Logarithms — Free IB MYP5 Mathematics (Extended) Questions

1QuestionBasic log function graphsConcept Practice

2 marks~3 minCriterion A

The diagram shows the graph of

y = \log_2 x

.

a

State the domain of

y = \log_2 x

. [1]

b

State the range of

y = \log_2 x

. [1]

Solutions

2QuestionCommon logarithms (base 10)Assessment Practice

6 marks~9 minCriterion D

The Richter scale measures earthquake magnitude using a logarithmic model. The intensity ratio of two earthquakes with magnitudes

M_1

and

M_2

is given by

\text{Intensity ratio} = 10^{(M_1 - M_2)}.

The 1906 San Francisco earthquake had magnitude

M_1 = 7.9

. A seismologist compares predicted intensity ratios (relative to the 1906 earthquake) with observed field measurements for three minor tremors.

Tremor magnitudes: A:

3.2

— B:

4.4

— C:

5.1

Predicted ratio: A: ? — B: ? — C: ?

Observed ratio: A:

48\,000

— B:

2900

— C:

640

a

Calculate the predicted intensity ratio for each tremor. Give each answer to 3 significant figures. [2]

b

For tremor B, calculate the percentage error between the predicted and observed intensity ratios using

\text{percentage error} = \frac{|\text{predicted} - \text{observed}|}{\text{observed}} \times 100\%.

Give your answer to 1 decimal place. [2]

c

A journalist claims the Richter scale model is equally reliable for all three tremors. Justify whether this claim is supported by your results, and identify one limitation of using the Richter scale formula to predict intensity ratios in the field. [2]

Solutions

3QuestionCommon logarithms (base 10)Assessment Practice

8 marks~12 minCriterion A

The diagram shows part of the graph of

f(x) = \log_{10}(x)

.

a

Find the value of

a

such that

f(a) = f(25) + f(4)

, giving your answer as an integer. [2]

b

Solve the equation

\log_{10}(x+3) + \log_{10}(x-1) = 1

, rejecting any value of

x

that is not in the domain of

f

. [4]

c

Justify whether the point

(50, 2)

lies above or below the graph of

f(x) = \log_{10}(x)

. [2]

Solutions

4QuestionCommon logarithms (base 10)Assessment Practice

5 marks~8 minCriterion B

The diagram shows the first four figures of a pattern made from small squares arranged in an L-shape.

a

Write down the number of small squares in Figure 4. [1]

b

Find a rule for the total number of squares

S

in Figure

n

, giving your answer in the form

S = an^2 + bn + c

, where

a

,

b

and

c

are integers to be found. [2]

c

Verify your rule for Figure 3. Hence justify why the rule must contain an

n^2

term by referring to the structure of the L-shape. [2]

Solutions

5QuestionCommon logarithms (base 10)Assessment Practice

5 marks~8 minCriterion C

The diagram shows part of the graph of

f(x) = \log_{10}(x)

.

a

Write down the value of

f(100)

. [1]

b

Given that

f(x) = -1.5

, find the exact value of

x

. [2]

c

A student claims that

\log_{10}(a^2) = 2\log_{10}(a)

holds for all real values of

a

. Justify whether this claim is correct, and state the values of

a

for which it is valid. [2]

Solutions

6QuestionCombining multiple log lawsAssessment Practice

7 marks~11 minCriterion A

Let

f(x) = \log_2(8x^3)

, where

x > 0

.

a

Show that

f(x) = 3 + 3\log_2 x

. [2]

b

Find the value of

x

for which

f(x) = 9

. [2]

c

The graphs of

y = f(x)

and

y = -3\log_2 x + 9

intersect at exactly one point. Find the coordinates of that point. [3]

Solutions

7QuestionCombining multiple log lawsAssessment Practice

5 marks~8 minCriterion B

The diagram shows three figures in a pattern made from small squares arranged in an L-shape.

