Matrices (DP IB Maths: AI HL)

Exam Questions

4 hours36 questions

4 marks

Consider the following matrices:

$A = (\begin{matrix} 1 & 0 & - 2 \\ - 1 & a & 3 \end{matrix}) B = (\begin{matrix} b & 2 \\ 1 & - 1 \end{matrix}) C = (\begin{matrix} 0 & c & - 3 \end{matrix})$

$D = (\begin{matrix} 0 & 1 & 0 \\ 3 & d & 1 \\ 0 & 3 & 0 \end{matrix}) E = (\begin{matrix} - 1 & 4 & e \\ - 2 & 1 & 0 \end{matrix}) F = (\begin{matrix} f \\ 2 \\ - 1 \end{matrix})$

where $a, b, c, d, e, f \in ℝ$ are constants.

Find each of the following matrix sums or differences, or if that is not possible then explain why:

(i)

A + B

(ii)

A + E

(iii)

E - 3 A

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5 marks

Find each of the following matrix products, or if that is not possible then explain why:

(i)

B E

(ii)

E B

(iii)

C D

(iv)

F C

(v)

A E

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3 marks

List any other matrix products of two of the above matrices that it would be possible to find, other than the ones included in part (b). You do not need to find the products, but do specify what the order of each of those products would be.

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3 marks

Find each of the following:

(i)

B (A + E)

(ii)

B A + B E

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7 marks

Consider the matrices:

$A = (\begin{matrix} 2 & - 1 \\ 4 & - 1 \end{matrix}) B = (\begin{matrix} 1 & 1 \\ - 4 & k \end{matrix})$

where $k \in ℝ$ is a constant.

Let $I$ be the $2 \times 2$ identity matrix, and let 0 be the $2 \times 2$ zero matrix.

Find the following

(i)

B^{2}

(ii)

A - 2 I

(iii)

A B

(iv)

B A

(v)

A^{- 1} B

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4 marks

Find the following:

(i)

\det A

(ii)

\det A^{- 1}

(iii)

\det B

(iv)

\det A B

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4 marks

$C$ and $D$ are matrices such that $A + C = 0$ and $B^{- 1} + D = 0$

(i)

Write down matrix

C

(ii)

Find matrix

D

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3 marks

Consider the matrix

$A = (\begin{matrix} 3 & - 8 \\ p & 7 \end{matrix})$

where $p \in ℝ$ is a constant.

Given that $\det A = - 3$ , find the value of $p$ .

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4 marks

Consider the two matrices

$B = (\begin{matrix} 4 & q \\ - 1 & 3 \end{matrix}) and C = (\begin{matrix} r & 1 \\ 1 & 3 \end{matrix})$

where $q, r \in ℝ$ are constants.

Given that $B C = C B$ , find the values of $q$ and $r$ .

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2 marks

Consider the matrices $P = (\begin{matrix} 2 & - 1 \\ 3 & 1 \end{matrix})$ and $D = (\begin{matrix} 1 & 0 \\ 0 & 0.8 \end{matrix})$ .

Find

(i) the determinant of

P

(ii)

P^{- 1}

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3 marks

Consider the matrix product $P D P^{- 1}$ .

(i)

Show that

{(P D P^{- 1})}^{2} = P D^{2} P^{- 1}

and

{(P D P^{- 1})}^{3} = P D^{3} P^{- 1}

(ii)

Use the results of part (b)(i) to suggest an expression for

{(P D P^{- 1})}^{n}

in terms of

P

, $D$ and

P^{- 1}

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1 mark

The transition matrix of a dynamic system is $T = (\begin{matrix} 0.88 & 0.08 \\ 0.12 & 0.92 \end{matrix})$ .

Find $T^{5}$ .

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4 marks

(i)

Show that

T = P D P^{- 1}

(ii)

Hence use the answer to part (c) to confirm the validity of your expression from part (b)(ii) for

n = 5

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1 mark

A family are buying burgers for dinner and wish to order three veggie burgers, one beef burger and two chicken burgers.

From burger store A three veggie burgers would cost a total of $15.45, one beef burger would cost $6.15, and 2 chicken burgers would cost a total of $11.90.

Write down

(i)

a row matrix,

Q

, to represent the quantities of each type of burger that the family wishes to purchase

(ii)

a column matrix,

P_{A}

, to represent the cost of each type of burger at store A.

