60 Chapter 2Algebraic functions
42.Explain howKandΛ
0
min Kohlrausch’s law (Exercise 33),
can be obtained graphically from the results of measurements ofΛ
mover a range of
concentration c.
43.The Debye equation
relates the relative permittivity (dielectric constant) ε
rof a pure substance to the
dipole moment μand polarizability αof the constituent molecules, where ρis the
density at temperature T, and M, N
A
, k, and ε
0
are constants. Explain how μand αcan
be obtained graphically from the results of measurements of ε
rand ρover a range of
temperatures.
Find the roots and sketch the graphs of the quadratic functions:
44.x
2
1 − 13 x 1 + 12 45.− 2 x
2
1 − 13 x 1 + 12 46. 3 x
2
1 − 13 x 1 − 11 47.−x
2
1 + 16 x 1 − 19
- 4 x
2
1 + 14 x 1 + 11 49.x
2
1 + 1 x 1 + 12 50.− 3 x
2
1 + 13 x 1 − 11
51.If find xas a function of y.
52.The acidity constantK
aof a weak acid at concentration cis
where αis the degree of ionization. Express αin terms ofK
aand c(remember that α, K
a,
and care positive quantities).
Given that x 1 − 11 is a factor of the cubic, (i) find the roots, (ii)sketch the graph:
53.x
3
1 + 14 x
2
1 + 1 x 1 − 16 54.x
3
1 − 16 x
2
1 + 19 x 1 − 14 55.x
3
1 − 13 x
2
1 + 13 x 1 − 11
- Given thatx
2
1 − 11 is a factor of the quarticx
4
1 − 15 x
3
1 + 15 x
2
1 + 15 x 1 − 16 , (i)find the roots,
(ii)sketch the graph.
Section 2.6
Use algebraic division to reduce the rational function to proper form:
Section 2.7
Express in terms of partial fractions:
xx
xx
2
2
21
12
+−
()( )−+
257
12
2
xx
xx x
−+
()( )−+
x
xx
−
++
2
32
2
x
xx
2
() 3
1
( )( ) 12
xx−+23456
22
432
2
xxxx
xx
−+ −+
−−
xxx
x
32
256
1
+−−
32 4
2
32
xxx
x
−−+
21
3
x
x
−
K
c
a=
−
α
α
2
1
y
xx
xx
=
++
+−
21
21
2
2
ε
ε
ρ
ε
α
μ
rrM
N
kT
−
=+
1
23 3
0
2
A
ΛΛ
mm=−
0
K c