Figure 1 has 3 squares, Figure 2 has 8 squares, and Figure 3 has 15 squares.

a

Write down the number of squares in Figure 4. [1]

b

Find a rule for the total number of squares

S

in Figure

n

, giving your answer in the form

S = an^2 + bn

, where

a

and

b

are integers. [2]

c

Verify your rule for Figure 3. Justify whether the

n^2

term is necessary by referring to the structure of the figures. [2]

Solutions

8QuestionCondensing logarithmic expressionsAssessment Practice

6 marks~9 minCriterion C

The diagram shows part of the graph of

f(x) = \log_2 x

for

x > 0

.

a

Write down the value of

f(8)

. [1]

b

Given that

\log_2 3 + \log_2 5 - \log_2 k = f(1)

, find the value of

k

. [2]

c

A student claims that

f(x) = 2

has the same solution as

f(x^2 - 3x) = f(4)

. Justify whether the student is correct. [3]

Solutions

9QuestionCombining multiple log lawsAssessment Practice

8 marks~12 minCriterion D

A sound engineer is designing an outdoor concert stage and needs to predict combined sound levels from multiple speaker stacks. The combined sound intensity level, in decibels (dB), from two independent sources is modelled by

L_{\text{total}} = 10\log_{10}\!\left(10^{L_1/10} + 10^{L_2/10}\right)

where

L_1

and

L_2

are the individual sound levels in dB. The main speaker stack measures

L_1 = 85

dB and the monitor stack measures

L_2 = 92

dB.

Predicted and actual combined levels at three stage positions are given below.

Position	A	B	C
Predicted $L_{\text{total}}$ (dB)	92.8	89.3	94.1
Actual measured (dB)	91.2	90.8	95.6

a

Calculate the percentage error between the predicted and actual combined level at position A, where the predicted value is 92.8 dB and the actual measured value is 91.2 dB. Give your answer correct to 2 decimal places.

\text{percentage error} = \frac{|\text{predicted} - \text{actual}|}{\text{actual}} \times 100\%

[2]

b

The engineer proposes adding a third speaker stack at 85 dB to the existing combination of 85 dB and 92 dB. Using the formula, find the new combined level of all three sources. Give your answer correct to 1 decimal place. [3]

c

The percentage errors at positions A, B and C are 1.75\%, 1.65\% and 1.00\% respectively. Advise the engineer whether this model is reliable enough to use for stage safety planning, and identify one limitation of using this logarithmic model to represent human perception of loudness. [3]

Solutions

10QuestionConverting exponential equations to logarithmic formAssessment Practice

7 marks~11 minCriterion B

The diagram shows three figures of a pattern made from small squares arranged in an L-shape.

Figure 1 has 3 squares, Figure 2 has 8 squares, and Figure 3 has 15 squares.

a

Write down the number of squares in Figure 4. [1]

b

Find a rule for the total number of squares

S

in Figure

n

, giving your answer in the form

S = an^2 + bn

, where

a

and

b

are integers. [3]

c

Justify why the rule must contain an

n^2

term by referring to the structure of the figures, and verify that your rule gives the correct count for Figure 3. [3]

Solutions

11QuestionConverting exponential equations to logarithmic formAssessment Practice

8 marks~12 minCriterion D

A savings account pays compound interest at a fixed annual rate. After

t

years, the balance

B

(in dollars) is modelled by

B(t) = 1000 \cdot r^{\,t}

where

r

is the growth factor. The account balance after 2 years is 1210 dollars.

a

Show that

r = 1.1

. [2]

b

Find the number of complete years it takes for the balance to first exceed 1600 dollars. Give a clear algebraic method using logarithms. [3]

c

A second account compounds interest every 6 months at half the annual rate, giving actual balances that differ from the original model. The percentage errors between the two models are:

t

(years): 2, 5, 10

Percentage error (\%): 0.46, 1.12, 2.25

Assess whether a long-term investor should rely on the original model to predict their balance, justifying your answer by citing the trend in the percentage errors above. [3]

Solutions

12QuestionConverting exponential equations to logarithmic formAssessment Practice

5 marks~8 minCriterion A

The diagram shows part of the graph of

f(x) = \log_{10} x

for

x > 0

.

