With the productive evolution of large language models (LLMs) in the field of natural language processing (NLP), tons of effort has been made to effectively fine-tune common pre-trained LLMs to fulfill a variety of tasks in one or multiple specific domain. In practice, there are two prevailing ways, in which the adaptation can be achieved: (i) Multiple Independent Models: Pre-trained LLMs are fine-tuned a few times independently using the corresponding training samples from each task. (ii) An Integrated Model: Samples from all tasks are employed to fine-tune a pre-trianed LLM unitedly. To address the high computing cost and seesawing issue simultaneously, we propose a unified framework that implements a 1 + N mutli-task fine-tuning pattern in LLMs using a novel Customized Gate Control (CGC) Low-rank Adaptation (LoRA) algorithm. Our work aims to take an advantage of both MTL (i.e., CGC) and PEFT (i.e., LoRA) scheme. For a given cluster of tasks, we design an innovative layer that contains two types of experts as additional trainable parameters to make LoRA be compatible with MTL. To comprehensively evaluate the proposed framework, we conduct well-designed experiments on two public datasets. The experimental results demonstrate that the unified framework with CGC-LoRA modules achieves higher evaluation scores than all benchmarks on both two datasets.