If you do not believe that I am the one I claim to be, you will indeed die in your sins. John 1.12 To all who received him, who believed in his name, he gave power to become children of God. This view makes faith purely intellectual and does seem a little arbitrary. But if we have a complete view of faith, it no longer seems odd. It should not be thought that repentance and faith are a dual prerequisite for salvation.
True faith entails true repentance.
Therefore it always results in good works. Every good tree bears good fruit, but a bad tree bears bad fruit. A good tree cannot bear bad fruit, and a bad tree cannot bear good fruit. James 2.17 Faith by itself, if it is not accompanied by action, is dead. But this is a slightly more complex question.