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SPM Additional Mathematics 2017, Paper 1 (Question 11 – 13)


Question 11 (4 marks):
The quadratic function f is defined by f(x) = x2 + 4x + h, where h is a constant.
(a) Express f(x) in the form (x + m)2 + n, where m and n are constants.

(b) Given the minimum value of f(x) is 8, find the value of h.

Solution:
(a)
f(x) = x2 + 4x + h
   = x2 + 4x + (2)2 – (2)2 + h
   = (x + 2)2 – 4 + h

(b)
Given the minimum value of f(x) = 8
– 4 + h = 8
h = 12


Question 12 (3 marks):
Find the range of values of x such that the quadratic function f(x) = 6 + 5x – x2 is negative.

Solution:
(a)
f(x) < 0
6 + 5x – x2 < 0
(6 – x)(x + 1) < 0
x < –1, x > 6




Question 13 (4 marks):
(a) It is given that one of the roots of the quadratic equation x2 + (p +3)x – p2 = 0, where p is a constant, is negative of the other.
Find the value of the product of roots.

(b) It is given that the quadratic equation mx2 – 5nx + 4m = 0, where m and n are constants, has two equal roots.
Find m : n.


Solution:
(a)
x 2 +( p+3 )x– p 2 =0 a=1, b=p+3, c=− p 2 Root 1=α, Root 2=−α SOR=− b a α+( −α )=− ( p+3 ) 1 −( p+3 )=0 p+3=0 p=−3 POR= c a = − p 2 1 =− ( −3 ) 2 =−9

(b)
m x 2 −5nx+4m=0 a=m, b=−5n, c=4m b 2 =4ac ( −5n ) 2 =4( m )( 4m ) 25 n 2 =16 m 2 m 2 n 2 = 25 16 ( m n ) 2 = ( 5 4 ) 2 m n = 5 4 m:n=5:4

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