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3 marks

Burger store B sells their veggie burger, beef burger and chicken burger for $4.75, $5.85 and $5.50, respectively.

(i)

Write down a column matrix,

P_{B}

, to represent the cost of each type of burger at store B.

(ii)

Hence write down a cost matrix, $C$ , to represent the costs for both stores.

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2 marks

By first calculating the matrix $Q C$ , compare the total cost of the family’s dinner at stores A and B.

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1 mark

A café is looking to hire a duty manager, two baristas, a dishwasher, and three waiters. They decide to advertise the jobs on social media, as well as using a hiring agency. They receive 123 applications from advertising the job on social media and 57 applications from the hiring agency.

Write down a column matrix, $C$ , to represent the number of applications they received from advertising the job on social media and from using the hiring agency.

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1 mark

Overall, 22% of the applicants applied for the duty manager job, 28% applied for the barista job, 18% applied for the dishwasher job, and the rest applied for the waiter job. Note that every applicant was only allowed to apply for one of the available jobs.

Write down a row matrix, $R$ , to represent the percentages of the applicants that applied for each of the different jobs.

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3 marks

(i)

Calculate the product

P = C R

(ii)

Use the elements of the matrix

P

to work out the total number of applicants for each of the positions, giving your answers to the nearest integer.

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2 marks

Once the café has selected the right candidate for each job, each new employee will work a total of 40 hours per week. The hourly wage for the duty manager and barista jobs is $20.00 per hour, and the hourly wage for the dishwasher and waiter jobs is $17.25.

Calculate the café’s weekly wage expenses for the new employees.

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2 marks

Amanda is a landlord who has new sets of tenants moving into three new unfurnished homes. Amanda receives a special deal at a small local furniture store and so she offers her tenants that she will buy some tables, chairs, and/or sofas on their behalf, given that they reimburse her for the cost. The quantities of different items ordered for each house are shown in the table below.

	Tables	Chairs	Sofas
House 1	2	5	2
House 2	1	3	0
House 3	1	4	1

(i)

Write down a

3 \times 3

matrix, $F$ , to represent the furniture orders for the three houses.

(ii)

Write down a

1 \times 3

row matrix, $S$ , to represent Amanda’s total shopping list at the furniture store.

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5 marks

The price Amanda pays for each item is shown in the table below.

	Tables	Chairs	Sofas
Price	$72.00	$14.50	$47.50

(i)

By performing an appropriate matrix multiplication with matrix

F

, find the total amount owed to Amanda by each house.

(ii)

By performing an appropriate matrix multiplication with matrix

S

, find the total amount paid by Amanda to the furniture store.

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2 marks

The bus fare a person pays in a city is dependent on whether the person is a student, an adult or a pensioner.

The total amount taken in on a particular day by three different buses, along with the numbers of each type of fare paid, are shown in the table below.

	Student	Adult	Pensioner	Total
Bus A	91	82	13	$348
Bus B	102	80	4	$355
Bus C	71	54	11	$247

Let $s, a$ and $p$ represent the amount paid by a student, an adult, and a pensioner respectively.

Write down a system of three linear equations in terms of $s, a$ and $p$ that represent the information shown in the table above.

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2 marks

Find the values of $s, a$ and $p$ using appropriate matrices and matrix inverses.

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2 marks

Bus D finished the day having sold 112 student tickets, 91 adult tickets and 22 pensioner tickets.

Calculate the total amount taken in by Bus D. Give your answer to 2 decimal places.

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3 marks

The number of times a gamer logged in to Call of Duty, FIFA, or Assassin’s Creed over 3 weeks is shown in the table below.

	Call of Duty	FIFA	Assassin’s Creed
Week 1	5	4	3
Week 2	7	2	4
Week 3	3	5	6

The total number of hours the gamer spent playing each week is shown in the table below.

	Week 1	Week 2	Week 3
Total hours	12.35	13.84	14.16

The gamer was never logged in to more than one game at the same time.

The gamer believes that, for each game, the average amount of time spent playing per log-in session was consistent over the three weeks.

Assuming that the gamer’s belief is true, use matrix multiplication to find the average number of hours and minutes per log-in session that the gamer spent playing each game.

