CDS 2026 (I) Elementary Mathematics, question 4

Topic: Algebra

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Q4

If 1x=1p+1q\frac{1}{x}=\frac{1}{p}+\frac{1}{q}, then what is pqp2−q2(x+px−p−x+qx−q)\frac{pq}{p^{2}-q^{2}}\left( \frac{x+p}{x-p}-\frac{x+q}{x-q} \right) equal to ?

  1. (a) −2-2
  2. (b) −1-1
  3. (c) 11
  4. (d) 22Correct answer

Solution.

Answer: (d) 22

  1. From 1/x=1/p+1/q1/x = 1/p + 1/q, obtain x=pq/(p+q)x = pq/(p + q).
  2. Substituting into the two fractions gives (x+p)/(x−p)=−(p+2q)/p(x + p)/(x - p) = - (p + 2q)/p and (x+q)/(x−q)=−(2p+q)/q(x + q)/(x - q) = - (2p + q)/q.
  3. Their difference is 2p/q−2q/p=2(p2−q2)/(pq)2p/q - 2q/p = 2\left( p^{2} - q^{2} \right)/(pq).
  4. Multiplication by pq/(p2−q2)pq/\left( p^{2} - q^{2} \right) leaves 2.