1. The boy cut his finger with a knife. 2. The boat was carried by the waves into the open sea. 3. The teacher was pleased with our work. 4. America was discovered by Columbus. 5. "Hamlet" was written by Shakespeare. 6. We eat soup with a spoon. 7. He was killed with a knife. 8. He was killed by the robbers. 9. He was knocked down by a big stick. 10. He was knocked down by a car. 11. He was taken to hospital by an ambulance. 12. He was treated with very effective drugs. 13. He was cured by a very skilful doctor. 14. He wrote his letter with a pencil. 15. He was scolded by his mother.

Коротко: by — когда есть «делатель» (агент: человек, группа, силы природы, машина как источник действия). with — когда это инструмент/средство, чем действие выполнено, либо устойчивые сочетания типа pleased with. Для транспорта: обычно by car/train/plane, но in an ambulance/taxi (внутри). Ниже исправленные варианты с пояснениями.

1. The boy cut his finger with a knife.
   with — нож как инструмент.

2. The boat was carried by the waves into the open sea.
   by — волны как действующая сила (агент).

3. The teacher was pleased with our work.
   Устойчивое: pleased with чем-то.

4. America was discovered by Columbus.
   by — агент (Колумб).

5. “Hamlet” was written by Shakespeare.
   by — агент (Шекспир).

6. We eat soup with a spoon.
   with — ложка как инструмент.

7. He was killed with a knife.
   with — нож как орудие.

8. He was killed by the robbers.
   by — агенты (грабители).

9. He was knocked down with a big stick.
   with — палка как орудие (не агент).

10. He was knocked down by a car.
    by — машина выступает как «источник действия» (агент/движущая сила).

11. He was taken to hospital in an ambulance. (вариант: by ambulance без артикля)
    Для кареты скорой — предлог in (внутри транспортного средства) или оборот by ambulance как способ передвижения.

12. He was treated with very effective drugs.
    with — лекарства как средство лечения.

13. He was cured by a very skilful doctor.
    by — агент (врач).

14. He wrote his letter with a pencil.
    with — карандаш как инструмент.

15. He was scolded by his mother.
    by — агент (мать).

Правило-подсказка: если можно мысленно подставить «кем?» — чаще by; если «чем?» — чаще with. Исключения — устойчивые фразы (pleased with, angry with и т. п.) и транспорт (in a car/taxi/ambulance, но by car/train/plane).