maximum for a $\_\_\_\_\_\_\_\_\_\_\_$ mixture while the specific

impulse, Isp, is maximum for a $\_\_\_\_\_\_\_\_\_\_\_\_$ mixture.

$6.$ Choose the correct answer:

In an axial turbine, larger discs and shorter blades are preferred because:

i$.$ Disc stresses are higher and hence larger discs are needed

ii$.$ Longer blades result in lower bending stresses over the blade

iii. Discs experience centrifugal stresses while blades experience bending, and

thermal stresses in addition to centrifugal stresses.

iv$.$ None of the above

$7.$ A reaction turbine is designed for a stage loading coefficient of $0\mathrm{.}5$ and a zero$-$exit swirl at the mean radius. Its degree of reaction at the mean radius, Rm$,$ is $\_\_\_\_\_\_\_\_\_\_$

$8.$ Fill in the blanks with under expansion$/$optimal expansion$/$over expansion:

A bell nozzle is designed such that the exhaust gases in the nozzle optimally expand at an altitude of $10$ km$.$ The nozzle operates in $\_\_\_\_\_\_\_\_\_$ mode at an altitude of $5$ km while it operates in $\_\_\_\_\_\_\_$ mode at an altitude of $15$ km$.$