632 Solutions to exercises
- (i) 67 (ii)− 3 (iii)− 31
(iv)− 197227
- (i) 2 a
2
1 + 14 a 1 + 13 (ii) 2 y
4
1 + 14 y
2
1 + 13
- (i)a
2
1 + 13 a 1 − 14 (ii)a
4
1 − 1 a
2
1 − 16
(iii)x
2
1 − 1 x 1 − 16
(iv)x
4
1 − 16 x
3
1 − 12 x
2
1 + 133 x 1 + 124
- 6 x 1 + 11
Section 2.3
- 2 y
2
(3x
2
1 − 1 xy 1 − 1 2) 10.(x 1 + 1 5)(x 1 + 1 1)
11.(x 1 + 1 3)(x 1 − 1 2) 12.(x 1 − 1 5)(x 1 − 1 3)
13.(x 1 + 1 2)(x 1 − 1 2) 14.(2x 1 + 1 3)(2x 1 − 1 3)
15.(2x 1 − 1 3)(x 1 + 1 2)
16.(x 1 − 1 1)(x 1 + 1 1)(x 1 − 1 3)(x 1 + 1 3)
- 19.x 1 + 12
20.x 1 + 11 21. 22.
Section 2.4
23.y 1 + 12 24.(2y 1 − 1 1) 23 25. 21 − 13 y
Section 2.5
- 11 + 12 x 1 + 13 x
2
- 11 + 12 x 1 + 13 x
2
- 11 + 1 x 1 + 12 x
4
- 16 x
9
40.y 1 = 13 x 1 + 11 41.y 1 = 1 − 2 x 1 + 14
42.PlotΛ
magainst for a straight line.
Then−Kis the slope andΛ
0
mis the
intercept.
43.Plot against for a straight
line. Thenμ
2
from the slope and αfrom
the intercept.
- 1 , 2 45. 122 ,− 2 46.
1
2
7
12
±
1
T
1 1
ρ 2
ε
ε
rr−
c
2612
23
x
xx++
p
K
=
−
θ
()1 θ
c=
−
ΛΛ
mm0
2
K
BV
pV
RT
=−
mm±± 1 y 1
±−y
2
1
±−
1
1
y
y
1
12
−
y
y
23
32
y
y
−
y
1 +y
21
2
x
x
−
−
x
x
−
3
2
x
x
2
4
1
32 x+
3 (double) 48.− 122 (double)
1 ,− 2 ,− 3 54. 1 (double), 4
1 (triple) 56.− 1 , 1 , 2 , 3
Section 2.6
59.(x 1 + 1 3)(x 1 − 1 2)
Section 2.7
Section 2.8
66.lines cross at(x,y) 1 = 1 (2,1)
67.lines cross at(x,y) 1 = 1 (7 213 , 42 13)
68.inconsistent; parallel lines
69.linear dependence; only one line
70.three lines cross at(x,y,z) 1 = 1 (2,− 2 ,−1)
71.line crosses ellipse at(x,y) 1 = 1 (0,−1)and
(x,y) 1 = 1 (1,1)
72.line touches (is tangential to) ellipse at
(x,y) 1 = 1 (1,0)
73.complex ,
line and ellipse do not touch
()xy,= ±i ,±i
()32 156 153
1
9
1
2
10
1
6
1
2
−
−
−
−
xx
()x1
6
21 8
1
25
2
−+
−
xx x
4
2
3
xx+ 1
−
1
3
21
xx 3
1
3
1
1
1
xx− 2
−
210
17 26
22
2
2
xx
x
xx
++ +
−−
3815
26
2
2
xx
x
−+−
2
7
3
−
x+
α=+−
K
c
c
K
aa2
1
4
1
1
4
118
1
1
−± +
−
y
y
1
2 12
±
i
12
−± 17
()i