| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Show stationary point exists or gradient has specific property |
| Difficulty | Moderate -0.8 This is a straightforward application of the product rule for differentiation. Part (i) requires finding dy/dx = x²e^(-x)(3-x) and verifying it equals zero at x=3 (simple substitution). Part (ii) requires finding the gradient and y-coordinate at x=1, then using y-y₁=m(x-x₁). Both parts are routine textbook exercises with no problem-solving insight required, making this easier than average. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use product rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Show that derivative is equal to zero when \(x = 3\) | A1 | [3] |
| (ii) Substitute \(x = 1\) into gradient function, obtaining \(2e^{-1}\) or equivalent | M1 | |
| State or imply required \(y\)-coordinate is \(e^{-1}\) | B1 | |
| Form equation of line through \((1, e^{-1})\) with gradient found (NOT the normal) | M1 | |
| Obtain equation in any correct form | A1 | [4] |
**(i)** Use product rule | M1 |
Obtain correct derivative in any form | A1 |
Show that derivative is equal to zero when $x = 3$ | A1 | [3]
**(ii)** Substitute $x = 1$ into gradient function, obtaining $2e^{-1}$ or equivalent | M1 |
State or imply required $y$-coordinate is $e^{-1}$ | B1 |
Form equation of line through $(1, e^{-1})$ with gradient found (NOT the normal) | M1 |
Obtain equation in any correct form | A1 | [4]5 The equation of a curve is $y = x ^ { 3 } \mathrm { e } ^ { - x }$.\\
(i) Show that the curve has a stationary point where $x = 3$.\\
(ii) Find the equation of the tangent to the curve at the point where $x = 1$.\\
\hfill \mbox{\textit{CAIE P2 2010 Q5 [7]}}