Number and Algebra — Free IB Maths AA HL Practice Questions

1FoundationSAQ-SDefinition and general term of arithmetic sequencess5 marksPaper 1~8 min

A car rental company charges a fixed initial fee of USD 200, and then a constant daily rate. The total cost, in dollars, after

n

days is modelled by an arithmetic sequence

u_n

, where

u_1 = 200

and

u_4 = 440

.

(a)

State the common difference

d

. [1 mark]

(b)

Calculate the total cost after 10 days. [2 marks]

(c)

A competitor charges no fixed fee and a daily rate of USD 95 per day, giving total cost

v_n = 95n

. Determine the number of complete days after which the competitor's scheme first becomes more expensive than the original company's scheme, and justify your answer. [2 marks]

Solutions

2MasterySAQ-SDefinition and general term of arithmetic sequencess5 marksPaper 1~8 min

Data

$

A water tank contains

1200

litres of water. Due to a leak, the volume that leaks out each day forms an arithmetic sequence. On day 1,

20

litres leak out; on day 2,

23

litres leak out; on day 3,

26

litres leak out.

(a)

Show that the volume of water, in litres, remaining in the tank after

n

days is given by $$V_n = 1200 - \frac{n}{2}(37 + 3n). \quad [3 marks]

(b)

Hence find the number of days until the tank is empty. $ [2 marks]

Solutions

3ChallengeSAQ-LDefinition and general term of arithmetic sequencess8 marksPaper 1~12 min

A company designs a storage tank whose cross-section is a trapezoid. The lengths of five parallel internal supports

L_1 < L_2 < L_3 < L_4 < L_5

form an arithmetic sequence, where

L_1 = 2.4\,\text{m}

and

L_5 = 4.8\,\text{m}

.

(a)

Calculate the common difference

d

of the arithmetic sequence. [2 marks]

(b)

Two trapezoids are constructed using the same perpendicular height

h = 3\,\text{m}

: - Trapezoid P has parallel sides

L_1

and

L_3

. - Trapezoid Q has parallel sides

L_2

and

L_5

. The area of a trapezoid with parallel sides

a

,

b

and height

h

is

A = \dfrac{h}{2}(a + b)

. Calculate the ratio of the area of Trapezoid P to the area of Trapezoid Q, giving your answer in the form

p:q

where

p,\,q \in \mathbb{Z}^+

. [3 marks]

(c)

Four trapezoids

T_1, T_2, T_3, T_4

are formed by taking consecutive pairs of supports

(L_1,\,L_2)

,

(L_2,\,L_3)

,

(L_3,\,L_4)

,

(L_4,\,L_5)

as parallel sides, each with height

h = 3\,\text{m}

. Determine whether the areas of

T_1, T_2, T_3, T_4

form an arithmetic sequence. Justify your answer algebraically. [3 marks]

Solutions

4FoundationSAQ-SDefinition and general term of arithmetic sequencess5 marksPaper 1~8 min

A gardener plants a row of

n

flowers in a straight line. The height of the first flower is

12\,\text{cm}

. Each subsequent flower is

3\,\text{cm}

taller than the previous one.

(a)

Write down an expression for

u_k

, the height of the

k

th flower, in terms of

k

. [1 mark]

(b)

Calculate the height of the 8th flower. [2 marks]

(c)

The gardener wants the total height of all flowers in the row to exceed

500\,\text{cm}

. Determine the minimum number of flowers required. [2 marks]

Solutions

5FoundationSAQ-SPolynomial expressions and their factorizationss5 marksPaper 1~8 min

A rectangular garden has length

(x+4)

metres and width

(x-1)

metres, where

x > 1

.

(a)

Show that the area of the garden, in square metres, is

x^2 + 3x - 4

. [1 mark]

(b)

Given that the area of the garden is

70\,\text{m}^2

, find the value of

x

. [3 marks]

(c)

Hence find the perimeter of the garden. [1 mark]

Solutions

6MasterySAQ-SPolynomial expressions and their factorizationss5 marksPaper 1~8 min

The graph of the polynomial

y = P(x)

passes through the points

(-1,\,0)

,

(2,\,0)

, and

(0,\,-12)

. The polynomial is of degree 3 with leading coefficient

2

.

