CAIE P1 2013 November — Question 5 7 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2013
SessionNovember
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStraight Lines & Coordinate Geometry
TypeRectangle or parallelogram vertices
DifficultyStandard +0.3 This is a straightforward coordinate geometry problem requiring finding the intersection of line AC with the perpendicular from B, then using vector/midpoint properties of rectangles. It involves standard techniques (perpendicular gradients, simultaneous equations, midpoint formula) with no novel insight required, making it slightly easier than average.
Spec1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships
5 \includegraphics[max width=\textwidth, alt={}, center]{d5f66324-e1fc-40e1-98e7-625187e24d3d-3_636_811_255_667} The diagram shows a rectangle \(A B C D\) in which point \(A\) is ( 0,8 ) and point \(B\) is ( 4,0 ). The diagonal \(A C\) has equation \(8 y + x = 64\). Find, by calculation, the coordinates of \(C\) and \(D\).
AnswerMarks Guidance
\(A(0, 8)\) \(B(4, 0)\) \(8y + x = 33\); \(m\) of \(AB = -2\); \(m\) of \(BC = \frac{1}{2}\); Eqn \(BC \to y - 0 = \frac{1}{2}(x - 4)\); Sim eqns \(\to\) \(C(16, 6)\)B1, M1, M1, M1 A1 [5] Use of \(m_1m_2 = -1\) for \(BC\) or \(AD\). Correct method for equation of \(BC\). Sim Eqns for \(BC\), \(AC\).
Vector step method \(\to\) \(D(12, 14)\) (or \(AD\) \(y = \frac{1}{2}x + 8\), \(CD\) \(y = -2x + 38\)) (or \(M = (8, 7) \to D = (12, 14)\))M1 A1 [7] M1 valid method. 
$A(0, 8)$ $B(4, 0)$ $8y + x = 33$; $m$ of $AB = -2$; $m$ of $BC = \frac{1}{2}$; Eqn $BC \to y - 0 = \frac{1}{2}(x - 4)$; Sim eqns $\to$ $C(16, 6)$ | B1, M1, M1, M1 A1 [5] | Use of $m_1m_2 = -1$ for $BC$ or $AD$. Correct method for equation of $BC$. Sim Eqns for $BC$, $AC$.

Vector step method $\to$ $D(12, 14)$ (or $AD$ $y = \frac{1}{2}x + 8$, $CD$ $y = -2x + 38$) (or $M = (8, 7) \to D = (12, 14)$) | M1 A1 [7] | M1 valid method.
5\\
\includegraphics[max width=\textwidth, alt={}, center]{d5f66324-e1fc-40e1-98e7-625187e24d3d-3_636_811_255_667}

The diagram shows a rectangle $A B C D$ in which point $A$ is ( 0,8 ) and point $B$ is ( 4,0 ). The diagonal $A C$ has equation $8 y + x = 64$. Find, by calculation, the coordinates of $C$ and $D$.

\hfill \mbox{\textit{CAIE P1 2013 Q5 [7]}}
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