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HSC by Topic 1995 to 2006 Polynomials
Page 1
Mathematics Extension 2 HSC Examination
Topic: Polynomials
06
3c
Two of the zeros of P(x) = x4– 12x3 + 59x2 – 138x + 130 are a + ib and a + 2ib,
where a and b are real and b > 0.
HSC
(i)
Find the values of a and b.
3
(ii)
Hence, or otherwise, express P(x) as the product of quadratic factors with real
1
coefficients.
06
4a
3
when divided by x + 1. Find a, b and c.
HSC
05
The polynomial p(x) = ax3 + bx + c has a multiple zero at 1 and has remainder 4
4b
Suppose α, β, γ and δ are the four roots of the polynomial equation
x4 + px3 + qx2 + rx + s = 0.
HSC
(i)
Find the values of α + β + γ + δ and α β γ + α β δ + α γ δ + β γ δ in terms
2
of p, q, r and s.
(ii)
(iii)
Show that α 2 + β 2 + γ 2 + δ 2 = p2– 2q.
2
4
3
2
Apply the result in part (ii) to show that x – 3x + 5x + 7x – 8 = 0 cannot
1
have four real roots.
(iv)
By evaluating the polynomial at x = 0 and x = 1, deduce that the polynomial
2
equation x4– 3x3 + 5x2 + 7x – 8 = 0 has exactly two real roots.
05
6b
Let n be an integer greater than 2. Suppose ω is an nth root of unity and ω ≠ 1.
(i)
HSC
By expanding the left-hand side, show that
2
3
(1 + 2ω + 3ω + 4ω + · · · + nω
(ii)
(iii)
(iv)
(v)
Using
the
4a
=
z −1
prove
that
By expressing the left-hand side of the equation in part (iv) in terms of cos
1
1
1
π
5
3
2π
π
, find the exact value, in surd form, of cos .
5
5
Let α, β and γ be the zeros of the polynomial p(x) = 3x3 + 7x2 + 11x + 51. Find
(i)
HSC
1
) (ω – 1) = n.
, or otherwise,
z2 − 1
z − z −1
1
cos θ − i sin θ
=
, provided that sin θ ≠ 0.
cos 2θ + i sin 2θ − 1
2i sin θ
2π
2π
1
Hence, if ω = cos
+ i sin
, find the real part of
.
n
n
ω −1
2π
4π
6π
8π
5
Deduce that 1 + 2cos
+ 3cos
+ 4cos
+ 5cos
=- .
2
5
5
5
5
and cos
04
identity,
2
n–1
Find α2β γ + α β 2 γ + α β γ2.
2
2
1
2
(ii)
Find α + β + γ .
2
(iii)
Using part (ii), or otherwise, determine how many of the zeros of p(x) are
1
real. Justify your answer.
04
7b
Let α be a real number and suppose that z is a complex number such that
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HSC by Topic 1995 to 2006 Polynomials
z+
HSC
(i)
(i)
(iii)
Page 2
1
= 2cos α.
z
By reducing the above equation to a quadratic equation in z, solve for z
1
and use de Moivre’s theorem to show that zn +
= 2cos nα.
zn
1
Let w = z +
.
z
1
1
1
Prove that w3 + w2 – 2w – 2 = (z +
) + (z2 +
) + (z3 +
)
2
z
z
z3
3
Hence, or otherwise, find all solutions of cos α + cos 2α + cos 3α = 0,
3
2
in the range 0 ≤ α ≤ 2π .
03
2b
Let α = −1 + i.
(i)
HSC
(ii)
(iii)
03
2d
5a
1
4
Hence, or otherwise, find a real quadratic factor of the polynomial z + 4.
5
By applying de Moivre’s theorem and by also expanding (cos θ + i sin θ ) ,
2
3
Let α, β and γ be the three roots of x3 + px + q = 0, and define sn by
(i)
Explain why s1 = 0, and show that s2= −2p and s3= −3q.
(ii)
Prove that for n > 3, sn −psn−2 − qsn−3 .
α2 + β 2 + γ 2
α5 + β5 + γ 5
Deduce that
= 