(a)

Show that

P(x) = 2x^3 + 4x^2 - 10x - 12

. [3 marks]

(b)

Find the

x

-coordinates of the turning points of the graph of

y = P(x)

, giving your answers in exact form. [2 marks]

Solutions

7ChallengeSAQ-LPolynomial expressions and their factorizationss8 marksPaper 1~12 min

A rectangular box has dimensions

x\,\text{cm}

,

(x+3)\,\text{cm}

, and

(2x-1)\,\text{cm}

, where

x > \dfrac{1}{2}

. The volume of the box is

V(x)

cubic centimetres.

(a)

Show that

V(x) = 2x^3 + 5x^2 - 3x

. [2 marks]

(b)

Find the three values of

x

for which

V(x) = 0

. State, with a reason, which of these values lie outside the physical domain

x > \dfrac{1}{2}

. [3 marks]

(c)

Given that

V(x) = 60

, show that

2x^3 + 5x^2 - 3x - 60 = 0

, and verify that a root lies in the interval

2.5 < x < 2.6

. [2 marks]

(d)

Use linear interpolation once on the interval

2.5 < x < 2.6

to find an estimate for

x

, giving your answer correct to three significant figures. [1 mark]

Solutions

8FoundationSAQ-SPolynomial expressions and their factorizationss17 marksPaper 1~26 min

A rectangular garden has a length of

(x+4)

metres and a width of

(x-1)

metres.

(a)

Given that the area of the garden is

60\,\text{m}^2

, show that

x^2 + 3x - 64 = 0

. [2 marks]

(b)

Find the value of

x

, giving your answer in exact form. [2 marks]

(c)

A path of uniform width

w

metres is built around the outside of the garden. The total area enclosed by the outer edge of the path is

\left(\frac{3}{2}\right)

times the area of the garden. Find the exact value of

w

. [3] (Total: 7 marks) > Note: mark total adjusted to 7 to accommodate the added AO3 discriminator sub-part; if 5 marks are required, omit (c) and award (a) [2] + (b) [3] with exact form and justification of sign. 5-mark version (if (c) is omitted): (a) Given that the area of the garden is

60\,\text{m}^2

, show that

x^2 + 3x - 64 = 0

. [2] (b) Find the value of

x

, giving your answer in exact form, justifying why only one solution is valid. (All subsequent sections use the 5-mark version.) [3 marks]

Solutions

9FoundationSAQ-SExponent laws and propertiess5 marksPaper 1~8 min

The number of bacteria in a culture is given by

N(t) = 250 \times 3^{kt}

, where

t

is the time in hours after the culture is first observed. After

2

hours, the number of bacteria is

4500

.

(a)

Calculate the value of

k

, giving your answer in the form

k = \log_3 a

where

a \in \mathbb{Q}

. [2 marks]

(b)

Calculate the number of bacteria present after

5

hours, giving your answer to the nearest integer. [3 marks]

Solutions

10MasterySAQ-SExponent laws and propertiess5 marksPaper 1~8 min

The number of bacteria in a culture,

N(t)

, after

t

hours is modelled by

N(t) = N_0 \cdot 2^{kt}

, where

N_0

is the initial number of bacteria and

k

is a positive constant.

(a)

Show that when

k = 1

, the population doubles every hour. [2 marks]

(b)

Hence, find the value of

k

such that the population triples every 4 hours. Give your answer in the form

\log_2 a

where

a \in \mathbb{Q}^+

. [3 marks]

Solutions

11ChallengeSAQ-LLogarithmic functions and their propertiess8 marksPaper 1~12 min

The graph of

y = \ln(ax + b)

passes through the points

P(1,\, \ln 5)

and

Q(3,\, \ln 13)

, where

a

and

b

are real constants with

ax + b > 0

for all

x

in the domain.

(a)

Determine the exact values of

a

and

b

. [4 marks]

(b)

Hence, find the exact coordinates of the point where the graph intersects the

x

-axis. [2 marks]

(c)

Show that

y = \ln(ax + b)

can be written in the form

y = \ln a + \ln\!\left(x + \dfrac{b}{a}\right)

, and hence describe fully the two transformations that map the graph of

y = \ln x

onto the graph of

y = \ln(ax + b)

, using your values of

a

and

b

. [2 marks]

Solutions

12FoundationSAQ-SExponent laws and propertiess5 marksPaper 1~8 min

The graph of

y = a \cdot b^x

passes through the points

(1,\, 6)

and

(3,\, 54)

.

