CDS 2026 (I) Elementary Mathematics, question 70

Topic: Mensuration

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Q70

Let x,y,zx,y,z (all prime) be the length, breadth and height respectively of a cuboid with x>y>zx>y>z. The volume of the cuboid is 30k330k^{3} cubic units, where kk is a natural number. What is the total surface area of the cuboid ?

  1. (a) 60 square units
  2. (b) 62 square unitsCorrect answer
  3. (c) 64 square units
  4. (d) Cannot be determined due to insufficient data

Solution.

Answer: (b) 62 square units

  1. The volume xyz is a product of three distinct primes, so every prime exponent is 1.
  2. In 30k3=2×3×5×k330k^{3} = 2 \times 3 \times 5 \times k^{3}, any k>1k > 1 would introduce a prime exponent of at least 3.
  3. Therefore k=1k = 1 and (x,y,z)=(5,3,2)(x,y,z) = (5,3,2).
  4. Surface area =2(xy+yz+zx)=2(15+6+10)=62= 2(xy + yz + zx) = 2(15 + 6 + 10) = 62.