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NDA 1 2025 Maths, question 44

Topic: Trigonometry

Passage · Q44 to Q45

Let 2sin⁡α+cos⁡α=22\sin\alpha + \cos\alpha = 2, where 0<α<90∘0 < \alpha < 90^{\circ}.

Q44

What is tan⁡α\tan\alpha equal to?

  1. (a) 1/21/2
  2. (b) 11
  3. (c) 3/43/4Correct answer
  4. (d) 22

Solution.

Answer: (c) 3/43/4

  1. Put s=sin⁡αs = \sin\alpha and use cos⁡α=2−2s\cos\alpha = 2 - 2s.
  2. Squaring and applying sin⁡2α+cos⁡2α=1\sin^{2}\alpha + \cos^{2}\alpha = 1 gives 5s2−8s+3=05s^{2} - 8s + 3 = 0, or (5s−3)(s−1)=0(5s - 3)(s - 1) = 0.
  3. The root s=1s = 1 would give α=90∘\alpha = 90^{\circ}, which is excluded.
  4. Thus sin⁡α=3/5\sin\alpha = 3/5, cos⁡α=4/5\cos\alpha = 4/5, and tan⁡α=3/4\tan\alpha = 3/4.