Year 10 Extension Mathematics Practice Exam PDF

    -- ----- ----- ----- ----- A  B  C  D  BJI QTR BJI CJO **Year 10 Extension Mathematics**  **Practice Examination Part A: CAS FREE**  **Reading time: 10 minutes for both Parts A and B**  **Writing time: 30 minutes**  **Structure of examinatio...

    -- ----- ----- ----- ----- A  B  C  D  BJI QTR BJI CJO **Year 10 Extension Mathematics**  **Practice Examination Part A: CAS FREE**  **Reading time: 10 minutes for both Parts A and B**  **Writing time: 30 minutes**  **Structure of examination** Number of questions Number of questions to be answered --------------------- ------------------------------------ ---- 5 5 20 +-----------------------------------------------------------------------+ | - Students are permitted to bring into the examination room: pens, | | pencils, highlighters, erasers, sharpeners, rulers. | | | | - For Part A: CAS FREE students are **NOT** permitted to use | | calculators but are **allowed** to use the bound reference or | | lecture pad of notes. | | | | - Students are not permitted to bring into the examination room: | | blank sheets of paper and/or white out liquid/tape. | | | | **Materials supplied** | | | | - - | | | | | | | | - - - - - | +-----------------------------------------------------------------------+ +-----------------------------------------------------------------------+ | **Students are NOT permitted to bring mobile phones or electronic | | devices that are capable of storing, receiving or transmitting | | information or electronic signals, such as recorded music and video | | players, organisers, dictionaries and computerised watches** **and/or | | any other unauthorised electronic devices into the examination | | room.** | | | | **Students must not disclose the contents of the task; to do so will | | be a breach of guidelines and will be dealt with according to school | | policies.** | +-----------------------------------------------------------------------+ +-----------------------------------------------------------------------+ | **Instructions** | | | | Answer **all** questions in the space provided. | | | | Unless otherwise specified an **exact** answer is required to a | | question. | | | | In questions where more than one mark is available, appropriate | | working **must** be shown. | | | | Unless otherwise indicated, the diagrams in this book are **not** | | drawn to scale. | +-----------------------------------------------------------------------+ **Question 1 (4 marks)** Simplify each of the following expressions, **expressing any answers with positive indices only**. a. -- -- b. [\$\\left( \\frac{8}{3} \\right)\^{- 1}\$]{.math.inline} -- -- c. [3*y*^0^]{.math.inline} -- -- d. [\$16\^{\\frac{3}{4}}\$]{.math.inline} -- -- **Question 2 (4 marks)** a. Factorise [*x*^2^ − 2*x* − 6]{.math.inline} by first completing the square 2 marks -- -- b. Solve the equation [*x*^2^ − 2*x* − 6 = 0]{.math.inline} 1 mark -- -- c. Determine the turning point for the parabola [*y* = *x*^2^ − 2*x* − 6]{.math.inline} 1 mark -- -- d. **Question 3 (3 marks)** Solve the equation [2*x*^3^ − *x*^2^ − 7*x* + 6 = 0]{.math.inline} for [*x*]{.math.inline}. -- -- **Question 4 (7 marks)** Solve for [*x*]{.math.inline} in the following: a. [\$3\^{x + 1} = 9\\sqrt{3}\$]{.math.inline}. -- -- b. [4 − log~5~(*x*) = log~5~(15)]{.math.inline} -- -- c. ***\ ***[\$9\^{x + 1} = \\frac{1}{27\\sqrt{3}}\$]{.math.inline}. -- -- ***Question 5 (2 marks)*** State the exact values for the following expressions: a. [\$\\sin\\left( \\frac{3\\pi}{2} \\right)\$]{.math.inline} 1 mark -- -- b. [cos (180^∘^ + *θ*)]{.math.inline}, where [0 ≤ *θ* ≤ 90^∘^]{.math.inline} and [\$\\cos\\left( \\theta \\right) = \\frac{2}{3}\$]{.math.inline} 1 mark -- -- **End of Part A** **\ ** **Year 10 Extension Mathematics** **PRACTICE Examination Part B: CAS ACTIVE** **Reading time: 10 minutes (for both Parts A and B)** **Writing time: 30 minutes** **Structure of examination** Section  Number of questions  Number of questions to be answered  Number of marks  ---------- ---------------------- ------------------------------------- ------------------ A  5  5  5 B 3  3  15 Total 20 +-----------------------------------------------------------------------+ | - Students are permitted to bring into the examination room: pens, | | pencils, highlighters, erasers, sharpeners, rulers, one bound | | reference, one approved CAS calculator and, if desired, one | | scientific calculator. | | | | - Students are not permitted to bring into the examination room: | | blank sheets of paper and/or white out liquid/tape. | | | | **Materials supplied** | | | | - Question and answer book of 9 pages. | | | | - **Working space is provided throughout the book.