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SPM Additional Mathematics 2018, Paper 1 (Question 20 – 22)


Question 20 (3 marks):
It is given that the curve y = (p – 2)x2x + 7, where p is a constant, intersects with the straight line y = 3x + 5 at two points.
Find the range of values of p.

Solution:
y=(p2)x2x+7 ……… (1)y=3x+5 ……………………… (2)Substitute (1) into (2):(p2)x2x+7=3x+5(p2)x24x+2=0a=(p2), b=4, c=2b24ac>0(4)24(p2)(2)>0168p+16>08p>328p<32p<4


Question 21 (3 marks):
It is given that the quadratic equation hx2 – 3x + k = 0, where h and k are constants has roots β and 2β.
Express h in terms of k.

Solution:
hx23x+k=0a=h, b=3, c=kSOR=ba=(3)h=3hPOR=ca=khGiven roots=β and 2β.SOR=β+2β=3β; POR=β(2β)=2β23h=3ββ=1h ………….. (1)kh=2β2 ……….. (2)Substitute (1) into (2):kh=2(1h)2kh=2h2h2h=2kh=2k


Question 22 (4 marks):
Diagram 9 shows the relation between set A, set B and set C.

Diagram 9

It is given that set A maps to set B by the function x+12 and maps to set C by fg : xx2 + 2x + 4.
(a) Write the function which maps set A to set B by using the function notation.
(b) Find the function which maps set B to set C.


Solution:

(a)
g:xx+12

(b)

g(x)=x+12fg(x)=x2+2x+4f[g(x)]=x2+2x+4f(x+12)=x2+2x+4Let x+12=yx+1=2yx=2y1f(y)=(2y1)2+2(2y1)+4f(y)=4y24y+1+4y2+4f(y)=4y2+3f(x)=4x2+3Thus, function which maps set B to set Cis f(x)=4x2+3

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