GATE Mechanical Engineering (ME) 2015 Shift 1 Solved Paper

For flow through a pipe of radius R, the velocity and temperature distribution are as follows:
$\mathit{u}\left(\mathit{r},\mathit{x}\right)={\mathit{C}}_1$, and $\mathit{T}\left(\mathit{r},\mathit{x}\right)={\mathit{C}}_2\left[1-{\left(\frac{\mathit{r}}{\mathit{R}}\right)}^3\right]$ , where C₁ and C₂ are constants. The bulk mean temperature is given by ${\mathit{T}}_{\mathit{m}}=\frac{2}{{\mathit{u}}_{\mathit{m}}{\mathit{R}}^2}\underset{0}{\overset{\mathit{R}}{\int }}\mathit{u}\left(\mathit{r},\mathit{x}\right)\mathit{T}\left(\mathit{r},\mathit{x}\right)\mathit{r}\mathit{d}\mathit{r},$
with U_m being the mean velocity of flow. The value of T_m is