4MA0 Higher June 2011 Paper 3 Q21
21
Diagram NOT accurately drawn
\(ABCD\) is a rectangle.
\(AB\) = 10 cm.
\(BC\) = 8 cm.
\(P\), \(Q\), \(R\) and \(S\) are points on the sides of the rectangle.
\(BP = CQ = DR = AS = x\) cm.
(a) Show that the area, \(A\) cm\(^2\), of the quadrilateral \(PQRS\) is given by the formula \[A = 2x^2 - 18x + 80\] (3)
(b) For \(A = 2x^2 - 18x + 80\)
(i) find \(\dfrac{\mathrm{d}A}{\mathrm{d}x}\),
(ii) find the value of \(x\) for which \(A\) is a minimum.
(iii) Explain how you know that \(A\) is a minimum for this value of \(x\).
(5)| Scheme | Marks |
|---|---|
Use of areas to obtain a correct expression for \(A\), which must be correctly punctuated. For example \((A =)\ 80 - 2 \times \frac{1}{2}x(10 - x) - 2 \times \frac{1}{2}x(8 - x)\) or \(10 \times 8 - \frac{1}{2}x(10 - x) - \frac{1}{2}x(10 - x) - \frac{1}{2}x(8 - x) - \frac{1}{2}x(8 - x)\) or \(80 - x(10 - x) - x(8 - x)\) or \(80 - 2\left(\dfrac{10x - x^2}{2}\right) - 2\left(\dfrac{8x - x^2}{2}\right)\) | B2 |
| Correct simplification of a correct expression for \(A\) to obtain an expression which is equivalent to \(2x^2 - 18x + 80\) For example \((A =)\ 80 - 10x + x^2 - 8x + x^2\) or \(80 - (10x - x^2) - (8x - x^2)\) or \(80 - (5x - \frac{1}{2}x^2) - (5x - \frac{1}{2}x^2) - (4x - \frac{1}{2}x^2) - (4x - \frac{1}{2}x^2)\) | B1 |
| (3) |
Notes
B2: B1 for expression for area of triangle or pair of congruent triangles, for example
\(\frac{1}{2}x(10 - x)\) or \(\frac{1}{2}x(8 - x)\)
or \(x(10 - x)\) or \(x(8 - x)\)
Condone omission of brackets for award of B1
B1: dep on B2
| Scheme | Marks |
|---|---|
| (i) \(4x - 18\) | B2 |
| (ii) “\(4x - 18\)” = 0 | M1 |
| 4.5 oe | A1 |
(iii) eg positive coefficient of \(x^2\) or U shape or \(\dfrac{\mathrm{d}^2A}{\mathrm{d}x^2} = 4\) which > 0 | B1 |
| (5) | |
| (Total 8 marks) |
Notes
B2: B1 for 2 of 3 terms differentiated correctly
A1: cao