IB Maths AA HL Sl 2 1 Straight Lines Questions

Practise IB Maths AA HL 2.1 through questions that extend lines to parameters, simultaneous intersections and geometric constraints.

[Maximum number: 4]

This question investigates a ratio of lengths found from the line passing through the points of inflexion of a quartic curve of the form $y=x^{4}-m x^{3}+n x$ .
The curve $y=x^{4}-3 x^{3}+3 x$ has points of inflexion at B and C . The line passing through B and C intersects the curve again at points A and D . This is shown in the following graph.

Figure for Question IB Maths AA HL Sl 2 1 Straight Lines Questions question 1 — IB Maths AA HL

(a)

Show that the equation of the line through B and C is y=-0.375 x.

[ 2 ]

$\quad y$ -intercept =0 (as line passes through (0,0) )

Note: Award R1 for correctly substituting point B or point C to show that y=0.

Award R1 for y=-0.375 x+0.

Note: Award AOFT if their answer to (b) doesn't lead to the given answer, but condone $\frac{-0.563}{1.5}$ .

so equation is y=-0.375 x
AG
Note: Award at most A1 R0 for working backwards to verify points B and C lie on the given line.

(b)

Show that the equation of the line through B and C is $y=\left(-\frac{m^{3}}{8}+n\right) x$ .

[ 2 ]

attempt to divide their $y_{C}$ by their $x_{C}$ OR substitute into $\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

\begin{aligned} & \text { e.g. }\left(-\frac{m^{4}}{16}+\frac{n m}{2}\right) \div \frac{m}{2} \\ & \left(-\frac{m^{4}}{16}+\frac{n m}{2}\right) \times \frac{2}{m} \text { OR equivalent manipulation leading to given answer } \end{aligned}

Note: Award AO FT if their answer does not lead to the AG.

=-\frac{m^{3}}{8}+n \text { and } y \text {-intercept }=0

\text { so equation is } y=\left(-\frac{m^{3}}{8}+n\right) x

Note: Award at most M1A0 for working backwards to verify points B and C lie on the given line.

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IB Maths AA HL Sl 2 1 Straight Lines Questions

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IB Maths AA HL Sl 2 1 Straight Lines Questions question 1

Question (a)

Question (b)