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HL Paper 3

This question will investigate power series, as an extension to the Binomial Theorem for negative and fractional indices.

A power series in $x$ is defined as a function of the form $f\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3} + ...$ where the ${a_i} \in \mathbb{R}$ .

It can be considered as an infinite polynomial.

This is an example of a power series, but is only a finite power series, since only a finite number of the ${a_i}$ are non-zero.

We will now attempt to generalise further.

Suppose ${\left( {1 + x} \right)^q}{\text{,}}\,\,q \in \mathbb{Q}$ can be written as the power series ${a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3} + ...$ .

Expand ${\left( {1 + x} \right)^5}$ using the Binomial Theorem.

[2]

Consider the power series $1 - x + {x^2} - {x^3} + {x^4} - ...$

By considering the ratio of consecutive terms, explain why this series is equal to ${\left( {1 + x} \right)^{ - 1}}$ and state the values of $x$ for which this equality is true.

[4]

Differentiate the equation obtained part (b) and hence, find the first four terms in a power series for ${\left( {1 + x} \right)^{ - 2}}$ .

[2]

Repeat this process to find the first four terms in a power series for ${\left( {1 + x} \right)^{ - 3}}$ .

[2]

Hence, by recognising the pattern, deduce the first four terms in a power series for ${\left( {1 + x} \right)^{ - n}}$ , $n \in {\mathbb{Z}^ + }$ .

[3]

By substituting $x = 0$ , find the value of ${a_0}$ .

[1]

By differentiating both sides of the expression and then substituting $x = 0$ , find the value of ${a_1}$ .

[2]

Repeat this procedure to find ${a_2}$ and ${a_3}$ .

[4]

Hence, write down the first four terms in what is called the Extended Binomial Theorem for ${\left( {1 + x} \right)^q}{\text{,}}\,\,q \in \mathbb{Q}$ .

[1]

Write down the power series for $\frac{1}{{1 + {x^2}}}$ .

[2]

Hence, using integration, find the power series for ${\text{arctan}}\,x$ , giving the first four non-zero terms.

[4]

This question asks you to investigate conditions for the existence of complex roots of polynomial equations of degree $3$ and $4$ .

The cubic equation $x^{3} + p x^{2} + q x + r = 0$ , where $p, q, r \in ℝ$ , has roots $α, β$ and $γ$ .

Consider the equation $x^{3} - 7 x^{2} + q x + 1 = 0$ , where $q \in ℝ$ .

Noah believes that if $p^{2} \geq 3 q$ then $α, β$ and $γ$ are all real.

Now consider polynomial equations of degree $4$ .

The equation $x^{4} + p x^{3} + q x^{2} + r x + s = 0$ , where $p, q, r, s \in ℝ$ , has roots $α, β, γ$ and $δ$ .

In a similar way to the cubic equation, it can be shown that:

$p = - (α + β + γ + δ)$

$q = α β + α γ + α δ + β γ + β δ + γ δ$

$r = - (α β γ + α β δ + α γ δ + β γ δ)$

$s = α β γ δ$ .

The equation $x^{4} - 9 x^{3} + 24 x^{2} + 22 x - 12 = 0$ , has one integer root.

By expanding $(x - α) (x - β) (x - γ)$ show that:

$p = - (α + β + γ)$

$q = α β + β γ + γ α$

$r = - α β γ$ .

[3]

Show that $p^{2} - 2 q = α^{2} + β^{2} + γ^{2}$ .

[3]

b.i.

Hence show that ${(α - β)}^{2} + {(β - γ)}^{2} + {(γ - α)}^{2} = 2 p^{2} - 6 q$ .

[3]

b.ii.

Given that $p^{2} < 3 q$ , deduce that $α, β$ and $γ$ cannot all be real.

[2]

Using the result from part (c), show that when $q = 17$ , this equation has at least one complex root.

