PENILAIAN AKHIR SEMESTER GENAP

QUESTIONS

1. a=i-8j+5k

b=3i+8j+2k

C=-2i-4j+3k

A+2b-3c = (1,-8, 5) + 2 (3,8,2) - 3 (-2,-4,3)

= (1,-8,5) + (6,16,4) - (-6,-12,9)

= 13i + 20j

2. (7,-4, p) x (1, 8, -5) = 0

7 + (-32) - 5p = 0

-25p = 5

p = -5

= a - b - c

= (7,-4,-5) - (1,-8,-4) - (1,-2,-3)

= (5,-10,-3)

3. PQ = Q - P

= (-4, 1) x (9, 11)

= ( -13, -10 )

QUESTIONS 2

4. AB = k × BC

B - A = k × C - B

(7, -14, 6) - (1, -5, -9) = k × (15, -26, 20) - (7, -14, 6 )

(6, -9, 15) = k × (8, -12, 14)

6 = k × 8

K = 6/8

K = 3 : 4

5. u = AB = B - A

= (2, -1, 8) - (-3, 5, 2)

= (5, -6, 6)

v = BC = C - B

= (3, -4, 6) - (2, -1, 8)

= (1, -3, -2)

U × V

= (5, -6, 6) x (1, -3, -2)

= 5 + 18 - 12

= 11

6. a = PQ = Q - P

= (-4, 20, 10) - (-8, -4, -15)

= (-3, 24, 25)

QR = R - Q

= (12, 22, -20) - (-11, 20, 10)

= (23, 2, -30)

PR = R - P

= (12, 22, -20) - (-8, -4, -15)

= (20, 26, -5)

b = QR - PR

= (23, 2, -30) - (20, 26, -5)

= (3, -2, -25)

a × b

= (-3, 24, 25) x (3, -24, -25)

= -9 - 576 - 625

= -1.210

7. a = PR = R - P

= (3, 15) - (-1, 10)

= (4, 5)

b = PQ = Q - P

= (5, 3) - (-1, 10)

= (6, -7)

c = QR = R - Q

= (3, 15) - (5, 3)

= (-2, 12)

c = ma + nb

(-2,12) = m (4, 5) + n(6, -7)

(-2,12) = (4m, 5m) + (6n, -7n)

4m + 6n = -2 | ×5 | 20m + 30n = -10

5m - 7n = 12 | ×4 | 20m - 28n = 48 -

58n = -58

n = -1

4m + 6n = -2

4m + 6(-1) = -2

4m - 6 = -2

4m = 4

m = 1

m+n = -1 + 1 = 0

8. u × v

= (5, -6, -2) x (1, p, -4)

= 5 + 6p + 8 = 0

= 6p = -13

= p = 2,1 ~ 2

9. c = a - b

= (5i + 6j - 2k) - (3i - 4j - 13k)

= 2i + 10j + 11k

= √ 2² + 10² + 11²

= √4 + 100 + 121

= √225 = 15

10. a = PQ = Q - P

= (6, 0, 12) - (3, -6, 9)

= (3, 6, 3)

b = QR = R - Q

= (3, -9, 18) - (6, 0, 12)

= (-3, -9, 6)

|a| = √3² + 6² + 3³

√9 + 36 + 9

√54 = 3√6

|b| = √-3³ + (-9²) + 6²

√9 + 81 + 36

√126 = 3√14

CosX = a × b / 3√6 × 3√14

= (3, 6, 3) x (-3, -9, 6) / 9√22

= -9 - 54 + 18 / 9√22

= 45 / 9√22 = -5/√22 × √22/√22

= -5√22 / 22

CosX = -1,06

= 122°

11. Cos0 = v × w / |v| × |w|

|v| = √4 + 16 + 4

√24 = 2√6

|w| = √1 + 1 + 4

= √6

(2, -4, -4) x (1, 1, 2) / 2√6 × √6

= 2 - 4 - 4 / 12

= -6 / 12

cos0 = - 1/2

= 120°

12. g = A - B

= (3, -9, -5) - (5, -8, -8)