a

Write down the value of

f(1000)

. [1]

b

Find the value of

x

such that

f(x) = f(25) + f(4)

, giving your answer as an exact integer. [2]

c

Given that

f(x^2) - f(x) = 2

, deduce the value of

x

and explain what this result reveals about the relationship between parts (b) and (c). [2]

Solutions

13QuestionConverting exponential equations to logarithmic formAssessment Practice

9 marks~14 minCriterion C

The diagram shows the graphs of

f(x) = 5^{x+2}

and

g(x) = 3^{2x-1}

for

-2 \le x \le 5

.

a

Show that the equation

5^{x+2} = 3^{2x-1}

can be written as

x = \frac{\log 3 + 2\log 5}{2\log 3 - \log 5}.

[3]

b

Hence find the value of

x

, giving your answer correct to 3 significant figures. Verify that your value satisfies

5^{x+2} = 3^{2x-1}

by evaluating both sides. [3]

c

A student claims that because

5^{x+2} = 3^{2x-1}

has exactly one solution, the graphs of

f

and

g

can intersect at most once for all real

x

. Justify whether this claim is correct, referring to the nature of exponential functions. [3]

Solutions

14QuestionChecking for extraneous solutionsAssessment Practice

6 marks~9 minCriterion D

A chemist monitors the acidity of a river water sample by measuring its pH. The pH of a solution is defined by

\text{pH} = -\log_{10}(c)

where

c

is the hydrogen ion concentration in mol/L. Three samples are recorded.

Sample A — Measured pH:

3

— Predicted

c

(mol/L): ? — Actual

c

(mol/L):

0.00120

Sample B — Measured pH:

5

— Predicted

c

(mol/L): ? — Actual

c

(mol/L):

0.0000085

Sample C — Measured pH:

7

— Predicted

c

(mol/L): ? — Actual

c

(mol/L):

0.000000950

a

Deduce the predicted hydrogen ion concentration for each sample. Give each answer in standard form to 3 significant figures with units mol/L. [2]

b

A student claims that a solution with

\text{pH} = -3

is chemically possible. Using

\text{pH} = -\log_{10}(c)

, find the value of

c

that corresponds to

\text{pH} = -3

, and justify whether this value represents a realistic hydrogen ion concentration. [2]

c

The chemist uses the model to predict concentrations outside the measured range. Assess whether

\text{pH} = -\log_{10}(c)

is a reliable predictive model for river water acidity, supporting your assessment with a specific limitation of the model in this context. [2]

Solutions

15QuestionChecking for extraneous solutionsAssessment Practice

5 marks~8 minCriterion A

The diagram shows part of the graph of

f(x) = \log_2(x + 3)

for

x > -3

.

a

Write down the

x

-intercept of the graph of

f

. [1]

b

Find the value of

x

for which

f(x) = 3

, showing your method clearly. [2]

c

A second function is defined as

g(x) = \log_2(x + 3) + k

, where

k

is a constant. The

x

-intercept of

g

is at

x = 5

. Justify whether the value of

k

you find is consistent with

g

being a vertical translation of

f

. [2]

Solutions

16QuestionChecking for extraneous solutionsAssessment Practice

6 marks~9 minCriterion B

The diagram shows three figures in a pattern made from unit squares arranged in an L-shape.

Figure 1 has 2 unit squares, Figure 2 has 6 unit squares, and Figure 3 has 12 unit squares.

a

Write down the number of unit squares in Figure 4. [1]

b

Find a rule for the number of unit squares

S

in Figure

n

, giving your answer in the form

S = an^2 + bn

where

a

and

b

are integers. [2]

c

Verify your rule for Figure 3. Justify why the rule must contain an

n^2

term by referring to the geometric structure of the figures. [3]

Solutions

17QuestionChecking for extraneous solutionsAssessment Practice

8 marks~12 minCriterion C

An audio engineer is designing a tuning system for a digital synthesiser. The relationship between a frequency ratio

r

and the number of semitones

s

separating two notes is modelled by

\log_2(r) = \frac{s}{12}.