[2]

By varying the value of $q$ in the equation $x^{3} - 7 x^{2} + q x + 1 = 0$ , determine the smallest positive integer value of $q$ required to show that Noah is incorrect.

[2]

e.i.

Explain why the equation will have at least one real root for all values of $q$ .

[1]

e.ii.

Find an expression for $α^{2} + β^{2} + γ^{2} + δ^{2}$ in terms of $p$ and $q$ .

[3]

f.i.

Hence state a condition in terms of $p$ and $q$ that would imply $x^{4} + p x^{3} + q x^{2} + r x + s = 0$ has at least one complex root.

[1]

f.ii.

Use your result from part (f)(ii) to show that the equation $x^{4} - 2 x^{3} + 3 x^{2} - 4 x + 5 = 0$ has at least one complex root.

[1]

State what the result in part (f)(ii) tells us when considering this equation $x^{4} - 9 x^{3} + 24 x^{2} + 22 x - 12 = 0$ .

[1]

h.i.

Write down the integer root of this equation.

[1]

h.ii.

By writing $x^{4} - 9 x^{3} + 24 x^{2} + 22 x - 12$ as a product of one linear and one cubic factor, prove that the equation has at least one complex root.

[4]

h.iii.

This question will explore connections between complex numbers and regular polygons.

The diagram below shows a sector of a circle of radius 1, with the angle subtended at the centre $O$ being $\alpha {\text{,}}\,\,0 < \alpha < \frac{\pi }{2}$ . A perpendicular is drawn from point $P$ to intersect the $x$ -axis at $Q$ . The tangent to the circle at $P$ intersects the $x$ -axis at $R$ .

By considering the area of two triangles and the area of the sector show that ${\text{cos}}\,\alpha \,{\text{sin}}\,\alpha < \alpha < \frac{{{\text{sin}}\,\alpha }}{{{\text{cos}}\,\alpha }}$ .

[5]

Hence show that $\mathop {{\text{lim}}}\limits_{\alpha \to 0} \frac{\alpha }{{{\text{sin}}\,\alpha }} = 1$ .

[2]

Let ${z^n} = 1{\text{,}}\,\,z \in \mathbb{C}{\text{,}}\,\,n \in \mathbb{N}{\text{,}}\,\,n \geqslant 5$ . Working in modulus/argument form find the $n$ solutions to this equation.

[8]

Represent these $n$ solutions on an Argand diagram. Let their positions be denoted by ${P_0}{\text{,}}\,\,{P_1}{\text{,}}\,\,{P_2}{\text{,}}\, \ldots {P_{n - 1}}$ placed in order in an anticlockwise direction round the circle, starting on the positive $x$ -axis. Show the positions of ${P_0}{\text{,}}\,\,{P_1}{\text{,}}\,\,{P_2}$ and ${P_{n - 1}}$ .

[1]

Show that the length of the line segment ${P_0}{P_1}$ is $2\,{\text{sin}}\frac{\pi }{n}$ .

[4]

Hence, write down the total length of the perimeter of the regular $n$ sided polygon ${P_0}{P_1}{P_2} \ldots {P_{n - 1}}{P_0}$ .

[1]

Using part (b) find the limit of this perimeter as $n \to \infty$ .

[2]

Find the total area of this $n$ sided polygon.

[3]

Using part (b) find the limit of this area as $n \to \infty$ .

[2]

In this question you will be exploring the strategies required to solve a system of linear differential equations.

Consider the system of linear differential equations of the form:

$\frac{d x}{d t} = x - y$ and $\frac{d y}{d t} = a x + y$ ,

where $x, y, t \in ℝ^{+}$ and $a$ is a parameter.

First consider the case where $a = 0$ .

Now consider the case where $a = - 1$ .

Now consider the case where $a = - 4$ .

From previous cases, we might conjecture that a solution to this differential equation is $y = F e^{λ t}$ , $λ \in ℝ$ and $F$ is a constant.

