CAIE A-Level Mathematics AS 2.6.1 Root Location Questions

Practise statistical hypothesis tests within the AS mathematics syllabus.

Exam points

use sign changes to bracket a root within a stated interval
sketch paired graphs to show the number of real roots
justify root-location conclusions from calculated values

[Maximum number: 2]

Figure for Question CAIE A-Level Mathematics AS 2.6.1 Root Location Questions question 1 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $y=\frac{\ln (2 x+1)}{x+3}$ . The curve has a maximum point M.

Show by calculation that the x-coordinate of M lies between 2.5 and 3.0 .

Consider sign of $x-\frac{x+3}{\ln (2 x+1)}+0.5$ or equivalent for 2.5 and 3.0
Obtain $-0.07(0.0696 \ldots)$ and $0.4(0.4166 \ldots)$ and justify
conclusion
Answer given - necessary detail needed.

2

Alternative Method for Question 6(c)

Consider the values of $\mathrm{f}(x)=\frac{x+3}{\ln (2 x+1)}-0.5$ and obtain
$f(2.5)=2.57(2.5696 \ldots)$ and $f(3)=2.58(2.58339 \ldots)$
Conclude f(2.5)<3 and f(3)>2.5 so root lies in given interval
Answer given - necessary detail needed.

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CAIE A-Level Mathematics AS 2.6.1 Root Location Questions question 1