Date | May 2009 | Marks available | 4 | Reference code | 09M.1.sl.TZ2.10 |
Level | SL only | Paper | 1 | Time zone | TZ2 |
Command term | Show that | Question number | 10 | Adapted from | N/A |
Question
The line L1 is parallel to the z-axis. The point P has position vector (810) and lies on L1.
Write down the equation of L1 in the form r=a+tb.
The line L2 has equation r=(24−1)+s(2−15) . The point A has position vector (629) .
Show that A lies on L2 .
Let B be the point of intersection of lines L1 and L2 .
(i) Show that →OB=(8114) .
(ii) Find →AB .
The point C is at (2, 1, − 4). Let D be the point such that ABCD is a parallelogram.
Find →OD .
Markscheme
L1:r=(810)+t(001) A2 N2
[2 marks]
evidence of equating r and →OA (M1)
e.g. (629)=(24−1)+s(2−15) , A=r
one correct equation A1
e.g. 6=2+2s , 2=4−s , 9=−1+5s
s=2 A1
evidence of confirming for other two equations A1
e.g. 6=2+4 , 2=4−2 , 9=−1+10
so A lies on L2 AG N0
[4 marks]
(i) evidence of approach M1
e.g. (24−1)+s(2−15)=(810)+t(001) L1=L2
one correct equation A1
e.g. 2+2s=8 , 4−s=1 , −1+5s=t
attempt to solve (M1)
finding s=3 A1
substituting M1
e.g. →OB=(24−1)+3(2−15)
→OB=(8114) AG N0
(ii) evidence of appropriate approach (M1)
e.g. →AB=→AO+→OB , →AB=→OB−→OA
→AB=(2−15) A1 N2
[7 marks]
evidence of appropriate approach (M1)
e.g. →AB=→DC
correct values A1
e.g. →OD+(2−15)=(21−4) , (xyz)+(2−15)=(21−4) , (2−15)=(2−x1−y−4−z)
→OD=(02−9) A1 N2
[3 marks]
Examiners report
Very few candidates gave a correct direction vector parallel to the z-axis. Provided they wrote down an equation here they were able to earn most of subsequent marks on follow through.
For (b), many found the correct parameter but neglected to confirm it in the other two equations.
In (c) some performed a trial and error approach to obtaining an integer parameter and thus did not "show" the mathematical origin of the result. Finding vector →AB proved accessible.
A good number of candidates had an appropriate approach to (d), although surprisingly many subtracted →OC from →AB in finding →OD .