Units & measurements practice · Medium

The correct answer is C (Assertion is correct, reason is incorrect) because Avogadro's number has units of mol^-1, making the reason incorrect.

Option A ($M{L^3}{T^{ - 1}}{Q^{ - 2}}$) is correct because it is derived from the definition of resistivity using Ohm's law and the dimensions of resistance, area, length, and charge.

The units of eV/T are the same as Boltzmann's constant because eV represents energy and T represents temperature, so eV/T is energy per temperature.

Option B ($\left( {\alpha a + \beta b + \gamma c} \right)\%$) is correct because the percentage error is the sum of the absolute values of the errors multiplied by their respective powers.

Option D (Pressure gradient) is correct because the dimensional formula of pressure gradient is force per unit volume per unit length, which matches the given formula.

Option A (${10^7}:1$) is correct because the SI unit of Planck's constant is $Js$ and the CGS unit is $erg \cdot s$, and $1 J = 10^7 erg$.

Option A (Taking large number of observations) is correct because averaging multiple measurements reduces the impact of random fluctuations.

Option C is correct. Angular momentum and Planck's constant have the same dimensions, while moment of force (torque) and linear momentum do not.

Option D ($\left( {10 \pm 0.1} \right)\,m{s^{ - 1}}$) is the closest to the correct answer because it correctly calculates the velocity and its associated uncertainty using error propagation.

Option A ($\mathrm{M}^1 \mathrm{L}^2 \mathrm{T}^{-1}$) is correct because angular impulse is the change in angular momentum, which has the dimensions of mass * velocity * radius.

Option C ($\left[ {M{L^{ - 1}}{T^{ - 2}}} \right]$) is correct because pressure is force per unit area, and its dimensions are derived accordingly.

Errors and mistakes in measurement are primarily due to defects in the measurement process itself, such as parallax error or incorrect reading of the instrument.

Option A (If both assertion and reason are true and reason is the correct explanation of assertion) is correct because the parallax method is limited to nearby stars due to the decreasing parallactic angle with increasing distance, making accurate measurement difficult.

Option B is correct because the least count of a spherometer is calculated by dividing the pitch by the number of divisions on the circular scale, which in this case is 0.01 cm / 100 = 10^-4 cm.

Option D (Dipole moment, electrical flux, electrical field) is correct because these quantities have different dimensional formulas compared to the others.

Option D ($\left[ {\mathrm{M}{\mathrm{L}^{ - 1}}{\mathrm{T}^{ - 2}}} \right]$) is correct because modulus of rigidity is defined as shear stress divided by shear strain, and its dimensional formula is derived accordingly.

Option C ($v$) is correct because the potential energy, and therefore the maximum kinetic energy and speed, is independent of the radius R.

Option A is correct because magnetic moment is defined as current times area, so the unit is Ampere-meter squared.

Option D (A screw gauge of least count $0.001 {~mm}$) is correct because the instrument with the smallest least count provides the most accurate measurement.

Option B is correct because dimensional analysis can only derive relations with up to three unknown variables.

Option C is correct because one-watt hour is equal to 3600 joules, which is $3.6 \times 10^3$ J.

Option D (Area) is correct because for the equation to be dimensionally consistent, B must have the dimensions of wavelength squared, which is area.

Option A ($3.9 \times 10^4$) is correct because the product is rounded to the least number of significant figures in the given quantities.

Option D (Least count of instrument) is correct because precision in measurement is directly related to the least count of the measuring instrument.

Option A ($AT$) is correct because charge is the product of current and time, so its dimensions are [A][T].

Option B is correct. The term $\frac{a}{V^2}$ has the dimensions of pressure, not energy, making it dimensionally different from PV, bP, and $\frac{ab}{V^2}$ which all have dimensions of energy.

Option C is correct because the dimensions of magnetic field intensity are derived from the force on a current-carrying conductor.

Option C is correct because spring constant and surface energy have the same dimensions of [ML⁰T⁻²].

Option A ($[M{L^2}{T^{ - \mathrm{2}}}]$) is correct because the dimensions of $Li^2$ represent energy, which has dimensions of mass times length squared divided by time squared.

Option B ($21.76 \times {10^{ - 19}}J$) is correct because it's the result of converting the binding energy from electron-volts to joules using the given conversion factor.