CAIE A-Level Further Math AS 1 Further Pure Mathematics 1 Questions

$b=-(\alpha+\beta+\gamma+\delta)=-3$$5=(-3)^{2}-2(\alpha \beta+\alpha \gamma+\alpha \delta+\beta \gamma+\beta \delta+\gamma \delta)$
Uses formula for sum of squares.

c=2
$6=\frac{\alpha \beta \gamma+\beta \gamma \delta+\gamma \delta \alpha+\delta \alpha \beta}{\alpha \beta \gamma \delta}=\frac{-d}{-2}$
Uses
$$\alpha^{-1}+\beta^{-1}+\gamma^{-1}+\delta^{-1}=\frac{\alpha \beta \gamma+\beta \gamma \delta+\gamma \delta \alpha+\delta \alpha \beta}{\alpha \beta \gamma \delta}.$$

d=12
Equation is $x^{4}-3 x^{3}+2 x^{2}+12 x-2=0$.

6

$\alpha^{4}+\beta^{4}+\gamma^{4}+\delta^{4}=3(-27)-2(5)-12(3)+2(4)$
Uses their quartic equation derived in (a).

-119
2

$x=-1, \quad x=2$
Vertical asymptotes.

y=1
Horizontal asymptote.

2

$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\left(x^{2}-x-2\right)(2 x)-\left(x^{2}+2\right)(2 x-1)}{\left(x^{2}-x-2\right)^{2}}$

M1*

Finds $\frac{\mathrm{d} y}{\mathrm{~d} x}$.

$x^{2}+8 x-2=0$
Sets equal to 0 and forms equation.

(-8.2,0.9), (0.2, -0.9).
Condone
$$\left(-4-3 \sqrt{2}, \frac{2}{3} \sqrt{2}\right),\left(-4+3 \sqrt{2},-\frac{2}{3} \sqrt{2}\right) .$$

4

Axes and all three asymptotes.
Correct shape and position, crossing horizontal
asymptote.

States (0,-1) coordinates of intersection with axes,
may be seen on diagram.

3

B1 FT

FT from sketch in (c)

B1

All correct.

2

$$\begin{aligned} \frac{x^{2}+2}{x^{2}-x-2}=1 \text { or } \frac{x^{2}+2}{x^{2}-x-2}=-1 x+4=0 \quad \text { or } \quad 2 x^{2}-x=0 \end{aligned}$$
Finds critical points, award M1 for each case.

$x=-4 \quad$ or $x=0, \quad x=\frac{1}{2}$$-4<x<-1, \quad 0<x<\frac{1}{2}, x>2$
Must have three distinct regions. Condone $\leqslant-1$
and $\geqslant 2$.

4

$\left|\begin{array}{ll}1 2 5\end{array}\right|-k\left|\begin{array}{ll}2 3 5\end{array}\right|+3\left|\begin{array}{ll}2 3 2\end{array}\right|=0 \Rightarrow-1-k+3=0 \Rightarrow k=2$
Sets determinant of A equal to zero.

$\left(\begin{array}{ccc}-2 -1 1 1 3\end{array}\right)\left(\begin{array}{ccc}1 2 2 1 3 2 5\end{array}\right)\left(\begin{array}{cc}0 -2 -1 0 0\end{array}\right)=\left(\begin{array}{ccc}-2 -1 1 1 3\end{array}\right)\left(\begin{array}{cc}-2 -1 -1 -2 0\end{array}\right)$
Multiplying two matrices correctly, correct
dimensions.

$\left(\begin{array}{cc}3 -7 -9 3\end{array}\right)$
Completing matrix multiplication, AG.

5

$\left(\begin{array}{cc}3 -7 -9 3\end{array}\right)\binom{x}{y}=\binom{3 x-7 y}{-9 x+3 y}$
Transforms $\binom{x}{y}$ to $\binom{X}{Y}$.

-9 x+3 m x=m(3 x-7 m x)
Uses y=m x and Y=m X.

$-9+3 m=3 m-7 m^{2} \Rightarrow 7 m^{2}=9$$y=\frac{3}{\sqrt{7}} x$ and $y=-\frac{3}{\sqrt{7}} x$
5

$\mathbf{D}=\left(\begin{array}{cc}\alpha 0 \alpha\end{array}\right)$$\mathbf{E}=\left(\begin{array}{ll}\beta 0 1\end{array}\right)$$\mathbf{F}=\left(\begin{array}{ll}0 1 0\end{array}\right)$$\left(\begin{array}{cc}3 -7 -9 3\end{array}\right)=\left(\begin{array}{ll}\alpha 0 \alpha\end{array}\right)-9\left(\begin{array}{ll}0 \beta 1 0\end{array}\right)$
Setting up simultaneous equations using their D and
E.

$\mathbf{D}=\left(\begin{array}{ll}3 0 3\end{array}\right) \quad \mathbf{E}=\left(\begin{array}{ll}\frac{7}{9} 0 1\end{array}\right)$
Condone $\alpha=3, \beta=\frac{7}{9}$ if it is clear that they refer to
the correct matrices.

5