CDS 2026 (I) Elementary Mathematics, question 1

Topic: Algebra

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Q1

If p+q+r=0p+q+r=0, then what is zp2qr×zq2rp×zr2pqz^{\frac{p^{2}}{qr}}\times z^{\frac{q^{2}}{rp}}\times z^{\frac{r^{2}}{pq}} equal to ?

  1. (a) 11
  2. (b) zz
  3. (c) z2z^{2}
  4. (d) z3z^{3}Correct answer

Solution.

Answer: (d) z3z^{3}

  1. Combine the powers: the exponent is (p3+q3+r3)/(pqr)\left( p^{3} + q^{3} + r^{3} \right)/(pqr).
  2. Since p+q+r=0p + q + r = 0, the identity p3+q3+r3−3pqr=(p+q+r)(p2+q2+r2−pq−qr−rp)p^{3} + q^{3} + r^{3} - 3pqr = (p + q + r)\left( p^{2} + q^{2} + r^{2} - pq - qr - rp \right) gives an exponent of 3.