JAWAB UAN IPA 2006/2007 P12

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JAWABAN UJIAN NASIONAL 2006 / 2007

MATEMATIKA IPA P12 - A

RABU, 18 APRIL 2007

1. Jawab: C

2. Jawab: B

2log 3 . 3log 5 = 2log 5 = ab

15log 20 =

3. Jawab: C

x2 – 5x + 6 = 0

x1 + x2 = - --

x1 . x2 = 61

6

a

c

y1 + y2 = (x1 – 3) + (x2 – 3)

= (x1 + x2) – 6 = 5 – 6

= – 1 y1 . y2 = (x1 – 3) (x2 – 3)

= x1.x2 – 3(x1 + x2) + 9

= 6 – 3(5) + 9 = 0

x2 – (y1 + y2)x + (y1 . y2) = 0 x2 – (–1)x + 0 = 0 x2 + x = 0

4. Jawab: E

Titik Puncak (1, 4)

Titik potong dengan sumbu X (–1, 0) dan (3, 0)

Titik potong dengan sumbu Y (0, 3) Cara 1: Gunakan persamaan y = a (x – x1) (x – x2)

Titik puncak y = 4, x = 1

Titik potong sumbu X x1 = – 1, x2 = 3

4 = a (1 + 1) (1 – 3) = a (–4)

4 = – 4a

a = – 1

y = – 1 (x + 1) (x - 3) = – (x2 – 2x - 3)

y = – x2 + 2x + 3

(1, 4)

(0, 3)

(-1, 0) (3, 0)



Cara 2: Gunakan persamaan y = a (x – xp)2 + yp

Titik potong sumbu x x = –1, y = 0

Titik puncak xp = 1, yp = 4

0 = a (– 1 – 1)2 + 4 = a (4) + 4

0 = 4a + 4 4a = – 4

a = – 1

y = –1 (x – 1)2 + 4 = – (x2 – 2x + 1) + 4 = – x2 + 2x – 1

y = –x2 + 2x + 3

5. Jawab: A

f(x) = 3x2 – 4x + 6 ; g(x) = 2x – 1 ; (f o g)(x) = 101

(f o g)(x) = f(g(x)) = 3(2x – 1)2 – 4(2x – 1) + 6

= 3(4x2 – 4x + 1) – 8x + 4 + 6

= 12x2 – 12x + 3 – 8x + 10

101 = 12x2 – 20x + 13

12x2 – 20x – 88 = 0 dibagi 4

3x2 – 5x – 22 = 0

(3x – 11) (x + 2) = 0

2 x ; 3

23

3

11x 21

6. Jawab: E

32x+1 – 28.3x + 9 = 0 32x . 31 – 28 . 3x + 9 = 0

Misal: 3x = A

3A2 – 28A + 9 = 0

(3A – 1) (A – 9) = 0 A = 1/3 ; A = 9

3x = 9 x1 = 2 ; 3x = 1/3 x2 = –1

3x1 – x2 = 3(2) – (–1) = 7

7. Jawab: D

(x – 2)2 + (y + 1)2 = 13 Pusat (2, –1) ; Jari-jari (r) = 13

x = –1 (–1 – 2)2 + (y + 1)2 = 13 9 + y2 + 2y + 1 = 13 y2 + 2y – 3 = 0 (y + 3) (y – 1) = 0 y = –3 ; y = 1 ada 2 titik pada lingkaran: (–1, –3) dan (–1, 1)

Gunakan rumus :(x – a) (x1 – a) + (y – b) (y1 – b) = r2

(–1, –3) : (x – 2) (–1 – 2) + (y + 1) (–3 + 1) = 13

–3 (x – 2) – 2 (y + 1) = 13

–3x + 6 – 2y – 2 = 13

3x + 2y + 9 = 0



(–1, 1) : (x – 2) (–1 – 2) + (y + 1) (1 + 1) = 13

–3 (x – 2) + 2 (y + 1) = 13

–3x + 6 + 2y + 2 = 13

3x – 2y + 5 = 0

8. Jawab: A

f(x) : (x – 2) sisa 24 f(2) = 24 f(x) : (2x – 3) sisa 20 f(3/2) = 20

–

-

–

- ; a = 2 ; b = 3/2 ; f(a) = 24 ; f(b) = 20

9. Jawab: E

2x + 2y + 1z = 67.000 3x + 1y + 1z = 61.000 1x + 3y + 2z = 80.000

D =

3

1

2

1

3

2

231

113

122

(4) + (2) + (9) – (1) – (6) – (12) = –4

Dx =

3

1

2

80

61

67

2380

1161

1267

(134) + (160) + (183) – (80) – (201) – (244) = –48

Dy =

80

61

67

1

3

2

2801

1613

1672

(244) + (67) + (240) – (61) – (160) – (402) = –72

Dz =

3

1

2

1

3

2

8031

6113

6722

(160) + (122) + (603) – (67) – (366) – (480) = –28

74

28

D

D z ; 18

4

72

D

Dy ; 12

4

48

D

Dx zyx

Harga: x + y + 4z = (12.000) + (18.000) + 4(7.000) = 58.000

10. Jawab: C

12

37 C

13

27 C ;

y3

2y x B ;

41

12A t

80

61

67

z

y

x

231

113

122



B – A = Ct 12

37

41

12

y3

2y x

y – 4 = 1 y = 5 x + (5) – 2 = 7 x = 4

x . y = (4)(5) = 20

11. Jawab: C

1x + 0 y < 0200 4x + 20y < 1760

Pendapatan maksimum: 1.000x + 2.000y = 1.000(140) + 2.000(60) = 260.000

12. Jawab: B

0

3

3

2

0

1

2

3

2

RQRQ ;

