IB Maths AA HL 1 Number and Algebra

Practise IB Maths AA HL number and algebra through shared-core and HL sequences, finance, proof, complex numbers and exact symbolic methods with clear reasoning.

Covered syllabus points

1.1 Number and algebra - SL content 1.2 Number and algebra - AHL content

[Maximum number: 11]

In this question, you will investigate the maximum product of positive real numbers with a given sum.
Consider the two numbers $x_{1}, x_{2} \in \mathbb{R}^{+}$ , such that $x_{1}+x_{2}=12$ .

(a)

Consider n positive real numbers, $x_{1}, x_{2}, \ldots, x_{n}$ .
The geometric mean is defined as $\left(x_{1} \times x_{2} \times \ldots \times x_{n}\right)^{\frac{1}{n}}$ . It is given that the geometric mean is always less than or equal to the arithmetic mean, so $\left(x_{1} \times x_{2} \times \ldots \times x_{n}\right)^{\frac{1}{n}} \leq \frac{\left(x_{1}+x_{2}+\ldots+x_{n}\right)}{n}$ .

[ 6 ]

(i)

Show that the geometric mean and arithmetic mean are equal when $x_{1}=x_{2}=\ldots=x_{n}$ .

[ 2 ]

let all $x_{i}$ be labelled as x (or $x_{1}$ or $x_{n}$ etc.)

Note: Do not accept use of a specific number for x or n.

\left(x^{n}\right)^{\frac{1}{n}}=x \text { and } \frac{n x}{n}=x

(ii)

Use this result to prove that $M_{n}(S)=\left(\frac{S}{n}\right)^{n}$ .

[ 4 ]

$x_{1}+x_{2}+\ldots+x_{n}=S$

\left(x_{1} \times x_{2} \times \ldots \times x_{n}\right)^{\frac{1}{n}} \leq \frac{S}{n}

Note: Award M1 for use of the inequality, which may be seen as an equality.
$x_{1} \times x_{2} \times \ldots \times x_{n} \leq\left(\frac{S}{n}\right)^{n}$ (as both sides are positive)

LHS and RHS are equal when all values of $x_{i}$ are equal (to $\frac{S}{n}$ )
$M_{n}(S)=\left(\frac{S}{n}\right)^{n}$

(b)

Verify that $g(x)=M_{x}(S)$ , when $x \in \mathbb{Z}^{+}$ .

[ 2 ]

$\quad \ln (g(x))=x \ln \left(\frac{S}{x}\right) \Rightarrow \ln (g(x))=\ln \left(\frac{S}{x}\right)^{x} \quad$ M1
$g(x)=\left(\frac{S}{x}\right)^{x}$ $=M_{x}(S)$ for $x \in \mathbb{Z}^{+}$
AG

(c)

Use your answer to part (h) to find the largest possible product of positive numbers whose sum is 100 . Give your answer in the form $a \times 10^{k}$ , where $1 \leq a<10$ and $k \in \mathbb{Z}^{+}$ .

[ 3 ]

$\frac{100}{\mathrm{e}}=36.8 \quad$ M1
$\left(\frac{100}{36}\right)^{36}=9.3996 \ldots \times 10^{15}$ AND $\left(\frac{100}{37}\right)^{37}=9.47406 \ldots \times 10^{15}$
largest possible product is $9.47 \times 10^{15}\left(9.47406 \ldots \times 10^{15}\right)$

Note: Award A1 independently of the R1 (but not independently of the M1).

[Maximum number: 4]

Consider a three-digit code a b c, where each of a, b and c is assigned one of the values 1,2,3,4 or 5 .

(a)

Find the total number of possible codes

[ 4 ]

(i)

assuming that each value can be repeated (for example, 121 or 444);

[ 2 ]

$5^{3}$
=125

(ii)

assuming that no value is repeated.

Let $P(x)=x^{3}+a x^{2}+b x+c$ , where each of a, b and c is assigned one of the values 1,2,3,4 or 5 . Assume that no value is repeated.

Consider the case where P(x) has a factor of $\left(x^{2}+3 x+2\right)$ .

[ 2 ]

$\quad{ }^{5} P_{3}=5 \times 4 \times 3$
=60

[Maximum number: 18]

In this question you will investigate series of the form

\sum_{i=1}^{n} \boldsymbol{i}^{q}=1^{q}+2^{q}+3^{q}+\ldots+n^{q} \text { where } n, q \in \mathbb{Z}^{+}

and use various methods to find polynomials, in terms of n, for such series.
When q=1, the above series is arithmetic.

(a)

Show that $\sum_{i=1}^{n} i=\frac{1}{2} n(n+1)$ .

Consider the case when q=2.

[ 1 ]

EITHER
$S_{n}=\frac{n}{2}(2 \times 1+(n-1) \times 1)$

OR
$u_{1}=1$ and either $u_{n}=n$ or d=1 stated explicitly

OR
$1+2+\ldots+n$ (or equivalent) stated explicitly

THEN
$S_{n}=\frac{n}{2}(1+n)$

Note: Award A0 for a numerical verification.

