Foundation June 2024 Paper 3 Q15

15 In this question all measurements are in centimetres.

The shaded shape is made by cutting a square from the corner of a rectangle.

The width of the rectangle is \(k\).
The length of the rectangle is \(2k\).
Each side of the square is \(g\).

(a) Write down the relationship between the length and the width of the rectangle. [1]

(b) Find an expression for the area of the shaded shape.
Give your answer in its simplest form. [2]

(c)

(i) Find an expression for the perimeter of the shaded shape.
Give your answer in its simplest form. [3]

(ii) Find the value of \(k\) when the perimeter of the shaded shape is 62.4. [2]

Mark scheme (a)

Answer	Marks	Part marks and guidance
The length is twice the width oe or The width is half the length	1		“Twice” or “double” or “half” is not enough unless it is clear that length = 2 × width Accept \(2k\) is double \(k\) oe or \(L = 2W\) oe Do not accept values or length = \(2 \times k\) Mark the best response as long as it is not contradictory or has an incorrect statement

Answer	Marks	Part marks and guidance
\(2k^2 - g^2\) final answer	2	M1 for \(2k \times k - g \times g\) oe	Ignore units if included oe for M1 may be e.g. \(2k \times k - g^2\)

Mark scheme (c)

Answer	Marks	Part marks and guidance
(i) \(6k\) final answer	3	B2 for correct answer unsimplified or B2 for \(6k - 2g\) [\(+ g + g\)] or M2 for \(2k + k + 2k - g + k - g\) [\(+ g + g\)] oe or M1 for [height =] \(k - g\) or [length =] \(2k - g\)	Condone \(6k + 0\)[\(g\)] for 3 marks Accept in any order Accept e.g. \(2g\) for \(g + g\) Identified or seen on diagram in correct position
(ii) 10.4 nfww	2	M1 for their part (i) = 62.4 or \(\frac{62.4}{6}\)	Their part (i) must be algebraic in terms of \(k\) or \(k\) and \(g\) Their (a) can be rearranged Note: \(6k - 2g = 62.4\) scores M1 but does not score the second mark as from wrong working