2

1

1

2

0

1

4

1

0

RPRP

026

0

0 (-3)3 . 211

(2)(0)(1)(-3) (1)(3)

RQ RP

RQ . RPθcos

22222

Θ = 90°

13. Jawab: A

2

2

0

0

0

0

2

2

0

ACAC ;

0

2

2

0

0

0

0

2

2

ABAB

Proyeksi vektor orthogonal AB pada AC :

k j

2

2

0

2

1

2

2

0

8

040

2

2

0

220

2

2

0

0

2

2

222

14. Jawab: D

y = x2 – 3 Persamaan kuadrat Carilah titik potong dengan sumbu X dan Y

X 0 + 3 – 3

y –3 0 0 A (0, –3) B (+ 3 , 0) C (– 3 , 0)

4x + 20y = 1760 4x + 24y = 0800 –

16y = 960 y = 60

x + (60) = 200 x = 200 – 60 = 140



Refleksi

(x, y) xsumbu (x, -y)

Dilatasi

(x, y) 2k (kx, ky)

(0, -3) (0, 3) (0, 6)

(+ 3 , 0) (+ 3 , 0) (+2 3 , 0)

(– 3 , 0) (– 3 , 0) (–2 3 , 0)

y = a (x – x1) (x – x2) 6 = a (0 - 2 3 ) (0 + 2 3 )

6 = a (-2 3 ) (2 3 )

6 = – 12 a a = – ½

y = –½ (x – 2 3 ) (x + 2 3 )

= –½ (x2 – 12) = –½x2 + 6

15. Jawab: B

U3 = a + 2b = 36 U5 + U7 = (a + 4b) + (a + 6b) 144 = 2a + 10b 72 = a + 5b

16. Jawab: E

a = 80.000.000 sederhanakan menjadi a = 80 r = ¾ U3 = a . r2 = (80) (¾)2 = (80) (9/16) = 45

17. Jawab: B

p = hari panas q = ani memakai topi r = memakai payung ~p = hari tidak panas

18. Jawab: B

ACH titik tengah P DP = 1/3 DF EGB titik tengah Q FQ = 1/3 DF DF Diagonal ruang

Jarak PQ = 1 – 1/3 – 1/3 = 1/3 . 6 3 = 2c

a + 2b = 36 a + 5b = 72 -

–3b = –36 b = 12

a + 2(12) = 36 a = 36 – 24 a = 12

p q p q p r ~q v r q r ~r (modus tolens) ~r p r ~p

D

P

Q

F



19. Jawab: -

BG = a 2 ; BDHF diwakili oleh garis BH ; BH = a 3

HG2 = BH2 + BG2 – 2(BH) (BG) cos B

a2 = (a 3 )2 + (a 2 )2 – 2(a 3 ) (a 2 ) cos B

a2 = 3a2 + 2a2 - 2 6 a2 . cos B

cos B = 63

1

a62

a4

a62

aa52

2

2

22

B = 35,3

20. Jawab: A

C = 45 ; a = p ; b = 2 2 p

c2 = a2 + b2 – 2ab cos C = (p)2 + (2 2 p)2 – 2 (p) (2 2 p) cos 45

= p2 + 8p2 – 4 2 p2 (½ 2 )

= 9p2 – 4p2 = 5p2

c = 5 p

21. Jawab: C

cos 40 + (cos 80 + cos 160) = cos 40 + [2 cos ½ (80 + 160) . cos ½ (80 – 160) = cos 40 + [2 cos 120 . cos 40 ] = cos 40 (1 + 2 cos 120) = cos 40 (1 + 2 (-½)) = 0

22. Jawab: A

A = x2 – x – 6 A’ = 2x – 1

B = 4 - 1x5 B’ = - ½ . 5 (5x + 1)-½ = - 5/2 (5x + 1) -½

85

1625

1352

5

132 .

)(

)(

23. Jawab: E

4

x2

1 tan . x

x sin . xsin . 2 Limit

x2

1tan . x

)xsin21(1 Limit

0x

2

0x

24. Jawab: C

6 2x cos

6 2x sin 4

6 2x cos 2

6 2x sin 2)x('f

332

1 .

2

1 . 4

6 2(0) cos

6 2(0) sin 4)0('f

B

H G a

a 3 a 2



25. Jawab: D

27 + 9 + 3 – a3 – a2 – a = 25 a3 + a2 + a – 14 = 0

Gunakan horner:

2 1 1 1 -14

+ + +

2 6 14

1 3 7 0

26. Jawab: B

Short-cut agar luas persegi panjang maksimum:

- panjang = ½ x = ½ (4) = 2

- lebar = ½ y = ½ (5) = 2½

- luas maksimum = ¼ xy = ¼ (4)(5) = 5

27. Jawab: C

y = x2 ; x + y = 6 y = 6 – x ykurva = ygaris x2 = 6 – x x2 + x – 6 = 0 (x + 3)(x – 2) = 0 x = -3, x = 2

- --

- - - - - - - -

28. Jawab: D

y = -x2 + 4 ; y = -2x + 1 ykurva = ygaris -x2 + 4 = -2x + 1 x2 – 2x – 3 = 0

(x – 3) (x + 1) = 0 x = 3 , x = -1

–-

– - – - - -

(a – 2) (a2 + 3a + 7) = 0

a = 2 ½ a = 1



29. Jawab: E

P(A) = 3/8 ; P(B) = 6/10

30. Jawab: D

Fmod = 14 ; L = 48,5 ; c = 6 ; fk = 4 + 6 + 9 = 19 ; n = 50

Mod = – –

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JAWAB UAN IPA 2006/2007 P12