Assertion (A): Light year and wavelength have same dimensions.
Reason (R): Light year represent time while wavelength represent distance.
Both light year and wavelength represent distance and have the dimension \([L]\). Since light year represents distance and not time, R is false.
Assertion (A): A displacement can be added with a distance.
Reason (R): Adding a scalar to a vector of the same dimensions is a meaningful algebraic operation.
Displacement is a vector quantity while distance is a scalar quantity. Scalars and vectors cannot be added directly even if they have the same dimensions, making both statements false.
Assertion (A): Angle and strain are dimensionless.
Reason (R): Angle and strain have no unit.
Both angle and strain are ratios of like quantities and are dimensionless. However, angle has a unit (radian), so R is false.
Assertion (A): Mass, length and time may be taken as fundamental quantities.
Reason (R): Mass, length and time are independent of one another.
Mass, length, and time are chosen as fundamental quantities because they are independent of each other and cannot be defined in terms of one another.
Assertion (A): If \(vec{r}\) is the position vector then dimensions of \(\frac{d^2vec{r}}{dt^2}\) is \([M^0L^1T^{-2}]\).
Reason (R): Dimensions of \(int \left(\frac{d^2vec{r}}{dt^2}\right) dt\) is \([M^0L^1T^{-1}]\) where \(\vec{r} \rightarrow\) position vector, \(t \rightarrow\) time.
The second derivative of position with respect to time represents acceleration with dimensions \([L T^{-2}]\). Integrating this acceleration with respect to time yields velocity with dimensions \([L T^{-1}]\). Both statements are true, but R is not the explanation of A.
Assertion (A): The error in measurement of radius of the sphere is 0.3%. The permissible error in its surface area is 1.2%.
Reason (R): Area of sphere, \(A = 4\pi r^2 \Rightarrow \frac{\Delta A}{A} = 4\frac{\Delta r}{r}\).
The surface area of a sphere is \(A = 4\pi r^2\), which gives the fractional error relation as \(\frac{\Delta A}{A} = 2\frac{\Delta r}{r}\). Thus, the error in area is \(2 \times 0.3% = 0.6%\), making both statements false.
Assertion (A): Mean absolute error of a measurement is always positive.
Reason (R): Mean absolute error is defined as the magnitude of difference between true value and measured value.
Mean absolute error is the arithmetic mean of all absolute errors and is always positive. The definition given in the reason describes individual absolute error rather than the mean absolute error, so R is false.
Assertion (A): In mechanics the method of dimensions can’t be applied to derive formula of a physical quantity which depends on more than three physical quantities.
Reason (R): We can derive relation of a physical quantity with other physical quantities out of which two have same dimensions.
In mechanics, we have only three base dimensions (M, L, T). Thus, we cannot determine more than three independent exponents. If two quantities have the same dimensions, they cannot be resolved independently, making R false.
Assertion (A): Only like quantities can be added or subtracted from each other.
Reason (R): Velocity can be subtracted from the velocity gradient.
By the principle of homogeneity, only physical quantities with identical dimensions can be added or subtracted. Velocity and velocity gradient have different dimensions, hence they cannot be subtracted.
Assertion (A): If a physical quantity has a unit it must have dimension.
Reason (R): There may exist a physical quantity which has dimension but no unit.
An angle has a unit (radian) but is dimensionless, so A is false. Any physical quantity that possesses dimensions must have a unit, so R is also false.