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HL Paper 2

Consider the identity $\frac{2 + 7 x}{(1 + 2 x) (1 - x)} \equiv \frac{A}{1 + 2 x} + \frac{B}{1 - x}$ , where $A, B \in ℤ$ .

Find the value of $A$ and the value of $B$ .

[3]

Hence, expand $\frac{2 + 7 x}{(1 + 2 x) (1 - x)}$ in ascending powers of $x$ , up to and including the term in $x^{2}$ .

[4]

Give a reason why the series expansion found in part (b) is not valid for $x = \frac{3}{4}$ .

[1]

Use mathematical induction to prove that $\frac{d^{n}}{d x^{n}} (x e^{p x}) = p^{n - 1} (p x + n) e^{p x}$ for $n \in ℤ^{+}, p \in ℚ$ .

At a gathering of $12$ teachers, seven are male and five are female. A group of five of these teachers go out for a meal together. Determine the possible number of groups in each of the following situations:

There are more males than females in the group.

[4]

Two of the teachers, Gary and Gerwyn, refuse to go out for a meal together.

[3]

Consider the complex numbers $z = 2 (\cos \frac{π}{5} + i \sin \frac{π}{5})$ and $w = 8 (\cos \frac{2 k π}{5} - i \sin \frac{2 k π}{5})$ , where $k \in ℤ^{+}$ .

Suppose that $z w \in ℤ$ .

Find the modulus of $z w$ .

[1]

Find the argument of $z w$ in terms of $k$ .

[2]

Find the minimum value of $k$ .

[3]

c.i.

For the value of $k$ found in part (i), find the value of $z w$ .

[1]

c.ii.

Solve the inequality ${x^2} > 2x + 1$ .

[2]

Use mathematical induction to prove that ${2^{n + 1}} > {n^2}$ for $n \in \mathbb{Z}$ , $n \geqslant 3$ .

[7]

Consider the set of six-digit positive integers that can be formed from the digits $0, 1, 2, 3, 4, 5, 6, 7, 8$ and $9$ .

Find the total number of six-digit positive integers that can be formed such that

the digits are distinct.

[2]

the digits are distinct and are in increasing order.

[2]

Consider the function $f (x) = \frac{x^{2} - x - 12}{2 x - 15}, x \in ℝ, x \neq \frac{15}{2}$ .

Find the coordinates where the graph of $f$ crosses the

$x$ -axis.

[2]

a.i.

$y$ -axis.

[1]

a.ii.

Write down the equation of the vertical asymptote of the graph of $f$ .

[1]

The oblique asymptote of the graph of $f$ can be written as $y = a x + b$ where $a, b \in ℚ$ .

Find the value of $a$ and the value of $b$ .

[4]

Sketch the graph of $f$ for $- 30 \leq x \leq 30$ , clearly indicating the points of intersection with each axis and any asymptotes.

[3]

Express $\frac{1}{f (x)}$ in partial fractions.

[3]

e.i.

Hence find the exact value of $\int_{0}^{3} \frac{1}{f (x)} d x$ , expressing your answer as a single logarithm.

[4]

e.ii.

Given that ${\log _{10}}\left( {\frac{1}{{2\sqrt 2 }}\left( {p + 2q} \right)} \right) = \frac{1}{2}\left( {{{\log }_{10}}p + {{\log }_{10}}q} \right),{\text{ }}p > 0,{\text{ }}q > 0$ , find $p$ in terms of $q$ .

Let $z = a + b{\text{i}}$ , $a$ , ${\text{b}} \in {\mathbb{R}^ + }$ and let ${\text{arg}}\,z = \theta$ .

Show the points represented by $z$ and $z - 2a$ on the following Argand diagram.

[1]

Find an expression in terms of θ for ${\text{arg}}\left( {z - 2a} \right)$ .

[1]

b.i.

Find an expression in terms of θ for ${\text{arg}}\left( {\frac{z}{{z - 2a}}} \right)$ .

[2]

b.ii.

Hence or otherwise find the value of θ for which ${\text{Re}}\left( {\frac{z}{{z - 2a}}} \right) = 0$ .