(a)

Show that

ab = 6

and

ab^3 = 54

. [2 marks]

(b)

Hence calculate the value of

a

and the value of

b

. [3 marks]

Solutions

13FoundationSAQ-SBinomial expansion and coefficientss4 marksPaper 1~6 min

In the expansion of

(2x + 3)^5

:

(a)

Write down the general term of the expansion in terms of

r

. [1 mark]

(b)

Find the coefficient of

x^3

. [3 marks]

Solutions

14MasterySAQ-SBinomial expansion and coefficientss5 marksPaper 1~8 min

Consider the binomial expansion of

\left(2x^3 - \dfrac{1}{x}\right)^n

, where

n \in \mathbb{Z}^+

.

(a)

Show that the coefficient of the term independent of

x

is

\dbinom{n}{\frac{n}{4}} 2^{\frac{n}{4}} (-1)^{\frac{3n}{4}}

, and state the condition

n

for this term to exist. [3 marks]

(b)

Hence, given that

n = 8

, find the value of the constant term. [2 marks]

Solutions

15ChallengeSAQ-LBinomial expansion and coefficientss8 marksPaper 1~12 min

Consider the binomial expansion of

(1 + kx)^n

, where

n \in \mathbb{Z}^+

and

k

is a positive constant. The coefficient of

x^2

is

112

and the coefficient of

x^3

is

448

.

(a)

Show that

(n-2)k = 12

. [2 marks]

(b)

Hence determine the value of

n

and the value of

k

. [3 marks]

(c)

Determine the coefficient of

x^4

in the expansion. [3 marks]

Solutions

16FoundationSAQ-SBinomial expansion and coefficientss5 marksPaper 1~8 min

The shows the first six rows (rows

0

to

5

) of Pascal's triangle.

\begin{array}{c} 1 \\ 1 \quad 1 \\ 1 \quad 2 \quad 1 \\ 1 \quad 3 \quad 1 \\ 1 \quad 4 \quad 6 \quad 4 \quad 1 \\ 1 \quad 5 \quad 10 \quad 5 \quad 1 \end{array}

(a)

Write down the value of

\dbinom{5}{3}

. [1 mark]

(b)

Find the coefficient of

x^3

in the expansion of

(2x + 3)^5

. [4 marks]

Solutions

17MasterySAQ-SDefinition and general term of arithmetic sequencess7 marksPaper 2~11 min

A company is developing a new recycling program. In the first month,

1200\,\text{kg}

of waste are collected. The monthly amounts collected form an arithmetic sequence. The company collects a total of

6000\,\text{kg}

over the first four months.

(a)

Calculate the common difference of the sequence. [3 marks]

(b)

Calculate the total waste collected over the first three months. [2 marks]

(c)

Determine the month in which the amount of waste collected first exceeds

2500\,\text{kg}

. [2 marks]

Solutions

18MasterySAQ-SDefinition and general term of arithmetic sequencess7 marksPaper 2~11 min

A biologist studies the population of a fish species in a lake. The population in the first year of observation is

4500

fish. The population decreases each year by a constant amount, forming an arithmetic sequence. In the sixth year, the population is

3200

fish.

(a)

Calculate the common difference of the sequence. [2 marks]

(b)

Determine the population in the tenth year. [2 marks]

(c)

The biologist estimates that the population will fall below

1000

fish in year

k

. Calculate the value of

k

. [3 marks]

Solutions

19MasterySAQ-SDefinition and general term of arithmetic sequencess10 marksPaper 2~15 min

Data

u_n = u_1 + (n-1)d \qquad S_n = \frac{n}{2}(2u_1 + (n-1)d), \quad n \in \mathbb{Z}^+

A construction company is building a series of houses along a straight road. The first house is located

20\,\text{m}

from the start of the road. Each subsequent house is placed

12\,\text{m}

further from the start of the road than the previous house.

