SPM Additional Mathematics 2017, Paper 2 (Question 7 – 9)


Question 8:
Diagram 5 shows a triangle ABC. The straight line AE intersects with the straight line BC at point D. Point V lies on the straight line AE.

It is given that  BD = 1 3 BC , AC =6 x ˜  and  AB =9 y ˜ . ( a ) Express in terms of  x ˜  and / or  y ˜ :    ( i )  BC ,    ( ii )  AD . ( b ) It is given that  AV =m AD  and  BV =n( x ˜ 9 y ˜ ), where m and n are constants.   Find the value of m and of n. ( c ) Given  AE =h x ˜ +9 y ˜ , where h is a constant, find the value of h.

Solution: 
(a)(i)
BC = BA + AC  =9 y ˜ +6 x ˜  =6 x ˜ 9 y ˜

(a)(ii)
AD = AB + BD  =9 y ˜ + 1 3 BC  =9 y ˜ + 1 3 ( 6 x ˜ 9 y ˜ )  =9 y ˜ +2 x ˜ 3 y ˜  =2 x ˜ +6 y ˜


(b)
Given  AV =m AD =m( 2 x ˜ +6 y ˜ ) =2m x ˜ +6m y ˜ AV = AB + BV    = 9 y ˜ +n( x ˜ 9 y ˜ )   =9 y ˜ +n x ˜ 9n y ˜   =n x ˜ +( 99n ) y ˜ By equating the coefficients of  x ˜  and  y ˜ 2m x ˜ +6m y ˜ =n x ˜ +( 99n ) y ˜ 2m=n n=2m………….( 1 ) 6m=99n………….( 2 ) Substitute (1) into (2), 6m=99( 2m ) 6m=918m 24m=9 m= 9 24 = 3 8 From ( 1 ): n=2( 3 8 )= 3 4


(c)
A, D and E are collinear. AD =k( AE ) AD =k( h x ˜ +9 y ˜ ) 2 x ˜ +6 y ˜ =kh x ˜ +9k y ˜ Equating the coefficients of  y ˜ : 9k=6 k= 6 9 k= 2 3 Equating the coefficients of  x ˜ : kh=2 ( 2 3 )h=2 h=2× 3 2 h=3




Question 9:
Use a graph to answer this question.
Table 1 shows the values of two variables, x and y, obtained from an experiment.
The variables x and y are related by the equation y h = hk x , where h and k are constants.


(a) Plot xy against x, using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the xy-axis.
Hence, draw the line of best fit.
(b) Using the graph in 9(a), find
(i) the value of h and of k,
(ii) the correct value of y if one of the values of y has been wrongly recorded during the experiment.

Solution: 
(a)




(b)
y h = hk x xy h x=hk xy= h x+hk Y=mX+C Y=xy, m= h , C=hk


(b)(i)
m= 36.5 5.1 h = 36.5 5.1 h =7.157 h=51.22 C=4 hk=4 k= 4 h k= 4 51.22 k=0.0781


(b)(ii)
xy=21 3.5y=21 y= 21 3.5 =6.0 Correct value of y is 6.0.


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