[3]

Solve ${z^2} = 4{{\text{e}}^{\frac{\pi }{2}{\text{i}}}}$ , giving your answers in the form

$r{{\text{e}}^{{\text{i}}\theta }}$ where $r$ , $\theta \in \mathbb{R}$ , $r > 0$ .

[3]

$a + {\text{i}}b$ where $a$ , $b \in \mathbb{R}$ .

[2]

Consider the differential equation $x^{2} \frac{d y}{d x} = y^{2} - 2 x^{2}$ for $x > 0$ and $y > 2 x$ . It is given that $y = 3$ when $x = 1$ .

Use Euler’s method, with a step length of $0.1$ , to find an approximate value of $y$ when $x = 1.5$ .

[4]

Use the substitution $y = v x$ to show that $x \frac{d v}{d x} = v^{2} - v - 2$ .

[3]

By solving the differential equation, show that $y = \frac{8 x + x^{4}}{4 - x^{3}}$ .

[10]

c.i.

Find the actual value of $y$ when $x = 1.5$ .

[1]

c.ii.

Using the graph of $y = \frac{8 x + x^{4}}{4 - x^{3}}$ , suggest a reason why the approximation given by Euler’s method in part (a) is not a good estimate to the actual value of $y$ at $x = 1.5$ .

[1]

c.iii.

Prove the identity ${(p + q)}^{3} - 3 p q (p + q) \equiv p^{3} + q^{3}$ .

[2]

The equation $2 x^{2} - 5 x + 1 = 0$ has two real roots, $α$ and $β$ .

Consider the equation $x^{2} + m x + n = 0$ , where $m, n \in ℤ$ and which has roots $\frac{1}{α^{3}}$ and $\frac{1}{β^{3}}$ .
Without solving $2 x^{2} - 5 x + 1 = 0$ , determine the values of $m$ and $n$ .

[6]

Consider a geometric sequence with a first term of 4 and a fourth term of −2.916.

Find the common ratio of this sequence.

[3]

Find the sum to infinity of this sequence.

[2]

A particle $P$ moves in a straight line such that after time $t$ seconds, its velocity, $v$ in ${m s}^{- 1}$ , is given by $v = e^{- 3 t} \sin 6 t$ , where $0 < t < \frac{π}{2}$ .

At time $t$ , $P$ has displacement $s (t)$ ; at time $t = 0$ , $s (0) = 0$ .

At successive times when the acceleration of $P$ is $0 {m s}^{- 2}$ , the velocities of $P$ form a geometric sequence. The acceleration of $P$ is zero at times $t_{1}, t_{2}, t_{3}$ where $t_{1} < t_{2} < t_{3}$ and the respective velocities are $v_{1}, v_{2}, v_{3}$ .

Find the times when $P$ comes to instantaneous rest.

[2]

Find an expression for $s$ in terms of $t$ .

[7]

Find the maximum displacement of $P$ , in metres, from its initial position.

[2]

Find the total distance travelled by $P$ in the first $1.5$ seconds of its motion.

[2]

Show that, at these times, $\tan 6 t = 2$ .

[2]

e.i.

Hence show that $\frac{v_{2}}{v_{1}} = \frac{v_{3}}{v_{2}} = - e^{- \frac{π}{2}}$ .

[5]

e.ii.

Find the term independent of $x$ in the expansion of $\frac{1}{x^{3}} {(\frac{1}{3 x^{2}} - \frac{x}{2})}^{9}$ .

Express the binomial coefficient $\left( \begin{gathered} 3n + 1 \hfill \\ 3n - 2 \hfill \\ \end{gathered} \right)$ as a polynomial in $n$ .

[3]

Hence find the least value of $n$ for which $\left( \begin{gathered} 3n + 1 \hfill \\ 3n - 2 \hfill \\ \end{gathered} \right) > {10^6}$ .

[3]

The following diagram shows part of the graph of $y = p + q \sin (r x)$ . The graph has a local maximum point at $(- \frac{9 π}{4}, 5)$ and a local minimum point at $(- \frac{3 π}{4}, - 1)$ .

Determine the values of $p$ , $q$ and $r$ .

[4]

Hence find the area of the shaded region.

