Challenging +1.2 This is a Further Maths question requiring understanding of hyperbola asymptotes and matrix transformations, but the approach is methodical rather than insightful. Students must find asymptotes of the hyperbola (y = ±3x/4), apply the reflection matrix (swapping x and y), and verify both methods agree. While it involves multiple Further Maths topics, the steps are standard and the 'show that' format provides clear direction, making it moderately above average difficulty.
Josh and Zoe are solving the following mathematics problem:
The curve \(C_1\) has equation
$$\frac{x^2}{16} - \frac{y^2}{9} = 1$$
The matrix \(\mathbf{M} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) maps \(C_1\) onto \(C_2\)
Find the equations of the asymptotes of \(C_2\)
Josh says that to solve this problem you must first carry out the transformation on \(C_1\) to find \(C_2\), and then find the asymptotes of \(C_2\)
Zoe says that you will get the same answer if you first find the asymptotes of \(C_1\), and then carry out the transformation on these asymptotes to obtain the asymptotes of \(C_2\)
Show that Zoe is correct.
[5 marks]
Question 5:
5 | States the correct asymptotes of
C
1 | 1.1b | B1 | Josh’s method
Reflection in y = x
y2 x2
− =1
C is
2
16 9
The asymptotes of C are y = ± 4 x
2 3
Zoe’s method
The asymptotes of C are y = ±3 x
1 4
The transformation is a reflection in
y = x
The asymptotes of C are y = ± 4 x
2 3
Both answers are the same, so Zoe is
correct.
States the correct equation of C
2 | 3.1a | B1
States the correct asymptotes of
C
2 | 1.1b | B1
Obtains the asymptotes of C by
2
both methods. | 3.1a | M1
Shows that both methods lead to
the same answer and concludes
that Zoe is correct. | 2.3 | R1
Question total | 5
Q | Marking Instructions | AO | Marks | Typical solution
Josh and Zoe are solving the following mathematics problem:
The curve $C_1$ has equation
$$\frac{x^2}{16} - \frac{y^2}{9} = 1$$
The matrix $\mathbf{M} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ maps $C_1$ onto $C_2$
Find the equations of the asymptotes of $C_2$
Josh says that to solve this problem you must first carry out the transformation on $C_1$ to find $C_2$, and then find the asymptotes of $C_2$
Zoe says that you will get the same answer if you first find the asymptotes of $C_1$, and then carry out the transformation on these asymptotes to obtain the asymptotes of $C_2$
Show that Zoe is correct.
[5 marks]
\hfill \mbox{\textit{AQA Further Paper 2 2023 Q5 [5]}}