CAIE A-Level Mathematics AS 2.2.2 Exponential and Ln Questions

Practise using eˣ and ln x as inverse functions and sketching exponential curves with correct intercepts, monotonic direction and asymptotic behaviour against other graphs.

Exam points

use vector notation and operations
solve geometric problems with vectors
interpret vector relationships

[Maximum number: 3]

The polynomial p(x) is defined by

\mathrm{p}(x)=2 x^{3}+a x^{2}-3 x-4

where a is a constant. It is given that ( x-4 ) is a factor of p(x).

Show that the equation $\mathrm{p}\left(\mathrm{e}^{3 y}\right)=0$ has only one real root and find its exact value.

Apply logarithms and use power law for $\mathrm{e}^{3 y}=4$
Obtain $\frac{1}{3} \ln 4$ or exact equivalent
Use discriminant [=1-8=-7] or equivalent to show no other roots
AG - necessary detail needed

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CAIE A-Level Mathematics AS 2.2.2 Exponential and Ln Questions question 1