CDS 2026 (I) Elementary Mathematics, question 47

Topic: Mensuration

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Q47

Let $x$ be the length of a diagonal of a face of a cube and $y$ be the length of a diagonal of the cube. If $x+y=(5+2\sqrt{6})$ units, then what is the total surface area of the cube ?

(a) $6(x+y)$ Correct answer
(b) $6xy$
(c) $3(x+y)$
(d) $3xy$

Solution.

Answer: (a) $6(x+y)$

If the cube edge is a, then $x = a\sqrt{2}$ and $y = a\sqrt{3}$ .
Also $5 + 2\sqrt{6} = \left( \sqrt{2} + \sqrt{3} \right)^{2}$ .
Therefore $a\left( \sqrt{2} + \sqrt{3} \right) = \left( \sqrt{2} + \sqrt{3} \right)^{2}$ , giving $a = \sqrt{2} + \sqrt{3}$ and $a^{2} = 5 + 2\sqrt{6} = x + y$ .
Total surface area $= 6a^{2} = 6(x + y)$ .