When does the statute of limitations expire? Why?

a$.$ Frannie Smithe timely files her return on April $15,20$x$1,$ stating that her income is $$60,000.$ Her actual income is $$100,000.$

b$.$ Bob Un banks has not filed a return for the past five years, because he does not believe the government is working for him.

c$.$ A taxpayer undergoing an audit sign a Form $872,$ Consent to Extend the Time to Assess Income Tax, and agrees to extend the period of limitations for another six months.

d$.$ ZEEE Company carries back a net operating loss to its third prior tax year by filing an amended return for the carryback year.