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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 + 5xx2 is negative.

Solution:
(a)
f(x) < 0
6 + 5xx2 < 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)xp2 = 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)
x2+(p+3)xp2=0a=1, b=p+3, c=p2Root 1=α, Root 2=αSOR=baα+(α)=(p+3)1(p+3)=0p+3=0p=3POR=ca=p21=(3)2=9

(b)
mx25nx+4m=0a=m, b=5n, c=4mb2=4ac(5n)2=4(m)(4m)25n2=16m2m2n2=2516(mn)2=(54)2mn=54m:n=5:4

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