= (-2, -1, 3)

h = C - D

= (11, -12, -2) - (10, -9, -4)

= (1, -3, 2)

|g| = √4 + 1 + 9 = √14

|h| = √1 + 9 + 4 = √14

(-2, -1, 3) x (1, -3, 2) / √14 × √14

= - 2 + 3 + 6 / 14

= 7/14 = 1/2

= 60°

13. ab = √7 × 2√3 = 2√21

ab = a² + 2ab + b²

a+b² = 7 + 2( 2√21) + 12

a+b = √19 + 4√21

= 19 + 84

= 103

14. AB = B - A

(7, 4, -9) = B - (8, -11, 3)

(4) = B - (-11)

B = (7, 4, -9) + (8, -11, 3)

= (15, -7, -6)

15. a = v - u

= (9, 3, -3) - (5, 3, -3)

= (4, 0, 0)

|a| = √16 = 4

b = w - v

(13, 7, -3) - (9, 3, -3)

= (4, 4, 0)

|b| = √32 = 4√2

Cos0 = a × b / |a| × |b|

= (4, 0, 0) x (4, 4, 0) / 4 × 4√2

= 16 + 0 + 0 / 16√2

= 16/16√2 × 16√2 / 16√2

= 1/√2 × √2/√2

= 32√2/32 = √5 × 1/2√2

= 45° atau 315°

Tan 45° = 1

Tan 315° = -1

16. |v| = √25 + 9 + p²

|v|² = 34 + p

|u| = 1/2 × v

1/2 × (34 + p)

17 + 1/2p

u × v / |v| = (-1, 4, 6) x (-5, 3, p) / 17 + 1/2p

= 5 + 12 + 6p / 17 + 1/2p

= 17 + 6p / 17 + 1/2p

17+6p = 17 + 1/2p

0 = 3p

p = 0/3 = 0

17. a x b / |b|

= (-4, 1, 2) x (4, -5, 3) / 5√2

= -16 - 5 + 6 / 5√2

= -15/ 5√2

= -3/2√2

QUESTIONS 3

15). 5^x+2 < 4^x

log5^x+2 < log4^x

(x+2) log5 < x (log4)

xlog5 + 2log5 < xlog4

xlog5 - xlog4 < -2 log5

x (log5 - log4) < -2log5

HP = {x < -2log5 /log5-log4}

16). (x-4)^4x < (x-4)^1+3x

4x < 1+3x

x < 1

HP : {x < 1 atau x < 4}

x-4 < 0

x < 4

17). 2^x³-x < 1

2^x³ - x < 2^0

x³ - x < 0

x (x - 1) (x + 1) < 0

x = 0, x = 1, x = -1

HP : {0 < x < 1 atau x < -1}

18). 5^2x+1 > 5^x+4

(5^x)² × 5¹ > 5^x + 4

5 p² > p + 4

5p² - p - 4 > 0

(p - 1) (5p + 4) > 0

p = 1 p = -4/5

HP : {x < -4/5 atau x > 1}

19). 2^x × 2¹ × 2^-4 ≤ 0 / 1-2^x

p - 2p - 4 / 1-2x ≤ 0

(p-2) (p+1) / p (1-p) ≤ 0

p = 2 p = -1

HP : { -1 < x < 2}

20). (4^x)² × 4¹ > 4^x + 3

4p² - p - 3 > 0

(4p - 4/4) (4p +3) > 0

(p - 1) (4p + 3) > 0

p = 1 p = -3/4

HP : {-3/4 < x < 1}

21). 3^x-2y = 3^-4

x - 2y = -4

x - y = 4

__________________ -

-y = 8

y = -8

x-8 = 4

x = 12

x+y = 12 + (-8) = 4

22). (2a^5 - b^-5)/3 × 2a^9 × b^-1) ^-1

(2¹a^5 b^-5/2^5 a^9 b^-1)

(2^-4 a^-4 b^-4)

(2^⁴ a^⁴ b^⁴)

