O-LevelAdditional MathematicsStraight line graphs4037/22 · May/June 2015 · Q912 Marks

Solutions by accurate drawing will not be accepted. [Figure 9.1] The line y = mx + 4 meets the lines x = 2 and x = -1 at the points P and Q respectively. The point R is such that QR is parallel to the y-axis and the gradient of RP is 1. The point P has coordinates (2, 10).

(i)Find the value of m.[2]
(ii)Find the y-coordinate of Q.[1]
(iii)Find the coordinates of R.[2]
(iv)Find the equation of the line through P, perpendicular to PQ, giving your answer in the form ax + by = c, where a, b and c are integers.[3]
(v)Find the coordinates of the midpoint, M, of the line PQ.[2]
(vi)Find the area of triangle QRM.[2]

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Mark scheme

Marking points from the official mark scheme (May/June 2015). The mark or mark type is shown in brackets.

[M1] 10 = 2m + 4 soi or [m =](10-4)/(2-0) oe soi
[A1] m = 3
[B1] 1 (for y-coordinate of Q)
[M1] 10 - yR / (2 - (-1)) = 1 oe soi or y = x + 8 oe
[A1] (-1, 7) or x = -1 and y = 7 if y = 7 only stated, provided that x = -1 is soi in working allow both marks if M0 then B1 for y = 7 only with no working
[M1] Use of m₁m₂ = -1 with their m from (i) may be implied by perpendicular gradient seen in equation
[A1] y - 10 = (-(1/3))(x - 2) or y = (-(1/3))x + c and 10 = (-(1/3))(2)+c
[A1] 3y + x = 32 isw allow for correct equation with integer coefficients in any simplified form
[B1,B1ft] (1/2, 11/2) oe isw ft their yQ
or M1 for ((2+(-1))/2, (10+1)/2) seen
[B2] 4.5 oe cao not from wrong working
or M1 for any correct method with correct coordinates

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