| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Show/verify a given line is a tangent |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard techniques: writing a line equation (trivial), finding tangent conditions by solving a discriminant equation (routine but requires careful algebra), and completing the square (basic skill). While part (ii) involves multiple steps and solving a quadratic, these are all textbook methods with no novel insight required, making it slightly easier than average. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(y = m(x-2)\) or \(oe\) | B1 [1] | Accept \(y = mx + c\), \(c = -2m\) |
| Answer | Marks | Guidance |
|---|---|---|
| \((3, 2), (1, 2)\) | M1, DM1, A1, DM1, A1, A1 [6] | Apply \(b^2 - 4ac\); Substitute their m and attempt to solve for x; Allow for a pair of x values or 1 x and 1 y; m=2,-2 also needed for final mark |
| Answer | Marks |
|---|---|
| M1, M1, M1, A1, A1, A1, B1,B1 [2] | Eliminating 2 variables from 3 equations; Obtaining a quadratic in x or y; Solving their quadratic correctly; A pair of x values or 1 x and 1 y..; m=2,-2 also needed for final mark |
| (iii) \((x-2)^2 + 1, (2, 1)\) | B1,B1 [2] |
**(i)** $y = m(x-2)$ or $oe$ | B1 [1] | Accept $y = mx + c$, $c = -2m$
**(ii)** $x^2 - 4x + 5 = mx - 2m \Rightarrow x^2 - x(4+m) + 5 + 2m = 0$
$(4+m)^2 - 4(5+2m) = 0 \Rightarrow m^2 - 4 = 0$
$m = \pm 2 \Rightarrow x^2 - 6x + 9 = 0 \Rightarrow x = 3$
$m = -2 \Rightarrow x^2 - 2x + 1 = 0 \Rightarrow x = 1$
$(3, 2), (1, 2)$ | M1, DM1, A1, DM1, A1, A1 [6] | Apply $b^2 - 4ac$; Substitute their m and attempt to solve for x; Allow for a pair of x values or 1 x and 1 y; m=2,-2 also needed for final mark
**OR** $m = 2x - 4$
$y = mx - 2m, y = x^2 - 4x + 5$
M1, M1, M1, A1, A1, A1, B1,B1 [2] | Eliminating 2 variables from 3 equations; Obtaining a quadratic in x or y; Solving their quadratic correctly; A pair of x values or 1 x and 1 y..; m=2,-2 also needed for final mark
**(iii)** $(x-2)^2 + 1, (2, 1)$ | B1,B1 [2]
---\begin{enumerate}[label=(\roman*)]
\item A straight line passes through the point $(2, 0)$ and has gradient $m$. Write down the equation of the line. [1]
\item Find the two values of $m$ for which the line is a tangent to the curve $y = x^2 - 4x + 5$. For each value of $m$, find the coordinates of the point where the line touches the curve. [6]
\item Express $x^2 - 4x + 5$ in the form $(x + a)^2 + b$ and hence, or otherwise, write down the coordinates of the minimum point on the curve. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2011 Q7 [9]}}