Spelling difference in middle name in school certificate and government ID proofs

There should not be any difference in your name between the different documents like school certificate, birth certificate or Aadhar card. If some change is there, even if some minor spelling change, then it will not be acceptable anywhere. So, in this particular case your middle name in Aadhar card is not correct and is to be rectified and corrected from 'Indrajit' to 'Indrajeet'. What I will suggest you is to apply for name correction in your Aadhar card for which you can visit the Aadhar centre near your place with all the supporting documents like high school certificate, birth certificate, voter's card, residential address proof etc. You can try it doing online also by giving the correct credential and getting it corrected online. Otherwise going to an Aadhar centre would be an easier option.

Once that change is done then all your documents will now be having the same name and there is absolutely no problem in it in future. Your father's name has nothing to do with your name. They are two different entities. The issue will only come if the name of your father is appearing wrong or incorrect in your documents and then it will also require correction. I think that is not the thing in your case so there is no worry.

The certificate issuing authorities cannot be blamed for spelling difference as many people do have their names with different spelling in the thought of numerology and practice of their own area. In Pondicherry area people write Mourougan for Murugan. In Kerala Mani is written as Moni. In Bengal area Vasu is written as Basu.
It is common to have some mistakes while entering name/s in certificates and it is our duty to chek while receiving. This may be due to the steps of both sides. While furnishing our name in application we should take utmost care as a)with or without space b) when we follow some numerology adjustments.
To correct this we have to approach the concerned authority IN PERSON with relevant documents without delay.
Subsequently we should have all certificates, Government IDs like Aadhaar etc., with similar name and spelling.

It would be best on your part to correct the middle name appearing in the Aadhar Card since there is proper reprentation of your name in other documents such as SSC, HSC and Engineering Certificates. Even your Voter ID card and PAN Card is free from errors. Uniformity in respect of your name in all the certificates is essential to avoid any embracement in case of document verification of an appropriate authority.
The same can be rectified by approaching the Aadhar Centre with your other supporting documents to be produced before the officer responsible for rectification of errors crept in while making the original document of the Aadhar Card. Your photo copy of Voter ID card and PAN Card would suffice the purpose.
You need not think unnecessarily about the ID card of your father and even if there is mismatch in his document, it is not likely to impact you in any way in course of your document verification.

Whenever you apply for a course or position of employment or present your certificates and other documents for document verification then the authorities check one thing and that is the consistency and uniformity of personal data from one certificate to another. If there is a mismatch even at one place, even of a minor nature, they will get a doubt on the credentials of the particular document and definitely report it for rectification or keep the things in pending till the candidate brings in the correct certificate. I have seen many cases where provisional admission or appointment is given by giving some time or so to bring the corrected document failing which the admission or appointment is cancelled. So, this is an important thing that our documents should be in order. In your case there is only the middle name that needs some correction in your Aadhar card and then it will become consistent with other documents. So, the only solution is to go to an Aadhar centre with all the related documents where your correct name is reflecting and get the name corrected in the Aadhar card. Nowadays, because of many banks doing these jobs for Aadhar authority, you can find one near your place and get the corrections done and submit the corrected document wherever you are planning to apply.

You have to get the name in the Aadhaar card get corrected. You can do this online also.
The following are the 3 steps:
1. log in to AADHAAR card Online Self Service Update Portal using your AADHAAR number
2. Submit your AADHAAR card name change request online. The site will guide you step by step;
3. Upload supporting self-attested Proof of Identity document (soft copy)
You have to submit this change request in English/local language. Once the change request is submitted you will be given your AADHAAR card Update Request Number. You have to save this number to track the status. Your request information will be verified by the UIDAI representative. Once it is successfully validated, the changes will be seen in your AADHAAR card.
Please see the spelling of your father in your Aadhar Card and if there is any change required you can get that change also so that in future there will not be any problem.

If you have your full name as 'Chandan Indrajeet Singh' in your SSC, HSC and Engineering certificates, there is no problem because all your academic certificates have the same name and now it is your name and it doesn't matter if your father's name in Indrajit Singh. you can't change your name in your academic certificates now. It's better to keep your name as it is in your academic documents.

But as your Identity proof documents like Aadhar card and PAN card are having a spelling difference in middle name only. Then in your case the simple solution to your problem is that you can get the rectify your name in these identity documents. You can correct your name in both of these documents (Aadhar card and PAN card) online.
For change in name you will have to log on to official website of UIDAI and can correct your name by uploading self attested supporting identity proof. For correction of name in PAN card you can raise a request online on the NSDL website.
After correction in your name you can apply anywhere without any worry and also open a bank account.