EXN.1.AHL.TZ0.12

pestleMathematicsAIHLPaper 1EXN· sl-4-4-pearsons-scatter-diagrams-eqn-of-y-on-xsource ↗

It is believed that the power $P$ of a signal at a point $d$ km from an antenna is inversely proportional to $d^{n}$ where $n \in ℤ^{+}$ .

The value of $P$ is recorded at distances of $1 m$ to $5 m$ and the values of $\log_{10} d$ and $\log_{10} P$ are plotted on the graph below.

The values of $\log_{10} d$ and $\log_{10} P$ are shown in the table below.

Explain why this graph indicates that $P$ is inversely proportional to $d^{n}$ .

[2]

Find the equation of the least squares regression line of $\log_{10} P$ against $\log_{10} d$ .

[2]

Use your answer to part (b) to write down the value of $n$ to the nearest integer.

[1]

c.i.

Find an expression for $P$ in terms of $d$ .

[2]

c.ii.

Markscheme / solution

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

a straight line with a negative gradient A1A1

[2 marks]

$\log P = - 2.040 \dots \log d - 0.12632 \dots \approx - 2.04 \log d - 0.126$ A1A1

Note: A1 for each correct term.

[2 marks]

$n = 2$ A1

[1 mark]

c.i.

$P = 10^{- 0.126 \dots} d^{- 2}$ (M1)

$\approx 0.748 d^{- 2}$ A1

[2 marks]

c.ii.

Examiners’ report

[N/A]

c.i.

[N/A]

c.ii.