By solving the differential equation $\frac{d y}{d t} = y$ , show that $y = A e^{t}$ where $A$ is a constant.

[3]

a.i.

Show that $\frac{d x}{d t} - x = - A e^{t}$ .

[1]

a.ii.

Solve the differential equation in part (a)(ii) to find $x$ as a function of $t$ .

[4]

a.iii.

By differentiating $\frac{d y}{d t} = - x + y$ with respect to $t$ , show that $\frac{d^{2} y}{d t^{2}} = 2 \frac{d y}{d t}$ .

[3]

b.i.

By substituting $Y = \frac{d y}{d t}$ , show that $Y = B e^{2 t}$ where $B$ is a constant.

[3]

b.ii.

Hence find $y$ as a function of $t$ .

[2]

b.iii.

Hence show that $x = - \frac{B}{2} e^{2 t} + C$ , where $C$ is a constant.

[3]

b.iv.

Show that $\frac{d^{2} y}{d t^{2}} - 2 \frac{d y}{d t} - 3 y = 0$ .

[3]

c.i.

Find the two values for $λ$ that satisfy $\frac{d^{2} y}{d t^{2}} - 2 \frac{d y}{d t} - 3 y = 0$ .

[4]

c.ii.

Let the two values found in part (c)(ii) be $λ_{1}$ and $λ_{2}$ .

Verify that $y = F e^{λ_{1} t} + G e^{λ_{2} t}$ is a solution to the differential equation in (c)(i),where $G$ is a constant.

[4]

c.iii.

A Gaussian integer is a complex number, $z$ , such that $z = a + b i$ where $a, b \in ℤ$ . In this question, you are asked to investigate certain divisibility properties of Gaussian integers.

Consider two Gaussian integers, $α = 3 + 4 i$ and $β = 1 - 2 i$ , such that $γ = α β$ for some Gaussian integer $γ$ .

Now consider two Gaussian integers, $α = 3 + 4 i$ and $γ = 11 + 2 i$ .

The norm of a complex number $z$ , denoted by $N (z)$ , is defined by $N (z) = {|z|}^{2}$ . For example, if $z = 2 + 3 i$ then $N (2 + 3 i) = 2^{2} + 3^{2} = 13$ .

A Gaussian prime is a Gaussian integer, $z$ , that cannot be expressed in the form $z = α β$ where $α, β$ are Gaussian integers with $N (α), N (β) > 1$ .

The positive integer $2$ is a prime number, however it is not a Gaussian prime.

Let $α, β$ be Gaussian integers.

The result from part (h) provides a way of determining whether a Gaussian integer is a Gaussian prime.

Find $γ$ .

[2]

Determine whether $\frac{γ}{α}$ is a Gaussian integer.

[3]

On an Argand diagram, plot and label all Gaussian integers that have a norm less than $3$ .

[2]

Given that $α = a + b i$ where $a, b \in ℤ$ , show that $N (α) = a^{2} + b^{2}$ .

[1]

By expressing the positive integer $n = c^{2} + d^{2}$ as a product of two Gaussian integers each of norm $c^{2} + d^{2}$ , show that $n$ is not a Gaussian prime.

[3]

Verify that $2$ is not a Gaussian prime.

[2]

Write down another prime number of the form $c^{2} + d^{2}$ that is not a Gaussian prime and express it as a product of two Gaussian integers.

[2]

Show that $N (α β) = N (α) N (β)$ .

[6]

Hence show that $1 + 4 i$ is a Gaussian prime.

[3]

Use proof by contradiction to prove that a prime number, $p$ , that is not of the form $a^{2} + b^{2}$ is a Gaussian prime.

[6]

In this question you will explore some of the properties of special functions $f$ and $g$ and their relationship with the trigonometric functions, sine and cosine.

Functions $f$ and $g$ are defined as $f (z) = \frac{e^{z} + e^{- z}}{2}$ and $g (z) = \frac{e^{z} - e^{- z}}{2}$ , where $z \in ℂ$ .

