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Answer all 10 questions.  \n2. Show your reasoning clearly. A final answer without clear working may receive no marks.  \n3. Each question is worth 10 marks. Total: 100 marks.  \n4. Do not use a calculator, phone, smartwatch, notes, Google, or any electronic aid.  \n5. Write exact answers where possible. Use decimals only when the question asks for them.  \n6. Manage your time. If one question is taking too long, move on and return later.  \nScholarship-level focus  \nThis paper is designed to test deeper problem solving, algebraic accuracy, proof skills, geometry reasoning, and clean no-calculator execution.  \nThis paper is designed for MathPro students. Do not use a calculator.  \nNo calculator. Show workings for every question.  \nMathPro | Scholarship Mock Test \\#1 Page 2 of 7  \nAnswer Sheet  \nFinal answers only  \nWrite the final result for each question in the boxes below. Full working must still be shown on the question pages or on extra paper.  \nScore  \nQ1 /10  \nQ6 /10  \nQ2 /10  \nQ7 /10  \nQ3 /10  \nQ8 /10  \nQ4 /10  \nQ9 /10  \nQ5 /10  \nQ10 /10  \nTotal score:  \n/100  \nNo calculator. Show workings for every question.  \nMathPro | Scholarship Mock Test \\#1 Page 3 of 7  \nPART A Geometry, Counting & Functions | 30 marks  \n1. 10 marks  \nIn the rectangle below, two equal circles are tangent to each other and to the sides of the rectangle. The red region is the part of the rectangle below the diagonal but outside the circles.  \nFind the total area of the red region.  \n20  \n2. 10 marks  \nA device has a 5-digit password. Each digit can be any number from 0 to 9, but digits cannot be repeated. After every block of 5 unsuccessful attempts, the hacker must wait 15 seconds before making the next attempt. The hacker can test 100 passwords per second while not waiting.  \na) How many different possible passwords are there?  \nb) If the hacker tries every possible password, how long will it take in total? Give your answer in seconds, and then approximately in hours and minutes.  \nNo calculator. Show workings for every question.  \nMathPro | Scholarship Mock Test \\#1 Page 4 of 7  \n3. 10 marks  \nA company’s monthly income 􀪨 (􀫐), in thousands of dollars, is modeled by  \n􀪨 (􀫐) = −2􀫐2 + 20􀫐 + 100,  \nwhere 􀫐 is the amount of money spent on advertising, in thousands of dollars.  \na) Find the maximum possible monthly income.  \nb) Determine how much the company should spend on advertising to achieve this maximum income.  \nNo calculator. Show workings for every question.  \nMathPro | Scholarship Mock Test \\#1 Page 5 of 7  \nPART B Algebra, Logarithms & Trigonometry | 30 marks  \n4. 10 marks  \nSolve the equation:  \n1 1 1  \n~~ ~~ − ~~ ~~ = ~~ ~~  \n􀫐3 + 2 􀫐3 + 3 12 .  \n5. 10 marks  \nLet 􀪺 > 0 , 􀪺 ≠ 1 . If  \nlog􀪺 27 = 􀪻,  \nfind  \nlog √3 √6 􀪺  \nin terms of 􀪻 .  \n6. 10 marks  \nSolve the trigonometric equation for 0 ≤ 􀫐 \u003C 2􀵝 :  \nsin 3􀫐 + sin 5􀫐 = sin 4􀫐 .  \nNo calculator. Show workings for every question.  \nMathPro | Scholarship Mock Test \\#1 Page 6 of 7  \nPART C 3D Geometry, Proof & Applied Problems | 40 marks  \n7. 10 marks  \nIn a cube, the centre of the lower base is connected to the centres of the four side faces. The four side-face centres are connected in order, forming the base of a pyramid. The height of the cube is 6 units.  \nFind the volume of the created pyramid.  \n8. 10 marks  \nSeven equal touching disks are arranged inside the smaller of two concentric circles, as shown in Fig. 1. The annulus is the region between the two concentric circles. The area of the annulus is equal to the sum of the areas of all seven disks.  \nProve that the width of the annulus is equal to the radius of one ","cbCaimS0jc2427IJ","https://ap.wps.com/l/cbCaimS0jc2427IJ","pdf",40590,"English","# Student Instructions\n# Scholarship-level focus\n# Part A: Geometry, Counting & Functions\n## Questions 1-3\n# Part B: Algebra, Logarithms & Trigonometry\n## Questions 4-6\n# Part C: 3D Geometry, Proof & Applied Problems\n## Questions 7-10","[{\"question\":\"How many questions and marks are included in MathPro Scholarship Mock Test #1?\",\"answer\":\"There are 10 questions worth 10 marks each, for a total of 100 marks.\"},{\"question\":\"What tools are allowed during the mock test?\",\"answer\":\"No calculator or electronic aids are allowed, including phones, smartwatches, notes, or Google.\"},{\"question\":\"What is required for scoring on each question?\",\"answer\":\"A final answer without clear working may receive no marks; students should show workings for every question.\"}]","MathPro - Scholarship Mock Test 1 - Student Instructions | PDF"]