Standard +0.8 This question requires finding roots of a cubic (likely factorisable), identifying multiple regions where the curve crosses the x-axis, setting up separate integrals with absolute values for areas above and below the axis, and summing their magnitudes. While the integration itself is routine, the multi-step process of root-finding, determining sign changes, and correctly handling absolute areas makes this moderately harder than a standard single-region area question.
10 In this question you must show detailed reasoning.
The equation of a curve is \(y = 12 x ^ { 3 } - 24 x ^ { 2 } - 60 x + 72\).
Determine the magnitude of the total area bounded by the curve and the \(x\)-axis.
e.g. divide by 12 and find \(f(k)\) where \(k = \pm1, \pm2, \pm3, \pm6\)
\((x-1)\), \((x-3)\) or \((x+2)\) identified
A1
or attempt at long division, allow sign errors
\((x-1)(x^2 - x - 6)\) oe
M1
\(x\)-values are \(-2\), \(1\) and \(3\)
A1
could be implied from the limits
\(F[x] = 3x^4 - 8x^3 - 30x^2 + 72x\)
M1
integration with at least 2 terms correct; may be unsimplified (+c not necessary)
all terms correct, may be unsimplified, can have \(+c\)
A1
\(F[b] - F[a]\) or \(F[c] - F[b]\)
M1
one subtraction attempted where \(a,b\) and \(c\) are their solutions to original equation
\(189\) or \(-64\) seen
A1
\(253\)
A1
## Question 10:
Valid attempt to solve $12x^3 - 24x^2 - 60x + 72 = 0$ | M1 | e.g. divide by 12 and find $f(k)$ where $k = \pm1, \pm2, \pm3, \pm6$
$(x-1)$, $(x-3)$ or $(x+2)$ identified | A1 | or attempt at long division, allow sign errors
$(x-1)(x^2 - x - 6)$ oe | M1 |
$x$-values are $-2$, $1$ and $3$ | A1 | could be implied from the limits
$F[x] = 3x^4 - 8x^3 - 30x^2 + 72x$ | M1 | integration with at least 2 terms correct; may be unsimplified (+c not necessary)
all terms correct, may be unsimplified, can have $+c$ | A1 |
$F[b] - F[a]$ or $F[c] - F[b]$ | M1 | one subtraction attempted where $a,b$ and $c$ are their solutions to original equation
$189$ or $-64$ seen | A1 |
$253$ | A1 |
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10 In this question you must show detailed reasoning.\\
The equation of a curve is $y = 12 x ^ { 3 } - 24 x ^ { 2 } - 60 x + 72$.\\
Determine the magnitude of the total area bounded by the curve and the $x$-axis.
\hfill \mbox{\textit{OCR MEI AS Paper 2 2022 Q10 [9]}}