(a)

Write down the values of

u_1

and

d

for this arithmetic sequence. [2 marks]

(b)

Calculate the distance from the start of the road to the

30

th house. [2 marks]

(c)

The local council requires that no house may be placed more than

400\,\text{m}

from the start of the road. Determine the maximum number of houses that can be built. [3 marks]

(d)

The company also requires that the total distance covered by all houses (i.e. the sum of all individual house distances from the start of the road) must not exceed

8000\,\text{m}

. Determine whether the constraint in part (c) or this new constraint is more restrictive, justifying your answer. [3 marks]

Solutions

20MasterySAQ-SSum of an arithmetic sequences6 marksPaper 2~9 min

A construction company is building a staircase. The first step has a height of

12\,\text{cm}

. Each subsequent step is

3\,\text{cm}

taller than the previous step. The total height of the staircase is

1050\,\text{cm}

.

(a)

Calculate the number of steps in the staircase. [3 marks]

(b)

Building regulations require that no single step in a staircase may exceed

80\,\text{cm}

in height. Determine whether this staircase complies with the regulation. [3 marks]

Solutions

21MasterySAQ-SPolynomial expressions and their factorizationss9 marksPaper 2~14 min

The polynomial

P(x) = 2x^3 + ax^2 + bx - 6

has factors

(x-1)

and

(x+2)

.

(a)

Show that

a = 5

and

b = -1

. [3 marks]

(b)

Hence factorize

P(x)

completely. [2 marks]

(c)

The curve

y = P(x)

is translated by vector

\begin{pmatrix} k \\ 0 \end{pmatrix}

, where

k \in \mathbb{Z}^+

, to give the curve

y = Q(x)

. Given that

Q(x)

has a repeated linear factor, find the value of

k

and write down

Q(x)

in fully factorized form. [4 marks]

Solutions

22MasterySAQ-SPolynomial expressions and their factorizationss6 marksPaper 2~9 min

The graph of

y = P(x)

, where

P(x)

is a cubic polynomial, passes through the points

(-2,\,0)

,

(1,\,0)

, and

(3,\,0)

. The point

(0,\,12)

also lies on the graph.

(a)

Write down the three linear factors of

P(x)

. [1 mark]

(b)

Determine the value of the leading coefficient and hence write

P(x)

in the form

k(x+2)(x-1)(x-3)

. [3 marks]

(c)

The equation

P(x) = c

has exactly one real solution. Determine all possible values of

c

, justifying your answer. [2 marks]

Solutions

23MasterySAQ-SPolynomial expressions and their factorizationss6 marksPaper 2~9 min

A rectangular box has a volume of

12\,\text{cm}^3

. The length of the box is

1\,\text{cm}

more than the width, and the height is

2\,\text{cm}

less than the width.

(a)

Show that the width

x\,\text{cm}

satisfies

x^3 - x^2 - 2x - 12 = 0

. [3 marks]

(b)

Hence find the exact dimensions of the box, justifying that no other real solution exists. [3 marks]

Solutions

24MasterySAQ-SRational expressions and their simplifications6 marksPaper 2~9 min

Consider the rational function

f(x) = \dfrac{3x^2 - 5x - 2}{x^2 - 4}

.

(a)

State the domain of

f(x)

. [1 mark]

(b)

Show that

f(x)

can be written in the form

A + \dfrac{B}{x + C}

, where

A, B, C \in \mathbb{Z}

, and state the values of

A

,

B

, and

C

. [3 marks]

(c)

Using your result from part (b), determine the equation of the horizontal asymptote of

f

and hence calculate

f(4)

, giving your answer as an exact fraction. [2 marks]

Solutions

25MasterySAQ-SExponent laws and propertiess6 marksPaper 2~9 min

The population of a certain species of bird on an island is modelled by

P(t) = 500 \times 2^{kt}

, where

t

is the time in years since the start of the study. After 3 years, the population is 800 birds.

(a)

Calculate the value of

k

, giving your answer in exact form. [3 marks]

(b)

Determine the number of years it will take for the population to first reach 2000 birds. Give your answer correct to 3 significant figures. [1 mark]

(c)

Find the rate at which the population is increasing at the moment it reaches 2000 birds. Give your answer correct to 3 significant figures. [2 marks]

Solutions

26MasterySAQ-SExponent laws and propertiess6 marksPaper 2~9 min

The intensity

I

(in watts per square metre) of a sound wave is related to its decibel level

D

by the formula

D = 10\log_{10}\!\left(\frac{I}{I_0}\right)

where

I_0 = 1.0 \times 10^{-12}\,\text{W\,m}^{-2}

is the threshold of hearing.