The engineer records observed frequency ratios for three intervals.

$s$ (semitones)	7	12	19
Observed $r$	1.498	2.000	2.973

a

Calculate the predicted frequency ratio

r

for each value of

s

, giving your answers correct to 3 decimal places. [2]

b

The percentage error between predicted and observed values is given by

\text{percentage error} = \frac{|\,\text{predicted} - \text{observed}\,|}{\text{observed}} \times 100\%.

Find the percentage error at

s = 7

and at

s = 19

, giving your answers correct to 2 decimal places. Hence determine at which of these two values the model is less accurate. [3]

c

The engineer claims that because the model predicts exactly

r = 2

when

s = 12

, it is reliable across all values of

s

. Advise the engineer whether this single result provides sufficient justification to use the model with confidence across all values of

s

in the table, referring to your results from parts (a) and (b). [3]

Solutions

18QuestionApplying logarithms to both sidesAssessment Practice

6 marks~9 minCriterion C

The table below shows four exponential equations of the form

a^x = a^2 + 1

.

Base

a

: 2, 3, 4, 5

Equation:

2^x = 5

,

3^x = 10

,

4^x = 17

,

5^x = 26

Approximate solution

x

:

2.322

,

2.096

,

2.044

,

2.024

a

Write down the exact solution of

3^x = 10

in the form

x = \dfrac{\log p}{\log q}

, where

p

and

q

are integers. [1]

b

Find the exact solution of

a^x = a^2 + 1

in terms of

a

, and hence show that

x > 2

for all valid bases

a

, and that

x \to 2

as

a \to \infty

. [3]

c

A student claims: "For a very large base

a

, the equation

a^x = a^2 + 1

is approximately the same as

a^x = a^2

, so

x \approx 2

."

Justify whether the student's reasoning is correct, and state one limitation of using

x = 2

as an approximation for small values of

a

. [2]

Solutions

19QuestionIsolating exponential expressionsAssessment Practice

9 marks~14 minCriterion B

The diagram shows the first four figures of a pattern. Each figure is made from small squares arranged in an L-shape: Figure

n

consists of a column of

n

squares on the left and a row of

n

squares along the bottom, sharing one corner square.

Figure ( $n$ )	1	2	3	4
Squares ( $S$ )	1	3	5	7

a

Find the number of squares in Figure 6 and Figure 10. [2]

b

Show that the number of squares in Figure

n

is given by

S(n) = 2n - 1

. Verify your rule for Figure 4. [3]

c

A second pattern is created where each Figure

n

is formed by placing two copies of the original L-shape together, sharing their corner squares. The number of squares in Figure

n

of the second pattern is

T(n)

.

Show that

T(n) = 4n - 3

, and hence justify whether

n = 26

is the only positive integer solution to

T(n) = S(n) + 50

. [4]

Solutions

20QuestionApplying logarithms to both sidesAssessment Practice

5 marks~8 minCriterion A

The diagram shows part of the graph of

f(x) = \log_5 x

for

x > 0

.

a

Write down the value of

f(125)

. [1]

b

The graph of

g

is obtained by reflecting

f

in the

y

-axis and then translating it by

\begin{pmatrix}3\\0\end{pmatrix}

. Find an expression for

g(x)

. [2]

c

Deduce the equation of the vertical asymptote of

g

and hence state the domain of

g

. [2]

Solutions

21QuestionApplying logarithms to both sidesAssessment Practice

11 marks~17 minCriterion D

The table below shows four data points believed to follow a relationship of the form

y = A \cdot 3^x + B

.

$x$	1	2	3	4
$y$	7	19	55	163

a

Show that the first differences of

y

have a common ratio of 3, and hence find the values of

A

and

B

by solving a pair of simultaneous equations. [4]

b

Show that plotting

\log_{10}(y - B)

against

x

produces a linear relationship with gradient

\log_{10} 3

, and verify this using at least two data points from the table. [3]

c

Justify why

\log_{10}

cannot be applied directly to both sides of

y = A \cdot 3^x + B

to linearise the model, and assess whether the model

y = 2 \cdot 3^x + 1

provides a reliable basis for predicting values beyond

x = 4

. [4]

Solutions

22QuestionComparing exponential and logarithmic graphsAssessment Practice

8 marks~12 minCriterion B

The diagram shows the graphs of

y = 2^x

and

y = \log_2 x

drawn on the same axes for

x > 0

.

