| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find tangent at given point (polynomial/algebraic) |
| Difficulty | Easy -1.2 This is a straightforward C2 integration question testing basic polynomial integration and finding a particular solution using a boundary condition. Part (i) is direct application of the power rule, and part (ii) adds one simple step of substituting a point to find the constant. Both are routine textbook exercises with no problem-solving required. |
| Spec | 1.08a Fundamental theorem of calculus: integration as reverse of differentiation1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\int(4x - 5)dx = 2x^2 - 5x + c\) | M1, A1 (2 marks) | Obtain at least one correct term; Obtain at least \(2x^2 - 5x\) |
| (ii) \(y = 2x^2 - 5x + c\) \(7 = 2 × 3^2 - 5 × 3 + c ⟹ c = 4\) So equation is \(y = 2x^2 - 5x + 4\) | B1∇, M1, A1 (3 marks) | State or imply \(y =\) their integral from (i); Use \((3,7)\) to evaluate \(c\); Correct final equation |
**(i)** $\int(4x - 5)dx = 2x^2 - 5x + c$ | M1, A1 (2 marks) | Obtain at least one correct term; Obtain at least $2x^2 - 5x$
**(ii)** $y = 2x^2 - 5x + c$ $7 = 2 × 3^2 - 5 × 3 + c ⟹ c = 4$ So equation is $y = 2x^2 - 5x + 4$ | B1∇, M1, A1 (3 marks) | State or imply $y =$ their integral from (i); Use $(3,7)$ to evaluate $c$; Correct final equation
**Total: 5 marks**\begin{enumerate}[label=(\roman*)]
\item Find $\int (4x - 5) dx$. [2]
\item The gradient of a curve is given by $\frac{dy}{dx} = 4x - 5$. The curve passes through the point $(3, 7)$. Find the equation of the curve. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR C2 2007 Q3 [5]}}