Consider $t$ and $u$ , such that $t, u \in ℝ$ .

Using $e^{i u} = \cos u + i \sin u$ , find expressions, in terms of $\sin u$ and $\cos u$ , for

The functions $\cos x$ and $\sin x$ are known as circular functions as the general point ( $\cos θ, \sin θ$ ) defines points on the unit circle with equation $x^{2} + y^{2} = 1$ .

The functions $f (x)$ and $g (x)$ are known as hyperbolic functions, as the general point ( $f (θ), g (θ)$ ) defines points on a curve known as a hyperbola with equation $x^{2} - y^{2} = 1$ . This hyperbola has two asymptotes.

Verify that $u = f (t)$ satisfies the differential equation $\frac{d^{2} u}{d t^{2}} = u$ .

[2]

Show that ${(f (t))}^{2} + {(g (t))}^{2} = f (2 t)$ .

[3]

$f (i u)$ .

[3]

c.i.

$g (i u)$ .

[2]

c.ii.

Hence find, and simplify, an expression for ${(f (i u))}^{2} + {(g (i u))}^{2}$ .

[2]

Show that ${(f (t))}^{2} - {(g (t))}^{2} = {(f (i u))}^{2} - {(g (i u))}^{2}$ .

[4]

Sketch the graph of $x^{2} - y^{2} = 1$ , stating the coordinates of any axis intercepts and the equation of each asymptote.

[4]

The hyperbola with equation $x^{2} - y^{2} = 1$ can be rotated to coincide with the curve defined by $x y = k, k \in ℝ$ .

Find the possible values of $k$ .

[5]

This question asks you to investigate and prove a geometric property involving the roots of the equation $z^{n} = 1$ where $z \in ℂ$ for integers $n$ , where $n \geq 2$ .

The roots of the equation $z^{n} = 1$ where $z \in ℂ$ are $1, ω, ω^{2}, \dots, ω^{n - 1}$ , where $ω = e^{\frac{2 πi}{n}}$ . Each root can be represented by a point $P_{0}, P_{1}, P_{2}, \dots, P_{n - 1}$ , respectively, on an Argand diagram.

For example, the roots of the equation $z^{2} = 1$ where $z \in ℂ$ are $1$ and $ω$ . On an Argand diagram, the root $1$ can be represented by a point $P_{0}$ and the root $ω$ can be represented by a point $P_{1}$ .

Consider the case where $n = 3$ .

The roots of the equation $z^{3} = 1$ where $z \in ℂ$ are $1$ , $ω$ and $ω^{2}$ . On the following Argand diagram, the points $P_{0}, P_{1}$ and $P_{2}$ lie on a circle of radius $1$ unit with centre $O (0, 0)$ .

Line segments $[P_{0} P_{1}]$ and $[P_{0} P_{2}]$ are added to the Argand diagram in part (a) and are shown on the following Argand diagram.

$P_{0} P_{1}$ is the length of $[P_{0} P_{1}]$ and $P_{0} P_{2}$ is the length of $[P_{0} P_{2}]$ .

Consider the case where $n = 4$ .

The roots of the equation $z^{4} = 1$ where $z \in ℂ$ are $1, ω, ω^{2}$ and $ω^{3}$ .

On the following Argand diagram, the points $P_{0}, P_{1}, P_{2}$ and $P_{3}$ lie on a circle of radius $1$ unit with centre $O (0, 0)$ . $[P_{0} P_{1}]$ , $[P_{0} P_{2}]$ and $[P_{0} P_{3}]$ are line segments.

For the case where $n = 5$ , the equation $z^{5} = 1$ where $z \in ℂ$ has roots $1, ω, ω^{2}, ω^{3}$ and $ω^{4}$ .

It can be shown that $P_{0} P_{1} \times P_{0} P_{2} \times P_{0} P_{3} \times P_{0} P_{4} = 5$ .