The table below records selected values of both functions.

$x$	1	2	4	8
$y = 2^x$	2	4	16	256
$y = \log_2 x$	0	1	2	3

a

Describe the pattern in the values of

y = \log_2 x

as

x

doubles from 1 to 2, from 2 to 4, and from 4 to 8. Hence deduce the value of

\log_2 32

. [2]

b

A student claims that when

x

doubles, the value of

y = 2^x

is squared. Using values from the table, verify this claim for two consecutive doublings of

x

, and find the value of

n

such that

2^n = 256^2

. [3]

c

The two graphs are reflections of each other in the line

y = x

. Using the identity

\log_2(2^x) = x

and a pair of corresponding table values, justify whether

y = \log_2 x

and

y = 2^x

are inverse functions, and state the domain and range of

y = \log_2 x

. [3]

Solutions

23QuestionComparing exponential and logarithmic graphsAssessment Practice

5 marks~8 minCriterion C

The diagram shows the graphs of

y = 2^x

and

y = \log_2 x

for

-3 \leq x \leq 5

.

a

Write down the equation of the asymptote of

y = \log_2 x

. [1]

b

Both

y = 2^x

and

y = \log_2 x

intersect the line

y = x

. Deduce the geometric relationship between the graphs of

y = 2^x

and

y = \log_2 x

. [2]

c

A student claims that

2^x > \log_2 x

for all

x > 0

. Justify whether this claim is correct. [2]

Solutions

24QuestionComparing exponential and logarithmic graphsAssessment Practice

6 marks~9 minCriterion D

A biologist records the size of a bacterial colony at hourly intervals. The observed counts (in thousands of bacteria) are shown below.

$x$ (hours)	1	2	4	8
Observed $B$ (thousands)	2	4	8	14

The biologist proposes the exponential model

B(x) = 2^x

, where

B

is the population in thousands and

x

is the time in hours. The diagram shows the curve

B(x) = 2^x

and the four observed data points.

a

Complete the table of predicted values using

B(x) = 2^x

.

$x$ (hours)	1	2	4	8
Predicted $B$ (thousands)	2	4	\_\_\_	\_\_\_ [1]

b

Calculate the percentage error between the predicted and observed values at

x = 4

and at

x = 8

, giving your answers to 1 decimal place.

\text{percentage error} = \frac{|\text{predicted} - \text{observed}|}{\text{observed}} \times 100\%

[2]

c

Using your results from part (b), justify whether the biologist should continue using the exponential model

B(x) = 2^x

to predict bacterial population beyond

x = 8

hours, and identify one biological reason why exponential growth cannot continue indefinitely. [3]

Solutions

25QuestionCompound interest applicationsAssessment Practice

8 marks~12 minCriterion A

A savings account pays compound interest at a fixed annual rate. The table below shows the account balance at the end of each of the first four years.

Year	1	2	3	4
Balance (USD)	1050.00	1102.50	1157.63	1215.51

The initial deposit made at the start of Year 1 (before any interest) is 1000 USD.

a

Show that the annual interest rate is 5%, and write down the balance at the end of Year 5. [2]

b

Find a formula for the balance

B_n

(in USD) at the end of year

n

, where

n

is a positive integer. Hence determine the first year in which the balance exceeds 1500 USD. Give your answer to the nearest dollar. [4]

c

A second account offers a one-time bonus of 150 USD added to the initial deposit of 1000 USD, but pays only 3% annual compound interest. Advise a customer, with supporting calculations, which account they should choose if their goal is to maximise their balance at the end of Year 10. [2]