(2 ab)⁴

23). 9^3x-4 = 81^½x-5

(3²)^3x-4 = (3^-4)^2x-5

6x - 8 = -8x + 20

14x = 28

x = 2

24). 4¹+² × 3^4x+1 < 432

4 × 4^2x × 3^4x × 3 < 432

4^2x × 3^4x < 36

16^x × 18^x < 36

1.296^x < 36

36^x < 36

x < 1

25). (1/3)^x+2 < (1/3)^x

x + 2 < x

0 < 2 (TIDAK MEMENUHI)

(1/3)^x+2 < (1/3)^-x

x +2 < -x

2x < -2

x < -1

QUESTIONS 4

32. g = ( 2, 1, -3 ) h = ( -1, 3, -2 )

ǀ g ǀ = √14 ǀ h ǀ = √14

Cos Ɵ = g.h / ǀ g ǀ. ǀ h ǀ

= -2 + 3+6 / √14. √14

= 7/14

= 1/2

33. ǀuǀ . ǀv ǀ + ǀu-vǀ = ǀu + v ǀ

45 + 4√5 = ǀu + v ǀ

ǀu + v ǀ = 53,944 = 54

34. 34. AB = B – A

AB + A = B

(-15, 12 ) + (-4, -6 ) = B

(-19, 6 ) = B

35. C = u.v / ǀvǀ . v

= -6 + 2 + (-3) / (√14)2 . ( -3i + 2j – k )

= -7/14 . ( -3i + 2j – k )

= -1/2 . ( -3i + 2j – k )

= ( 3/2i -j + 1/236 4

= 14 + 90 -11m / √4+100+121

4 = 104 – 11m / √225

4 = 104 – 11m / 15

60 = 104 – 11m

11m = 104 – 60

11m = 44

m = 4k

37. p . q = | p | | q | cos Θ

p . q = (-3i - 3aj - bk ) . ( -ai + bj - 2ak )

p . q = 3a - 3ab + 2ab

p . q = 3a

| p |² = (-3)² + (-3a)² + (-b)² = 9 + 9a² + b²

| p | = √ (9 + 9a² + b² )

| q |² = (-a)² + b² + (-2a)² = a² + b² + 4a²

| q | = √ (a² + b² + 4a² ) = √(5a²+b²)

r = [ ( p . q ) / (| q |²) . q

i - j + 2k = [ 3a /(5a²+b²) . ( -ai + bj - 2ak )

-3 = 3a² /( 5a² + b² )

-3a² - b² = 3a²

-b² = a²

cos Θ = p . q / ( | p | | q | )

(√6)/8 = 3a / ( √ (9 + 9a² + b² ) . √(5a²+b²)

[ (√6)/8 ]² = (-3b)² / (9 - 9b² + b² ) . (-5b²+b²)

( 6 / 64 ) (9 - 8b²) . (-4b²) = 9b²

( -24 / 64 ) (9 - 8b²) = 9

-8b²+ 9 = -24

-8b² = -33

b² = 4,125

b = √4,125

b = 2,03

maka, b = -a = -2,03. Jadi, nilai a = -2,03

38). Cos x = a.b/|a|.|b|

½√2 = -12 + 2x / √20 . √9+x²

½√2 = -12 + 2x / √180 + 20x²

½√2 . √180 + 2x² = -12 + 2x

½ √360 + 40x² = -12 + 2x

(√360 + 40x²)² = (-24 + 4x)²

360 + 40x² = 16x² - 192 + 576

24x² + 192x - 216 = 0

x² + 8x - 9 = 0

(x+9) (x-1) = 0

x = -9 x = 1

Jadi, HP nya bilangan yang positif yaitu x = 1

39). |r| = p.q / |p|

2 = 6 + b + 2 / 3

12 = 8 + b

4 = b

Jadi, b = 4

40). AB = (5,1,1)

BC = (-3, -2, 2)

D = AB.BC / |BC| ×BC

= -15 + (-2) + (-2) / (√9+4+4)² × (-3,-2,2)

= -19/17 (-3, -2, 2)

= 57/17, 38/17, -38/17

Jadi, HP = {57/17, 38/17, -38/17}

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