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P1 Q6
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09 P1 Q6
2009 H2 Mathematics Paper 1 Question 6
Graphs and Transformations
Answers
(i)
(ii)
(iii)
x
=
−
0.515
{x=-0.515 \;}
x
=
−
0.515
or
x
=
2.45
{\; x=2.45}
x
=
2.45
Full solutions
(i)
(ii)
x
2
6
+
y
2
3
=
1
x
2
+
2
y
2
=
6
\begin{gather*} \frac{x^2}{6} + \frac{y^2}{3} = 1 \\ x^2 + 2 y^2 = 6 \\ \end{gather*}
6
x
2
+
3
y
2
=
1
x
2
+
2
y
2
=
6
Substituting equation of
C
1
,
{C_1,}
C
1
,
x
2
+
2
(
x
−
2
x
+
2
)
2
=
6
2
(
x
−
2
x
+
2
)
2
=
6
−
x
2
2
(
x
−
2
)
2
=
(
x
+
2
)
2
(
6
−
x
2
)
■
\begin{gather*} x^2 + 2 \left( \frac{x-2}{x+2} \right)^2 = 6 \\ 2 \left( \frac{x-2}{x+2} \right)^2 = 6 - x^2 \\ 2 \left( x-2 \right)^2 = (x+2)^2(6 - x^2) \; \blacksquare \end{gather*}
x
2
+
2
(
x
+
2
x
−
2
)
2
=
6
2
(
x
+
2
x
−
2
)
2
=
6
−
x
2
2
(
x
−
2
)
2
=
(
x
+
2
)
2
(
6
−
x
2
)
■
(iii)
Solving the equation with a GC,
x
=
−
0.515
(3 sf)
■
or
x
=
2.45
(3 sf)
■
\begin{align*} x &= -0.515 \textrm{ (3 sf)} \; \blacksquare \\ \textrm{or } \; x &= 2.45 \textrm{ (3 sf)} \; \blacksquare \\ \end{align*}
x
or
x
=
−
0.515
(3 sf)
■
=
2.45
(3 sf)
■
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