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HL Paper 3

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.

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]

Consider the functions $f$ , $g$ : $\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}$ defined by

$f\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = \left( {x + y,\,\,x - y} \right)$ and $g\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = \left( {xy,\,\,x + y} \right)$ .

Find $\left( {f \circ g} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right)$ .

[3]

a.i.

Find $\left( {g \circ f} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right)$ .

[2]

a.ii.

State with a reason whether or not $f$ and $g$ commute.

[1]

Find the inverse of $f$ .

[3]

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 asks you to explore the behaviour and some key features of the function $f_{n} (x) = x^{n} (a - x)^{n}$ , where $a \in ℝ^{+}$ and $n \in ℤ^{+}$ .

In parts (a) and (b), only consider the case where $a = 2$ .

Consider $f_{1} (x) = x (2 - x)$ .

Consider $f_{n} (x) = x^{n} {(2 - x)}^{n}$ , where $n \in ℤ^{+}, n > 1$ .

Now consider $f_{n} (x) = x^{n} {(a - x)}^{n}$ where $a \in ℝ^{+}$ and $n \in ℤ^{+}, n > 1$ .

By using the result from part (f) and considering the sign of ${f_{n}}^{'} (- 1)$ , show that the point $(0, 0)$ on the graph of $y = f_{n} (x)$ is

Sketch the graph of $y = f_{1} (x)$ , stating the values of any axes intercepts and the coordinates of any local maximum or minimum points.

[3]

Use your graphic display calculator to explore the graph of $y = f_{n} (x)$ for

• the odd values $n = 3$ and $n = 5$ ;

• the even values $n = 2$ and $n = 4$ .

Hence, copy and complete the following table.

[6]

Show that ${f_{n}}^{'} (x) = n x^{n - 1} (a - 2 x) {(a - x)}^{n - 1}$ .

[5]

State the three solutions to the equation ${f_{n}}^{'} (x) = 0$ .

[2]

Show that the point $(\frac{a}{2}, f_{n} (\frac{a}{2}))$ on the graph of $y = f_{n} (x)$ is always above the horizontal axis.

[3]

Hence, or otherwise, show that ${f_{n}}^{'} (\frac{a}{4}) > 0$ , for $n \in ℤ^{+}$ .

[2]

a local minimum point for even values of $n$ , where $n > 1$ and $a \in ℝ^{+}$ .

[3]

g.i.

a point of inflexion with zero gradient for odd values of $n$ , where $n > 1$ and $a \in ℝ^{+}$ .

[2]

g.ii.

Consider the graph of $y = x^{n} {(a - x)}^{n} - k$ , where $n \in ℤ^{+}$ , $a \in ℝ^{+}$ and $k \in ℝ$ .

State the conditions on $n$ and $k$ such that the equation $x^{n} {(a - x)}^{n} = k$ has four solutions for $x$ .

[5]

This question asks you to explore the behaviour and key features of cubic polynomials of the form $x^{3} - 3 c x + d$ .

Consider the function $f (x) = x^{3} - 3 c x + 2$ for $x \in ℝ$ and where $c$ is a parameter, $c \in ℝ$ .

The graphs of $y = f (x)$ for $c = - 1$ and $c = 0$ are shown in the following diagrams.

$c = - 1$ $c = 0$

On separate axes, sketch the graph of $y = f (x)$ showing the value of the $y$ -intercept and the coordinates of any points with zero gradient, for

Hence, or otherwise, find the set of values of $c$ such that the graph of $y = f (x)$ has

Given that the graph of $y = f (x)$ has one local maximum point and one local minimum point, show that

Hence, for $c > 0$ , find the set of values of $c$ such that the graph of $y = f (x)$ has

$c = 1$ .

[3]

a.i.

$c = 2$ .

[3]

a.ii.

Write down an expression for $f' (x)$ .

[1]

a point of inflexion with zero gradient.

[1]

c.i.

one local maximum point and one local minimum point.

[2]

c.ii.

no points where the gradient is equal to zero.

[1]

c.iii.

the $y$ -coordinate of the local maximum point is $2 c^{\frac{3}{2}} + 2$ .

[3]

d.i.

the $y$ -coordinate of the local minimum point is $- 2 c^{\frac{3}{2}} + 2$ .

[1]

d.ii.

exactly one $x$ -axis intercept.

