Question 20 (3 marks):
It is given that the curve y = (p – 2)x2 – x + 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=(p−2)x2−x+7 ……… (1)y=3x+5 ……………………… (2)Substitute (1) into (2):(p−2)x2−x+7=3x+5(p−2)x2−4x+2=0a=(p−2), b=−4, c=2b2−4ac>0(−4)2−4(p−2)(2)>016−8p+16>0−8p>−328p<32p<4
It is given that the curve y = (p – 2)x2 – x + 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=(p−2)x2−x+7 ……… (1)y=3x+5 ……………………… (2)Substitute (1) into (2):(p−2)x2−x+7=3x+5(p−2)x2−4x+2=0a=(p−2), b=−4, c=2b2−4ac>0(−4)2−4(p−2)(2)>016−8p+16>0−8p>−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:
hx2−3x+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
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:
hx2−3x+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 : x → x2 + 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+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=2y−1∴f(y)=(2y−1)2+2(2y−1)+4f(y)=4y2−4y+1+4y−2+4f(y)=4y2+3f(x)=4x2+3Thus, function which maps set B to set Cis f(x)=4x2+3
Diagram 9 shows the relation between set A, set B and set C.

It is given that set A maps to set B by the function x+12 and maps to set C by fg : x → x2 + 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:

g:x→x+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=2y−1∴f(y)=(2y−1)2+2(2y−1)+4f(y)=4y2−4y+1+4y−2+4f(y)=4y2+3f(x)=4x2+3Thus, function which maps set B to set Cis f(x)=4x2+3