IB Mathematics Analysis and Approaches Topic AHL…

For a real number $t$ , consider the system of equations

$\begin{cases} 2x - y + z = t \\ x + 3y - 2z = 2^{t} \\ 3x + 2y - z = t^{3} \end{cases}$

(a) Show that this system can never have exactly one solution.

(b) Find the values of $t$ for which the system has a solution, giving your answers correct to three significant figures.

(c) For the smaller of these values of $t$ , find the general solution, giving the coefficients correct to three significant figures.

The planes $\Pi_{1}$ , $\Pi_{2}$ and $\Pi_{3}$ have equations

$\begin{cases} x + 2y - z = 1 \\ 2x - y + 3z = 2 \\ x + 7y - 6z = k \end{cases}$

where $k$ is a real constant.

A till contains 60 coins, each of value 5 cents, 10 cents or 20 cents, and their total value is 700 cents.

Let $x$ , $y$ and $z$ be the numbers of 5-cent, 10-cent and 20-cent coins.

(a) Show that the two conditions on the coins give infinitely many real solutions, and that they are the triples $(x,\,y,\,z) = (2t-20,\,80-3t,\,t)$ .

(b) Find the number of ways the 60 coins could be made up.

(c) Three of the 60 coins are drawn at random, without replacement. Show that the probability that they have three different values is $\frac{(2t-20)(80-3t)t}{34220}$ .

(d) This probability is found to be $0.230$ , correct to three significant figures. Find all possible values of $x$ , $y$ and $z$ .

Three quantities $x$ , $y$ , $z$ are believed to satisfy the system

$\begin{cases} 1.8x + 2.6y + 3.1z = 12.4 \\ 2.4x + 1.7y + 5.2z = 15.9 \\ 3.9x + 4.75y + cz = k \end{cases}$

in which the constants $c$ and $k$ are obtained from measurements.

(a) Find the value of $c$ for which the system does not have a unique solution.

(b) Using that value of $c$ , find the value of $k$ for which the system is consistent, and find the general solution in that case.

A factory produces three products P, Q and R. Each unit of each product requires time on Machine A, time on Machine B and a fixed quantity of raw material, as shown in the table below.

Product	Machine A (hours)	Machine B (hours)	Raw material (kg)
P	2	1	4
Q	3	2	1
R	5	4	2

On a particular day, the factory uses all of its available $31$ hours on Machine A, $21$ hours on Machine B and $28$ kg of raw material. Let $p$ , $q$ and $r$ denote the number of units of P, Q and R produced on that day.

(a) Write down a system of three equations relating $p$ , $q$ and $r$ .

(b) Solve the system to find the number of units of each product produced.

(a) Show that the first two equations force $y = z$ and $x = 1-z$ , and that the three planes have a common point only when $k = 1$ .

(b) For $k = 1$ , find the solution set, giving it in the form $\mathbf{r} = \mathbf{a} + t\mathbf{d}$ , and the configuration of the three planes.

(c) the configuration of the three planes when $k \neq 1$ , justifying your description, and what happens to that configuration as $k$ approaches $1$ .

(c) The measured value of $k$ is $26.70$ . Solve the system for $c = 7.20$ and for $c = 7.30$ , and on what this shows about the reliability of the solution.

(c) Each constraint can be interpreted as a plane in $\mathbb{R}^3$ . what your answer in part b) tells you about the geometric arrangement of these three planes.

IB Mathematics Analysis and Approaches AHL 1.16—solution of Systems of Linear Equations Questionbank

IB Mathematics Analysis and Approaches AHL 1.16—solution of Systems of Linear Equations Questionbank

IB Mathematics Analysis and Approaches AHL 1.16—solution of Systems of Linear Equations Questionbank

IB Mathematics Analysis and Approaches AHL 1.16—solution of Systems of Linear Equations Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 1.16—solution of Systems of Linear Equations Questionbank