Units & measurements practice · Mixed difficulty

Option A (SI) is correct because the SI system includes units beyond mass, length, and time, such as electric current (ampere), temperature (kelvin), luminous intensity (candela), and amount of substance (mole).

Option D (If both Assertion & Reason are false) is correct because unit vectors are dimensionless and the product of a scalar and a vector is a vector.

Option A ($\left[{M}^{-1} {L}^3 {T}^{-2}\right]$) is correct because it is derived from the formula for gravitational force and isolating the gravitational constant G.

Option B is correct because a femtometer is defined as 10 to the power of -15 meters.

Option D (${10^{ - 9}}\mathrm{\;s\;of\;Cs} - \mathrm{clock\;of\;}9192631770\mathrm{\;oscillations\;}$) is correct because it accurately defines a nanosecond based on Cesium-133 oscillations.

Option D ($kg/m - s$) is correct because the SI unit of the coefficient of viscosity is kilogram per meter-second (kg/m·s) or Pascal-second (Pa·s).

Option D (Dipole moment, electric flux, electric field) has different dimensions because each quantity has a unique dimensional formula.

Option D (0.0812) is correct because the value of R is converted from J/K-mol to atm L/K-mol using the appropriate conversion factor.

Option B ($1.4\, \mathrm{kWm}^{-2}$) is correct because it is the solar constant value after converting from $\mathrm{cal} / \mathrm{cm}^2 / \mathrm{min}$ to S.I. units.

Option A (Faulty construction of instrument) is correct because instrumental errors arise from inherent limitations or defects in the instrument itself, such as faulty construction.

The correct answer is 4 % because the radius and diameter are directly proportional, so the percentage error is the same.

Option A (Sec) is correct because the ratio of inductance (L) to resistance (R) has the dimensions of time, hence the unit is seconds.

Option A (1.67) is correct because it is the percentage change in length calculated using the bulk modulus and the relationship between volume and length for a cube.

Option A (500) is correct because converting 0.5 gm/cc to MKS units (kg/m³) results in 500 kg/m³.

Option D ($5.5 \%$) is correct because the percentage error is calculated by summing the absolute values of the percentage errors multiplied by their respective powers.

Option B ($0.003 \mathrm{~m}^2$) is correct because it has only one significant figure, while the others have more.

The correct answer is C (If assertion is true but reason is false) because while the phase (ωt-kx) is dimensionless, the dimension of angular frequency ω is [T^-1], not [M⁰ L⁰ T⁰].

Option A (Momentum) is correct because both impulse and momentum have the same dimensional formula: [MLT^-1].

Option B (Units but no dimensions) is correct because plane and solid angles are dimensionless quantities but have units (radians and steradians, respectively).

Option C ($\mathrm{Kg}\,{\mathrm{C}^{ - 1}}$) is correct because the electrochemical equivalent (Z) is defined as the mass of the substance deposited per unit charge, so its unit is kg/C or Kg C⁻¹.

Option C is correct because it is the only equation that is dimensionally consistent, with all terms having the dimension of length [L].

Option B (2) is correct because leading zeros are not significant, but trailing zeros after the decimal point are significant.

Option B (Moment of inertia and force) is correct because moment of inertia has dimensions of $ML^2$ while force has dimensions of $MLT^{-2}$, which are not identical.

Option D ($\frac{{3 \pi}}{{2}}$) is correct because 270 degrees is equal to $\frac{{3 \pi}}{{2}}$ radians.

Option B ($5 \alpha^{-1} \beta^{-2} \gamma^2$) is correct because it correctly converts the units of energy from the SI system to the new system.

Option C (Statement-I is false and statement-II is true) is incorrect. Both statements are true. The product of numerical value and unit is constant, and the magnitude of a physical quantity remains the same regardless of the system of units.

The correct answer is $\mathrm{LT}{ }^{-2}$ because $\frac{d^2 y}{d x^2}$ represents the second derivative of distance with respect to time, which is acceleration.

Option D (Gravitational constant) is correct because its dimensional formula includes a negative dimension of mass, specifically [M⁻¹L³T⁻²].

Option C is correct because the dimensions of mutual inductance are [ML²T⁻²A⁻²] and the dimensions of capacitance are [M⁻¹L⁻²T⁴A²].

Option A (${{\mathrm{M}^0}{\mathrm{L}^2}{\mathrm{T}^0}}$) is correct because it is the dimensional formula derived by equating dimensions in the given relation.