[2]

e.i.

exactly two $x$ -axis intercepts.

[2]

e.ii.

exactly three $x$ -axis intercepts.

[2]

e.iii.

Consider the function $g (x) = x^{3} - 3 c x + d$ for $x \in ℝ$ and where $c, d \in ℝ$ .

Find all conditions on $c$ and $d$ such that the graph of $y = g (x)$ has exactly one $x$ -axis intercept, explaining your reasoning.

[6]

This question asks you to explore properties of a family of curves of the type $y^{2} = x^{3} + a x + b$ for various values of $a$ and $b$ , where $a, b \in ℕ$ .

On the same set of axes, sketch the following curves for $- 2 \leq x \leq 2$ and $- 2 \leq y \leq 2$ , clearly indicating any points of intersection with the coordinate axes.

Now, consider curves of the form $y^{2} = x^{3} + b$ , for $x \geq - \sqrt[3]{b}$ , where $b \in ℤ^{+}$ .

Next, consider the curve $y^{2} = x^{3} + x, x \geq 0$ .

The curve $y^{2} = x^{3} + x$ has two points of inflexion. Due to the symmetry of the curve these points have the same $x$ -coordinate.

$P (x, y)$ is defined to be a rational point on a curve if $x$ and $y$ are rational numbers.

The tangent to the curve $y^{2} = x^{3} + a x + b$ at a rational point $P$ intersects the curve at another rational point $Q$ .

Let $C$ be the curve $y^{2} = x^{3} + 2$ , for $x \geq - \sqrt[3]{2}$ . The rational point $P (- 1, - 1)$ lies on $C$ .

$y^{2} = x^{3}, x \geq 0$

[2]

a.i.

$y^{2} = x^{3} + 1, x \geq - 1$

[2]

a.ii.

Write down the coordinates of the two points of inflexion on the curve $y^{2} = x^{3} + 1$ .

[1]

b.i.

By considering each curve from part (a), identify two key features that would distinguish one curve from the other.

[1]

b.ii.

By varying the value of $b$ , suggest two key features common to these curves.

[2]

Show that $\frac{d y}{d x} = \pm \frac{3 x^{2} + 1}{2 \sqrt{x^{3} + x}}$ , for $x > 0$ .

[3]

d.i.

Hence deduce that the curve $y^{2} = x^{3} + x$ has no local minimum or maximum points.

[1]

d.ii.

Find the value of this $x$ -coordinate, giving your answer in the form $x = \sqrt{\frac{p \sqrt{3} + q}{r}}$ , where $p, q, r \in ℤ$ .

[7]

Find the equation of the tangent to $C$ at $P$ .

[2]

f.i.

Hence, find the coordinates of the rational point $Q$ where this tangent intersects $C$ , expressing each coordinate as a fraction.

[2]

f.ii.

The point $S (- 1, 1)$ also lies on $C$ . The line $[QS]$ intersects $C$ at a further point. Determine the coordinates of this point.

[5]

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]

Let $A$ be the set $\{ x|x \in \mathbb{R},{\text{ }}x \ne 0\}$ . Let $B$ be the set $\{ x|x \in ] - 1,{\text{ }} + 1[,{\text{ }}x \ne 0\}$ .

A function $f:A \to B$ is defined by $f(x) = \frac{2}{\pi }\arctan (x)$ .

Let $D$ be the set $\{ x|x \in \mathbb{R},{\text{ }}x > 0\}$ .

A function $g:\mathbb{R} \to D$ is defined by $g(x) = {{\text{e}}^x}$ .

(i) Sketch the graph of $y = f(x)$ and hence justify whether or not $f$ is a bijection.

(ii) Show that $A$ is a group under the binary operation of multiplication.

(iii) Give a reason why $B$ is not a group under the binary operation of multiplication.

(iv) Find an example to show that $f(a \times b) = f(a) \times f(b)$ is not satisfied for all $a,{\text{ }}b \in A$ .

[13]

(i) Sketch the graph of $y = g(x)$ and hence justify whether or not $g$ is a bijection.

(ii) Show that $g(a + b) = g(a) \times g(b)$ for all $a,{\text{ }}b \in \mathbb{R}$ .

(iii) Given that $\{ \mathbb{R},{\text{ }} + \}$ and $\{ D,{\text{ }} \times \}$ are both groups, explain whether or not they are isomorphic.

[8]

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.