CAIE P1 2010 June — Question 4 6 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2010
SessionJune
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStraight Lines & Coordinate Geometry
TypeIntersection of two lines
DifficultyModerate -0.3 This is a standard coordinate geometry question requiring finding equations of lines using parallel/perpendicular conditions, then solving simultaneous equations. It involves multiple routine steps (gradient of OB, gradient perpendicular to AB, two line equations, intersection) but each step uses well-practiced techniques with no novel insight required. Slightly easier than average due to being purely procedural.
Spec1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships

\includegraphics{figure_4} In the diagram, \(A\) is the point \((-1, 3)\) and \(B\) is the point \((3, 1)\). The line \(L_1\) passes through \(A\) and is parallel to \(OB\). The line \(L_2\) passes through \(B\) and is perpendicular to \(AB\). The lines \(L_1\) and \(L_2\) meet at \(C\). Find the coordinates of \(C\). [6]

AnswerMarks Guidance
Gradient of \(L_1\) is \(\frac{1}{4}\).
Equation of \(L_1\) is \(y - 3 = \frac{1}{4}(x+1)\)M1 A1 M1 for equation for his \(m\). A1 co.
Gradient of \(AB\) is \(-\frac{1}{4}\). Perp \(= 2\).M1 Use of \(m_m_2 = -1\).
Equation of \(L_2\) is \(y - 1 = 2(x - 3)\).A1 co.
Sim eqns \(3y = x + 10\), \(y = 2x - 5\). \(\rightarrow (5, 5)\)M1, A1 [6] Method of solution. co.
Gradient of $L_1$ is $\frac{1}{4}$. | | 

Equation of $L_1$ is $y - 3 = \frac{1}{4}(x+1)$ | M1 A1 | M1 for equation for his $m$. A1 co.

Gradient of $AB$ is $-\frac{1}{4}$. Perp $= 2$. | M1 | Use of $m_m_2 = -1$.

Equation of $L_2$ is $y - 1 = 2(x - 3)$. | A1 | co.

Sim eqns $3y = x + 10$, $y = 2x - 5$. $\rightarrow (5, 5)$ | M1, A1 [6] | Method of solution. co.
\includegraphics{figure_4}

In the diagram, $A$ is the point $(-1, 3)$ and $B$ is the point $(3, 1)$. The line $L_1$ passes through $A$ and is parallel to $OB$. The line $L_2$ passes through $B$ and is perpendicular to $AB$. The lines $L_1$ and $L_2$ meet at $C$. Find the coordinates of $C$. [6]

\hfill \mbox{\textit{CAIE P1 2010 Q4 [6]}}