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2 marks

Write down a system of linear equations that could be used to find the answers in part (a).

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10a

2 marks

The graph of the quadratic function $f (x) = a x^{2} + b x + c$ passes through the points $(- 1, 11), (3, - 5)$ and $(6, 4)$ .

Show that $a$ , $b$ and $c$ must satisfy the following system of linear equations:

$a - b + c = 11$

$9 a + 3 b + c = - 5$

$36 a + 6 b + c = 4$

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10b

1 mark

Represent the system of equations in part (a) in matrix form.

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10c

3 marks

Hence use a matrix method to find the values of $a$ , $b$ and $c$ .

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11a

2 marks

The graph of the function $f (x) = a x^{2} + b x + c$ passes through the points $(- 1, 5), (3, 1)$ and $(4, - 5)$ .

Write down a system of linear equations that $a, b$ and $c$ must satisfy.

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11b

4 marks

Hence use a matrix method to determine the values of $a, b$ and $c$ .

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12a

4 marks

The amounts of wheat, soybeans and sugar produced by three different farms in a given week, along with the respective total revenues for each farm, are shown in the table below.

	Wheat, kg	Soybeans, kg	Sugar, kg	Revenue, $
Farm A	820	532	535	835.54
Farm B	1210	641	274	948.75
Farm C	922	211	503	716.11

Let $x, y$ and $z$ represent the prices, in $/kg, for wheat, soybeans and sugar respectively.

(i)

Write down a system of linear equations that represents the information in the table above.

(ii)

Solve the system of linear equations using matrices.

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12b

3 marks

In the same week, Farm D produced a fifth of the amount of wheat as Farm A, a quarter of the amount of soybeans as Farm B, and half the amount of sugar as Farm C.

Calculate the revenue made by Farm D from selling these crops. Give your answer correct to two decimal places.

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13a

6 marks

Grace has decided that she wants to invest $10 000 split between three companies: company A, company B, and company C. She creates three different portfolio options based on risk levels, and calculates what each option’s value would be today if the identical amounts had been invested one year ago.

	Company A	Company B	Company C	Value
Safe	$1500	$8000	$500	$10,620.00
Middle	$2000	$6750	$1250	$10,827.50
Risky	$2500	$2500	$5000	$11,725.00

Use a matrix method to find the annual percentage return (i.e., the percentage increase or decrease of an investment in the company) for the previous year for each company.

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13b

2 marks

Grace hears some good news about the growth of company A before she invests her money, and so she decides to put 78% of it into company A and split the rest evenly between company B and company C.

Compared with the previous year, the annual percentage return for company A for the coming year is expected to increase by 26 percentage points (so if the previous year’s return was $x$ %, then the return is expected to be $(x + 26)$ % for the coming year). For company B the return is expected to remain the same, while for company C it is expected to decrease by 8 percentage points.

Find the expected value of Grace’s investment at the end of the coming year.

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2 marks

Consider the following matrices:

$A = (\begin{matrix} x & 7 \\ - 4 & 3 \end{matrix}) B = (\begin{matrix} 2 & y \\ 4 & 8 \end{matrix})$

$C = (\begin{matrix} 1 & a \\ 4 & 8 \end{matrix}) D = (\begin{matrix} b & 2 \\ - 9 & - 10 \end{matrix}) E = (\begin{matrix} - 34 & 62 \\ - 66 & - 52 \end{matrix})$

$M = (\begin{matrix} 3 & 2 c \\ - c & 6 \end{matrix}) N = (\begin{matrix} 10 & d + 24 \\ d & 22 \end{matrix})$

where $a, b, c, d, x, y \in ℝ$ are constants.

Given that $A + q B = z I$ , find the values of x, y, z, and q.

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3 marks

Given that $\frac{1}{2} E = r C + s D$ find the values of a, b, r, and s.

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3 marks

Given that $e M - f N = I$ find the values of c, d, e, and f.

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3 marks

Consider the two matrices

$A = (\begin{matrix} 3 & 2 \\ 2 & 0 \end{matrix})$

$B = (\begin{matrix} 13 & 6 \\ 6 & q \end{matrix})$

where $q \in ℝ$ is a constant.

Given that $A$ and $B$ are commutative, find the value of $q$ .