(a)

Calculate the decibel level of a sound with intensity

I = 3.2 \times 10^{-4}\,\text{W\,m}^{-2}

, giving your answer correct to the nearest decibel. [2 marks]

(b)

Show that a decibel level of

D

corresponds to an intensity

I = I_0 \times 10^{D/10}.

[2 marks]

(c)

Two sounds have decibel levels

D_1 = 95\,\text{dB}

and

D_2 = 88\,\text{dB}

. Determine the decibel level of the combined sound, whose intensity equals

I_1 + I_2

. Give your answer correct to the nearest decibel. [2 marks]

Solutions

27MasterySAQ-SExponent laws and propertiess6 marksPaper 2~9 min

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

M(t) = M_0 e^{-kt}

, where

M_0

is the initial mass in grams,

t

is the time in years, and

k

is a positive constant. After 10 years, the mass of the sample is 80% of its initial mass.

(a)

Show that

k = \dfrac{1}{10}\ln\!\left(\dfrac{5}{4}\right)

. [2 marks]

(b)

Calculate the time, in years, for the mass to decrease to 25% of its initial mass. Give your answer correct to 3 significant figures. [4 marks]

Solutions

28MasterySAQ-SLogarithmic functions and their propertiess6 marksPaper 2~9 min

A chemist studies the decomposition of a compound. The mass

M(t)

grams remaining after

t

hours is modelled by

M(t) = M_0\,e^{-kt}

where

M_0

is the initial mass in grams and

k

is a positive constant. Initially the mass is

120

grams; after

24

hours,

90

grams remain.

(a)

Calculate the value of

k

, giving your answer correct to 3 significant figures. [3 marks]

(b)

Hence calculate the time, in hours, for the mass to reduce to

50

grams, giving your answer correct to the nearest hour. [3 marks]

Solutions

29MasterySAQ-SBinomial expansion and coefficientss6 marksPaper 2~9 min

A pharmaceutical company is developing a new drug. In a clinical trial, the probability that a patient experiences a particular side effect is

p = 0.15

. A random sample of

n

patients is selected, where

n \geq 2

.

(a)

Write down an expression, in terms of

n

, for

P(X = 2)

, the probability that exactly

2

patients experience the side effect. [1 mark]

(b)

Show that the condition

P(X = 2) > 0.25

is equivalent to

n(n-1)(0.85)^{n-2} > 22.2\overline{2}

[2 marks]

(c)

By testing integer values of

n

, find the smallest value of

n

for which

P(X = 2) > 0.25

. [3 marks]

Solutions

30MasterySAQ-SBinomial expansion and coefficientss6 marksPaper 2~9 min

Consider the binomial expansion of

(3 - 2x)^n

, where

n \in \mathbb{Z}^+

.

(a)

Write down the general term

T_{r+1}

of the expansion. [1 mark]

(b)

Given that the coefficient of

x^2

is 12 times the magnitude of the coefficient of

x

, find the value of

n

. [5 marks]

Solutions

31MasterySAQ-SBinomial expansion and coefficientss6 marksPaper 2~9 min

A pharmaceutical company models the concentration of a drug in a patient's bloodstream,

t

hours after administration, using

C(t) = k(2t-1)^5

, where

k

is a positive integer and concentration is measured in mg/L. The coefficient of the

t^3

term in the expansion of

C(t)

is

160

.

(a)

Determine the value of

k

. [3 marks]

(b)

Hence, find the coefficient of the

t^4

term in the expansion of

C(t)

. [3 marks]

Solutions

32MasterySAQ-SApplications of binomial expansionss6 marksPaper 2~9 min

A company produces electronic chips. The probability that a randomly selected chip is defective is

0.02

. A quality control inspector examines a batch of

12

chips.

(a)

State the distribution of

X

, the number of defective chips in the batch, giving the values of all parameters. [1 mark]

(b)

Calculate

P(X \leq 1)

, giving your answer correct to three significant figures. [3 marks]

(c)

Given that least one chip in the batch is defective, find the probability that exactly one chip is defective. Give your answer correct to three significant figures. [2 marks]

Solutions

33ChallengeLAQSum of an arithmetic sequences10 marksPaper 3~15 min

A manufacturing company produces cylindrical metal rods. The rods are produced in batches, and the quality control department inspects them for defects. The number of rods inspected each day forms an arithmetic sequence. On the first day,

15

rods are inspected. On each subsequent day, the number inspected increases by a constant amount

d

, where

d > 0

.

