The value of $T\left(\frac{1}{2}\right)$ is :
The question asks for the value of $T\left(\frac{1}{2}\right)$. Given the options and the common mathematical context where such questions appear, $T(x)$ typically represents the Gamma function, denoted as $\Gamma(x)$.
The Gamma function $\Gamma(x)$ is a generalization of the factorial function to complex and real numbers. A fundamental property of the Gamma function is its value at $x = \frac{1}{2}$:
$ \Gamma\left(\frac{1}{2}\right) = \sqrt{\pi} $
Assuming $T(x) = \Gamma(x)$, we can directly apply this property:
$ T\left(\frac{1}{2}\right) = \Gamma\left(\frac{1}{2}\right) = \sqrt{\pi} $
Therefore, the value of $T\left(\frac{1}{2}\right)$ is $\sqrt{\pi}$.
| List-I | List-II |
| Electronic Configuration | First Ionisation energy (kJ mol$^{-1}$) |
| (A). ns$^2$ | (I). 2100 |
| (B). ns$^2$np$^1$ | (II). 1400 |
| (C). ns$^2$np$^3$ | (III). 800 |
| (D). ns$^2$np$^6$ | (IV). 900 |
| List-I | List-II |
| Spectroscopy | Property |
| (A). Raman | (I). Polarizability |
| (B). FTIR | (II). Dipole Moment |
| (C). UV-Visible | (III). Absorbance |
| (D). NMR | (IV). Spin |