| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find tangent at given point (polynomial/algebraic) |
| Difficulty | Moderate -0.3 This is a straightforward multi-part calculus question covering standard C2 techniques: differentiation (power rule), tangent equations (point-slope form), verifying roots by substitution, and definite integration. All parts are routine applications with no problem-solving insight required, though the integration in part (iv) requires careful handling of the sign to find area below the axis. Slightly easier than average due to the guided structure and standard methods. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks |
|---|---|
| 1 | i |
| Answer | Marks |
|---|---|
| iv | 3x2 − 20x + 12 |
| Answer | Marks |
|---|---|
| 341(.3..) cao | 2 |
| Answer | Marks |
|---|---|
| A1 | B1 if one error “+c” is an error |
| Answer | Marks |
|---|---|
| Must have integrated f(x) | 2 |
Question 1:
1 | i
ii
iii
iv | 3x2 − 20x + 12
y − 64 = −16(x − 2) o.e.
eg y = −16x + 96
Factorising f(x) ≡(x+2)(x−6)2
OR Expanding (x+2)(x−6)2
x4 10x3
− +6x2 +72x
4 3
value at (x = 6) ~ value at (x = −2)
341(.3..) cao | 2
4
B3
M2
E1
B2
M1
A1 | B1 if one error “+c” is an error
M1 for subst x = 2 in their y′
A1 for y′ = −16 and B1 for y = 64
or B1 for f(-2) = -8-40-24+72 =0 and
B1 for f ′(6) = 0 and
B1dep for f(6)=0
-1 for each error
Must have integrated f(x) | 2
4
3
4\includegraphics{figure_1}
Fig. 9 shows a sketch of the graph of $y = x^3 - 10x^2 + 12x + 72$.
\begin{enumerate}[label=(\roman*)]
\item Write down $\frac{dy}{dx}$. [2]
\item Find the equation of the tangent to the curve at the point on the curve where $x = 2$. [4]
\item Show that the curve crosses the $x$-axis at $x = -2$. Show also that the curve touches the $x$-axis at $x = 6$. [3]
\item Find the area of the finite region bounded by the curve and the $x$-axis, shown shaded in Fig. 9. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C2 Q1 [13]}}