CAIE P1 2014 June — Question 11 9 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2014
SessionJune
Marks9
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Mark schemeDownload PDF ↗
TopicStraight Lines & Coordinate Geometry
TypeRectangle or parallelogram vertices
DifficultyStandard +0.8 This question requires finding the intersection of two lines to get A, then using the property that diagonals of a parallelogram bisect each other at E to find C from A and subsequently B and D. It involves multiple coordinate geometry techniques (line intersection, midpoint formula, parallelogram properties) across several steps, making it moderately challenging but still within standard A-level scope.
Spec1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships
11 \includegraphics[max width=\textwidth, alt={}, center]{0b047754-84f2-46ea-b441-7c68cef47641-4_995_867_260_639} The diagram shows a parallelogram \(A B C D\), in which the equation of \(A B\) is \(y = 3 x\) and the equation of \(A D\) is \(4 y = x + 11\). The diagonals \(A C\) and \(B D\) meet at the point \(E \left( 6 \frac { 1 } { 2 } , 8 \frac { 1 } { 2 } \right)\). Find, by calculation, the coordinates of \(A , B , C\) and \(D\).
AnswerMarks Guidance
Sim eqns \(\rightarrow A(1, 3)\)M1 A1 co Allow answer only B2
Vectors or mid-point \(\rightarrow C(12, 14)\)M1 A1✓ Allow answer only B2✓
Eqn of \(BC\) \(4y = x + 44\) or \(CD\) \(y = 3x - 22\)M1 equation ok – unsimplified
Sim eqns \(\rightarrow B(4, 12)\) or \(D(9, 5)\)DM1A1 Sim eqns. co
Vectors or mid-point \(\rightarrow B(4, 12)\) or \(D(9, 5)\)DM1A1 Valid method (or sim eqns) co
[9] 
Sim eqns $\rightarrow A(1, 3)$ | M1 A1 | co Allow answer only B2 |
Vectors or mid-point $\rightarrow C(12, 14)$ | M1 A1✓ | Allow answer only B2✓ |
Eqn of $BC$ $4y = x + 44$ or $CD$ $y = 3x - 22$ | M1 | equation ok – unsimplified |
Sim eqns $\rightarrow B(4, 12)$ or $D(9, 5)$ | DM1A1 | Sim eqns. co |
Vectors or mid-point $\rightarrow B(4, 12)$ or $D(9, 5)$ | DM1A1 | Valid method (or sim eqns) co |
| | [9] |
11\\
\includegraphics[max width=\textwidth, alt={}, center]{0b047754-84f2-46ea-b441-7c68cef47641-4_995_867_260_639}

The diagram shows a parallelogram $A B C D$, in which the equation of $A B$ is $y = 3 x$ and the equation of $A D$ is $4 y = x + 11$. The diagonals $A C$ and $B D$ meet at the point $E \left( 6 \frac { 1 } { 2 } , 8 \frac { 1 } { 2 } \right)$. Find, by calculation, the coordinates of $A , B , C$ and $D$.

\hfill \mbox{\textit{CAIE P1 2014 Q11 [9]}}
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