Exercises

PRELIMINARY ALGEBRA
1.8 Exercises
Polynomial equations
1.1
Continue the investigation of equation (1.7), namely
g(x) = 4x3 + 3x2 − 6x − 1,
as follows.
(a) Make a table of values of g(x) for integer values of x between −2 and 2. Use
it and the information derived in the text to draw a graph and so determine
the roots of g(x) = 0 as accurately as possible.
(b) Find one accurate root of g(x) = 0 by inspection and hence determine precise
values for the other two roots.
(c) Show that f(x) = 4x3 + 3x2 − 6x − k = 0 has only one real root unless
−5 ≤ k ≤ 74 .
1.2
Determine how the number of real roots of the equation
g(x) = 4x3 − 17x2 + 10x + k = 0
1.3
depends upon k. Are there any cases for which the equation has exactly two
distinct real roots?
Continue the analysis of the polynomial equation
f(x) = x7 + 5x6 + x4 − x3 + x2 − 2 = 0,
investigated in subsection 1.1.1, as follows.
(a) By writing the fifth-degree polynomial appearing in the expression for f (x)
in the form 7x5 + 30x4 + a(x − b)2 + c, show that there is in fact only one
positive root of f(x) = 0.
(b) By evaluating f(1), f(0) and f(−1), and by inspecting the form of f(x) for
negative values of x, determine what you can about the positions of the real
roots of f(x) = 0.
1.4
Given that x = 2 is one root of
g(x) = 2x4 + 4x3 − 9x2 − 11x − 6 = 0,
1.5
1.6
use factorisation to determine how many real roots it has.
Construct the quadratic equations that have the following pairs of roots:
(a) −6, −3; (b) 0, 4; (c) 2, 2; (d) 3 + 2i, 3 − 2i, where i2 = −1.
Use the results of (i) equation (1.13), (ii) equation (1.12) and (iii) equation (1.14)
to prove that if the roots of 3x3 − x2 − 10x + 8 = 0 are α1 , α2 and α3 then
(a)
(b)
(c)
(d)
−1
−1
α−1
1 + α2 + α3 = 5/4,
α21 + α22 + α23 = 61/9,
α31 + α32 + α33 = −125/27.
Convince yourself that eliminating (say) α2 and α3 from (i), (ii) and (iii) does
not give a simple explicit way of finding α1 .
Trigonometric identities
1.7
Prove that
cos
π
=
12
by considering
36
√
3+1
√
2 2
1.8 EXERCISES
(a) the sum of the sines of π/3 and π/6,
(b) the sine of the sum of π/3 and π/4.
1.8
1.9
The following exercises are based on the half-angle formulae.
√
(a) Use the fact that sin(π/6) = 1/2 to prove that tan(π/12) = 2 − 3.
(b) Use the √
result of (a) to show further that tan(π/24) = q(2 − q) where
q 2 = 2 + 3.
Find the real solutions of
(a) 3 sin θ − 4 cos θ = 2,
(b) 4 sin θ + 3 cos θ = 6,
(c) 12 sin θ − 5 cos θ = −6.
1.10
If s = sin(π/8), prove that
1.11
8s4 − 8s2 + 1 = 0,
√
and hence show that s = [(2 − 2)/4]1/2 .
Find all the solutions of
sin θ + sin 4θ = sin 2θ + sin 3θ
that lie in the range −π < θ ≤ π. What is the multiplicity of the solution θ = 0?
Coordinate geometry
1.12
Obtain in the form (1.38) the equations that describe the following:
(a) a circle of radius 5 with its centre at (1, −1);
(b) the line 2x + 3y + 4 = 0 and the line orthogonal to it which passes through
(1, 1);
(c) an ellipse of eccentricity 0.6 with centre (1, 1) and its major axis of length 10
parallel to the y-axis.
1.13
Determine the forms of the conic sections described by the following equations:
(a)
(b)
(c)
(d)
1.14
x2 + y 2 + 6x + 8y = 0;
9x2 − 4y 2 − 54x − 16y + 29 = 0;
2x2 + 2y 2 + 5xy − 4x + y − 6 = 0;
x2 + y 2 + 2xy − 8x + 8y = 0.
For the ellipse
x2
y2
+ 2 =1
a2
b
with eccentricity e, the two points (−ae, 0) and (ae, 0) are known as its foci. Show
that the sum of the distances from any point on the ellipse to the foci is 2a. (The
constancy of the sum of the distances from two fixed points can be used as an
alternative defining property of an ellipse.)
Partial fractions
1.15
Resolve the following into partial fractions using the three methods given in
section 1.4, verifying that the same decomposition is obtained by each method:
(a)
2x + 1
,
x2 + 3x − 10
37
(b)
4
.
x2 − 3x
PRELIMINARY ALGEBRA
1.16
Express the following in partial fraction form:
1.17
2x3 − 5x + 1
x2 + x − 1
,
(b) 2
.
2
x − 2x − 8
x +x−2
Rearrange the following functions in partial fraction form:
(a)
x−6
x3 + 3x2 + x + 19
,
(b)
.
2
− x + 4x − 4
x4 + 10x2 + 9
Resolve the following into partial fractions in such a way that x does not appear
in any numerator:
(a)
1.18
(a)
x3
2x2 + x + 1
,
(x − 1)2 (x + 3)
(b)
x2 − 2
,
+ 8x2 + 16x
x3
(c)
x3 − x − 1
.
(x + 3)3 (x + 1)
Binomial expansion
1.19
1.20
Evaluate those of the following that are defined: (a) 5 C3 , (b) 3 C5 , (c) −5 C3 , (d)
−3
C5 .
√
Use a binomial expansion to evaluate 1/ 4.2 to five places of decimals, and
compare it with the accurate answer obtained using a calculator.
Proof by induction and contradiction
1.21
Prove by induction that
n
r = 12 n(n + 1)
and
r=1
1.22
n
r3 = 14 n2 (n + 1)2 .
r=1
Prove by induction that
1 − r n+1
.
1−r
2n
Prove that 3 + 7, where n is a non-negative integer, is divisible by 8.
If a sequence of terms, un , satisfies the recurrence relation un+1 = (1 − x)un + nx,
with u1 = 0, show, by induction, that, for n ≥ 1,
1 + r + r2 + · · · + rk + · · · + rn =
1.23
1.24
un =
1.25
1.26
1
[nx − 1 + (1 − x)n ].
x
Prove by induction that
n
1
1
θ
θ
= n cot
− cot θ.
tan
r
r
n
2
2
2
2
r=1
The quantities ai in this exercise are all positive real numbers.
(a) Show that
a1 a2 ≤
a1 + a2
2
2
.
(b) Hence prove, by induction on m, that
p
a1 + a2 + · · · + ap
,
a1 a2 · · · ap ≤
p
where p = 2m with m a positive integer. Note that each increase of m by
unity doubles the number of factors in the product.
38