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

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

Practice IB Mathematics Analysis and Approaches (AA) Topic AHL 1.16—solution of Systems of Linear Equations with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 1.16—solution of Systems of Linear Equations and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 2

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.

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

[3]

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

[7]

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

[3]

Question 2

HLPaper 1

Consider the system of linear equations

$x + y + z 2 x + 3 y + z 3 x + 5 y + (k + 1) z = 1, = 4, = 7,$

where $k \in R$ .

Write the system as an augmented matrix.

[1]

Use row reduction to reduce the augmented matrix to row-echelon form, in terms of $k$ .

[6]

Find the value of $k$ for which the system does not have a unique solution.

[2]

For this value of $k$ , state whether the system has no solution or infinitely many solutions, and justify your answer.

[4]

Find the unique solution when $k = 5$ .

[4]

Question 3

HLPaper 1

Consider the system of equations

$⎩ ⎨ ⎧ x + 2 y + z = 4 2 x + 5 y + 3 z = 11 3 x + 7 y + a z = b$

where $a, b \in R$ .

Show that the system can be reduced to

$⎩ ⎨ ⎧ x + 2 y + z = 4 y + z = 3 (a - 4) z = b - 15$

[6]

Hence find conditions on $a$ and $b$ for which the system has a unique solution.

[2]

Hence find conditions on $a$ and $b$ for which the system has no solution.

[2]

Hence find conditions on $a$ and $b$ for which the system has infinitely many solutions.

[2]

For the case in part 4, express the solution in parametric form.

[3]

Question 4

HLPaper 1

Consider the system of equations

$⎩ ⎨ ⎧ x + y + z = 6 2 x - y + z = 3 x + 2 y - z = 2$

Solve the system.

[5]

State the geometric significance of your answer in terms of the three planes in $R^{3}$ .

[2]

Question 5

HLPaper 1

Factorise $x^{3} - 7 x^{2} + 14 x - 8$ completely.

[4]

Hence express $\frac{3 x + 5}{x ^{3} - 7 x ^{2} + 14 x - 8}$ in partial fractions.

[6]

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

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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

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