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NDA 2 2024 Maths, question 52

Topic: Vectors

Q52

For what value of the angle between the vectors a→\overrightarrow{a} and b→\overrightarrow{b} is the quantity ∣a→×b→∣+3∣a→⋅b→∣|\overrightarrow{a}\times \overrightarrow{b}|+\sqrt{3}|\overrightarrow{a}\cdot \overrightarrow{b}| maximum?

  1. (a) 0∘0^{\circ}
  2. (b) 30∘30^{\circ}Correct answer
  3. (c) 45∘45^{\circ}
  4. (d) 60∘60^{\circ}

Solution.

Answer: (b) 30∘30^{\circ}

  1. For fixed nonzero magnitudes, divide by |a||b| to maximise sinθ+√3|cosθ| on 0≤θ≤π.
  2. In the acute half this is 2sin(θ+60°), reaching 2 at 30°.
  3. Reflection gives another maximum at 150°, which is not listed.
  4. Among the supplied angles, 30° is correct.