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3 marks

Consider the matrix

$M = (\begin{matrix} 3 & 0 \\ 0 & - 2 \end{matrix})$

Find an expression for $M^{k}$ .

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8 marks

Consider the matrices:

$A = (\begin{matrix} 2 & a \\ 3 & 4 \end{matrix}) B = (\begin{matrix} 1 & 3 & 4 \\ 3 & - 2 & b \end{matrix}) C = (\begin{matrix} 1 & 2 & 3 \\ 4 & c & 5 \\ - 5 & - 3 & - 2 \end{matrix})$

$D = (\begin{matrix} d & 1 \\ 2 & 3 \end{matrix}) E = (\begin{matrix} 3 & 2 & - 1 \\ e & - 2 & 4 \\ 1 & - 3 & 2 \end{matrix}) F = (\begin{matrix} 2 & f \\ - 2 & - 1 \\ 3 & 2 \end{matrix})$

where $a, b, c, d e, f \in ℝ$ are constants.

Find the following products in terms of the appropriate constants. If it is not possible to do so, explain why.

(i)

A B

(ii)

C A

(iii)

E B

(iv)

B E

(v)

E A + F

(vi)

F (C - E)

(vii)

(C - E) F

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5 marks

Consider the following matrices:

$M = (\begin{matrix} a & b \\ c & d \end{matrix}) N = (\begin{matrix} e & f \\ g & h \end{matrix}) P = (\begin{matrix} i & j \\ k & l \end{matrix})$

Show that $(M N) P = M (N P)$ and state the name of this property.

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2 marks

For each of the following matrices,

(i) find the values of $x$ for which $M^{- 1}$ does not exist, and

(ii) for the cases where $M^{- 1}$ does exist, find $M^{- 1}$ in terms of $x$ .

$M = (\begin{matrix} x & - 2 \\ 1 & 3 \end{matrix})$

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2 marks

$M = (\begin{matrix} x - 2 & 3 \\ 4 & 2 x \end{matrix})$

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2 marks

$M = (\begin{matrix} x^{2} & x - 1 \\ 3 x & 5 \end{matrix})$

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4 marks

A message is encoded using a matrix. Letters in the message are represented by numbers as given in the table below.

A	B	C	D	E	F	G	H	I	J	K	L	M
1	2	3	4	5	6	7	8	9	10	11	12	13

N	O	P	Q	R	S	T	U	V	W	X	Y	Z
14	15	16	17	18	19	20	21	22	23	24	25	26

Messages are encoded by splitting the message into pairs of letters and writing in a $2 \times n$ matrix. For example, “encode” becomes $(\begin{matrix} E & C & D \\ N & O & E \end{matrix})$ or $(\begin{matrix} 5 & 3 & 4 \\ 14 & 15 & 5 \end{matrix})$ . Then the message is multiplied, on the left, by an encryption matrix.

Here is a message which has been encoded using the encryption matrix $(\begin{matrix} 3 & 2 \\ - 1 & 4 \end{matrix})$ :

$(\begin{matrix} 41 & 96 & 33 & 53 & 96 & 46 \\ - 9 & 52 & 3 & 71 & 66 & 8 \end{matrix})$

Decode the message.

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1 mark

A spy (whose name cannot be given) notices that if the message has an odd number of letters, it will not completely fill a $2 \times n$ matrix. The spy suggests putting the message into a $3 \times n$ matrix if the number of letters in the message is a multiple of three.

Assuming the same encryption matrix is used as was used for the above message, explain the problem with the spy’s suggestion.

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2 marks

Using the properties of matrices, explain why the following misconceptions are incorrect.

“I know that $(a + b) (a - b) = a^{2} - b^{2}$ , therefore $(A + B) (A - B) = A^{2} - B^{2}$ must also be true if $A$ and $B$ are matrices.”

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2 marks

“I know that ${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$ , therefore ${(A + B)}^{2} = A^{2} + 2 A B + B^{2}$ must also be true if $A$ and $B$ are matrices.”

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5 marks

In this question $A$ and $B$ are $2 \times 2$ matrices, $k, p, q \in ℝ$ are constants, and $n$ is a positive integer.

(i)

Show that

(p A) (q B) = p q (A B)

(ii)

Hence show that

{(k A)}^{n} = k^{n} A^{n}

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2 marks

A student claims it is always true that $(A B)'' = A'' B''$ .