(b)

The following table gives values of $n^{2}$ and $\sum_{i=1}^{n} i^{2}$ for n=1,2,3.

[ 5 ]

(i)

The sum of the first n square numbers can be expressed as a cubic polynomial with three terms:

\sum_{i=1}^{n} i^{2}=a_{1} n+a_{2} n^{2}+a_{3} n^{3} \text { where } a_{1}, a_{2}, a_{3} \in \mathbb{Q}^{+}

Hence, write down a system of three linear equations in $a_{1}, a_{2}$ and $a_{3}$ .

[ 3 ]

$\quad a_{1}+a_{2}+a_{3}=1$ $2 a_{1}+4 a_{2}+8 a_{3}=5$ $3 a_{1}+9 a_{2}+27 a_{3}=14$
Note: For the third A mark, award A 1FT for $3 a_{1}+9 a_{2}+27 a_{3}=p$ where p is their answer to part (b) (i).

(ii)

Hence, find the values of $a_{1}, a_{2}$ and $a_{3}$ .

You will now consider a method that can be generalized for all values of q.
Consider the function $f(x)=1+x+x^{2}+\ldots+x^{n}, n \in \mathbb{Z}^{+}$ .

[ 2 ]

$\quad a_{1}=\frac{1}{6}(=0.166666 \ldots \approx 0.167), a_{2}=\frac{1}{2}(=0.5)$ ,

Note: Award A1 if only two of $a_{1}, a_{2}, a_{3}$ are correct.
Only award FT if three linear equations, each in $a_{1}, a_{2}$ and $a_{3}$ are stated in part (b) (ii) or (iii).

Award A2 F T for their $a_{1}, a_{2}$ and $a_{3}$ .
Award A1FT for their $a_{1}, a_{2}$ and $a_{3}=0$ .

(c)

Prove by mathematical induction that $f_{q}(x)=\sum_{i=1}^{n} i^{q} x^{i}, q \in \mathbb{Z}^{+}$ .

[ 6 ]

consider q=1

f_{1}(x)=x+2 x^{2}+\ldots+n x^{n} \text { (reference to part (c)) and } f_{1}(x)=\sum_{i=1}^{n} i x^{i}

assume true for $q=k,\left(f_{k}(x)=\sum_{i=1}^{n} i^{k} x^{i}\right)$

Note: Do not award M1 for statements such as "let q=k " or " q=k is true".
Subsequent marks after this M1 are independent of this mark and can be awarded.
consider q=k+1

\begin{aligned} & f_{k+1}(x)=x f_{k}^{\prime}(x) \\ & =x \sum_{i=1}^{n} i^{k+1} x^{i-1} \quad \text { OR } \quad x\left(1+2^{k+1} x+3^{k+1} x^{2}+\ldots+n^{k+1} x^{n-1}\right) \end{aligned}

Note: Award the above M1 if $f_{k+1}(x)=x \sum_{i=1}^{n} i^{k+1} x^{i-1}$ or $x f_{k}^{\prime}(x)=x \sum_{i=1}^{n} i^{k+1} x^{i-1}$ (or equivalent) is stated.

=\sum_{i=1}^{n} i^{k+1} x^{i} \text { OR } x+2^{k+1} x^{2}+3^{k+1} x^{3}+\ldots+n^{k+1} x^{n}

since true for q=1 and true for q=k+1 if true for q=k, hence true for all $q\left(\in \mathbb{Z}^{+}\right)$

Note: To obtain the final R1, three of the previous five marks must have been awarded.

(d)

Using sigma notation, write down an expression for $f_{q}(1)$ .

[ 1 ]

$f_{q}(1)=1^{q}+2^{q}+3^{q}+\ldots+n^{q}$

=\sum_{i=1}^{n} i^{q}\left(=\sum_{i=1}^{n} 1^{i} i^{q}\right)

(e)

By considering $f(x)=1+x+x^{2}+\ldots+x^{n}$ as a geometric series, for $x \neq 1$ , show that $f(x)=\frac{x^{n+1}-1}{x-1}$ .

[ 2 ]

uses $S_{n}=\frac{u_{1}\left(r^{n}-1\right)}{r-1}$ with r=x and $u_{1}=1$
clear indication there are ( n+1 ) terms

f(x)=\frac{x^{n+1}-1}{x-1}

(f)

For $x \neq 1$ , show that $f_{1}(x)=\frac{n x^{n+2}-(n+1) x^{n+1}+x}{(x-1)^{2}}$ .

[ 3 ]

METHOD 1

\begin{aligned} & f_{1}(x)=x f^{\prime}(x) \\ & =x \frac{(x-1)(n+1) x^{n}-1 \times\left(x^{n+1}-1\right)}{(x-1)^{2}} \end{aligned}

Note: Award M1 for attempting to use the quotient or the product rule to find $f^{\prime}(x)$ .

=x \frac{(n x+x-n-1) x^{n}-\left(x^{n+1}-1\right)}{(x-1)^{2}}\left(=x \frac{n x^{n+1}-n x^{n}-x^{n}+1}{(x-1)^{2}}\right)

Note: Award A1 for any correct manipulation of the derivative that leads to the AG.