Now consider the general case for integer values of $n$ , where $n \geq 2$ .

The roots of the equation $z^{n} = 1$ where $z \in ℂ$ are $1, ω, ω^{2}, \dots, ω^{n - 1}$ . On an Argand diagram, these roots can be represented by the points $P_{0}, P_{1}, P_{2}, \dots, P_{n - 1}$ respectively where $[P_{0} P_{1}], [P_{0} P_{2}], \dots, [P_{0} P_{n - 1}]$ are line segments. The roots lie on a circle of radius $1$ unit with centre $O (0, 0)$ .

$P_{0} P_{1}$ can be expressed as $| 1 - ω |$ .

Consider $z^{n} - 1 = (z - 1) (z^{n - 1} + z^{n - 2} + \dots + z + 1)$ where $z \in ℂ$ .

Show that $(ω - 1) (ω^{2} + ω + 1) = ω^{3} - 1$ .

[2]

a.i.

Hence, deduce that $ω^{2} + ω + 1 = 0$ .

[2]

a.ii.

Show that $P_{0} P_{1} \times P_{0} P_{2} = 3$ .

[3]

By factorizing $z^{4} - 1$ , or otherwise, deduce that $ω^{3} + ω^{2} + ω + 1 = 0$ .

[2]

Show that $P_{0} P_{1} \times P_{0} P_{2} \times P_{0} P_{3} = 4$ .

[4]

Suggest a value for $P_{0} P_{1} \times P_{0} P_{2} \times \dots \times P_{0} P_{n - 1}$ .

[1]

Write down expressions for $P_{0} P_{2}$ and $P_{0} P_{3}$ in terms of $ω$ .

[2]

f.i.

Hence, write down an expression for $P_{0} P_{n - 1}$ in terms of $n$ and $ω$ .

[1]

f.ii.

Express $z^{n - 1} + z^{n - 2} + \dots + z + 1$ as a product of linear factors over the set $ℂ$ .

[3]

g.i.

Hence, using the part (g)(i) and part (f) results, or otherwise, prove your suggested result to part (e).

[4]

g.ii.

This question asks you to explore some properties of polygonal numbers and to determine and prove interesting results involving these numbers.

A polygonal number is an integer which can be represented as a series of dots arranged in the shape of a regular polygon. Triangular numbers, square numbers and pentagonal numbers are examples of polygonal numbers.

For example, a triangular number is a number that can be arranged in the shape of an equilateral triangle. The first five triangular numbers are $1, 3, 6, 10$ and $15$ .

The following table illustrates the first five triangular, square and pentagonal numbers respectively. In each case the first polygonal number is one represented by a single dot.

For an $r$ -sided regular polygon, where $r \in ℤ^{+}, r \geq 3$ , the $n$ th polygonal number $P_{r} (n)$ is given by

$P_{r} (n) = \frac{(r - 2) n^{2} - (r - 4) n}{2}$ , where $n \in ℤ^{+}$ .

Hence, for square numbers, $P_{4} (n) = \frac{(4 - 2) n^{2} - (4 - 4) n}{2} = n^{2}$ .

The $n$ th pentagonal number can be represented by the arithmetic series

$P_{5} (n) = 1 + 4 + 7 + \dots + (3 n - 2)$ .

For triangular numbers, verify that $P_{3} (n) = \frac{n (n + 1)}{2}$ .

[2]

a.i.

The number $351$ is a triangular number. Determine which one it is.

[2]

a.ii.

Show that $P_{3} (n) + P_{3} (n + 1) \equiv {(n + 1)}^{2}$ .

[2]

b.i.

State, in words, what the identity given in part (b)(i) shows for two consecutive triangular numbers.

[1]

b.ii.

For $n = 4$ , sketch a diagram clearly showing your answer to part (b)(ii).

[1]

b.iii.

Show that $8 P_{3} (n) + 1$ is the square of an odd number for all $n \in ℤ^{+}$ .