** | | | | **Instructions** | | | | - Write your name in the space provided above on this page. | | | | - Circle your teacher's initials above. | | | | - Calculators of any type must **not** be used during reading time. | | | | - No dictionaries are allowed. | | | | - All responses must be written in English. | +-----------------------------------------------------------------------+ +-----------------------------------------------------------------------+ | **Students are NOT permitted to bring mobile phones or electronic | | devices that are capable of storing, receiving or transmitting | | information or electronic signals, such as recorded music and video | | players, organisers, dictionaries and computerised watches** **and/or | | any other unauthorised electronic devices into the examination | | room.** | | | | **Students must not disclose the contents of the task; to do so will | | be a breach of guidelines and will be dealt with according to school | | policies.** | +-----------------------------------------------------------------------+ **\ ** **SECTION A** +-----------------------------------------------------------------------+ | **Instructions for Section A** | | | | Answer **all** questions in pencil on the **answer sheet provided on | | page 4.** | | | | Choose the response that is **correct** for the question. | | | | A correct answer scores 1, an incorrect answer scores 0. | | | | Marks will **not** be deducted for incorrect answers. | | | | No marks will be given if more than one answer is selected for any | | question. | | | | Only the answers on the **answer sheet** will be marked. | +-----------------------------------------------------------------------+ **Question 1** The function with rule [*f*(*x*) = 5sin (*αx*)]{.math.inline} has a period of [\$\\frac{5\\pi}{2}\$]{.math.inline}. The value of [*α*]{.math.inline} is: A. [\$\\frac{1}{5}\$]{.math.inline} B. [\$\\frac{4}{5}\$]{.math.inline} C. 4 D. 5 **Question 2** The solution(s) to [8*x*^2^ − 14*x* + 3 = 0]{.math.inline} are: ------------------------------------------------------------------------------ **A.** \ [\$\$x = \\frac{1}{8},\\ x = - \\frac{1}{3}\$\$]{.math.display}\ -------- --------------------------------------------------------------------- **B.** \ [\$\$x = \\frac{3}{4},\\ x = - \\frac{1}{2}\$\$]{.math.display}\ **C.** \ [\$\$x = \\frac{1}{4},x = \\frac{3}{2}\$\$]{.math.display}\ **D.** \ [\$\$x = - \\frac{1}{2},\\ x = - \\frac{3}{8}\$\$]{.math.display}\ ------------------------------------------------------------------------------ **Question 3** Which of the following expressions are equivalent to [\$4\\sqrt{5}?\$]{.math.inline} ------------------------------------------------- **A.** \ [\$\$\\sqrt{100}\$\$]{.math.display}\ -------- ---------------------------------------- **B.** \ [\$\$\\sqrt{80}\$\$]{.math.display}\ **C.** \ [\$\$2\\sqrt{10}\$\$]{.math.display}\ **D.** \ [\$\$\\sqrt{40}\$\$]{.math.display}\ ------------------------------------------------- **\ ** **Question 4** Triangle ABC is constructed according to the diagram below: A triangle with a number of angles Description automatically generated with medium confidence What are the two possible angles for [∠*A*]{.math.inline} (in degrees), rounded to two decimal places? **A.** [33.89^∘^ ]{.math.inline}or[ 146.11^∘^]{.math.inline} -------- --------------------------------------------------------- **B.** [126.59^∘^ ]{.math.inline}or[ 53.41^∘^]{.math.inline} **C.** [108^∘^ ]{.math.inline}or[ 30^∘^]{.math.inline} **D.** [8.10^∘^ ]{.math.inline}or[ 171.90^∘^]{.math.inline} **Question 5** Which of the following could be the graph of [*y* =  − 2^ − *x*^ + 2 ]{.math.inline}? +-----------------------------------+-----------------------------------+ | **A.** | **B.