Find the value of \(a\). You must show all your working.[3]
(b) Another rectangle has a width of \(4x-10\).
(i) Given that \(x\) is a whole number, explain why the value of \(x\) cannot be less than 3. [1]
(ii) The perimeter of the rectangle is \(14x-4\).
Diagram not drawn to scale
Find the length of the rectangle in terms of \(\boldsymbol{x}\). [4]
Mark scheme (a)
Scheme
Marks
Comments
\(3y \times (y + a)\) or \(3y^2 + 3ay\) or equivalent
B1
Award B1 for intention of width × length. Allow \(3y \times y + a\) for B1.
(\(a\) =) 4
B2
Award B1 for one of the following:
\(3y^2 + 3ay = 3y^2 + 12y\)
\(3a = 12\)
\(3ay = 12y\).
Award B1B2 for unsupported answer of (\(a\) =) 4. Allow B1B2 for a correct embedded answer BUT only B1B1 if contradicted by \(a \neq 4\) or equivalent.
Alternative Method
Scheme
Marks
Comments
(Area =) \(3y\,(y + 4)\)
B2
Accept \(3y(1y + 4)\). Award B1 for one of the following:
\(3y(y \pm \ldots..)\)
\(3y(\ldots.. + 4)\)
\(3(y^2 + 4y)\)
\(y(3y + 12)\)
(\(a\) =) 4
B1
CAO
Mark scheme (b)
(i)
Scheme
Marks
Comments
Valid written explanation referring to negative values representing sides e.g. “ (4 × 2 – 10 = -2) you can’t have a negative value for a side” “\(x\) must be greater than 2·5 to have a positive value for the side”
E1
Allow “you can’t have a negative length”.
Do not allow
calculations only
“it can’t be negative”
“the value can’t be negative”.
(ii)
Scheme
Marks
Comments
\(14x - 4 - 2 \times (4x - 10)\) OR \(14x - 4 - 8x + 20\) or equivalent
M2
May be seen on diagram. Award M1 for intention for a method e.g. \(4x - 10 + 4x - 10 + \boldsymbol{?} = 14x - 4\) e.g. \(14x - 4 = \boldsymbol{?} + 2 \times (4x - 10)\) e.g. incorrect use of brackets but a clear attempt at the correct calculation is seen: \(14x - 4 - 8x - 20\).
(sum of both lengths=) \(6x + 16\)
A1
FT from M1: \(14x - 4\) – ‘their \(2 \times [4x - 10]\)’ or equivalent, provided ‘their \(8x - 20\)’ can be expressed in the form \(ax + b\), with a & b ≠ 0. Note: \(14x - 4 - 8x - 20 = 6x - 24\) is awarded M1A1.
(length=) \(3x + 8\)
B1
May be seen on diagram. Mark final answer. FT ‘their \(6x + 16\)’ ÷ 2, provided in the form \(ax + b\), with \(a\) & \(b\) ≠ 0.
Unsupported \(3x + 8\) is awarded M2 A1 B1.
If no marks awarded, award SC1 for a final answer of:
\(3x + c\) (\(c\) ≠ 8)
\(kx + 8\) (\(k\) ≠ 3 and positive).
Alternative method
Scheme
Marks
Comments
Sight of an appropriate \(7x - 2\)
B1
May be seen on diagram.
\(7x - 2 - 4x + 10\)
M2
Award M1 for intention for a method e.g. \(4x - 10 + \boldsymbol{?} = 7x - 2\) \([14x - 4] \div 2 - [4x - 10]\) or equivalent e.g. incorrect use of brackets but a clear attempt at the correct calculation is seen
(length=) \(3x + 8\)
A1
May be seen on diagram. Mark final answer. Allow FT from M1.
Unsupported \(3x + 8\) is awarded B1 M2 A1.
If no marks awarded, award SC1 for a final answer of:
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