Fetal brain segmentation and gestational age prediction have been under active research in the field of medical image processing for a long time. However, both these tasks are challenging due to factors like difficulty in acquiring a proper fetal brain image owing to the fetal movement during the scan. With the recent advancements in deep learning, many models have been proposed for performing both the tasks, individually, with good accuracy. In this paper, we present Multi-Tasking Single Encoder U-Net, MTSE U-Net, a deep learning architecture for performing three tasks on fetal brain images. The first task is the segmentation of the fetal brain into its seven components: intracranial space and extra-axial cerebrospinal fluid spaces, gray matter, white matter, ventricles, cerebellum, deep gray matter, and brainstem, and spinal cord. The second task is the prediction of the type of the fetal brain (pathological or neurotypical). The third task is the prediction of the gestational age of the fetus from its brain. All of this will be performed by a single model. The fetal brain images can be obtained by segmenting it from the fetal magnetic resonance images using any of the previous works on fetal brain segmentation, thus showing our work as an extension of the already existing segmentation works. The Jaccard similarity and Dice score for the segmentation task by this model are 77 and 82%, respectively, accuracy for the type of prediction task is 89% and the mean absolute error for the gestational age task is 0.83 weeks. The salient region identification by the model is also tested and these results show that a single model can perform multiple, but related, tasks simultaneously with good accuracy, thus eliminating the need to use separate models for each task.