(a)

Starting from the definition of an arithmetic sequence

u_k = 15 + (k-1)d

, use the identity

\displaystyle S_n = \sum_{k=1}^{n} u_k

and the standard result

\displaystyle\sum_{k=1}^{n} k = \frac{n(n+1)}{2}

to prove that

S_n = \frac{n}{2}\bigl(30 + (n-1)d\bigr).

[5 marks]

(b)

After

20

days, the total number of rods inspected is

2100

. Calculate the value of

d

. [2 marks]

(c)

The factory's inspection line has a maximum daily capacity of

200

rods. Determine the day on which the number of rods inspected first reaches or exceeds

200

, and hence calculate the total number of rods inspected up to and including that day. [3 marks]

Solutions

34ChallengeLAQSum of an arithmetic sequences10 marksPaper 3~15 min

A financial analyst models the depreciation of a specialised machine. The machine is purchased for USD 50000. At the end of the first year its value is USD 47000, and the value decreases by the same fixed amount each year, forming an arithmetic sequence.

(a)

Show that the total value of the machine (the sum of its values at the end of each year) over the first

n

years is given by

S_n = \frac{n}{2}(97000 - 3000n).

[4 marks]

(b)

Calculate

S_{10}

, the total value accumulated over the first 10 years. [2 marks]

(c)

The analyst claims the model is only physically valid while the machine retains a non-negative value. (i) Determine the maximum number of complete years for which the model is valid. [2]
(ii) Hence find the value of

n

that maximises

S_n

within the valid domain, and state this maximum total value. [2 marks]

Solutions

35ChallengeLAQDivision of polynomialss10 marksPaper 3~15 min

Let

f(x) = \dfrac{x^3 + 2x^2 - 5x - 6}{x^2 - 4}

.

(a)

Show that

f(x)

can be written in the form

f(x) = Ax + B + \frac{Cx + D}{x^2 - 4}

where

A, B, C, D \in \mathbb{Z}

, and state the values of

A

,

B

,

C

, and

D

. [4 marks]

(b)

State and justify the equation of the oblique asymptote of the graph of

y = f(x)

. [3 marks]

(c)

Determine whether the graph of

y = f(x)

intersects its oblique asymptote, and if so, find the coordinates of any such point(s). [3 marks]

Solutions

36ChallengeLAQPolynomial expressions and their factorizationss10 marksPaper 3~15 min

Consider the polynomial

P(x) = x^4 + ax^3 + bx^2 + cx + d

, where

a, b, c, d \in \mathbb{R}

. It is given that

P(x)

has two distinct real roots

\alpha

and

\beta

(with

\alpha < \beta

) and two non-real complex conjugate roots

\gamma

and

\bar{\gamma}

.

(a)

By writing

P(x)

as a product of two quadratic factors with real coefficients, prove that

b = \alpha\beta + \gamma\bar{\gamma} - (\alpha+\beta)(\gamma + \bar{\gamma}).

[3 marks]

(b)

Using the factorisation from part (a), show that

c = -\alpha\beta(\gamma+\bar{\gamma}) - \gamma\bar{\gamma}(\alpha+\beta).

[2 marks]

(c)

Let

s = \alpha + \beta

,

p = \alpha\beta

,

t = \gamma + \bar{\gamma}

, and

q = \gamma\bar{\gamma}

. Using the results of parts (a) and (b), together with Vieta's formulae applied to

P(x)

, show that the four Vieta relations can be written as

a = -(s+t), \quad b = p + q - st, \quad c = -(pt + qs), \quad d = pq.

[2 marks]

(d)

Given that

P(x)

has a repeated real root (so

\alpha = \beta

, denoted

\alpha

), use the relations from part (c) to eliminate

t

and

q

and hence derive a single polynomial constraint relating

a

,

b

,

c

, and

d

that must be satisfied. Verify your constraint is consistent with the specific case

P(x) = (x-1)^2(x^2+1)

. [3 marks]

Solutions

37ChallengeLAQExponent laws and propertiess13 marksPaper 3~20 min

The number of bacteria in a laboratory culture is modelled by

N(t) = N_0 e^{kt}

, where

N_0

is the initial population,

t

is time in hours, and

k > 0

.

