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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

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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 arep and 2p, find the values of h.

Answer: h =

3 Given that the quadratic equation x2

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

Answer: m =

4Given that one of the roots of the quadratic equation 2x

2

+ 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 y

2 + 4x = 20.

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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 ofm 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).

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MODUL BIMBINGAN EMaS 2007 ADDITIONAL MATHEMATICS FORM 4

2007 All Rights Reserved JABATAN PELAJARAN TERENGGANU 4

Answer: ..

10 Find the range of values ofk such that the quadratic equationx2 +x + 8 = k(2x k) has two real

roots.

Answer: ..

PAPER 2

11 The quadratic equation xqpxpx 10222 has roots

1

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 quadratic

equation with roots (+ 1) and (+ 1).

(b) Find the value ofp 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 ofp obtained.

13 (a) Given that 2 and m 1 are the roots of the equation x2 + 3x = k, find the values ofm and k.

(b) Find the range of values of p if the straight liney = 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

.

15 (a) Find the range of values of p wherepx2 + 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.

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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 ofp which satisfies the inequality 2p2 + 7p 4.

Answer: ..

3 Find the range of values ofm 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: ..

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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 ofx so that g(x) is always positive.

Answer: ..

7 The expression x2

x + p, where p is a constant, has a minimum value . Find the value of p.

Answer: p =

8 The quadratic functions2 3

( ) 3 ( 1)2

kf x x

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

Answer: k =

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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 ofy,(ii) the corresponding value ofx.

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 curve2( )y p x q r with the turning point at (4, 1).

Find the values of p, q and r.

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MODUL BIMBINGAN EMaS 2007 ADDITIONAL MATHEMATICS FORM 4

2007 All Rights Reserved JABATAN PELAJARAN TERENGGANU 8

11 Given the function f (x) = 7 mx x2 = 16 (x + n)2 for all real values ofx where m and n arepositive, find(a) the values ofm and n,

(b) the maximum point off(x),(c) the range of values ofx 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) expressf(x) in the form m(x + n)2 + p, where m, n andp 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 + kfor all real values of x, vherep, h and k are constants.

(a) State the values ofp, h and k,

(b) Find the minimum or maximum value of x2 3x + 5 and the corresponding value of x.(c) Sketch a graph off(x) = x

2 3x + 5.

(d) Find the range of values ofm such that the equation x2 3x + 5 = 2m has two different roots.