1. My mom and I go to the cinema at the weekends.

Correct
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2. My family and I watch sports programmes in the evening.

Correct
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3. We do not play board games in the mornings.

Correct
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4. My brother do not help to do the dishes I do them in the kitchen.

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“auxiliary does”

5. I arrives at home at 3 p.m. after I finishes my extracurricular activities.

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Both verbs are incorrect, check the subject.

6. Do my father and my mother goes to dance on Saturdays?

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It is a question and the auxiliary is “do”, therefore the verb “goes” must be in simple form: go

7. Some friends and my dog play with a ball in the park in the afternoon.

Correct
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