| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2003 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Perpendicular line through point |
| Difficulty | Moderate -0.8 This is a straightforward two-part question testing basic coordinate geometry: finding a perpendicular line equation (requiring gradient manipulation m₁ × m₂ = -1 and point-slope form) and then finding intersection point and distance. These are standard textbook exercises with routine procedures and no problem-solving insight required, making it easier than average but not trivial since it requires multiple steps and careful arithmetic. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships1.10f Distance between points: using position vectors |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Gradient of \(L_1 = -2\); Gradient of \(L_2 = 1/2\); Eqn of \(L_2\): \(y - 4 = 1/2(x - 7)\) | B1, M1, M1A1∨ [4] | Co – anywhere. Use of \(m_1m_2 = -1\). Use of line eqn – or \(y = mx + c\). Line must be through (7, 4) and non-parallel |
| (ii) Sim Eqns \(\rightarrow x = 3, y = 2\) | M1, A1 [2] | Solution of 2 linear eqns. Co |
| \(AB = \sqrt{(2^2 + 4^2)} = \sqrt{20}\) or 4.47 | M1A1 [4] | Correct use of distance formula. Co |
**(i)** Gradient of $L_1 = -2$; Gradient of $L_2 = 1/2$; Eqn of $L_2$: $y - 4 = 1/2(x - 7)$ | B1, M1, M1A1∨ [4] | Co – anywhere. Use of $m_1m_2 = -1$. Use of line eqn – or $y = mx + c$. Line must be through (7, 4) and non-parallel
**(ii)** Sim Eqns $\rightarrow x = 3, y = 2$ | M1, A1 [2] | Solution of 2 linear eqns. Co
$AB = \sqrt{(2^2 + 4^2)} = \sqrt{20}$ or 4.47 | M1A1 [4] | Correct use of distance formula. Co7 The line $L _ { 1 }$ has equation $2 x + y = 8$. The line $L _ { 2 }$ passes through the point $A ( 7,4 )$ and is perpendicular to $L _ { 1 }$.\\
(i) Find the equation of $L _ { 2 }$.\\
(ii) Given that the lines $L _ { 1 }$ and $L _ { 2 }$ intersect at the point $B$, find the length of $A B$.
\hfill \mbox{\textit{CAIE P1 2003 Q7 [8]}}