[3]

Hence show that $P_{5} (n) = \frac{n (3 n - 1)}{2}$ for $n \in ℤ^{+}$ .

[3]

By using a suitable table of values or otherwise, determine the smallest positive integer, greater than $1$ , that is both a triangular number and a pentagonal number.

[5]

A polygonal number, $P_{r} (n)$ , can be represented by the series

$Σ_{m = 1}^{n} (1 + (m - 1) (r - 2))$ where $r \in ℤ^{+}, r \geq 3$ .

Use mathematical induction to prove that $P_{r} (n) = \frac{(r - 2) n^{2} - (r - 4) n}{2}$ where $n \in ℤ^{+}$ .

[8]

This question asks you to explore cubic polynomials of the form $(x - r) (x^{2} - 2 a x + a^{2} + b^{2})$ for $x \in ℝ$ and corresponding cubic equations with one real root and two complex roots of the form $(z - r) (z^{2} - 2 a z + a^{2} + b^{2}) = 0$ for $z \in ℂ$ .

In parts (a), (b) and (c), let $r = 1, a = 4$ and $b = 1$ .

Consider the equation $(z - 1) (z^{2} - 8 z + 17) = 0$ for $z \in ℂ$ .

Consider the function $f (x) = (x - 1) (x^{2} - 8 x + 17)$ for $x \in ℝ$ .

Consider the function $g (x) = (x - r) (x^{2} - 2 a x + a^{2} + b^{2})$ for $x \in ℝ$ where $r, a \in ℝ$ and $b \in ℝ, b > 0$ .

The equation $(z - r) (z^{2} - 2 a z + a^{2} + b^{2}) = 0$ for $z \in ℂ$ has roots $r$ and $a \pm b i$ where $r, a \in ℝ$ and $b \in ℝ, b > 0$ .

On the Cartesian plane, the points $C_{1} (a, \sqrt{g' (a)})$ and $C_{2} (a, - \sqrt{g' (a)})$ represent the real and imaginary parts of the complex roots of the equation $(z - r) (z^{2} - 2 a z + a^{2} + b^{2}) = 0$ .

The following diagram shows a particular curve of the form $y = (x - r) (x^{2} - 2 a x + a^{2} + 16)$ and the tangent to the curve at the point $A (a, 80)$ . The curve and the tangent both intersect the $x$ -axis at the point $R (- 2, 0)$ . The points $C_{1}$ and $C_{2}$ are also shown.

Consider the curve $y = (x - r) (x^{2} - 2 a x + a^{2} + b^{2})$ for $a \neq r, b > 0$ . The points $A (a, g (a))$ and $R (r, 0)$ are as defined in part (d)(ii). The curve has a point of inflexion at point $P$ .

Consider the special case where $a = r$ and $b > 0$ .

Given that $1$ and $4 + i$ are roots of the equation, write down the third root.

[1]

a.i.

Verify that the mean of the two complex roots is $4$ .

[1]

a.ii.

Show that the line $y = x - 1$ is tangent to the curve $y = f (x)$ at the point $A (4, 3)$ .

[4]

Sketch the curve $y = f (x)$ and the tangent to the curve at point $A$ , clearly showing where the tangent crosses the $x$ -axis.

[2]

Show that $g' (x) = 2 (x - r) (x - a) + x^{2} - 2 a x + a^{2} + b^{2}$ .

[2]

d.i.

Hence, or otherwise, prove that the tangent to the curve $y = g (x)$ at the point $A (a, g (a))$ intersects the $x$ -axis at the point $R (r, 0)$ .

[6]

d.ii.

Deduce from part (d)(i) that the complex roots of the equation $(z - r) (z^{2} - 2 a z + a^{2} + b^{2}) = 0$ can be expressed as $a \pm i \sqrt{g' (a)}$ .

