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

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 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

Unlock all 8 questions with a free account