Here are the mistakes and their corrections for each sentence:

1. **Mistake:** "your mother's date of birth"
**Correction:** No correction needed; this sentence is correct.

2. **Mistake:** "In spite of they were tired"
**Correction:** "In spite of being tired"

3. **Mistake:** "Did he a doctor or a manager"
**Correction:** "Was he a doctor or a manager"

4. **Mistake:** "less TV programs"
**Correction:** "fewer TV programs"

5. **Mistake:** "kilometre"
**Correction:** "kilometers" (if using American English) or keep "kilometre" for British English, but 1.877 should be plural if it's above one.
**Correction:** "about 1.877 kilometres"

In summary:
1. Correct
2. Change "In spite of they were tired" to "In spite of being tired"
3. Change "Did he a doctor or a manager" to "Was he a doctor or a manager"
4. Change "less TV programs" to "fewer TV programs"
5. Change "kilometre" to "kilometers" or "kilometres" depending on the dialect.