(a)

Prove that the time

T_n

for the population to increase by a factor of

2^n

(where

n

is a positive integer) satisfies

T_n = \frac{n \ln 2}{k}.

[4 marks]

(b)

A culture has

k = 0.35

. (i) Using the result from part (a), find the least positive integer

n

such that

T_n > 10

hours. [2 marks]
(ii) Show that

T_{2m} = 2T_m

for any positive integer

m

, and state what this result means in the context of the model. [2 marks]

(c)

A researcher proposes the modified model

M(t) = N_0 e^{pt + qt^2}

, where

p

and

q

are real constants. (i) Find

M'(t)

in terms of

N_0

,

p

,

q

, and

t

. [2 marks]
(ii) Hence determine the conditions on

p

and

q

for which

M(t)

is strictly increasing for all

t > 0

. Justify your answer. [Total: 13 marks — recalibrated to match independently markable steps] > Recalibrated to 13 marks to match step count; if constrained to 10 marks, remove (b)(ii) and reduce (c)(ii) to 2 marks. [3 marks]

Solutions

38ChallengeLAQLogarithmic functions and their propertiess10 marksPaper 3~15 min

A researcher models the growth of a bacterial colony using the function

N(t) = A \cdot 10^{kt}

, where

N(t)

is the number of bacteria at time

t

hours, and

A

and

k

are positive constants.

(a)

Show that

\log_{10} N(t)

is a linear function of

t

, stating its gradient and vertical intercept in terms of

A

and

k

. [4 marks]

(b)

The researcher records

N(2) = 5000

and

N(5) = 500\,000

. Using the result from part (a), calculate the values of

A

and

k

, giving your answers correct to 3 significant figures. [4 marks]

(c)

A second colony is modelled by

M(t) = B \cdot 10^{mt}

, where

\log_{10} B = \log_{10} A + 1

and

m = \frac{k}{2}

, with

A

and

k

as found in part (b). Determine the time

t^*

at which

M(t^*) = N(t^*)

, giving your answer in exact form, and hence comment on the long-term relationship between the two populations. [2 marks]

Solutions

39ChallengeLAQBinomial expansion and coefficientss10 marksPaper 3~15 min

The binomial theorem states that for any positive integer

n

,

(1+x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k.

Consider the sum

S_n

defined for integers

n \geq 1

by

S_n = \sum_{k=0}^{n} \frac{(-1)^k}{k+1}\binom{n}{k}.

(a)

Prove that

S_n = \dfrac{1}{n+1}

for all integers

n \geq 1

. [5 marks]

(b)

Let

T_n = \displaystyle\sum_{k=0}^{n} \dfrac{(-1)^k}{k+2}\binom{n}{k}

. By writing

\dfrac{1}{k+2}\binom{n}{k}

in terms of

\binom{n+1}{k+1}

or otherwise, express

T_n

in terms of

S_{n+1}

and hence find a closed form for

T_n

in terms of

n

. Justify each step of your reasoning. [5 marks]

Solutions

40ChallengeLAQBinomial expansion and coefficientss10 marksPaper 3~15 min

Pascal's triangle is generated by the recurrence relation

\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}

for

1 \leq k \leq n-1

, with boundary conditions

\binom{n}{0} = \binom{n} = 1

. The rising-diagonal sums of Pascal's triangle are defined for

n \geq 0

by

F_n = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k}.

(a)

Prove that

F_n = F_{n-1} + F_{n-2}

for all integers

n \geq 2

, given that

F_0 = 1

and

F_1 = 1

. [6 marks]

(b)

(i) Using the recurrence relation from part (a), show that

F_{10} = 89

.
(ii) Verify this result by evaluating

\displaystyle\sum_{k=0}^{5} \binom{10-k}{k}

directly from Pascal's triangle.
(iii) The

n

-th Fibonacci number

\phi_n

is defined by

\phi_1 = 1,\ \phi_2 = 1,\ \phi_n = \phi_{n-1}+\phi_{n-2}

. Given that

F_n

satisfies the same recurrence with the same initial values, deduce a general relationship between

F_n

and

\phi_n

, and hence find the smallest

n

for which

F_n > 200

. [4 marks]

Solutions

Number and Algebra — Free Maths AA HL Practice Questions