2021 JC1 H2 Math MYA Solution and Comments.pdf - 1 1 A...

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1 TJC H2 JC1 MYA 2021 9758/01 1 A particular country ordered 11 million doses of vaccines to protect its population against rubella, hepatitis A and chicken pox. The country spent 244.25 million dollars in purchasing r million doses of rubella vaccines, a million doses of hepatitis A vaccines and c million doses of chicken pox vaccines. The cost of each dose of rubella vaccine is \$6.50, hepatitis A vaccine \$21 and chicken pox vaccine \$27.50. Furthermore, it is known that the country had purchased 0.5 million more doses of hepatitis A vaccines than of rubella vaccines. Find how much more the country spent on chicken pox vaccines than on the other two types of vaccines combined. [4] Solution Marker’s Comments 1 r + a + c = 11 ----- (1) 6.5 r + 21 a + 27.5 c = 244.25 ----- (2) a = r + 0.5 r + a = 0.5 ----- (3) From GC, 2, 2.5, 6.5 r a c 6.5 (27.5) – 2 (6.5) – 2.5 (21) = 113.25 The country spent \$113.25 million more on chicken pox vaccines than on the other two types combined. Note: If 2 000 000 or 2 million doses of rubella vaccines has been purchased, then r = 2, not 2x10 6 . Need to answer to the question in context.
2 TJC H2 JC1 MYA 2021 9758/01 2 (a) Find the vector u which has a magnitude of 2 units and is in the same direction as the vector i + k , where i , j and k are unit vectors along the x -axis, y -axis and z -axis respectively. State the geometrical relationship between u and the y -axis. [2] (b) The position vectors of points A and B , relative to the origin O , are a and b respectively and the angle AOB is . The point M divides AB in the ratio :1. (i) Write down OM  in terms of , a and b . [1] (ii) The points C and D lie on OA and OB extended respectively such that OC = pOA and OD = qOB , where p and q are real numbers. It is also given that a and b are both unit vectors. If OM is perpendicular to CD , show that cos , where 0. p q q p q p [3] Solution Marker’s Comment (a) 1 1 1 2 0 2 0 1 1 1 1 u Since 1 2 0 1 u is parallel to the x - z plane, hence u is perpendicular to the y –axis. Common errors: 1 1 2 0 or 2 1 1 0 u Many students wrote u incorrectly as vector 1. parallel to the y -axis; 2. lies on plane contain y -axis; or 3. make an angle of 45 degree with the y -axis. (b)(i) 1 ( ) 1 1 b a OM b a  Majority of the students answer this correctly. Common errors include or 1 OM b a b a  Alternative Thinking Unit vector along y -axis = j 1 0 2 0 1 0 1 0 u j , hence u is perpendicular to the y –axis. Did not convert to unit vector. Carelessly write down the vector.
3 TJC H2 JC1 MYA 2021 9758/01 (b)(ii) Given OC p OA   and OD q OB   Since OM is perpendicular to CD , 0 OM CD   1 ( ) ( ) 0 1 b a qb pa 2 2 1 ( ) ( ) 0 1 q b p b a q a b p a 2 2 ( cos ) ( cos ) 0 q b p a b q a b p a     Since 1 (unit vectors) a b ( ) ( )cos 0 q p p q   cos , where 0.