[4]

Mary, three female friends, and her brother, Peter, attend the theatre. In the theatre there is a row of $10$ empty seats. For the first half of the show, they decide to sit next to each other in this row.

For the second half of the show, they return to the same row of $10$ empty seats. The four girls decide to sit at least one seat apart from Peter. The four girls do not have to sit next to each other.

Find the number of ways these five people can be seated in this row.

[3]

Find the number of ways these five people can now be seated in this row.

[4]

Write down and simplify the first three terms, in ascending powers of $x$ , in the Extended Binomial expansion of ${\left( {1 - x} \right)^{\frac{1}{3}}}$ .

[3]

By substituting $x = \frac{1}{9}$ find a rational approximation to $\sqrt[3]{9}$ .

[3]

Consider the equation ${x^5} - 3{x^4} + m{x^3} + n{x^2} + px + q = 0$ , where $m$ , $n$ , $p$ , $q \in \mathbb{R}$ .

The equation has three distinct real roots which can be written as ${\text{lo}}{{\text{g}}_2}\,a$ , ${\text{lo}}{{\text{g}}_2}\,b$ and ${\text{lo}}{{\text{g}}_2}\,c$ .

The equation also has two imaginary roots, one of which is $d{\text{i}}$ where $d \in \mathbb{R}$ .

The values $a$ , $b$ , and $c$ are consecutive terms in a geometric sequence.

Show that $abc = 8$ .

[5]

Show that one of the real roots is equal to 1.

[3]

Given that $q = 8{d^2}$ , find the other two real roots.

[9]

The following diagram shows part of the graph of $2{x^2} = {\text{si}}{{\text{n}}^3}\,y$ for $0 \leqslant y \leqslant \pi$ .

The shaded region $R$ is the area bounded by the curve, the $y$ -axis and the lines $y = 0$ and $y = \pi$ .

Using implicit differentiation, find an expression for $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ .

[4]

a.i.

Find the equation of the tangent to the curve at the point $\left( {\frac{1}{4}{\text{, }}\frac{{5\pi }}{6}} \right)$ .

[4]

a.ii.

Find the area of $R$ .

[3]

The region $R$ is now rotated about the $y$ -axis, through $2\pi$ radians, to form a solid.

By writing ${{\text{si}}{{\text{n}}^3}\,y}$ as $\left( {1 - {\text{co}}{{\text{s}}^2}\,y} \right){\text{sin}}\,y$ , show that the volume of the solid formed is $\frac{{2\pi }}{3}$ .

[6]

Phil takes out a bank loan of $150 000 to buy a house, at an annual interest rate of 3.5%. The interest is calculated at the end of each year and added to the amount outstanding.

To pay off the loan, Phil makes annual deposits of $P at the end of every year in a savings account, paying an annual interest rate of 2% . He makes his first deposit at the end of the first year after taking out the loan.

David visits a different bank and makes a single deposit of $Q , the annual interest rate being 2.8%.

Find the amount Phil would owe the bank after 20 years. Give your answer to the nearest dollar.

[3]

Show that the total value of Phil’s savings after 20 years is $\frac{{({{1.02}^{20}} - 1)P}}{{(1.02 - 1)}}$ .

[3]

Given that Phil’s aim is to own the house after 20 years, find the value for $P$ to the nearest dollar.

[3]

David wishes to withdraw $5000 at the end of each year for a period of $n$ years. Show that an expression for the minimum value of $Q$ is

$\frac{{5000}}{{1.028}} + \frac{{5000}}{{{{1.028}^2}}} + \ldots + \frac{{5000}}{{{{1.028}^n}}}$ .

[3]

d.i.

Hence or otherwise, find the minimum value of $Q$ that would permit David to withdraw annual amounts of $5000 indefinitely. Give your answer to the nearest dollar.

[3]

d.ii.

Eight boys and two girls sit on a bench. Determine the number of possible arrangements, given that

the girls do not sit together.

[3]

the girls do not sit on either end.

[2]

the girls do not sit on either end and do not sit together.

[3]

The complex numbers $w$ and $z$ satisfy the equations

$\frac{w}{z} = 2{\text{i}}$

${{\text{z}}^ * } - 3w = 5 + 5{\text{i}}$ .

