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Exponential and Logarithmic Functions · Paper 1 — Free Maths AA SL Practice Questions

1FoundationSAQ-SExponent laws and propertiess5 marksPaper 1~8 min

The mass of a sample of a radioactive isotope is modelled by

M(t) = M_0 \cdot 2^{-\frac{t}{8}}

, where

M_0

is the initial mass in grams and

t

is the time in years.

(a)

Show that the half-life of the isotope is 8 years. [1 mark]

(b)

Find the value of

t

for which

M(t) = \dfrac{M_0}{4}

. [3 marks]

(c)

A second isotope has the same initial mass

M_0

and a decay model

N(t) = M_0 \cdot 2^{-\frac{t}{k}}

, where

k > 0

. Given that

N(t) > M(t)

for all

t > 0

, determine the set of possible values of

k

. [1 mark]

Solutions

2FoundationSAQ-SExponent laws and propertiess5 marksPaper 1~8 min

Consider the expression

\dfrac{2x^3 \times 4x^5}{8x^2}

(a)

Simplify the expression, giving your answer in the form

kx^n

where

k \in \mathbb{Z}

and

n \in \mathbb{Z}

. [3 marks]

(b)

The simplified expression equals

\dfrac{1}{x^p}

for some

p \in \mathbb{Z}^+

. Determine the value of

x

for which this possible, and find the corresponding value of

p

. [2 marks]

Solutions

3MasterySAQ-SExponent laws and propertiess5 marksPaper 1~8 min

A radioactive isotope has initial mass

100\,\text{g}

. The mass

m

(in grams) remaining after

t

years is modelled by

m = 100 \times 2^{-\frac{t}{10}}

(a)

Show that the mass remaining after

30

years is

12.5\,\text{g}

. [2 marks]

(b)

Using your answer to part (a), or otherwise, find the value of

t

for which the mass remaining is

3.125\,\text{g}

. [3 marks]

Solutions

4ChallengeSAQ-LExponent laws and propertiess6 marksPaper 1~9 min

The number of bacteria in a laboratory culture,

N(t)

, after

t

hours is modelled by

N(t) = A \cdot 2^{kt}

where

A

and

k

are positive constants. Initially there are

500

bacteria. After

3

hours there are

4000

bacteria.

(a)

Write down the value of

A

. [1 mark]

(b)

Determine the exact value of

k

. [2 marks]

(c)

The culture reaches a critical threshold when

N(t) = 500 \times 2^{10}

. (i) Show that the time

T

to reach the critical threshold satisfies

T = \dfrac{10}{k}

. [2]
(ii) Hence determine the exact value of

T

, and state what this reveals about the effect of doubling

k

on the time to reach the critical threshold. [3 marks]

Solutions

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