QUESTIONS 1
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
