| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2007 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find curve equation from derivative (extended problem with normals, stationary points, or further geometry) |
| Difficulty | Standard +0.3 This is a straightforward multi-part integration and coordinate geometry question. Part (i) requires basic integration of a linear function and using a point to find the constant. Part (ii) needs finding the gradient at P and using perpendicular gradient formula. Part (iii) involves solving simultaneous equations (line and parabola), which is routine but requires careful algebra. All techniques are standard P1 material with no novel insight required, making it slightly easier than average. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.07m Tangents and normals: gradient and equations1.08a Fundamental theorem of calculus: integration as reverse of differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(y = 4x - \frac{1}{2}x^2 + c\); Uses \((2,9)\) → \(c = 3\) | B1, M1 A1 [3] | For \(y = 4x - \frac{1}{2}x^2\); Introduces +c and attempts to evaluate |
| (ii) grad of tan = 2, normal = \(-\frac{1}{2}\); Eqn \(y - 9 = -\frac{1}{2}(x - 2)\) | M1, M1 A1 [3] | Uses \(m_1m_2 = -1\), \(m_1 = dy/dx = \) number; Any correct method – not for tangent. |
| (iii) \(y = 4x - \frac{1}{2}x^2 + 3\), \(2y + x = 20\); eliminates \(y\) → \(x^2 - 9x + 14 = 0\); eliminates \(x\) → \(2y^2 - 31y + 117 = 0\); Soln of quadratic → \(x = 7, y = 6.5\) | M1, DM1 A1 [3] | Eliminates one variable completely – needs a linear and quadratic eqn. Correct method for quad. co. |
**(i)** $y = 4x - \frac{1}{2}x^2 + c$; Uses $(2,9)$ → $c = 3$ | B1, M1 A1 [3] | For $y = 4x - \frac{1}{2}x^2$; Introduces +c and attempts to evaluate
**(ii)** grad of tan = 2, normal = $-\frac{1}{2}$; Eqn $y - 9 = -\frac{1}{2}(x - 2)$ | M1, M1 A1 [3] | Uses $m_1m_2 = -1$, $m_1 = dy/dx = $ number; Any correct method – not for tangent.
**(iii)** $y = 4x - \frac{1}{2}x^2 + 3$, $2y + x = 20$; eliminates $y$ → $x^2 - 9x + 14 = 0$; eliminates $x$ → $2y^2 - 31y + 117 = 0$; Soln of quadratic → $x = 7, y = 6.5$ | M1, DM1 A1 [3] | Eliminates one variable completely – needs a linear and quadratic eqn. Correct method for quad. co.9 A curve is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 - x$ and the point $P ( 2,9 )$ lies on the curve. The normal to the curve at $P$ meets the curve again at $Q$. Find\\
(i) the equation of the curve,\\
(ii) the equation of the normal to the curve at $P$,\\
(iii) the coordinates of $Q$.
\hfill \mbox{\textit{CAIE P1 2007 Q9 [9]}}