CAIE P1 2014 June — Question 7 6 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2014
SessionJune
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStraight Lines & Coordinate Geometry
TypeParameter from distance condition
DifficultyModerate -0.3 This is a straightforward coordinate geometry problem requiring simultaneous equations from two standard formulas (distance and gradient). Students apply the gradient formula to get b = 2a - 4, substitute into the distance formula to get a quadratic, then solve for both variables. While it involves multiple steps, the techniques are routine and the algebraic manipulation is standard for A-level, making it slightly easier than average.
Spec1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.10c Magnitude and direction: of vectors
7 The coordinates of points \(A\) and \(B\) are \(( a , 2 )\) and \(( 3 , b )\) respectively, where \(a\) and \(b\) are constants. The distance \(A B\) is \(\sqrt { } ( 125 )\) units and the gradient of the line \(A B\) is 2 . Find the possible values of \(a\) and of \(b\).
AnswerMarks Guidance
\((a - 3)^2 + (2 - b)^2 = 125\) oeB1
\(\frac{2 - b}{a - 3} = -2\) oeB1
\((a - 3)^2 + (2a - 6)^2 = 125\) (sub for \(a\) or \(b\))M1 Or \(1/4(2 - b)^2 + (2 - b)^2 = 125\) Or \((5)(b - 12)(b + 8) (= 0)\) Answers (no working) after 2 correct eqns score SCB1B1 for each correct pair (\(a, b\))
\((5)(a + 2)(a - 8) (= 0)\) Attempt factorise/solveM1 A1A1 [6]
\(a = -2\) or \(8\), \(b = 12\) or \(-8\) 
$(a - 3)^2 + (2 - b)^2 = 125$ oe | B1 | 

$\frac{2 - b}{a - 3} = -2$ oe | B1 | 

$(a - 3)^2 + (2a - 6)^2 = 125$ (sub for $a$ or $b$) | M1 | Or $1/4(2 - b)^2 + (2 - b)^2 = 125$ Or $(5)(b - 12)(b + 8) (= 0)$ Answers (no working) after 2 correct eqns score SCB1B1 for each correct pair ($a, b$)

$(5)(a + 2)(a - 8) (= 0)$ Attempt factorise/solve | M1 A1A1 [6] |

$a = -2$ or $8$, $b = 12$ or $-8$ |
7 The coordinates of points $A$ and $B$ are $( a , 2 )$ and $( 3 , b )$ respectively, where $a$ and $b$ are constants. The distance $A B$ is $\sqrt { } ( 125 )$ units and the gradient of the line $A B$ is 2 . Find the possible values of $a$ and of $b$.

\hfill \mbox{\textit{CAIE P1 2014 Q7 [6]}}
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