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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.

[2]
a.

The line L2 has equation r=(241)+s(215) . The point A has position vector (629) .

Show that A lies on L2 .

[4]
b.

Let B be the point of intersection of lines L1 and L2 .

(i)     Show that OB=(8114) .

(ii)    Find AB .

[7]
c.

The point C is at (2, 1, − 4). Let D be the point such that ABCD is a parallelogram.

Find OD .

[3]
d.

Markscheme

L1:r=(810)+t(001)     A2     N2

[2 marks]

a.

evidence of equating r and OA     (M1)

e.g. (629)=(241)+s(215) , A=r

one correct equation     A1

e.g. 6=2+2s , 2=4s , 9=1+5s

s=2     A1

evidence of confirming for other two equations     A1

e.g. 6=2+4 , 2=42 , 9=1+10

so A lies on L2      AG     N0

[4 marks]

b.

(i) evidence of approach     M1

e.g. (241)+s(215)=(810)+t(001) L1=L2

one correct equation     A1

e.g. 2+2s=8 , 4s=1 , 1+5s=t

attempt to solve     (M1)

finding s=3     A1

substituting     M1

e.g. OB=(241)+3(215)

OB=(8114)     AG     N0

(ii) evidence of appropriate approach     (M1)

e.g. AB=AO+OB , AB=OBOA

AB=(215)     A1     N2  

[7 marks]

c.

evidence of appropriate approach    (M1)

e.g. AB=DC

correct values     A1

e.g. OD+(215)=(214) ,  (xyz)+(215)=(214) , (215)=(2x1y4z)

OD=(029)     A1     N2

[3 marks]

d.

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.

a.

For (b), many found the correct parameter but neglected to confirm it in the other two equations.

b.

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.

c.

A good number of candidates had an appropriate approach to (d), although surprisingly many subtracted OC from AB in finding OD .

d.

Syllabus sections

Topic 4 - Vectors » 4.3 » Vector equation of a line in two and three dimensions: r=a+tb .
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