Q35 / 59 · KVPY 2013 (SA) · Mathematics

On each face of a cuboid, the sum of its perimeter and its area is written. Among the six numbers so written, there are three distinct numbers and they are $16,24$ and $31$. The volume of the cuboid lies between

  • A
    $7$ and $14$
  • B
    $14$ and $21$
  • C
    $21$ and $28$
  • Correct answer
    D
    $28$ and $35$

Solution

$2(a+b)+a b=16 \quad\dots(1)$ $2(b+c)+b c=24 \quad\dots(2)$ $2(c+a)+c a=31 \quad\dots(3)$ From equation $(2)$, equation $(3) \Rightarrow(a-b)(2+c)=7 \quad\dots(4)$ From equation $(2)$ and equation $(4) \Rightarrow 4 a=2+5 b \quad\dots(5)$ Solve equation $(1)$ and $(5)$ $b=2 a=3, c=5$ Volume $=30$

Hence, the correct answer is option: 'D'.

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