1. MODUL BIMBINGAN EMaS 2007 ADDITIONAL MATHEMATICS FORM 4
2007 All Rights Reserved JABATAN PELAJARAN TERENGGANU 1
ADDITIONAL MATHEMATICS
FORM 4
MODULE 2
QUADRATIC EQUATIONS
QUADRATIC FUNCTIONS
PANEL
EN. KAMARUL ZAMAN BIN LONG – SMK SULTAN SULAIMAN, K. TRG.
EN. MOHD. ZULKIFLI BIN IBRAHIM – SMK KOMPLEKS MENGABANG TELIPOT, K. TRG
EN. OBAIDILLAH BIN ABDULLAH – SM TEKNIK TERENGGANU, K. TRG
PUAN NORUL HUDA BT. SULAIMAN – SM SAINS KUALA TERENGGANU, K. TRG.
PUAN CHE ZAINON BT. CHE AWANG – SBP INTEGRASI BATU RAKIT, K. TRG.
MODUL KECEMERLANGAN AKADEMIK
TERENGGANU TERBILANG 2007
PROGRAM PRAPEPERIKSAAN SPM

2. MODUL BIMBINGAN EMaS 2007 ADDITIONAL MATHEMATICS FORM 4
2007 All Rights Reserved JABATAN PELAJARAN TERENGGANU 2
2 QUADRATIC EQUATIONS
 PAPER 1
1 One of the roots of the quadratic equation 2x2
+ kx – 3 = 0 is 3, find the value of k.
Answer : k = …………….…………….
2 Given that the roots of the quadratic equation x2
– hx + 8 = 0 are p and 2p, find the values of h.
Answer : h = …………………………
3 Given that the quadratic equation x
2
+ (m – 3)x = 2m – 6 has two equal roots, find the values
of m.
Answer : m = …………………………
4 Given that one of the roots of the quadratic equation 2x2
+ 18x = 2 – k is twice the other root, find
the value of k.
Answer : k = …………………………
5 Find the value of p for which 2y + x = p is a tangent to the curve y2
+ 4x = 20.

3. MODUL BIMBINGAN EMaS 2007 ADDITIONAL MATHEMATICS FORM 4
2007 All Rights Reserved JABATAN PELAJARAN TERENGGANU 3
Answer : p = …………………………
6 Solve the equation 2(3x – 1)2
= 18.
Answer : …..…………………………
7 Solve the equation (x + 1)(x – 4) = 7. Give your answer correct to 3 significant figures.
Answer : …..…………………………
8 Find the range of values of m such that the equation 2x
2
– x = m – 2 has real roots.
Answer : …..…………………………
9 Find the range of values of x for which (2x + 1)(x + 3) > (x + 3)(x – 3).

4. MODUL BIMBINGAN EMaS 2007 ADDITIONAL MATHEMATICS FORM 4
2007 All Rights Reserved JABATAN PELAJARAN TERENGGANU 4
Answer : …..…………………………
10 Find the range of values of k such that the quadratic equation x2
+ x + 8 = k(2x – k) has two real
roots.
Answer : …..…………………………
 PAPER 2
11 The quadratic equation xqpxpx 10222
 has roots
1
p
and q.
(a) Find the values of p and q.
(b) Hence, form a quadratic equation which has the roots p and 3q.
12 (a) Given that and are the roots of the quadratic equation 2x2
+ 7x – 6 = 0, form a quadratic
equation with roots (+ 1) and (+ 1).
(b) Find the value of p such that (p – 4)x2
+ 2(2 – p)x + p + 1 = 0 has equal roots. Hence, find the
root of the equation based on the value of p obtained.
13 (a) Given that 2 and m – 1 are the roots of the equation x2
+ 3x = k, find the values of m and k.
(b) Find the range of values of p if the straight line y = px – 5 does not intersect the curve
y = x2
– 1.
14 (a) Given that 3 and m are the roots of the quadratic equation 2(x + 1)(x + 2) = k(x – 1).
Find the values of m and k .
(b) Prove that the roots of the equation x2
+ (2a – 1)x + a2
= 0 is real when a 
1
4
.
15 (a) Find the range of values of p where px2
+ 2(p + 2)x + p + 7 = 0 has real roots.
(b) Given that the roots of the equation x
2
+ px + q = 0 are and 3, show that 3p
2
= 16q.

