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 ) x 2 x+7 ……… ( 1 ) y=3x+5 ……………………… ( 2 ) Substitute ( 1 ) into ( 2 ): ( p2 ) x 2 x+7=3x+5 ( p2 ) x 2 4x+2=0 a=( p2 ), b=4, c=2 b 2 4ac>0 ( 4 ) 2 4( p2 )( 2 )>0 168p+16>0 8p>32 8p<32 p<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:
h x 2 3x+k=0 a=h, b=3, c=k SOR= b a = ( 3 ) h = 3 h POR= c a = k h Given roots=β and 2β. SOR=β+2β=3β;  POR=β( 2β )=2 β 2 3 h =3β β= 1 h  ………….. ( 1 ) k h =2 β 2  ……….. ( 2 ) Substitute ( 1 ) into ( 2 ): k h =2 ( 1 h ) 2 k h = 2 h 2 h 2 h = 2 k h= 2 k


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+1 2 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:x x+1 2

(b)

g( x )= x+1 2 fg( x )= x 2 +2x+4 f[ g( x ) ]= x 2 +2x+4 f( x+1 2 )= x 2 +2x+4 Let  x+1 2 =y x+1=2y x=2y1 f( y )= ( 2y1 ) 2 +2( 2y1 )+4 f( y )=4 y 2 4y+1+4y2+4 f( y )=4 y 2 +3 f( x )=4 x 2 +3 Thus, function which maps set B to set C is f( x )=4 x 2 +3

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