Grade 10 Double Math Exam Review January 2025 PDF

Math 10 Exam Review - January 2025 1- Linear Systems Summary Classifying Systems How many solutions could there be? How do we find out? Examples: 𝑦 = 2𝑥 + 5...

Math 10 Exam Review - January 2025 1- Linear Systems Summary Classifying Systems How many solutions could there be? How do we find out? Examples: 𝑦 = 2𝑥 + 5 𝑦 = 2𝑥 + 1 𝑦 = 2𝑥 + 1 𝑦 = 3𝑥 − 10 𝑦 = 2𝑥 + 5 6𝑥 − 3𝑦 + 3 = 0 Solving By Graphing How do you graph lines? How do you find the POI? What is important to remember? 𝑦 = 2𝑥 − 1 1 𝑦= 3𝑥+6 2𝑥 − 𝑦 = 2 2𝑥 + 𝑦 = 6 (rearrange)  Solving by Substitution How do we solve by substitution? When does this method make the most sense? Isolate, substitute, solve, substitute, solve again Example - Solve by substitution 𝑦 =− 3𝑥 + 5 5𝑥 − 4𝑦 =− 3 Solving by Elimination How do we solve by elimination? When does this method make the most sense? What do you need to remember? Match coefficients, add/subtract, solve, substitute, solve again Example - Solve by elimination 3𝑥 − 4𝑦 =− 5 5𝑥 − 2𝑦 =− 6  Applications (concentrations, interest) Ms. Johannsen has $3300 in her safe consisting of $50 bills and $100 bills! If there are 42 bills in total, how many of each type of bill does she have? Solve algebraically. Ms. Gardner invested $9000 in a combination of low risk bonds yielding 3% per year and high risk bonds yielding 8% per year. If the interest after one year was $420, calculate the amount invested in each bond type. Solve algebraically. No Calculator Problems (Paper 1) 1. Find the solutions to the following systems by graphing. a) b) c) 2. Classify the following systems and state the number of solutions. The table below will help you justify your work. Linear System Slopes y-intercepts Number of Solutions a) b) c) d) e) f) 3. Find the solutions to the following systems by substitution or elimination. a) b) c) d) Linear Systems Word Problems - no calculator 4. Sally finds $3.25 in nickels and dimes in her piggy bank. She has eight more nickels than dimes. How many of each coin does she have? 5. A group of kids went to Mayfair and bought 10 bags of treats at the concession stand, which cost $54. If the donuts cost $5 a bag and the cotton candy $6 a bag, how many bags of donuts did the kids buy? Solve algebraically. 6. The table shows the flying times taken by a small plane to make a round trip. Part of the trip was down wind and the other part was directly into the wind. If x represents the speed of the plane in kilometers per hour and y represents the speed of the wind in kilometers per hour, find the speed of the wind and the speed of the plane. Speed Time Distance With the Wind 4h 1600 km Against the Wind 5h 1600 km Total 9h 3200 km Calculator Problems (Paper 2) 1. Find the solutions to the following systems by substitution or elimination. a) b) Linear Systems Word Problems - calculator allowed 2. Kaya’s ski lodge can provide accommodation for 18 people. The profit for the lodge is $21.50 for each child and $35.50 for each adult per day. How many adults and children were at the lodge if the profit was $555? 3. Melissa took a Sunday drive with her grandmother into the country. Going into the country, they averaged 60 km/h. They returned over the same road and averaged 48 km/h. If the round-trip took 3 h, how many kilometers did they travel in all? Show the steps of your solution. 4. Adam invested $24 000. He invested part of it at 8% interest per year and the rest at 10% interest per year. His total interest for the year was $2300. How much was invested at 10%? 5. Hydrochloric acid is the solution of hydrogen chloride (HCl) in water. It is highly corrosive and therefore has major industrial use. There are two types of hydrochloric acid – a 25% solution and a 50% solution. How many millilitres of each solution must be mixed to make 800mL of a 35% HCl solution? 6. A tour group traveled along a river in the Amazon to spot wildlife. They traveled down the 6km river in one hour and back in one and a half hours. Find the speed of the boat and the current.  2 - Analytic Geometry Summary Finding the equation of a line Working with Perpendicular Lines 𝑦 = 𝑚𝑥 + 𝑏 or 𝐴𝑥 + 𝐵𝑦 = 𝐶 What are perpendicular slopes? Find the equation of the line that passes Find the equation of the line that passes through (3, -2) and is through (3, 4) and (-2, -6). perpendicular to the line. Midpoint between two points Distance between two points Formula: Formula: Find the midpoint between A(-3, 4) and Find the distance between A(-3, 4) and B(-8, 10) B(-8, 10) Classifying Triangles How would you determine if a triangle is… Isosceles: Equilateral: Scalene: Right-angled: Classify the triangle with vertices A(1, 4), B(-5, -10) and C(5, -6).  Classifying Quadrilaterals How would you determine if a quadrilateral is a…. Square? Kite? Rectangle? Parallelogram? Rhombus? Trapezoid? Quadrilateral TUNE has vertices T(0, 10), U(4, 2), N(−2, −1), and E(−6, 7). