Higher January 2020 Paper 1 Q15
15
Diagram NOT accurately drawn
The diagram shows a cuboid of volume \(V\) cm³
There is a value of \(x\) for which the volume of the cuboid is a maximum.
Show your working clearly.
Give your answer correct to 3 significant figures. (5)
| Scheme | Marks |
|---|---|
| \((2x + 5)(x + 1) = 2x^2 + 2x + 5x + 5\) \((= 2x^2 + 7x + 5)\) or \((x + 1)(3 - x) = -x^2 + 3x - x + 3\) \((= -x^2 + 2x + 3)\) or \((3 - x)(2x + 5) = -2x^2 + 6x - 5x + 15\) \((= -2x^2 + x + 15)\) | M1 |
| E.g. \([(2x^2 + 7x + 5)(3 - x) =]\) \(-2x^3 - 7x^2 - 5x + 6x^2 + 21x + 15\) or \([(-x^2 + 2x + 3)(2x + 5) =]\) \(-2x^3 - 5x^2 + 10x + 4x^2 + 6x + 15\) or \([(-2x^2 + x + 15)(x + 1) =]\) \(-2x^3 - 2x^2 + 15x + x^2 + x + 15\) | M1 |
| Shown | A1 |
| (3) |
Notes
M1: for multiplying out two brackets correctly at least 3 terms correct
OR M2 (for both M1 marks) for at least 4 terms correct out of a maximum of 8 terms
\(6x^2 - 2x^3 + 6x - 2x^2 + 15x - 5x^2 + 15 - 5x\)
M1: for at least 3 terms correct out of a maximum of 6 terms
or
for at least 4 terms correct out of a maximum of 8 terms
| Scheme | Marks |
|---|---|
| \(\left(\dfrac{\mathrm{d}V}{\mathrm{d}x} =\right) 16 - 2x + (3 \times -2x^2)\) oe | M1 |
| \(\left(\dfrac{\mathrm{d}V}{\mathrm{d}x} =\right) 16 - 2x - 6x^2\) oe | A1 |
| \(\text{‘}16 - 2x - 6x^2\text{’} = 0\) oe | M1 |
E.g. \((x =)\; \dfrac{-2 \pm \sqrt{2^2 - 4 \times 6 \times -16}}{2 \times 6}\) oe (accept + in place of ±) or E.g. \(6\left(\left(x + \dfrac{1}{6}\right)^2 - \left(\dfrac{1}{6}\right)^2\right) - 16\;(= 0)\) oe | M1 |
| 1.47 | A1 |
| (5) | |
| (8 marks) |
Notes
M1: for the correct differentiation of at least 2 correct terms from
16 or \(-2x\) or \((3 \times -2x^2)\)
A1: for a correct differentiated expression
M1: (dep on M1) for equating their differentiated expression to zero
M1: (dep on M1) for a complete method to solve their 3-term quadratic equation (allow one sign error and some simplification – allow as far as \(\dfrac{-2 \pm \sqrt{4 + 384}}{12}\))
A1: dep on M1 for answer in range 1.47 – 1.5 from correct working
(Must reject −1.80 to −1.81 if calculated)