Find $w$ and $z$ in the form $a + b{\text{i}}$ where $a$ , ${\text{b}} \in \mathbb{Z}$ .

Boxes of mixed fruit are on sale at a local supermarket.

Box A contains 2 bananas, 3 kiwifruit and 4 melons, and costs $6.58.

Box B contains 5 bananas, 2 kiwifruit and 8 melons and costs $12.32.

Box C contains 5 bananas and 4 kiwifruit and costs $3.00.

Find the cost of each type of fruit.

A random variable $X$ has probability density function

$f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {3a}&{\text{,}}&{0 \leqslant x < 2}&{} \\ {a\left( {x - 5} \right)\left( {1 - x} \right)}&{\text{,}}&{2 \leqslant x \leqslant b}&{a{\text{, }}b \in {\mathbb{R}^ + }{\text{, }}3 < b \leqslant 5.} \\ 0&{\text{,}}&{{\text{otherwise}}}&{} \end{array}} \right.$

Consider the case where $b = 5$ .

Find the value of

Find, in terms of $a$ , the probability that $X$ lies between 1 and 3.

[4]

Sketch the graph of $f$ . State the coordinates of the end points and any local maximum or minimum points, giving your answers in terms of $a$ .

[4]

$a$ .

[4]

c.i.

${\text{E}}\left( X \right)$ .

[3]

c.ii.

the median of $X$ .

[4]

c.iii.

Eight runners compete in a race where there are no tied finishes. Andrea and Jack are two of the eight competitors in this race.

Find the total number of possible ways in which the eight runners can finish if Jack finishes

in the position immediately after Andrea.

[2]

in any position after Andrea.

[3]

Prove by contradiction that $\log_{2} 5$ is an irrational number.

A geometric sequence has ${u_4} = - 70$ and ${u_7} = 8.75$ . Find the second term of the sequence.

Consider the complex number $z = \frac{{2 + 7{\text{i}}}}{{6 + 2{\text{i}}}}$ .

Express $z$ in the form $a + {\text{i}}b$ , where $a,\,b \in \mathbb{Q}$ .

[2]

Find the exact value of the modulus of $z$ .

[2]

Find the argument of $z$ , giving your answer to 4 decimal places.

[2]

It is known that the number of fish in a given lake will decrease by 7% each year unless some new fish are added. At the end of each year, 250 new fish are added to the lake.

At the start of 2018, there are 2500 fish in the lake.

Show that there will be approximately 2645 fish in the lake at the start of 2020.

[3]

Find the approximate number of fish in the lake at the start of 2042.

[5]

The coefficient of ${x^2}$ in the expansion of ${\left( {\frac{1}{x} + 5x} \right)^8}$ is equal to the coefficient of ${x^4}$ in the expansion of ${\left( {a + 5x} \right)^7},{\text{ }}a \in \mathbb{R}$ . Find the value of $a$ .

Write down the first three terms of the binomial expansion of $(1 + t)^{- 1}$ in ascending powers of $t$ .

[1]

By using the Maclaurin series for $\cos x$ and the result from part (a), show that the Maclaurin series for $sec x$ up to and including the term in $x^{4}$ is $1 + \frac{x^{2}}{2} + \frac{5 x^{4}}{24}$ .

[4]

By using the Maclaurin series for $arctan x$ and the result from part (b), find $\lim_{x \to 0} (\frac{x arctan 2 x}{sec x - 1})$ .

[3]

A biased coin is weighted such that the probability, $p$ , of obtaining a tail is $0.6$ . The coin is tossed repeatedly and independently until a tail is obtained.

Let $E$ be the event “obtaining the first tail on an even numbered toss”.

Find $P (E)$ .

Consider $z = \cos θ + i \sin θ$ where $z \in ℂ, z \neq 1$ .

Show that $Re (\frac{1 + z}{1 - z}) = 0$ .

Find the roots of the equation ${w^3} = 8{\text{i}}$ , $w \in \mathbb{C}$ . Give your answers in Cartesian form.

[4]

One of the roots ${w_1}$ satisfies the condition ${\text{Re}}\left( {{w_1}} \right) = 0$ .

