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Date May 2008 Marks available 4 Reference code 08M.1.sl.TZ2.8
Level SL only Paper 1 Time zone TZ2
Command term Find Question number 8 Adapted from N/A

Question

Consider the points A (1 , 5 , 4) , B (3 , 1 , 2) and D (3 , k , 2) , with (AD) perpendicular to (AB) .

The point O has coordinates (0 , 0 , 0) , point A has coordinates (1 , – 2 , 3) and point B has coordinates (– 3 , 4 , 2) .

Find

(i)     AB ;

(ii)    AD giving your answer in terms of k .

[3 marks]

[3]
a(i) and (ii).

Show that k=7 .

[3]
b.

The point C is such that BC=12AD .

Find the position vector of C.

[4]
c.

Find cosAˆBC .

[3]
d.

Markscheme

(i) evidence of combining vectors     (M1)

e.g. AB=OBOA  (or AD=AO+OD in part (ii)) 

AB=(242)    A1     N2

(ii) AD=(2k52)     A1     N1

[3 marks]

a(i) and (ii).

evidence of using perpendicularity scalar product = 0     (M1)

e.g. (242)(2k52)=0

44(k5)+4=0     A1

4k+28=0 (accept any correct equation clearly leading to k=7 )    A1

k=7     AG     N0

[3 marks]

b.

AD=(222)     (A1)

 BC=(111)    A1

evidence of correct approach     (M1)

e.g. OC=OB+BC , (312)+(111) , (x3y1z2)=(111)

OC=(421)     A1     N3

[4 marks]

 

c.

METHOD 1

choosing appropriate vectors, BA , BC     (A1)

finding the scalar product     M1

e.g. 2(1)+4(1)+2(1) , 2(1)+(4)(1)+(2)(1)

cosAˆBC=0     A1     N1

METHOD 2

BC parallel to AD (may show this on a diagram with points labelled)     R1

BCAB (may show this on a diagram with points labelled)     R1

AˆBC=90

cosAˆBC=0     A1     N1

[3 marks]

d.

Examiners report

This question was well done by many candidates. Most found AB and AD correctly.

a(i) and (ii).

The majority of candidates correctly used the scalar product to show k=7 .

b.

Some confusion arose in substituting k=7 into AD , but otherwise part (c) was well done, though finding the position vector of C presented greater difficulty.

c.

Owing to AB and BC being perpendicular, no problems were created by using these two vectors to find cosAˆBC=0 , and the majority of candidates answering part (d) did exactly that.

d.

Syllabus sections

Topic 4 - Vectors » 4.1 » Algebraic and geometric approaches to the sum and difference of two vectors; the zero vector, the vector v.

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