The correct option is
D Irrespective
The word, 'irrespective' is an adjective which means 'regardless'. When used in the given sentence, the sentence means: 'Regardless of your performance in the exam, you will get admission to the course'. In other words, it means: 'Whether you perform well or not (regardless of that), you will get admission to the course'. As this word completes the sentence meaningfully, option A is correct.
The word, 'despite' is a preposition which means the same as 'in spite of'. As it is a preposition, it cannot be followed by another preposition. It has to be followed by a noun/ noun phrase or a pronoun. In the given sentence, the blank is followed by the preposition, 'of'. The phrase, 'despite of' is grammatically incorrect. The correct usage is: 'Despite your performance in the exam, you will get admission to the course'. Hence, option B is incorrect.
The words, 'nevertheless' in option C and 'notwithstanding' in option D mean the same. They mean: 'in spite of that'.
'Nevertheless' is used to contrast something which is said earlier. An example of its usage is: 'You may not have done well in the exam. Nevertheless, you will get admission to the course'. The phrase, 'Nevertheless of your performance in the exam', is grammatically incorrect. Hence, option C is incorrect.
'Notwithstanding' is usually used as a preposition and is preceded or followed by the object of the sentence (a noun/ noun phrase/ pronoun). It cannot be followed by another preposition (like 'of'). An example of its usage is: 'Notwithstanding your performance in the exam, you will get admission to the course'. Given the usage as above, this word cannot be used in the given sentence. Hence, option D is incorrect.