Scholarship Mock Test 3 - Geometry, Algebra, Logarithms & 3D Geometry Practice

Q: What topics are covered in Scholarship Mock Test 3?

It covers geometry, counting & functions, algebra, logarithms, trigonometry, and 3D geometry with proof and applied problems.

Q: What types of questions appear in the test?

Questions include area calculations from geometric diagrams, password counting with constraints, function optimization for maximum income, equation solving, trigonometric equation solving, pyramid volume computation, proof about an annulus width, and alloy ratio mixing.

Q: Is a calculator allowed for these questions?

No calculator is allowed, and workings must be shown for every question.

Aug 25, 2026

Uploaded by Levi

Scholarship Mock Test 3 provides a mixed set of exam-style problems across geometry, counting and functions, algebra and logarithms, trigonometry, and 3D geometry with proof and applied problem solving. It includes tasks such as finding an area in a circle-and-rectangle configuration, counting passwords with wait-time constraints, optimizing an advertising-based income function, solving logarithmic and trigonometric equations, computing a pyramid volume from a cube structure, and ratio mixing for alloys. No calculator is required and full working is expected.

LanguageEnglish

CategoryExam

FormatPDF

Pages5

Reading Time13 min

AccessFree to download

Part A Geometry, Counting & Functions

Question 1

Question 2

Question 3

Part B Algebra, Logarithms & Trigonometry

Question 4

Question 5

Question 6

Part C 3D Geometry, Proof & Applied Problems

Question 7

Question 8

Question 9

Question 10

PART A Geometry, Counting & Functions | 30 marks 1. 10 marks In 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. Find the total area of the red region. 20 2. 10 marks A 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. a) How many different possible passwords are there? b) 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. 3. 10 marks A company’s monthly income 􀪨(􀫐) , in thousands of dollars, is modeled by 􀪨(􀫐) = −2􀫐2 + 20􀫐 + 100, where 􀫐 is the amount of money spent on advertising, in thousands of dollars. a) Find the maximum possible monthly income. b) Determine how much the company should spend on advertising to achieve this maximum income. PART B Algebra, Logarithms & Trigonometry | 30 marks 4. 10 marks Solve the equation: 1 1 1 ~~ ~~ − ~~ ~~ = ~~ ~~ 􀫐3 + 2 􀫐3 + 3 12 . 5. 10 marks Let 􀪺 > 0 , 􀪺 ≠ 1 . If log􀪺 27 = 􀪻, find log √3 √6 􀪺 in terms of 􀪻 . 6. 10 marks Solve the trigonometric equation for 0 ≤ 􀫐 < 2􀵝 : sin 3􀫐 + sin 5􀫐 = sin 4􀫐 . PART C 3D Geometry, Proof & Applied Problems | 40 marks 7. 10 marks In 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. Find the volume of the created pyramid. 8. 10 marks Seven 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. Prove that the width of the annulus is equal to the radius of one small disk. Fig. 1 9. 10 marks One alloy contains two metals A and B in the ratio 1 ∶ 2 . Another alloy contains the same metals in the ratio 2 ∶ 3. How many parts of each alloy must be mixed to obtain a third alloy containing metals A and B in the ratio 17 ∶ 27? 10. 10 marks Several students decided to buy a speaker costing between 170 and 195 dollars. At the last moment, two students backed out, so each remaining student had to pay 1 dollar more than originally planned. How much did the speaker cost? No calculator. Show workings for every question.

FAQs

What topics are covered in Scholarship Mock Test 3?

It covers geometry, counting & functions, algebra, logarithms, trigonometry, and 3D geometry with proof and applied problems.

What types of questions appear in the test?

Questions include area calculations from geometric diagrams, password counting with constraints, function optimization for maximum income, equation solving, trigonometric equation solving, pyramid volume computation, proof about an annulus width, and alloy ratio mixing.

Is a calculator allowed for these questions?

No calculator is allowed, and workings must be shown for every question.

Aug 25, 2026

Uploaded by Levi