Vectors, SPM Practice (Paper 2)


Question 7:
Diagram below shows quadrilateral OPBC. The straight line AC intersects the straight line PQ at point B.

It is given that  OP = a ˜ ,  OQ = b ˜ ,  OA =4 AP ,  OC =3 OQ ,  PB =h PQ  and AB =k AC . (a) Express  OB  in terms of h a ˜  and  b ˜ . (b) Express  OB  in terms of k a ˜  and  b ˜ . (c)(i) Find the value of h and of k. (ii) Hence, state  OB  in terms of  a ˜  and  b ˜ .

Solution:
(a)
OB = OP + PB  = a ˜ +h PQ  = a ˜ +h( PO + OQ )  = a ˜ +h( a ˜ + b ˜ )  = a ˜ h a ˜ +h b ˜ OB =( 1h ) a ˜ +h b ˜


(b)
OB = OP + PB  = a ˜ + PA + AB  = a ˜ +( 1 5 OP )+k AC  = a ˜ +( 1 5 a ˜ )+k( AO + OC )  = 4 5 a ˜ +k( 4 5 OP +3 OQ )  = 4 5 a ˜ +k( 4 5 a ˜ +3 b ˜ )  = 4 5 a ˜ 4 5 k a ˜ +3k b ˜ OB = 4 5 ( 1k ) a ˜ +3k b ˜


(c)(i)
( 1h ) a ˜ +h b ˜ = 4 5 ( 1k ) a ˜ +3k b ˜ 1h= 4 5 4 5 k……….( 1 ) h=3k……….( 2 ) Substitute ( 2 ) into the ( 1 )  13k= 4 5 4 5 k 515k=44k 11k=1 k= 1 11 Substitute k= 1 11  into ( 2 ) h=3( 1 11 )   = 3 11


(c)(ii)
OB =( 1h ) a ˜ +h b ˜ when h= 3 11 =( 1 3 11 ) a ˜ +( 3 11 ) b ˜ = 8 11 a ˜ + 3 11 b ˜


Question 8:
Diagram below shows quadrilateral OPQR. The straight line PR intersects the straight line OQ at point S.

It is given that  OP =7 x ˜ ,  OR =5 y ˜ , PS:SR=3:1 and  OR  is parallel to  PQ . (a) Express in terms of  x ˜  and  y ˜ , (i)  PR (ii)  OS (b) Using  PQ =m OR  and  SQ =n OS , where m and n are constants,   Find the value of m and of n. (c) Given that | y ˜ |=4 units and the area of ORS  is 50 cm 2 , find the   perpendicular distance from point S to OR.

Solution:
(a)(i)
PR = PO + OR   =7 x ˜ +5 y ˜


(a)(ii)
OS = OP + PS   =7 x ˜ + 3 4 PR   =7 x ˜ + 3 4 ( 7 x ˜ +5 y ˜ )   =7 x ˜ 21 4 x ˜ + 15 4 y ˜   = 7 4 x ˜ + 15 4 y ˜


(b)
PS = PQ SQ 3 4 PR =m OR n OS 3 4 ( 7 x ˜ +5 y ˜ )=m( 5 y ˜ )n( 7 4 x ˜ + 15 4 y ˜ ) 21 4 x ˜ + 15 4 y ˜ =5m y ˜ 7 4 n x ˜ 15 4 m y ˜ 21 4 x ˜ + 15 4 y ˜ = 7 4 n x ˜ + 5 4 m y ˜ 21 x ˜ +15 y ˜ =7n x ˜ +5m y ˜ 7n=21 n=3 5m=15 m=3


(c)
Area of ΔORS=50 1 2 ×( 5 y ˜ )×t=50 1 2 ×5( 4 )×t=50 10t=50 t=5  Perpendicular distance from point S to OR=5 units.

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