For a FIRST-ORDER reaction with rate constant $k$, the half-life is given by

1. Integrated first-order rate law: $[A] = [A]_0\,e^{-kt}$ (NCERT §3.3.2). 2. At the half-life, $[A] = [A]_0/2$ by definition. 3. Substitute: $[A]_0/2 = [A]_0\,e^{-k t_{1/2}}$, so $1/2 = e^{-k t_{1/2}}$. 4. Take the natural log: $\ln(1/2) = -k\,t_{1/2}$, i.e. $t_{1/2} = \dfrac{\ln 2}{k} = \dfrac{0.693}{k}$. 5. KEY FEATURE: first-order half-life is INDEPENDENT of initial concentration — a defining characteristic of first-order decay (also seen in radioactive decay). 6. Option B is the second-order half-life. Option C is the zero-order half-life. Option D inverts the formula (multiplied instead of divided). _Source: NCERT Class 12 Chemistry Part 1, Ch 3, §3.3.2 (Integrated Rate Equation — first order half-life), p. 13–14._