Assertion (A): When we change the unit of measurement of a quantity, its numerical value changes.
Reason (R): Smaller the unit of measurement smaller is its numerical value.
The physical magnitude is invariant, expressed as \(n u = \text{constant}\). Therefore, numerical value is inversely proportional to the unit size. A smaller unit leads to a larger numerical value, making R false.
Assertion (A): If the measuring instruments used are perfect, then measurements made will be perfect.
Reason (R): Measurements depend upon only on the instruments.
Even with perfect instruments, errors due to observation, environmental factors, or experimental setup can occur. Both statements are false.
Assertion (A): When an algebraic equation has been derived, it is advisable to check it for dimensional consistency.
Reason (R): This guarantees that the equation is correct.
A dimensionally consistent equation is not guaranteed to be physically correct, as dimensionless constants cannot be verified through dimensional analysis. Thus, R is false.
Assertion (A): eV and joule are the S.I. units of energy used in modern physics and mechanics respectively.
Reason (R): Different types of energies require different units in S.I. units.
The SI unit of all forms of energy is the joule. Electron-volt (eV) is not an SI unit, and different forms of energy do not require different SI units.
Assertion (A): Pressure and energy density have same units in SI.
Reason (R): Dimensions of energy density and pressure are same.
Pressure has dimensions of \([M L^{-1} T^{-2}]\). Energy density (energy per unit volume) also simplifies to \([M L^{-1} T^{-2}]\). Since their dimensions are identical, they share the same SI units.
Assertion (A): The dimensions of base (fundamental) quantity in other base quantities is always zero.
Reason (R): All derived quantities may be represented dimensionally in terms of fundamental quantities.
Base quantities are mutually independent and cannot be defined in terms of each other, making the exponent of one base quantity in another zero. Both statements are true, but R is not the explanation of A.
Assertion (A): A unitless quantity never has a non-zero dimension.
Reason (R): A dimensionless quantity never has a unit.
A quantity without a unit is always dimensionless, so A is true. However, a dimensionless quantity can have a unit (for example, plane angle has the unit radian), making R false.
Assertion (A): In SHM let \(x\) be the maximum speed, \(y\) the frequency of oscillation and \(z\) the maximum acceleration, then \(\frac{xy}{z}\) is a constant quantity.
Reason (R): This is because \(\frac{xy}{z}\) becomes a dimensionless quantity
For SHM, \(x=A\omega\), \(y=\frac{\omega}{2\pi}\), \(z=A\omega^2\). Thus, \(\frac{xy}{z} = \frac{(A\omega)(\omega/(2\pi))}{(A\omega^2)} = \frac{1}{2\pi}\), which is a constant. So (A) is true. The dimensions are \([x]=LT^{-1}\), \([y]=T^{-1}\), \([z]=LT^{-2}\), making \([xy/z]=1\), dimensionless. So (R) is true. However, being dimensionless does not explain why it's a constant.