State whether the following statements are true or false:
$(a)$ $A$ physical quantity can have units but still be dimensionless.
$(b)$ Impulse and the gradient of energy have the same units.
$(c)$ The absolute error in every measurement is equal to the least count of the measuring instrument.

The coefficients of thermal expansion of steel and a metal $X$ are respectively $12 \times 10^{-6} /^{\circ}C$ and $2 \times 10^{-6} /^{\circ}C$. At $40^{\circ}C$,the side of a cube of metal $X$ was measured using a steel vernier callipers. The reading was $100 \, mm$. Assuming that the calibration of the vernier was done at $0^{\circ}C$,then the actual length of the side of the cube at $0^{\circ}C$ will be:

The length of a cylinder is measured with a meter rod having a least count of $0.1 \, cm$. Its diameter is measured with vernier calipers having a least count of $0.01 \, cm$. Given that the length is $5.0 \, cm$ and the radius is $2.0 \, cm$,the percentage error in the calculated value of the volume will be ......... $\%$

State whether the following statements are true or false:
$(a)$ $A$ dimensionally incorrect equation can be correct.
$(b)$ $1 \ AU = 9.46 \times 10^{15} \ m$
$(c)$ The dimensional formula of force and stress are the same.
$(d)$ The dimensions of a physical quantity are different in different systems of units.

Fill in the blanks:
$(a)$ To measure the direct distance of a mountain,....... is used.
$(b)$ For the measurement of time,the ........... clock is the most accurate.
$(c)$ Rounding off $15.753$ to three significant figures gives ....... .

Match List-$I$ with List-$II$.

List-$I$	List-$II$
$(a)$ ${R}_{H}$ (Rydberg constant)	$(i)$ ${kg} {m}^{-1} {s}^{-1}$
$(b)$ $h$ (Planck's constant)	$(ii)$ ${kg} {m}^{2} {s}^{-1}$
$(c)$ $u_{B}$ (Magnetic field energy density)	$(iii)$ ${m}^{-1}$
$(d)$ $\eta$ (Coefficient of viscosity)	$(iv)$ ${kg} {m}^{-1} {s}^{-2}$

Choose the most appropriate answer from the options given below: