Will HR check my exact diploma title and fake internships in my resume?

I started bachelor of computer science, but it was too challenging,my GPA went down, it was very low and I was failing to meet min.GPA requirements. So I switched to Math, graduated with "3 y general degree in Math". Though I started as Computer science student and completed some CS courses, my last year was pure math courses.
I have been working as front-end web dev for 1.5y.
I want to work in a tech giant company(FB, google etc)
Also, I did not do any internships during my bachelor's. Can I add in my resume that I had some 1-2 internships in startups?
Will they run some check?
1)check exactly diploma title
2)check my internships?
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