=\frac{n x^{n+2}-(n+1) x^{n+1}+x}{(x-1)^{2}}

METHOD 2
attempts to form $(x-1) f_{1}(x)$

\begin{aligned} & (x-1) f_{1}(x)=n x^{n+1}-\left(x+x^{2}+x^{3}+\ldots+x^{n}\right) \\ & f_{1}(x)=\frac{1}{x-1}\left(n x^{n+1}-\left(\frac{x^{n+1}-1}{x-1}-1\right)\right) \\ & f_{1}(x)=\frac{1}{x-1}\left(\frac{n x^{n+1}(x-1)-x^{n+1}+x}{x-1}\right)\left(=\frac{1}{x-1}\left(\frac{n x^{n+2}-n x^{n+1}-x^{n+1}+x}{x-1}\right)\right) \end{aligned}

Note: Award A1 for any correct manipulation of the derivative that leads to the AG.

f_{1}(x)=\frac{n x^{n+2}-(n+1) x^{n+1}+x}{(x-1)^{2}}

[Maximum number: 7]

The following question explores features of a family of curves. The family is then linked to a homogeneous differential equation.
Consider the curve given by $y=\frac{x\left(x^{2}-16\right)}{x^{2}+16}$ .

(a)

Show that $x-\frac{2 A x}{x^{2}+A} \equiv \frac{x\left(x^{2}-A\right)}{x^{2}+A}$ .

[ 2 ]

METHOD 1

\begin{aligned} & x-\frac{2 A x}{x^{2}+A} \equiv \frac{x\left(x^{2}+A\right)-2 A x}{x^{2}+A}\left(\equiv \frac{x^{3}+A x-2 A x}{x^{2}+A}\right) \\ & \equiv \frac{x^{3}-A x}{x^{2}+A}\left(\equiv \frac{x\left(x^{2}+A-2 A\right)}{x^{2}+A}\right) \\ & x-\frac{2 A x}{x^{2}+A} \equiv \frac{x\left(x^{2}-A\right)}{x^{2}+A} \end{aligned}

Note: Award no marks for numerical approaches.

METHOD 2

EITHER
attempt to do long division on $\frac{x\left(x^{2}-A\right)}{x^{2}+A}$ or $\frac{x^{3}-A x}{x^{2}+A}$
quotient of x and remainder of -2 A x
$x-\frac{2 A x}{x^{2}+A} \equiv \frac{x\left(x^{2}-A\right)}{x^{2}+A}$

OR
$\frac{x^{3}}{x^{2}+A}-\frac{A x}{x^{2}+A} \equiv \frac{x^{3}+A x}{x^{2}+A}-\frac{A x+A x}{x^{2}+A}$ $\equiv \frac{x\left(x^{2}+A\right)}{\left(x^{2}+A\right)}-\frac{2 A x}{\left(x^{2}+A\right)}$ $\equiv x-\frac{2 A x}{x^{2}+A} \equiv \frac{x\left(x^{2}-A\right)}{x^{2}+A}$

Note: Award no marks for numerical approaches.

(b)

Using partial fractions, show that $\int \frac{1}{1-v^{2}} \mathrm{~d} v=\frac{1}{2} \ln \left|\frac{A(1+v)}{1-v}\right|$ , where A is a positive

[ 5 ]

Note: Award no marks for attempts at integration that do not involve partial fractions.
attempt to write $\frac{1}{1-v^{2}}$ in the form $\frac{B}{1+v}+\frac{C}{1-v}$

\begin{aligned} & B=\frac{1}{2} \text { and } C=\frac{1}{2} \\ & \int \frac{1}{1-v^{2}} \mathrm{~d} v=\int \frac{\frac{1}{2}}{1+v}+\frac{\frac{1}{2}}{1-v} \mathrm{~d} v \end{aligned}

attempt to integrate their partial fractions

EITHER

\frac{1}{2} \ln |1+v|-\frac{1}{2} \ln |1-v|+\frac{1}{2} \ln A

Note: Award A1 for first two terms. The moduli must be present for this mark. Award A1 for $\frac{1}{2} \ln A$ .

OR

\begin{aligned} & \frac{1}{2} \ln |1+v|-\frac{1}{2} \ln |1-v|(+c) \\ & =\frac{1}{2} \ln \left|\frac{\mathrm{e}^{2 c}(1+v)}{1-v}\right| \text { where } \mathrm{e}^{2 c}=A \end{aligned}

Note: The moduli must be present for each of these marks.

THEN

=\frac{1}{2} \ln \left|\frac{A(1+v)}{1-v}\right|

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Covered syllabus points

1 Number and algebra question 1

Question (a)

Question (i)

Question (ii)

Question (b)

Question (c)

1 Number and algebra question 2

Question (a)

Question (i)

Question (ii)

1 Number and algebra question 3

Question (a)

Question (b)

Question (i)

Question (ii)

Question (c)

Question (d)

Question (e)

Question (f)

1 Number and algebra question 4

Question (a)

Question (b)