2
5

(iii)
2c
3
2




α



3
3
+β +γ
2
3




2
It is given that 2 + i is a root of P(z) = z3 + rz2 + sz + 20, where r and s are real
numbers.
HSC
02
Show that α is a root of the equation z + 4 = 0.
sn = α n + β n + γn for n = 1, 2, 3,…
HSC
02
2
4
express cos 5θ as a polynomial in cos θ .
HSC
03
Express α in modulus-argument form.
(i)
State why 2 − i is also a root of P(z).
1
(ii)
Factorise P(z) over the real numbers.
2
3
5a
The equation 4x − 27x + k = 0 has a double root. Find the possible values of k.
5b
Let α, β, and γ be the roots of the equation x3 − 5x2 + 5 = 0.
2
HSC
02
(i)
HSC
Find a polynomial equation with integer coefficients whose roots are
2
α – 1, β – 1, and γ – 1.
(ii)
Find a polynomial equation with integer coefficients whose roots are
2
2
2
2
α , β , and γ .
(iii)
01
HSC
3b
Find the value of α 3 + β 3 + γ 3.
The numbers α, β and γ satisfy the equations
2
α+β+γ=3
α2 + β2 + γ2 = 1
1
1
1
+
+
=2
α
β
γ
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HSC by Topic 1995 to 2006 Polynomials
(i)
Page 3
Find the values of α β + β γ + γ α and α β γ.
3
Explain why α, β and γ are the roots of the cubic equation x3 – 3x2 + 4x – 2 = 0
(ii)
01
7b
2
Consider the equation x3 – 3x – 1 = 0, which we denote by (*).
(i)
HSC
Find the values of α, β and γ.
Let where p and q are integers having no common divisors other than +1 and
3
3
–1. Suppose that x is a root of the equation ax – 3x + b = 0, where a and b
are integers. Explain why p divides b and why q divides a. Deduce that (*)
does not have a rational root.
(ii)
Suppose that r, s and d are rational numbers and that
2
d is irrational. Assume
2
3
that r + s d is a root of (*). Show that 3r s + s d – 3s = 0 and show
r – s d that must also be a root of (*). Deduce from this result and part (i),
that no root of (*) can be expressed in the form r + s d with r, s and d
rational.
(iii)
Show that one root of (*) is 2 cos
π
9
2
.
(You may assume the identity cos 3θ = 4cos3 θ – 3cos θ.)
00
2b
2
Find the complex number a, given that i is a root of the equation.
HSC
00
Consider the equation z2 + az + (1 + i) = 0.
5b
Consider the polynomial p(x) = ax4 + bx3 + cx2 + dx + e, where a, b, c, d and e are
integers. Suppose α is an integer such that p(α) = 0.
HSC
(i)
(ii)
Prove that α divides e.
2
4
3
2
Prove that the polynomial q(x) = 4x – x + 3x + 2x – 3 does not have an
2
integer root.
99
2d
(i)
HSC
99
Consider the equation 2z3 – 3z2 + 18z + 10 = 0.
4b
Given that 1 – 3 i is a root of the equation, explain why 1 + 3i is another root.
3
2
2
(ii)
Find all roots of the equation 2z – 3z + 18z + 10 = 0.
2
(i)
Suppose the polynomial P(x) has a double root at x = α. Prove that P'(x) also
2
has a root at x = α.
HSC
(ii)
The polynomial A(x) = x4 + ax2 + bx + 36 has a double root at x = 2.
2
Find the values of a and b.
(iii)
99
5a
3
(ii)
98
2e
HSC
98
4a
2
2
The roots of x + 5x + 11 = 0, are α, β and γ.
(i)
HSC
Factorise the polynomial A(x) of part (ii) over the real numbers.
