Vectors, SPM Practice (Paper 2)


Question 5:
Diagram below shows a triangle KLM.

It is given that KP:PL=1:2, LR:RM=2:1,  KP =2 x ˜ ,  KM =3 y ˜ . (a) Express in terms of  x ˜  and  y ˜ , (i)  MP (ii)  MR (b) Given  x ˜ =2 i ˜  and  y ˜ = i ˜ +4 j ˜ , find  | MR | . (c) Given  MQ =h MP  and  QR =n KR , where h and n are constants,    find the value of h and of n.

Solution:
(a)(i)
MP = MK + KP   =3 y ˜ +2 x ˜   =2 x ˜ 3 y ˜

(a)(ii)
MR = 1 3 ML   = 1 3 ( MK + KL )   = 1 3 ( 3 y ˜ +6 x ˜ )   =2 x ˜ y ˜

(b)
MR =2( 2 i ˜ )( i ˜ +4 j ˜ )   =4 i ˜ + i ˜ 4 j ˜   =5 i ˜ 4 j ˜ | MR |= 5 2 + ( 4 ) 2    = 41  units

(c)
MQ + QR = MR h MP +n KR = MR h( 2 x ˜ 3 y ˜ )+n( KM + MR )=2 x ˜ y ˜ h( 2 x ˜ 3 y ˜ )+n( 3 y ˜ +2 x ˜ y ˜ )=2 x ˜ y ˜ 2h x ˜ 3h y ˜ +2n x ˜ +2n y ˜ =2 x ˜ y ˜ ( 2h+2n ) x ˜ +( 3h+2n ) y ˜ =2 x ˜ y ˜ 2h+2n=2……….(1) 3h+2n=1……….(2) ( 1 )( 2 ):5h=3  h= 3 5 From ( 1 ):h+n=1 3 5 +n=1    n=1 3 5    n= 2 5


Question 6:
Diagram below shows a trapezium OABC and point D lies on AC.


It is given that  OC =18 b ˜ ,  OA =6 a ˜  and  OC =2 AB . (a) Express in terms of  a ˜  and  b ˜ , (i)  AC (ii)  OB (b) It is given that  AD =k AC , where k is a constant. Find the value of k if the points OD and B are collinear.

Solution
:

(a)(i)
AC = AO + OC       =6 a ˜ +18 b ˜       =18 b ˜ 6 a ˜


(a)(ii)
OC =2 AB 18 b ˜ =2( AO + OB ) 18 b ˜ =2( 6 a ˜ + OB ) 18 b ˜ =12 a ˜ +2 OB OB =6 a ˜ +9 b ˜


(b)
OD =h OB =h( 6 a ˜ +9 b ˜ ) =6h a ˜ +9h b ˜ AD = OD OA =6h a ˜ +9h b ˜ 6 a ˜ = a ˜ ( 6h6 )+9h b ˜ AD =k AC a ˜ ( 6h6 )+9h b ˜ =k( 18 b ˜ 6 a ˜ ) a ˜ ( 6h6 )+9h b ˜ =6k a ˜ +18k b ˜ 6h6=6k h1=k h=1k……….( 1 ) 9h=18k h=2k From ( 1 ), 1k=2k 3k=1 k= 1 3

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