Solutions

26QuestionpH scale (log scale)Assessment Practice

14 marks~21 minCriterion B

A river is contaminated by an industrial acid spill. Its pH is measured daily at noon, starting at pH 6.8 immediately after the spill. The pH drops by exactly 0.3 each day for the first 5 days. The relationship between pH and hydrogen ion concentration [H+] (in mol/L) is pH = -log10[H+].

a

[3 marks] Calculate the hydrogen ion concentration [H+] for each of the 5 days (days 0 to 4). Present your results in a table with columns for Day, pH, and [H+] (in scientific notation to 3 significant figures).

b

[2 marks] Plot the data from part (a) on a graph of [H+] (y-axis) against Day (x-axis). Draw a smooth curve through the points.

c

[3 marks] The company responsible claims that the pH will continue to drop by 0.3 each day indefinitely. Using your graph or calculations, evaluate the validity of this claim. In your answer, comment on one limitation of modelling the hydrogen ion concentration with a constant daily pH drop.

Solutions

27QuestionpH scale (log scale)Assessment Practice

10 marks~15 minCriterion C

A chemist models the acidity of water samples using the formula

\text{pH} = -\log_{10}[\text{H}^+]

where

[\text{H}^+]

is the hydrogen ion concentration in mol/L.

Three water samples have the following measured

[\text{H}^+]

values and recorded actual pH values from a calibrated probe.

Sample A:

[\text{H}^+] = 4.2 \times 10^{-5}

mol/L, predicted pH

= p

, actual pH

= 4.21

Sample B:

[\text{H}^+] = 3.0 \times 10^{-7}

mol/L, predicted pH

= q

, actual pH

= 6.84

Sample C:

[\text{H}^+] = 8.5 \times 10^{-10}

mol/L, predicted pH

= r

, actual pH

= 9.04

a

Calculate the predicted pH values

p

,

q

, and

r

for samples A, B, and C respectively. Give each answer to 2 decimal places. [3]

b

The percentage error between the predicted and actual pH is defined as

\text{percentage error} = \frac{|\,\text{predicted} - \text{actual}\,|}{\text{actual}} \times 100\%.

The chemist claims the percentage error for sample C is less than

0.5\%

. Show that this claim is correct. [3]

c

A fourth sample D has an actual pH of

5.60

. The chemist uses the model to find

[\text{H}^+]

for sample D, obtaining

[\text{H}^+] \approx 2.51 \times 10^{-6}

mol/L, and concludes that sample D is acidic. Justify whether the chemist's conclusion is correct. [4]

Solutions

28QuestionCompound interest applicationsAssessment Practice

8 marks~12 minCriterion D

A technology company tracks the value of a server purchased for 8000 dollars. The value

V

(in dollars) after

t

years is modelled by

V(t) = 8000 \times (0.78)^{t}.

Year ( $t$ )	0	2	4
Predicted $V$ (USD)	8000	4867	2961
Actual $V$ (USD)	8000	5200	3400

a

Calculate the percentage error between the predicted and actual values at

t = 2

, giving your answer correct to 1 decimal place.

\text{percentage error} = \frac{|\text{predicted} - \text{actual}|}{\text{actual}} \times 100\%

[2]

b

Determine whether the model is acceptable at

t = 4

, given that the company requires the percentage error to be less than 15%. Show all working. [2]

c

A financial analyst suggests replacing the model with

V(t) = 8000 \times r^{t}

, where

r

is chosen so that the predicted value exactly matches the actual value of 5200 dollars at

t = 2

. Find the value of

r

, correct to 3 significant figures. Justify whether this revised model is appropriate for predicting the server's long-term value, referring to the behaviour of the model for large values of

t

. [4]

Solutions

29QuestionApplying change of base formulaAssessment Practice

7 marks~11 minCriterion A

Let

f(x) = \log_3 x

and

g(x) = \log_3(x - 2)

.

a

State the equation of the vertical asymptote of

g(x)

and write down its domain. [2]

b

Using the change of base formula

\log_3 a = \frac{\log a}{\log 3},

find the exact value of

x

such that

f(x) = g(x) + 2

, showing all steps clearly. [3]

c

A student claims that because

g(x) = \log_3(x - 2)

, the graph of

g

is the graph of

f

translated 2 units to the right. Justify whether this claim is correct, and state the value of

x

at which

g(x) = 0

. [2]

Solutions

30QuestionApplying change of base formulaAssessment Practice

8 marks~12 minCriterion C

The diagram shows four points plotted on a coordinate grid, where the horizontal axis represents

n

and the vertical axis represents

\log_{2^n}(8)

, for

n = 1, 2, 3, 4

.

