SULIT 3472/1/2

School of BIG BANG

MIDYEAR EXAMINATION

ADDITIONAL MATHEMATICS

Answer and Marking Scheme

Prepared by : Checked by: Verified by:

........................................ ........................................... ..………………………… (Mr.G.Dragon) (Mr. T.O.P) (Mr Yang Ju Hua )

3472/1/2 SULIT

FORM IV 2015

This question paper consists of 7 printed pages and 1 blank page

SECTION 1

BAHAGIAN 1[ 44 marks/markah ]

Answer all questions /Jawab semua soalan.

1. Diagram 1 shows the relation between set A and set B. Rajah 1 menunjukkan hubungan antara set A dan set B. Set A Set B

(a) Express the relation in the form of ordered pairs. Ungkapkan hubungan itu dalam bentuk pasangan tertib.

(b) State the type of the relation. Nyatakan jenis hubungan itu [ 3 marks/markah ]

Answer / Jawapan:

(a) {(1,p), (2,r), (3,s), (4,p)} √ 2 Any one pair correct √ B1 DIAGRAM 1 Rajah 1 (b) many – to – one √ 1

2. Diagram 2 shows the linear function f. Rajah 2 menunjukkan fungsi linear f.

(a) State the value of w. Nyatakan nilai bagi w.

(b) Using the function notation, express f in terms of x.

Dengan menggunakan tatatanda fungsi, ungkapkan f dalam sebutan x.

[ 2 marks/markah ]

Answer / Jawapan: DIAGRAM 2 Rajah 2

(a) 7 √ 1

(b) f(x) = x + 2 or f : + 2 √ 1

2

22

2

For examiner’s

use only

x f(x)f

2

3

w

10

0

1

5

8

3

1

1

23

4

p

q

r

s

3. Diagram 3 shows the graph of the function f(x) = |3x – 2|, for the domain 0 ≤ x ≤ 5. Rajah 3 menunjukkan graf bagi fungsi f(x) = |3x – 2| untuk domain 0 ≤ x ≤ 5. [ 3 marks/markah ] State / Nyatakan

(a) the value of m. y nilai m.

(b) the range of f(x) corresponding to the given domain. f(x) = |3x – 2|

julat bagi f(x) sepadan dengan domain yang diberikan.

Answer / Jawapan: 2

(a) √ 1

0 m 5

(b) 0 ≤ f(x) ≤ 13 √ 2 DIAGRAM 3 Rajah 3

13 √ B1 4. Two functions are defined by f : x 2x + 1 and g :x x2 + 2x – 6. Given that gf : x 4x2 + px + q, find the value of p and of q.

Dua fungsi ditakrifkan sebagai f : x 2x + 1 dan g : x x2 + 2x – 6. Diberi gf : x 4x2 + px + q, cari nilai p dan nilai q. [ 3 marks/markah ]

Answer / Jawapan:

p = 8, q = −3 √ 3 (both correct)

4x2 + 8x – 3 √ B2

(2x + 1)2 + 2(2x + 1) – 6 √ B1

5. The function of f is defined as f (x) . Find,

Fungsi f ditakrifkan sebagai f(x) = . Cari [ 3 marks/markah ]

(a) the value of m, nilai m,

(b) .

Answer / Jawapan:

(a) √ 1

3

For examiner’s

use only

33

4

33

3

33

5

(b) f -1(x) = √ 2 √ B1

6. Determine the roots of the quadratic equation 5x = 3x + 2. Tentukan punca-punca bagi persamaan kuadratik 5x2 = 3x + 2. [ 3 marks/markah ]

Answer / Jawapan:

x = or − 0.4 , x = 1 √ 3 (both correct)

(5x + 2) (x – 1) = 0 √ B2 or √ B2

5x2 – 3x – 2 = 0 √ B1

7. Find the range of values of k if the following quadratic equation (k + 1)x2 + 6x + 3 = 0 which has two different roots.

Cari julat bagi nilai k jika persamaan kuadratik berikut (k + 1)x2 + 6x + 3 = 0 mempunyai dua punca yang berbeza.

[ 3 marks/markah ] Answer / Jawapan:

k < 2 √ 3

6 2 – 4(k + 1)(3) > 0 √ B2

62 – 4(k + 1)(3) or b2 – 4ac > 0 or a = (k+1), b = 6, c = 3 √ B1

8. Given that and are the roots of the quadratic equation 2x2 – 5x + 3 = 0. Form the quadratic equation whose roots are and . Diberi dan adalah punca-punca bagi persamaan kuadratik 2x2 – 5x + 3 = 0. Bentuk satu persamaan

kuadratik yang mempunyai punca-punca 2 dan . [ 3 marks/markah ] Answer / Jawapan:

x2 – 5x + 12 = 0 √ 3

4αβ = 12 2(α + β) = 5 √ B2 (both)

α + β = √ B1 (both

4

33

7

For examiner’s

use only

33

6

33

8

αβ = 3

9. Diagram 3 shows the graph of a curve y = a(x + p) ² + q that passes through the point (0, 3) and has the minimum point (2, −1). Find the values of a, p and q.

