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8.7.2 Vector, Long Questions (Question 3 & 4)


Question 3:
In diagram below, PQRS is a quadrilateral. PTS and TUR are straight lines.
 
 
It is given that PQ=20x˜,  PT=8y˜,  SR=25x˜24y˜,  PT=14PS  and  TU=35TR
(a) Express in terms of x˜ and/or   y˜ :
 (i)   QS
 (ii) TR
(b) Show that  the points  Q, U and S  are collinear.
(c) If   |x˜| = 2  and |y˜| = 3, find   |QS|



Solution:

(a)(i)
QS=QP+PSQS=20x˜+32y˜Given PT=14PSPS=4PT=4(8y˜)=32y˜


(a)(ii)
TR=TS+SRTR=34PS+25x˜24y˜TR=34(32y˜)+25x˜24y˜TR=24y˜+25x˜24y˜TR=25x˜


(b)
QU=QP+PT+TUQU=20x˜+8y˜+35(25x˜)GivenTU=35TRQU=20x˜+8y˜+15x˜QU=5x˜+8y˜From (a)(i) QS=20x˜+32y˜QSQU=20x˜+32y˜5x˜+8y˜QSQU=4(5x˜+8y˜)(5x˜+8y˜)QSQU=4QS=4QU Q, U and S are collinear.


(c)









PS=32y˜|PS|=32|y˜||PS|=32×3=96PQ=20x˜|PQ|=20|x˜||PQ|=20×2=40|QS|=962+402|QS|=104


Question 4:
Diagram below shows quadrilateral ABCD. The straight line AC intersects the straight line BD at point E.

It is given that BE:ED=2:3, AB=10x˜, AD=25y˜ and BC=x˜+15y˜.(a) Express in terms of x˜ and y˜,(i) BD(ii) AE(b) Find the ratio AE:EC.


Solution:
(a)(i)
BD=BA+AD  =ADAB  =25y˜10x˜

(a)(ii)
AE=AB+BE  =AB+25BD  =10x˜+25(25y˜10x˜)  =10x˜+25(25y˜10x˜)  =10x˜+10y˜4x˜  =6x˜+10y˜  =2(3x˜+5y˜)

(b)
EC=EB+BC  =BCBE  =BC23ED  =BC23(EA+AD)  =x˜+15y˜23(6x˜10y˜+25y˜)  =x˜+15y˜23(6x˜+15y˜)  =x˜+15y˜+4x˜10y˜  =3x˜+5y˜AEEC=2(3x˜+5y˜)1(3x˜+5y˜)AE:EC=2:1

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