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Question 1

HLPaper 1

Given that M = $\left( {\begin{array}{*{20}{c}} 2&{ - 1} \\ { - 3}&4 \end{array}} \right)$ and that M²– 6M + kI = 0 find k.

Question 2

HLPaper 1

The rate, $A$ , of a chemical reaction at a fixed temperature is related to the concentration oftwo compounds, $B$ and $C$ , by the equation

$A = k{B^x}{C^y}$ , where $x$ , $y$ , $k \in \mathbb{R}$ .

A scientist measures the three variables three times during the reaction and obtains thefollowing values.

Find $x$ , $y$ and $k$ .

Question 3

SLPaper 1

Yejin plans to retire at age 60. She wants to create an annuity fund, which will pay her a monthly allowance of $4000 during her retirement. She wants to save enough money so that the payments last for 30 years. A financial advisor has told her that she can expect to earn 5% interest on her funds, compounded annually.

1.

Calculate the amount Yejin needs to have saved into her annuity fund, in order to meet her retirement goal.

[3]

2.

Yejin has just turned 28 years old. She currently has no retirement savings. She wants to save part of her salary each month into her annuity fund.

Calculate the amount Yejin needs to save each month, to meet her retirement goal.

[3]

Question 4

HLPaper 2

Long term experience shows that if it is sunny on a particular day in Vokram, then the probabilitythat it will be sunny the following day is $0.8$ . If it is not sunny, then the probability that it will besunny the following day is $0.3$ .

The transition matrix $T$ is used to model this information, where $T = (\begin{matrix} 0.8 & 0.3 \\ 0.2 & 0.7 \end{matrix})$ .

The matrix $T$ can be written as a product of three matrices, $P D P^{- 1}$ , where $D$ is adiagonal matrix.

1.

It is sunny today. Find the probability that it will be sunny in three days’ time.

[2]

2.

Find the eigenvalues and eigenvectors of $T$ .

[5]

3.

Write down the matrix $P$ .

[1]

4.

Write down the matrix $D$ .

[1]

5.

Hence find the long-term percentage of sunny days in Vokram.

[4]

Question 5

HLPaper 1

The following Argand diagram shows a circle centre $0$ with a radius of $4$ units.

A set of points, $\{z_{θ}\}$ ,on the Argand plane are defined by the equation

$z_{θ} = \frac{1}{2} θ e^{θ i}$ , where $θ \geq 0$ .

Plot on the Argand diagram the points corresponding to

Consider the case where $|z_{θ}| = 4$ .

1.

$θ = \frac{π}{2}$ .

[1]

2.

$θ = π$ .

[1]

3.

$θ = \frac{3 π}{2}$ .

[1]

4.

Find this value of $θ$ .

[2]

5.

For this value of $θ$ , plot the approximate position of $z_{θ}$ on the Argand diagram.

[2]

Question 6

HLPaper 1

Let A = $\left( {\begin{array}{*{20}{c}} 1&2&3 \\ 2&{ - 1}&2 \\ 3&{ - 3}&2 \end{array}} \right)$ ,D= $\left( {\begin{array}{*{20}{c}} { - 4}&{13}&{ - 7} \\ { - 2}&7&{ - 4} \\ 3&{ - 9}&5 \end{array}} \right)$ , andC= $\left( {\begin{array}{*{20}{c}} 5 \\ 7 \\ {10} \end{array}} \right)$ .

1.

Given matrices A, B, C for which AB = C and det A ≠ 0, express B in terms of A and C.

[2]

2.

Find the matrix DA.

[1]

3.

Find B if AB = C.

[2]

4.

Find the coordinates of the point of intersection of the planes $x + 2y + 3z = 5$ , $2x - y + 2z = 7$ , $3x - 3y + 2z = 10$ .

[2]

Question 7

SLPaper 1

A triangular field $ABC$ is such that $AB = 56 m$ and $BC = 82 m$ , each measured correct to thenearest metre, and the angle at $B$ is equal to $105 °$ , measured correct to the nearest $5 °$ .

Calculate the maximum possible area of the field.

Question 8

SLPaper 1

Give your answers in this question correct to the nearest whole number.

Imon invested $25 000$ Singapore dollars ( $SGD$ ) in a fixed deposit account with a nominalannual interest rate of $3.6 %$ , compounded monthly.

1.

Calculate the value of Imon’s investment after $5$ years.

[3]

2.

At the end of the $5$ years, Imon withdrew $x SGD$ from the fixed deposit account andreinvested this into a super-savings account with a nominal annual interest rate of $5.7 %$ , compounded half-yearly.

The value of the super-savings account increased to $20 000 SGD$ after $18$ months.

Find the value of $x$ .

[3]

Question 9

HLPaper 2

Let A = $\left( {\begin{array}{*{20}{c}} a&b \\ c&0 \end{array}} \right)$ and B = $\left( {\begin{array}{*{20}{c}} 1&0 \\ d&e \end{array}} \right)$ . Giving your answers in terms of $a$ , $b$ , $c$ , $d$ and $e$ ,

1.

write down A + B.

[2]

2.

find AB.

[4]

Question 10

HLPaper 2

Let $\gamma = \frac{{1 + {\text{i}}\sqrt 3 }}{2}$ .

The matrix A is defined by A = $\left( {\begin{array}{*{20}{c}} \gamma &1 \\ 0&{\frac{1}{\gamma }} \end{array}} \right)$ .

Deduce that

1.

Show that ${\gamma ^2} = \gamma - 1$ .

[2]

2.

Hence find the value of ${\left( {1 - \gamma } \right)^6}$ .

[4]

3.

A³= –I.

[3]

4.

A^–1= I – A.

[2]

Topic 1 - Number and Algebra

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