Find the polynomial equation whose roots are α 2, β 2, γ 2.
2
Find the value of α + β
2
2
+γ .
1
2
3
(i)
By solving the equation z + 1 = 0, find the three cube roots of –1.
(ii)
Let λ be a cube root of –1, where λ is not real. Show that λ2= l – λ.
(iii)
Hence simplify (1 - λ)6
(i)
Suppose that k is a double root of the polynomial equation f(x).
6
7
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HSC by Topic 1995 to 2006 Polynomials
Page 4
Show that f ’(k) = 0. .
HSC
(ii)
What feature does the graph of a polynomial have at a root of multiplicity 2?
(iii)
The polynomial P(x) = ax7 + bx6 + 1 is divisible by (x – 1)2. Find the
coefficients a and b.
(iv)
98
6a
HSC
Let E(x) = 1 + x +
x2
x3
x4
+
+
. Prove E(x) has no double roots.
2
24
6
Consider the following statements about a polynomial Q(x).
(i)
If Q(x) is even, then Q’(x) is odd.
(ii)
If Q’(x) is even, then Q(x) is odd.
2
Indicate whether each of these statements is true or false. Give reasons for your
answers.
97
5c
4
7
2
P(z) = z + bz + d. The polynomial has a double root α.
HSC
96
Suppose that b and d are real numbers and d ≠ 0. Consider the polynomial
5b
(i)
Prove that P’(z) is an odd function.
(ii)
Prove that -α is also a double root of P(z).
(iii)
Prove that d =
(iv)
For what values of b does P(z) have a double root equal to
(v)
For what values of b does P(z) have real roots?
b2
.
4
3 i?
Consider the polynomial equation x4 + ax3 + bx2 + cx + d = 0,
7
where a, b, c, and d are all integers. Suppose the equation has a root of the form ki,
HSC
where k is real, and k ≠ 0 .
95
4a
(i)
State why the conjugate -ki is also a root.
(ii)
Show that c = k2a.
(iii)
Show that c2 + a2d = abc.
(iv)
If 2 is also a root of the equation, and b = 0, show that c is even.
 4π 
 4π 
Find the least positive integer k such that cos 
 + i sin 
 is a solution of
 7 
 7 
(i)
4
HSC
zk = 1.
(ii)
Show that if the complex number w is a solution of zn = 1, then so is wm,
where m and n are arbitrary integers.
95
HSC
5b
Let f(t) = t3 + ct + d, where c and d are constants. Suppose that the equation
f (t) = 0 has three distinct real roots, t1, t2, and t3.
(i)
Find t1 + t2 + t3.
(ii)
Show that t12 + t22+ t32 = -2c.
(iii)
Since the roots are real and distinct, the graph of y=f(t) has two turning points,
at t = u and t = v, and f(u).f(v)<0.
Show that 27d2 + 4c3 < 0.
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A
HSC
Page 5
2006 3c.(i)a=3,b=1 (ii) (x2-6x+10)(x2-6x+13) 4a.a=1,b=-3,c=2
2
8
1
π 3π 5π 7π
6b.(iii)2004 4a.(i)39
(ii)-1
(iii)1 7b.(iii) ,
,
,
2
4
4
4
4
3
9
2005 4b.(i)-p,-r
3π
2003 2b.(i) 2 cis
4
(iii)z2+2z+2
2002 2c.(ii)(z+4)(z2-4z+5) 5a. ±27 5b.(i)x 3-2x 2-7x+1=0 (ii)x3-25x2+50x1
25=0 (iii)110
2001 3b.(i)4,2 (ii)1,1+i,1-i
2000 2b.-1
1999 2d.(ii)1+3i,1-3i,2
4b.(ii)a=-3,b=-20 (iii)(x-2)2(x2+4x+9) 5a.(i)x3-25x2-110x-121=0 (ii)25
1998 2e.(i)1 ± 3i
1,
(iii)1 4a.(ii)stat pt at root (iii)a=6,b=-7
1997 5c.(iv)6 (v)b<0
1995
2
4a.(i)k=7 5b.(i)0
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