| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Normal meets curve/axis — further geometry |
| Difficulty | Standard +0.3 This is a standard multi-part differentiation application question requiring sketching a quadratic, finding a tangent equation using the derivative, verifying a normal equation, and solving a simultaneous equation. While it has multiple steps (4 marks total based on typical MEI marking), each component uses routine techniques with no novel insight required—slightly above average difficulty only due to the algebraic intersection in part (iii). |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02q Use intersection points: of graphs to solve equations1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
The point $A$ has $x$-coordinate 5 and lies on the curve $y = x^2 - 4x + 3$.
(i) Sketch the curve. [2]
(ii) Use calculus to find the equation of the tangent to the curve at $A$. [4]
(iii) Show that the equation of the normal to the curve at $A$ is $x + 6y = 53$. Find also, using an algebraic method, the $x$-coordinate of the point at which this normal crosses the curve again. [6]11 The point A has $x$-coordinate 5 and lies on the curve $y = x ^ { 2 } - 4 x + 3$.\\
(i) Sketch the curve.\\
(ii) Use calculus to find the equation of the tangent to the curve at A .\\
(iii) Show that the equation of the normal to the curve at A is $x + 6 y = 53$. Find also, using an algebraic method, the $x$-coordinate of the point at which this normal crosses the curve again.
\hfill \mbox{\textit{OCR MEI C2 2012 Q11 [12]}}