[1]

Use this diagram to determine the roots of the corresponding equation of the form $(z - r) (z^{2} - 2 a z + a^{2} + 16) = 0$ for $z \in ℂ$ .

[4]

f.i.

State the coordinates of $C_{2}$ .

[1]

f.ii.

Show that the $x$ -coordinate of $P$ is $\frac{1}{3} (2 a + r)$ .

You are not required to demonstrate a change in concavity.

[2]

g.i.

Hence describe numerically the horizontal position of point $P$ relative to the horizontal positions of the points $R$ and $A$ .

[1]

g.ii.

Sketch the curve $y = (x - r) (x^{2} - 2 a x + a^{2} + b^{2})$ for $a = r = 1$ and $b = 2$ .

[2]

h.i.

For $a = r$ and $b > 0$ , state in terms of $r$ , the coordinates of points $P$ and $A$ .

[1]

h.ii.

This question investigates some applications of differential equations to modeling population growth.

One model for population growth is to assume that the rate of change of the population is proportional to the population, i.e. $\frac{{{\text{d}}P}}{{{\text{d}}t}} = kP$ , where $k \in \mathbb{R}$ , $t$ is the time (in years) and $P$ is the population

The initial population is 1000.

Given that $k = 0.003$ , use your answer from part (a) to find

Consider now the situation when $k$ is not a constant, but a function of time.

Given that $k = 0.003 + 0.002t$ , find

Another model for population growth assumes

there is a maximum value for the population, $L$ .
that $k$ is not a constant, but is proportional to $\left( {1 - \frac{P}{L}} \right)$ .

Show that the general solution of this differential equation is $P = A{{\text{e}}^{kt}}$ , where $A \in \mathbb{R}$ .

[5]

the population after 10 years

[2]

b.i.

the number of years it will take for the population to triple.

[2]

b.ii.

$\mathop {{\text{lim}}}\limits_{t \to \infty } P$

[1]

b.iii.

the solution of the differential equation, giving your answer in the form $P = f\left( t \right)$ .

[5]

c.i.

the number of years it will take for the population to triple.

[4]

c.ii.

Show that $\frac{{{\text{d}}P}}{{{\text{d}}t}} = \frac{m}{L}P\left( {L - P} \right)$ , where $m \in \mathbb{R}$ .

[2]

Solve the differential equation $\frac{{{\text{d}}P}}{{{\text{d}}t}} = \frac{m}{L}P\left( {L - P} \right)$ , giving your answer in the form $P = g\left( t \right)$ .

[10]

Given that the initial population is 1000, $L = 10000$ and $m = 0.003$ , find the number of years it will take for the population to triple.

[4]

In parts (b) and (c), ${\left( {abc \ldots } \right)_n}$ denotes the number ${abc \ldots }$ written in base $n$ , where $n \in {\mathbb{Z}^ + }$ . For example, ${\left( {359} \right)_n} = 3{n^2} + 5n + 9$ .

State Fermat’s little theorem.

[2]

a.i.

Find the remainder when ${15^{1207}}$ is divided by $13$ .

[5]

a.ii.

Convert ${\left( {7A2} \right)_{16}}$ to base $5$ , where ${\left( A \right)_{16}} = {\left( {10} \right)_{10}}$ .

[4]

Consider the equation ${\left( {1251} \right)_n} + {\left( {30} \right)_n} = {\left( {504} \right)_n} + {\left( {504} \right)_n}$ .

Find the value of $n$ .

[4]

Write down the remainder when $14^{2022}$ is divided by $7$ .

[1]

a.i.

Use Fermat’s little theorem to find the remainder when $14^{2022}$ is divided by $17$ .

[4]

a.ii.

Prove that a number in base $13$ is divisible by $6$ if, and only if, the sum of its digits is divisible by $6$ .

[4]

b.i.

The base $13$ number $1 y 93 y 25$ is divisible by $6$ . Find the possible values of the digit $y$ .

[4]

b.ii.