Higher June 2025 Paper 2 Q21
21 The diagram shows a cuboid.
Diagram NOT accurately drawn
The cuboid measures \(3x\) cm by \(2x\) cm by \(y\) cm
The volume of the cuboid is 1014 cm3
The total surface area of the cuboid is \(A\) cm2
Show that \(A = 12x^2 + \dfrac{1690}{x}\)
You must show all the stages of your working.
(3)
| Scheme | Marks |
|---|---|
| \(3x \times 2x \times y = 1014\) oe or \(6x^2y = 1014\) oe or \(x^2y = 169\) oe | M1 |
| \(2 \times 3x \times y + 2 \times 2x \times y + 2 \times 3x \times 2x\) oe or \(2 \times (3xy + 2xy + 6x^2)\) oe or \(6xy + 4xy + 12x^2\) oe or \(10xy + 12x^2\) oe | M1 |
Using \(y = \dfrac{1014}{6x^2}\left(= \dfrac{169}{x^2}\right)\) in formula for surface area to obtain correct expression eg (SA =) \(2 \times 5x \times \dfrac{169}{x^2} + 2 \times 6x^2 = 12x^2 + \dfrac{1690}{x}\) or equating their surface area equations eg \(10xy + 12x^2 = 12x^2 + \dfrac{1690}{x}\) leading to \(x^2y = 169\) oe Working required Answer: Shown | A1 |
| (3) | |
| (3 marks) |
Notes
M1: for an equation for volume in terms of \(x\) and \(y\)
M1: (indep) for a correct expression for the surface area
NB May not explicitly see the surface area expression in terms of \(y\) eg
\(2 \times 3x \times \dfrac{169}{x^2} + 2 \times 2x \times \dfrac{169}{x^2} + 2 \times 3x \times 2x\) oe
May be fully substituted using \(y = \dfrac{169}{x^2}\) oe
A1: dep on M2
For completing the ‘show that’ by clearly showing the stages that lead to the given expression for the surface area.