What is the accuracy of an atomic clock?

$(a)$ How many astronomical units $(AU)$ make $1$ parsec?
$(b)$ Consider a sun-like star at a distance of $2$ parsecs. When it is seen through a telescope with $100$ magnification,what should be the angular size of the star? The Sun appears to be $(\frac{1}{2})^{\circ}$ from the Earth. Due to atmospheric fluctuations,the eye cannot resolve objects smaller than $1$ arc minute.
$(c)$ Mars has approximately half of the Earth's diameter. When it is closest to the Earth,it is at about $\frac{1}{2} AU$ from the Earth. Calculate what size it will appear when seen through the same telescope.

What is parallax? Explain the parallax method to measure the distance between the Earth and a planet.

Match List-$I$ (Event) with List-$II$ (Order of the time interval for happening of the event) and select the correct option from the options given below the lists:

List-$I$	List-$II$
$(1)$ Rotation period of earth	$(i)$ $10^5\, s$
$(2)$ Revolution period of earth	$(ii)$ $10^7\, s$
$(3)$ Period of light wave	$(iii)$ $10^{-15}\, s$
$(4)$ Period of sound wave	$(iv)$ $10^{-3}\, s$

$(a)$ The earth-moon distance is about $60$ earth radii. What will be the angular diameter of the earth (approximately in degrees) as seen from the moon?
$(b)$ The moon is seen to have an angular diameter of $(1/2)^{\circ}$ from the earth. What is its relative size compared to the earth?
$(c)$ From parallax measurement,the sun is found to be at a distance of about $400$ times the earth-moon distance. Estimate the ratio of sun-earth diameters.

$A$ tachometer is a device used to measure: