| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2005 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Line and curve intersection |
| Difficulty | Standard +0.3 This is a standard coordinate geometry question involving line-curve intersection and tangent angles. Part (i) requires solving a quadratic equation, part (ii) uses discriminant analysis (b²-4ac < 0), and part (iii) involves finding a tangent gradient via implicit differentiation and calculating an angle between lines. All techniques are routine for P1 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02g Inequalities: linear and quadratic in single variable1.07m Tangents and normals: gradient and equations1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(2x^2 + 12 = 11x\) or \(y^2-11y+24=0\) | M1 | Elimination of one variable completely |
| Solution \(\rightarrow (1\frac{1}{2}, 8)\) and \((4, 3)\) | DM1 A1 | Correct method for soln of quadratic=0 co |
| Guesswork B1 for one, B3 for both. | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| \(-\sqrt{96} < k < \sqrt{96}\) or \( | k | <\sqrt{96}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dy}{dx} = -12/x^2\) (\(= -3\)) | B1 B1 | Anywhere For differentiation only – unsimplified |
| Answer | Marks | Guidance |
|---|---|---|
| Difference = \(8.1°\) or \(8.2°\) | M1 A1 | Used with either line or tangent Co |
(i) $2x^2 + 12 = 11x$ or $y^2-11y+24=0$ | M1 | Elimination of one variable completely
Solution $\rightarrow (1\frac{1}{2}, 8)$ and $(4, 3)$ | DM1 A1 | Correct method for soln of quadratic=0 co
Guesswork B1 for one, B3 for both. | [3]
(ii) $2x^2 - kx +12 = 0$
Use of $b^2 - 4ac$
$k^2 < 96$
$-\sqrt{96} < k < \sqrt{96}$ or $|k|<\sqrt{96}$ | M1 A1 DM1 A1 | Used on quadratic=0. Allow =$0, >0 etc For $k^2 - 96$ Definite use of $b^2-4ac<0$ Co. (condone inclusion of ≤) | [4]
(iii) gradient of $2x + y = k = -2$
$\frac{dy}{dx} = -12/x^2$ ($= -3$) | B1 B1 | Anywhere For differentiation only – unsimplified
Use of tangent for an angle
Difference = $8.1°$ or $8.2°$ | M1 A1 | Used with either line or tangent Co | [4]9 The equation of a curve is $x y = 12$ and the equation of a line $l$ is $2 x + y = k$, where $k$ is a constant.\\
(i) In the case where $k = 11$, find the coordinates of the points of intersection of $l$ and the curve.\\
(ii) Find the set of values of $k$ for which $l$ does not intersect the curve.\\
(iii) In the case where $k = 10$, one of the points of intersection is $P ( 2,6 )$. Find the angle, in degrees correct to 1 decimal place, between $l$ and the tangent to the curve at $P$.
\hfill \mbox{\textit{CAIE P1 2005 Q9 [11]}}