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HSC 2002 - Mathematics Extension 2
Exemplar Sample
Marks
Question 5 (15 marks) Use a SEPARATE writing booklet.
(a)
The equation 4x3 − 27x + k = 0 has a double root. Find the possible values of k.
(b)
Let α, β, and γ be the roots of the equation x3 − 5x2 + 5 = 0.
2
(i)
Find a polynomial equation with integer coefficients whose roots are
α – 1, β – 1, and γ – 1.
2
(ii)
Find a polynomial equation with integer coefficients whose roots are
α 2, β 2, and γ 2.
2
(iii)
Find the value of α 3 + β 3 + γ 3 .
2
(c)
y
T(x0, y0)
P(x1, y1)
Directrix D
S
O
Q
x
R
E
x2
+
y2
= 1 , and focus S and directrix D as shown
a2 b2
in the diagram. The point T (x0, y0) lies outside the ellipse and is not on the x axis.
The ellipse E has equation
The chord of contact PQ from T intersects D at R, as shown in the diagram.
(i)
Show that the equation of the tangent to the ellipse at the point P(x1, y1) is
x1 x
a2
(ii)
y1y
b2
=1.
Show that the equation of the chord of contact from T is
x0 x
a2
(iii)
+
+
y0 y
b2
Show that TS is perpendicular to SR.
–8–
2
2
= 1.
3
HSC 2002 - Mathematics Extension 2
Exemplar Sample
Question 5
HSC 2002 - Mathematics Extension 2
Exemplar Sample
HSC 2002 - Mathematics Extension 2
Exemplar Sample
HSC 2002 - Mathematics Extension 2
Exemplar Sample