CAIE A-Level Mathematics 2.2.2 Exponential and Natural Logarithm Functions

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

sketch y = Aeᵏˣ through its y-intercept with the correct growth or decay direction
use ln(eˣ) = x and e^(ln x) = x to convert exact exponential and logarithmic forms
compare curve placement and intersections with a line or modulus graph over the stated domain

[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 $\mathrm{e}^{3 y}=4$

M1

Obtain $\frac{1}{3} \ln 4$ or exact equivalent

A1

Use discriminant [=1-8=-7] or equivalent to show no other roots

B1

AG - necessary detail needed

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2.2.2—Exponential and ln question 1