CDS 2026 (I) Elementary Mathematics, question 100

Topic: Trigonometry

Answers by SolvedPaper. UPSC has not released the official answer key for this paper yet. We will update it when they do.

Passage · Q99 to Q100

p=sin⁡θ1+cos⁡θ+sin⁡θp=\frac{\sin \theta}{1+\cos \theta+\sin \theta} and

q=1+sin⁡θ1+sin⁡θ−cos⁡θq=\frac{1+\sin \theta}{1+\sin \theta-\cos \theta}

Q100

What is (p+1q)(q+1p)\left( p+\frac{1}{q} \right)\left( q+\frac{1}{p} \right) equal to ?

  1. (a) 12\frac{1}{2}
  2. (b) 174\frac{17}{4}
  3. (c) 92\frac{9}{2}Correct answer
  4. (d) 214\frac{21}{4}

Solution.

Answer: (c) 92\frac{9}{2}

  1. The definitions give pq=1/2pq = 1/2.
  2. Expand the requested product: (p+1/q)(q+1/p)=pq+1+1+1/(pq)(p + 1/q)(q + 1/p) = pq + 1 + 1 + 1/(pq).
  3. Substitute pq=1/2pq = 1/2 to obtain 1/2+2+2=9/21/2 + 2 + 2 = 9/2.