GATE 2026 IN Q.49: solution and answer

Official answer: BSource: Official answer key, GATE 2026 IN (IIT Guwahati)

1. In the present SI (revised in 2019), each base unit is defined by fixing the numerical value of a defining constant. A unit depends on every other base unit that appears in the SI unit of its constant. So an arrow $X\to Y$ is valid only if unit $X$ appears in the unit of the constant that defines $Y$.

2. Second: fixed by the caesium-133 hyperfine transition frequency, $\Delta {\nu }_{\text{Cs}}=9192631770\text{Hz}$, where $\text{Hz}={\text{s}}^{-1}$. No other base unit is needed, so the second comes first.

3. Metre: fixed by the speed of light in vacuum, $c=299792458{\text{m s}}^{-1}$. The second is needed, so $\text{second}\to \text{metre}$.

4. Kilogram: fixed by the Planck constant, $h=6.62607015\times 10^{-34}\text{J s}$, where $\text{J s}={\text{kg m}}^2{\text{s}}^{-1}$. Both the metre and the second are needed, so $\text{metre}\to \text{kilogram}$.

5. Kelvin: fixed by the Boltzmann constant, $k=1.380649\times 10^{-23}{\text{J K}}^{-1}$, where ${\text{J K}}^{-1}={\text{kg m}}^2{\text{s}}^{-2}{\text{K}}^{-1}$. It needs the kilogram, the metre and the second, so $\text{kilogram}\to \text{kelvin}$.

6. Every arrow in the chain is therefore a genuine dependency: $$\text{second}\to \text{metre}\to \text{kilogram}\to \text{kelvin}$$

7. Checking the other options arrow by arrow: (A) and (C) both begin with $\text{metre}\to \text{second}$, but the second is defined by $\Delta {\nu }_{\text{Cs}}$ alone and does not need the metre (it is the other way round). (C) also has $\text{kelvin}\to \text{kilogram}$, but $h$ contains no kelvin, so the kilogram does not need the kelvin. (D) ends with $\text{kelvin}\to \text{metre}$, but $c$ contains no kelvin, so the metre does not need it; in fact the kelvin needs the metre. The answer is (B).