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: 3]

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 $y=\frac{2 \ln x}{3 x+1}$ . The curve crosses the x-axis at the point A and has a maximum point B. The shaded region is bounded by the curve and the lines x=3 and y=0.

Show by calculation that the x-coordinate of B lies between 3.0 and 3.1.

Equate numerator of first derivative to zero
May be implied.

Consider sign of $\frac{2}{x}(3 x+1)-6 \ln x$ for 3.0 and 3.1
OE

Obtain $0.074 \ldots$ and $-0.14 \ldots$ or equivalents and justify conclusion
AG - necessary detail needed.
0.00075 and -0.001275 .

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