\

SPM Additional Mathematics 2023, Paper 1 (Question 6 & 7)


Question 6:
 (a) Diagram 5 shows the graph of the function f(x)=|x3|+1 for domain 2x7

 Find the value of the other object of h. [2 marks]
 (b) Given that the functions f:x4x1 and g:x2x+3, find 
 (i) the value of p if f 1(p)=2
 (ii) the value of x if fg(x)=5f(2). [4 marks]

Solution:
(a)
f(7)=h|73|+1=h|4|+1=hh=5
f(x)=5|x3|+1=5|x3|=4
x3=4x=7
x3=4x=1
 Object of h is 1

(b)(i)
f(x)=4x1 Let 4x1=y4x=y+1x=y+14f1(x)=x+14
f1(p)=2p+14=2p+1=8p=7

(b)(ii)
fg(x)=5f(2)f(2x+3)=5[4(2)1]4(2x+3)1=5(81)8x+121=5(7)8x+11=358x=24x=3

Question 7:
(a) Diagram 6 shows a curve and a tangent to the curve.

Express p in terms of c.
[3 marks]

(b) Find the range of values of x for (x + 1)(-3x – 3) < (x – 1) by using the table method.
[2 marks]


(c) It is given that p and 2p are the roots of the quadratic equation 2x2 + 6x + 4p2 = 0.
Find the quadratic equation with roots (p – 1) and (p + 1).
[3 marks]

Solution:
(a)
y=x+c(1)y=2x2+3x+p(2)(1)=(2):2x2+3x+p=x+c2x2+3x+p+xc=02x24xp+c=0a=2,b=4,c=p+c
 Tangent to the curve, b24ac=0(4)24(2)(p+c)=0168(p+c)=016+8p8c=08p=8c16p=c2

(b)
(x + 1)(-3x – 3) < x – 1
-3x2 – 3x – 3x – 3 – x + 1 < 0
-3x2 – 7x -2 < 0
3x2 – 7x – 2 > 0
(3x + 1)(x + 2) > 0
When (3x + 1)(x + 2) = 0,
x = -1/3 or   -2


From the table, 
(3x + 1)(x + 2) > 0  when  x < -2  or  x > -1/3
x < -2   or  x > -1/3


(c)
2x2+6x+4p2=0 Roots =p,2p SOR: p+2p=623p=3p=1

 New roots: p1=11,=2p+1=1+1=0
 SOR: 2+0=2 POR: (2)(0)=0
x2(SOR)x+(POR)=0x2(2)x+0=0x2+4x=0

Leave a Comment