Given that ${w_1} = \frac{z}{{z - {\text{i}}}}$ , express $z$ in the form $a + b{\text{i}}$ , where $a$ , $b \in \mathbb{Q}$ .

[3]

The function $f$ has a derivative given by $f' (x) = \frac{1}{x (k - x)}, x \in ℝ, x \neq o, x \neq k$ where $k$ is a positive constant.

Consider $P$ , the population of a colony of ants, which has an initial value of $1200$ .

The rate of change of the population can be modelled by the differential equation $\frac{d P}{d t} = \frac{P (k - P)}{5 k}$ , where $t$ is the time measured in days, $t \geq 0$ , and $k$ is the upper bound for the population.

At $t = 10$ the population of the colony has doubled in size from its initial value.

The expression for $f' (x)$ can be written in the form $\frac{a}{x} + \frac{b}{k - x}$ , where $a, b \in ℝ$ . Find $a$ and $b$ in terms of $k$ .

[3]

Hence, find an expression for $f (x)$ .

[3]

By solving the differential equation, show that $P = \frac{1200 k}{(k - 1200) e^{- \frac{t}{5}} + 1200}$ .

[8]

Find the value of $k$ , giving your answer correct to four significant figures.

[3]

Find the value of $t$ when the rate of change of the population is at its maximum.

[3]

Let $P\left( z \right) = a{z^3} - 37{z^2} + 66z - 10$ , where $z \in \mathbb{C}$ and $a \in \mathbb{Z}$ .

One of the roots of $P\left( z \right) = 0$ is $3 + {\text{i}}$ . Find the value of $a$ .

The population, $P$ , of a particular species of marsupial on a small remote island can be modelled by the logistic differential equation

$\frac{d P}{d t} = k P (1 - \frac{P}{N})$

where $t$ is the time measured in years and $k, N$ are positive constants.

The constant $N$ represents the maximum population of this species of marsupial that the island can sustain indefinitely.

Let $P_{0}$ be the initial population of marsupials.

In the context of the population model, interpret the meaning of $\frac{d P}{d t}$ .

[1]

Show that $\frac{d^{2} P}{d t^{2}} = k^{2} P (1 - \frac{P}{N}) (1 - \frac{2 P}{N})$ .

[4]

Hence show that the population of marsupials will increase at its maximum rate when $P = \frac{N}{2}$ . Justify your answer.

[5]

Hence determine the maximum value of $\frac{d P}{d t}$ in terms of $k$ and $N$ .

[2]

By solving the logistic differential equation, show that its solution can be expressed in the form

$k t = \ln \frac{P}{P_{0}} (\frac{N - P_{0}}{N - P})$ .

[7]

After $10$ years, the population of marsupials is $3 P_{0}$ . It is known that $N = 4 P_{0}$ .

Find the value of $k$ for this population model.

[2]

In a trial examination session a candidate at a school has to take 18 examination papers including the physics paper, the chemistry paper and the biology paper. No two of these three papers may be taken consecutively. There is no restriction on the order in which the other examination papers may be taken.

Find the number of different orders in which these 18 examination papers may be taken.

Consider the polynomial $P\left( z \right) \equiv {z^4} - 6{z^3} - 2{z^2} + 58z - 51,\,\,z \in \mathbb{C}$ .

Sketch the graph of $y = {x^4} - 6{x^3} - 2{x^2} + 58x - 51$ , stating clearly the coordinates of any maximum and minimum points and intersections with axes.

[6]

Hence, or otherwise, state the condition on $k \in \mathbb{R}$ such that all roots of the equation $P\left( z \right) = k$ are real.

[2]

Use mathematical induction to prove that ${\left( {1 - a} \right)^n} > 1 - na$ for $\left\{ {n\,{\text{:}}\,n \in {\mathbb{Z}^ + },\,n \geqslant 2} \right\}$ where $0 < a < 1$ .

The 3rd term of an arithmetic sequence is 1407 and the 10th term is 1183.

Calculate the number of positive terms in the sequence.

Consider the expansion of ${\left( {2 + x} \right)^n}$ , where $n \geqslant 3$ and $n \in \mathbb{Z}$ .

The coefficient of ${x^3}$ is four times the coefficient of ${x^2}$ . Find the value of $n$ .