Rajah 3 menunjukkan geraf bagi satu lengkung y = a(x + p) 2 + q yang melalui satu titik (0.3) dan mempunyai titik minima (2 , −1). Cari nilai-nilai a, p dan q.

[ 3 marks/markah ]

Answer / Jawapan:

p = −2 √ 1

q = −1 √ 1

a = 1 √ 1

10. Find the range of values of x for which x(x − 3) ≤ 4. Cari julat bagi nilai x yang mana x(x − 3) ≤ 4. [ 3 marks/markah ] Answer / Jawapan:

−1 ≤ x ≤ 4 or x ≥ −1 , x ≤ 4 √ 3

(x + 1)(x – 4) ≤ 0 √ B2 or

11. Solve

[ 4 marks markah ] Answer / Jawapan:

x = 3 √ 4

4x – 4 = 5 + x or 3x – 4 = 5 √ B3

3 4 x – 4 = 3 5 + x or 33 x – 4 = 35 √ B2

34 or 35 √ B1

5

33

9

43

11

DIAGRAM 3Rajah 3

For examiner’s

use only

(2, −1)

(0, 3)x

y

0

33

10

12. Given that and lg 5 = 0.7. Find, without using scientific calculator or mathematical tables, the value of log 2 20 .

Diberi lg 2 = 0.3 dan lg 5 = 0.7. Cari, tanpa menggunakan kalkulator saintifik atau jadual sifir matematk, nilai bagi log2 20. [ 4 marks/markah ] Answer / Jawapan:

or 4.333 √ 4

√ B3

13. Given that log5 x = k, find logx 125x2 in terms of k. Diberi log5 x = k, find logx 125x2 dalam sebutan k. [ 4 marks/markah ] Answer / Jawapan:

√ 4

√ B3

B2

14. Express in the simplest form. [ 3 marks/markah ] Ungakapkan 2n + 3 −2n + 5(2n – 1 ) dalam sebutan paling ringkas.

2 n -1(19) √ 3

2 n(8 –) √

2 n . 2 3 – 2 n +5 √ B1

6

44

13

For examiner’s

use only

44

12

33

14

11

SECTION 2BAHAGIAN 2

[ 26 marks/markah ]

Answer all questions / Jawab semua soalan.

1. Solve the simultaneous equations. Give your answers correct to three decimal places. Selesaikan persamaan serentak. Berikan jawapan anda betul kepada tiga angka perpuluhan.

. [ 5 marks/markah ]

Answer / Jawapan:

2. Given that f : x x2 − 2 and g : x 3x + 4. Diberi f : x x2 − 2 dan g : x 3x + 4.

(a) (i) Determine f −1(x). [ 2 marks/markah ] Tentukan f −1(x).

(ii) State whether the f −1(x) exist. Give a reason to your answer by showing the evidence of the reason given. Nyatakan sama ada f −1(x) itu wujud. Berikan sebab kepada jawapan anda dengan menunjukkan bukti kepada sebab yang anda berikan. [ 3 marks/markah ]

(b) Given gh(x) = 6x + 7, determine h(x). Diberi gh(x) = 6x + 7, tentukan h(x). [ 2 marks/markah ]

Answer / Jawapan:

(a) x = y2 - 2 √ K1

y2 = x + 2 , √ K1 an object has two images or it’s a one-to-many relation or the function undefined

Therefore the invest function does not exist. √ N1

7

3. (a) Simplify: Permudahkan:

4 x + 2 – 2 2x + 3 [ 4 marks/markah ]

(b) Hence, solve the equation:Seterusnya, selesaikan persamaan;

4 x + 2 – 2 2x + 3 = 64 [ 2 marks/markah ]

Answer / Jawapan:

(a) 2 2(x + 2) – 2 2x + 3 √ P1

2 2x . 2 4 − 2 2x . 2 3 √ K1

2 2x (2 4 – 2 3) √ K1

2 2x (8)

4. The curve of a quadratic function f(x) = x2 + 2hx – 5 has minimum point of (2, k).

Lengkung bagi satu fungsi kuadratik f(x) = x2 + 2hx – 5 mempunyai titik minimum (2, k).

(a) State the equation of the axis of symmetry of the curve. Nyatakan pesamaan paksi simmetri bagi lengkung itu. [ 1 mark/markah ]

(b) By using the method of completing the square, determine the value of h and of k. Dengan menggunakan kaedah penyempurnaan kuasadua, tentukan nilai bagi h dan bagi k.

[ 4 marks/markah ] (c) Hence, sketch the graph of the curve.

Seterusnya, lakarkan graf bagi lengkung itu. [ 3 marks/markah ]

Answer / Jawapan:

8

b) 2 2x + 3 =

= 6

x = √ N1

DON’T BE A LOSER ooooooo!!!!

END OF THE ANSWER & MARKING SCHME

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