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NDA 2 2024 Maths, question 75

Topic: Differential calculus

Q75

What is the maximum value of acos⁡x+bsin⁡x+ca\cos x+b\sin x+c?

  1. (a) a2+b2+c\sqrt{a^{2}+b^{2}+c}
  2. (b) a2+b2+c\sqrt{a^{2}+b^{2}}+cCorrect answer
  3. (c) a2+b2−c\sqrt{a^{2}+b^{2}}-c
  4. (d) a2+b2\sqrt{a^{2}+b^{2}}

Solution.

Answer: (b) a2+b2+c\sqrt{a^{2}+b^{2}}+c

  1. By Cauchy–Schwarz, a cos x+b sin x≤√(a²+b²)√(cos²x+sin²x)=√(a²+b²).
  2. Equality is attained by aligning (cos x,sin x) with (a,b), unless both are zero, when equality holds everywhere.
  3. Adding c gives √(a²+b²)+c.