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NDA 2 2024 Maths, question 53

Topic: Vectors

Q53

Let θ\theta be the angle between two unit vectors a→\overrightarrow{a} and b→\overrightarrow{b}. If a→+2b→\overrightarrow{a}+2\overrightarrow{b} is perpendicular to 5a→−4b→5\overrightarrow{a}-4\overrightarrow{b}, then what is cos⁡θ+cos⁡2θ\cos \theta+\cos 2\theta equal to?

  1. (a) 00Correct answer
  2. (b) 1/21/2
  3. (c) 11
  4. (d) 3+12\frac{\sqrt{3}+1}{2}

Solution.

Answer: (a) 00

  1. Take the dot product: (a+2b)·(5a−4b)=5−8+6cosθ=0.
  2. Hence cosθ=1/2 and cos2θ=2(1/2)²−1=−1/2.
  3. Their sum is zero.