modul 2 add maths 07

MODUL BIMBINGAN EMaS 2007 ADDITIONAL MATHEMATICS FORM 4

2007 All Rights Reserved JABATAN PELAJARAN TERENGGANU 1

ADDITIONAL MATHEMATICSFORM 4

MODULE 2QUADRATIC EQUATIONSQUADRATIC FUNCTIONS

PANEL

EN. KAMARUL ZAMAN BIN LONG – SMK SULTAN SULAIMAN, K. TRG.EN. MOHD. ZULKIFLI BIN IBRAHIM – SMK KOMPLEKS MENGABANG TELIPOT, K. TRGEN. OBAIDILLAH BIN ABDULLAH – SM TEKNIK TERENGGANU, K. TRGPUAN NORUL HUDA BT. SULAIMAN – SM SAINS KUALA TERENGGANU, K. TRG.PUAN CHE ZAINON BT. CHE AWANG – SBP INTEGRASI BATU RAKIT, K. TRG.

MODUL KECEMERLANGAN AKADEMIKTERENGGANU TERBILANG 2007

PROGRAM PRAPEPERIKSAAN SPM



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 x2 + (m – 3)x = 2m – 6 has two equal roots, find the valuesof m.

Answer : m = …………………………

4 Given that one of the roots of the quadratic equation 2x2 + 18x = 2 – k is twice the other root, findthe 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.



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 2x2 – 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).



Answer : …..…………………………

10 Find the range of values of k such that the quadratic equation x2 + x + 8 = k(2x – k) has two realroots.

Answer : …..…………………………

PAPER 2

11 The quadratic equation xqpxpx 10222 has roots1

pand 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 quadraticequation 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 theroot 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 curvey = 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 x2 + px + q = 0 are and 3, show that 3p2 = 16q.



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 ofvalues of k.

Answer : …..…………………………



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 value9

4. Find the value of p.

Answer : p = …………………………

8 The quadratic functions 2 3( ) 3 ( 1)

2

kf x x

has a minimum value of 6. Find the value of k.

Answer : k = …………………………


2007 All Rights Reserved JABATAN PELAJARAN TE

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

0

33

(4, 1)

x

y
The diagram on the left shows the graph of the curve2( )y p x q r with the turning point at (4, 1).
RENGGANU 7

Jawapan : p = ……………………………

q = ……………………………

r = ……………………………

Find the values of p, q and r .



PAPER 2

11 Given the function f (x) = 7 mxx2 = 16 (x + n)2 for all real values of x where m and n arepositive, 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.

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modul 2 add maths 07