5. MODUL BIMBINGAN EMaS 2007 ADDITIONAL MATHEMATICS FORM 4
2007 All Rights Reserved JABATAN PELAJARAN TERENGGANU 5
3 QUADRATIC FUNCTIONS
 PAPER 1
1 Solve the inequality 2(x – 3)2
> 8.
Answer : …..…………………………
2 Find the range of values of p which satisfies the inequality 2p2
+ 7p 4.
Answer : …..…………………………
3 Find the range of values of m if the equation (2 – 3m)x2
+ (4 – m)x + 2 = 0 has no real roots.
Answer : …..…………………………
4 The quadratic function 4x2
+ (12 – 4k)x + 15 – 5k = 0 has two different roots, find the range of
values of k.
Answer : …..…………………………

6. MODUL BIMBINGAN EMaS 2007 ADDITIONAL MATHEMATICS FORM 4
2007 All Rights Reserved JABATAN PELAJARAN TERENGGANU 6
5 Without using differentiation method find the minimum value of the function f(x) = 3x2
+ x + 2.
Answer : f (x)min = ……………………
6 Given that g(x) = 3x2
– 2x – 8, find the range of values of x so that g(x) is always positive.
Answer : …..…………………………
7 The expression x2
– x + p, where p is a constant, has a minimum value
9
4
. Find the value of p.
Answer : p = …………………………
8 The quadratic functions 2 3
( ) 3 ( 1)
2
k
f x x  
 
 
 
has a minimum value of 6. Find the value of k.
Answer : k = …………………………

7. MODUL BIMBINGAN EMaS 2007 ADDITIONAL MATHEMATICS FORM 4
2007 All Rights Reserved JABATAN PELAJARAN TERENGGANU 7
9 (a) Express y = 1 + 20x – 2x2
in the form y = a(x + p)2
+ q.
(b) Hence, state
(i) the minimum value of y,
(ii) the corresponding value of x.
Answer : (a) …………….……………..
(b) (i) ……….……………...
(ii) ………………………
10
Jawapan : p = ……………………………
q = ……………………………
r = ……………………………
0
33
(4, 1)


x
y The diagram on the left shows the graph of the curve
2
( )y p x q r   with the turning point at (4, 1).
Find the values of p, q and r .

8. MODUL BIMBINGAN EMaS 2007 ADDITIONAL MATHEMATICS FORM 4
2007 All Rights Reserved JABATAN PELAJARAN TERENGGANU 8
 PAPER 2
11 Given the function f (x) = 7 mx x2
= 16 (x + n)2
for all real values of x where m and n are
positive, find
(a) the values of m and n,
(b) the maximum point of f(x),
(c) the range of values of x so that f(x) is negative. Hence, sketch the graph of f(x) and state the
axis of symmetry.
12 Given that the quadratic function f (x) = –2x2
– 12x – 23,
(a) express f (x) in the form m(x + n)2
+ p, where m, n and p are constants.
(b) Determine whether the function f(x) has the minimum or maximum value and state its value.
13 Given that x2
– 3x + 5 = p(x – h)2
+ k for all real values of x, vhere p, h and k are constants.
(a) State the values of p, h and k,
(b) Find the minimum or maximum value of x2
– 3x + 5 and the corresponding value of x.
(c) Sketch a graph of f (x) = x2
– 3x + 5.
(d) Find the range of values of m such that the equation x2
– 3x + 5 = 2m has two different roots.