Either explain why the student’s claim is always true, or else show that it is not always true by providing a counterexample.

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10a

3 marks

Consider the following matrices:

$A = (\begin{matrix} 5 & p \\ p & 7 \end{matrix}) B = (\begin{matrix} 11 & q \\ q & 3 \end{matrix})$

where $p, q \in ℝ$ are constants, $q > 0 .$

Given that $A$ and $B$ are commutative, find $q$ in terms of $p .$

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10b

2 marks

Given that the determinant of $A$ is 26, find $p$ and $q$ .

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11a

3 marks

A couple are planning their wedding reception and wish to buy a bow for every chair, with a matching tablecloth for each table.

There are a total of 120 chairs and 15 tables, all of which need bows and tablecloths respectively.

Company A charges £1.03 per chair bow and £14 per tablecloth.

Company B charges £0.85 per chair bow and £16 per tablecloth.

By setting up one matrix equation that includes both companies, compare the overall prices that would be charged by the two companies.

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11b

3 marks

Married friends of the couple recommend Company C, whom they used for their wedding. The friends can remember that they paid £13 per tablecloth, but cannot remember the price per chair bow.

Set up and solve a matrix equation to find the maximum price per chair bow that Company C could charge so as still to be cheaper overall than companies A and B.

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12a

4 marks

Veterinarians, veterinary nurses, and animal care assistants are paid fixed salaries according to an industry standard. The totals of the annual payrolls for three veterinary practices that pay according to the industry standard are summarised in the table below.

Practice	Veterinarians	Veterinary Nurses	Animal Care Assistants	Total Salary Spend
Aspen Road Vets	3	5	2	$ 294000
Broadoak Way Vets	2	2	1	$ 158000
Cats n Dogs Vets	7	10	4	$ 634000

Using matrices, set up and solve a system of equations to find the fixed salaries that are paid for each of the three roles.

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12b

2 marks

Vicky is setting up a new veterinary practice, and to help recruit staff she is planning to pay 5% above the industry standard for all job roles. She uses the following matrix multiplication to help find the total cost of her staffing, where $p, q$ and $r$ represent the salaries for a veterinarian, a veterinary nurse, and an animal care assistant respectively:

$a \times (\begin{matrix} 3 & 4 & 2 \end{matrix}) \times (\begin{matrix} p \\ q \\ r \end{matrix})$ = Total salary spend in thousands

(i)

Write down the value of the constant

a

that Vicky should use

(ii)

Interpret the meaning of the element ‘4’ in the row matrix.

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13a

4 marks

Consider a curve with equation $y = a x^{3} + b x^{2} + c x - 8$ , where a, b, and c are real constants. The graph passes through the points $A (2, 116), B (- 4, - 712)$ and $C (3, 394) .$

(i)

Use a matrix method to find the values of

a, b

and

c

(ii)

Hence sketch the curve.

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13b

2 marks

Consider a second curve with equation $a x^{5} + b x^{4} + c x^{3} + d x^{2} + e x + f$ , where a, b, c, d, e and f are real constants with $a \neq 0 .$

By considering the method used to solve part (a), suggest the number of coordinates that would need to be known to determine the values of all the constants in the equation of the second curve.

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4 marks

Consider the following matrices:

$A = (\begin{matrix} a & 4 & 5 \\ 1 & 2 & 6 b \end{matrix}) B = (\begin{matrix} 1 & 3 \end{matrix}) C = (\begin{matrix} - 2 \\ - 1 \\ 0 \end{matrix})$

The matrix product $(M N) P$ is calculated, where M, N, and P can each be any one of the matrices A, B or C.

Find the possible dimensions of the resulting products.

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2 marks

The matrix sum M + N + P is calculated, where M, N, and P can each be any one of the matrices A, B or C.

Find the possible sums that could result from such an addition.

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5 marks

It is given that $(C B) A = t (\begin{matrix} 2 & 4 & 9.2 \\ 1 & 2 & 4.6 \\ 0 & 0 & 0 \end{matrix}) .$

Find the values of a, b, and t.

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7 marks

In this question A, B, and C are arbitrary square matrices.