The table below gives the values of

\log_{2^n}(8)

for

n = 1, 2, 3, 4

.

n

: 1, 2, 3, 4

\log_{2^n}(8)

:

3,\ \dfrac{3}{2},\ 1,\ \dfrac{3}{4}

a

Using the change of base formula

\log_a(b) = \frac{\log_2(b)}{\log_2(a)},

show that

\log_{2^n}(8) = \dfrac{3}{n}

for any positive integer

n

. [2]

b

Describe the pattern in the values of

\log_{2^n}(8)

as

n

increases, and find the value of

n

for which

\log_{2^n}(8) = \dfrac{3}{7}

. [3]

c

A student claims: "Since

\log_{2^n}(8) = \dfrac{3}{n}

, the value of

\log_{2^n}(8)

can never equal zero, no matter how large

n

becomes." Justify whether this claim is correct, and state what value

\log_{2^n}(8)

approaches as

n

grows very large. [3]

Solutions

31QuestionConverting between basesAssessment Practice

6 marks~9 minCriterion B

The diagram shows the first four figures of a pattern made from small squares arranged in an L-shape.

Figure 1 has 2 squares, Figure 2 has 5 squares, Figure 3 has 10 squares, and Figure 4 has 17 squares.

a

Write down the number of squares in Figure 5. [1]

b

Find a rule for the number of squares

S

in Figure

n

, giving your answer in the form

S = n^2 + c

, where

c

is an integer. [2]

c

The total number of squares across two consecutive figures, Figure

n

and Figure

n+1

, is 223. Determine whether

n = 10

is a valid figure number for this pattern, justifying your answer with full algebraic working. [3]

Solutions

32QuestionApplying change of base formulaAssessment Practice

6 marks~9 minCriterion D

A student invests 1000 dollars in two savings accounts.

Account A offers an annual interest rate of 5%, compounded annually.
Account B offers an annual interest rate of 4.8%, compounded quarterly.

The balance after

t

years in Account A is

A(t) = 1000(1.05)^t

and in Account B is

B(t) = 1000\!\left(1 + \dfrac{0.048}{4}\right)^{4t}

.

Observed balances (dollars):

$t$ (years)	5	10	15
Account A actual	1276	1629	2079
Account B actual	1268	1610	2045

a

Complete the predicted balances from the models at

t = 5

,

t = 10

, and

t = 15

. Give each answer to the nearest dollar. [1]

t

(years): 5, 10, 15
Predicted A (USD):
Predicted B (USD):

b

The number of years for an account to double is given by

t = \frac{\log 2}{\log(1 + r_{\text{eff}})},

where

r_{\text{eff}}

is the effective annual interest rate. For Account B,

r_{\text{eff}} = \left(1 + \frac{0.048}{4}\right)^4 - 1.

Calculate

r_{\text{eff}}

for Account B, and hence find the doubling time for each account. Give your answers to 2 decimal places. [3]

c

The percentage error between predicted and actual balance is

\text{percentage error} = \frac{|\text{predicted} - \text{actual}|}{\text{actual}} \times 100\%.

For Account A at

t = 10

the percentage error is approximately 0.06%, and for Account B at

t = 10

it is approximately 0.12%. A student claims that because both errors are below 1%, both models are equally reliable for predicting when each balance will reach 2000 dollars.

Assess this claim, referring to the doubling times found in part (b) and one limitation of using these models. [2]

Solutions

Logarithms — Free MYP5 Mathematics (Extended) Practice Questions