SPM Additional Mathematics (Model Test Paper)


SPM Additional Mathematics (Model Test Paper)

Section A
[40 marks]
 Answer all questions.


Question 1
Solve the simultaneous equations:
2(xy) = x + y – 1 = 2x2 – 11y2                                                                                                              [5 marks]

Answer and Solution:
2(xy) = x + y – 1
2x – 2y = x + y – 1
x= 3y – 1 —- (1)

x + y – 1 = 2x2 – 11y2
2x2– 11y2 x y+ 1 = 0 —- (2)

Substitute (1) into (2):
2(3y – 1)2– 11y2 – (3y – 1) y + 1 = 0
2(9y2– 6y + 1) – 11y2 – 3y + 1 y+ 1 = 0
18y2– 12y + 2 – 11y2 – 4y + 2 = 0
7y2– 16y + 4 = 0
(7y – 2)(y – 2) = 0
y= 2 7 or y=2

From (1):
when y= 2 7 , x=3( 2 7 )1 x= 6 7 1= 1 7

When y = 2,
x= 3(2) – 1 = 5

Hence the solutions are, x= 1 7 ,y= 2 7 or x=5,y=2



Question 2
Given that f:x 5x mx3 ,x 3 2 and g:x34x. Find
(a)    the value of m,                                                                                                        [2 marks]
(b)   gf-1(–2),                                                                                                                 [3 marks]
(c)    function h if hg (x) = 12x + 5                                                                                 [3 marks]

Answer and Solution:
(a)
mx3=0 m( 3 2 )3=0 m=3× 2 3 =2

(b)
Let y= 5x 2x3 2xy3y=5x 2xy5x=3y x( 2y5 )=3y x= 3y 2y5 f 1 ( y )= 3y 2y5 f 1 ( x )= 3x 2x5 f 1 ( 2 )= 3( 2 ) 2( 2 )5 = 2 3

g f 1 ( 2 )=g( 2 3 ) =34( 2 3 ) = 1 3

(c)
hg (x) = 12x + 5
h [g(x)] = 12x + 5
h(3 – 4x) = 12x + 5
Let u = 3 – 4x
x= 3u 4 h( u )=12( 3u 4 )+5 h( u )=93u+5 h( u )=143u h( x )=143x


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