Prove the following matrix results, stating any necessary assumptions:

(i)

A B = C

then

B = A^{- 1} C

(ii)

{(A B)}^{- 1} = B^{- 1} A^{- 1}

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2 marks

Using the result from part (a)(ii), simplify ${(A^{- 1} B)}^{- 1} A^{- 1}$ .

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5 marks

Consider the matrices:

$M = (\begin{matrix} 3 & 2 \\ 2 & 0 \end{matrix})$

$N = (\begin{matrix} a & b \\ c & d \end{matrix})$

where $a, b, c, d \in ℝ$ are constants.

Given that M and N are commutative, find an expression for N in terms of b and d only.

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4 marks

Given that the inverse matrix ${(M N)}^{- 1}$ exists

(i)

determine a relationship that the constants

a, b, c

and

d

must satisfy

(ii)

find

{(M N)}^{- 1}

in terms of

a, b, c

and

d

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6 marks

Given that

$A = (\begin{matrix} 2 & 8 \\ 2 & - 4 \end{matrix})$

$A^{- 1} B A = (\begin{matrix} 2 & 0 \\ 5 & 6 \end{matrix})$

$A B C = 3 I$

find matrices B and C.

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3 marks

For any $2 \times 2$ matrices M, A or B:

Prove that $\det (k M) = k^{2} \det M$ , where $k$ is a real constant.

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5 marks

Prove that $\det (A B) = \det A \times \det B$ .

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7 marks

Consider the matrix

$M = (\begin{matrix} a & 0 \\ c & - a \end{matrix})$

where $a, c \in ℝ$ . Find expressions for $M^{2 k}$ and $M^{2 k + 1}$ where k is a positive integer.

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4 marks

Consider the matrix

$A = (\begin{matrix} i & 0 \\ 3 & i \end{matrix})$

where $i = \sqrt{- 1}$ .

(i)

Find

A^{2}, A^{3}, A^{4}

and

A^{5}

(ii)

Hence determine

A^{15}

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3 marks

Find the general term for $A''$ where $n$ is a positive integer.

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7 marks

Consider the $2 \times 2$ matrix

$M = (\begin{matrix} a & b \\ c & d \end{matrix})$

Use algebra to find the requirements that must be satisfied by $a, b, c$ and $d$ in order for $M^{2} = (\begin{matrix} a^{2} & b^{2} \\ c^{2} & d^{2} \end{matrix})$ to be true.

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5 marks

A professional Football team are looking to buy new players. Their scouts have returned a shortlist containing 23 English, 17 German, 18 Spanish, and 8 Italian players.

The shortlisted players are in the following proportions for each playing position:

11% are goalkeepers
29% are wingers
39% are defenders
21% are strikers

A given player only plays in one of the listed positions.

(i)

Write a column matrix, N, representing the numbers of players from each country, and a row matrix, P, containing the proportions of players in each position.

(ii)

Hence find the total number of players in each position on the shortlist.

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1 mark

Explain why it would be incorrect in general to say that the elements of the matrix NP represent the numbers of players in each position by nationality. For example to say that (NP)_1,1 (the entry in the first row and first column of matrix NP ) might represent the number of English goalkeepers.

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10a

6 marks

A ball is thrown vertically downwards from the top of a cliff, and its position is tracked from when it is first thrown until it hits the ground (at which point it may be assumed that the ball comes instantaneously to rest).

The height of the ball above the ground after $t$ seconds is modelled by the equation $s (t) = a t^{2} + b t + c$ , where a, b and c are real constants and the height s is measured in metres. After 1 second, the ball is 175.4 m above the ground; after 5 seconds, its height is 105 m; and after 6 seconds it is 74.4 m.

Set up and solve a matrix equation to find

(i)

the height of the cliff

(ii)

the time taken for the ball to reach the ground.

How did you do?

10b

2 marks

Another experiment studies the motion of another object that only moves in one dimension. A quartic equation of the form $s = a t^{4} + b t^{3} + c t^{2} + d t + e$ is used to model the displacement of the object, where $a, b, c, d$ and $e$ are all real constants with $a \neq 0 .$

(i)

State the number of measurements of the object’s displacement at different times that would need to be taken, in order to find the explicit values of the constants

a, b, c, d

and

e

(ii)

State the dimensions of the matrices involved in forming a matrix equation, as in part (a), to determine the constants for the quartic model.

How did you do?

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