** | | | | |  | | +===================================+===================================+ | **C.** | **D.** | | | | |  | Graph Preview | +-----------------------------------+-----------------------------------+ **Multiple Choice Answer Sheet** *This answer sheet must not be detached from your examination paper.* +-----------------------------------------------------------------------+ | **INSTRUCTIONS FOR MULTIPLE CHOICE ANSWER SHEET** | +=======================================================================+ | Use a **PENCIL** for **ALL** entries. For each question, shade the | | box which indicates your answer. | | | | Marks will **NOT** be deducted for incorrect answers. | | | | **NO MARK** will be given if more than **one** answer is completed | | for any question. | | | | If you make a mistake, **ERASE** the incorrect answer - **DO NOT** | | cross it out. | +-----------------------------------------------------------------------+ ANSWER ONE PER LINE ------- --------------------- --- -- --- -- --- -- --- -- -- -- **1** A B C D **2** A B C D **3** A B C D **4** A B C D **5** A B C D **SECTION B** +-----------------------------------------------------------------------+ | **Instructions for Section B** | | | | Answer **all** questions in the space provided. | | | | Unless otherwise specified an **exact** answer is required to a | | question. | | | | Students that draw a happy cat on the front cover will get one bonus | | mark. All students will lose this mark if this is mentioned aloud. | | | | In questions where more than one mark is available, appropriate | | working **must** be shown. | | | | Unless otherwise indicated, the diagrams in this book are not drawn | | to scale. | +-----------------------------------------------------------------------+ **Question 1 (4 marks)** a. Solve the equation [8 = 10 + 4cos (*x*)]{.math.inline} for [0 ≤ *x* ≤ 2*π*]{.math.inline} *2 marks* -- -- b. Sketch the graph of the rule [*y* = 10 + 4cos (*x*)]{.math.inline} for [0 ≤ *x* ≤ 2*π*]{.math.inline}. Label any endpoints, axial intercepts and turning points. 2 marks  **Question 2 (5 marks)** The Siamese Fireback and Tufted Puffin are two different kind of bird species whose populations are tracked by researchers. [*S*]{.math.inline} is used to track the Siamese Fireback's population.\ Initially, researchers count an initial population of 800 and notice that the population grows at a rate of 4% per year. [*T*]{.math.inline} is used to track the Tufted Puffin's population.\ Initially, researchers count an initial population of 300 and notice that the population grows at a rate of 15% per year. a. Write two equations to represent the populations of each bird after [*t*]{.math.inline} years. 2 marks -- -- b. How many years does it take for the Tufted Puffin's population to be greater than that of the Siamese Fireback? Give your answer to the nearest whole number. 2 marks -- -- A third bird, the pūteketeke, has a population [*P*]{.math.inline} that is modelled by the following equation: \ [*P* = *P*~0~ × *k*^*t*^]{.math.display}\ - After 2 years, the population of pūteketeke is 1000. - After 5 years, the population of pūteketeke is 1850. c. State the value of [*k*]{.math.inline}, correct to two decimal places, and [*P*~0~]{.math.inline} to the nearest whole number. 1 mark -- -- **Question 3 (6 marks)** A tunnel is mapped from above whose path is shown in the graph below: Graph Preview The tunnel takes the shape of a cubic function where [*y*]{.math.inline} is the distance north (in km) of [(0, 0)]{.math.inline} and [*x*]{.math.inline} is the distance east (in km) of [(0, 0).]{.math.inline} a. State the [*x*]{.math.inline}-intercepts of the cubic function 1 marks -- -- b. Given that the cubic function passes through the coordinate [\$\\left( 0,\\ \\frac{9}{5} \\right)\$]{.math.inline}, state the equation for the cubic in the form: [*y* = *a*(*x*−*m*)(*x*−*n*)^2^]{.math.inline} 2 marks -- -- An access tunnel whose position can be modelled by the equation [\$T = - \\frac{1}{2}x - 2\$]{.math.inline} is created where [*T*]{.math.inline} is the access tunnel's distance north of [(0, 0)]{.math.inline}. c. Find the point where the two tunnels meet. Give your answer to two decimal places. 3 marks -- -- **End of Part B and Exam**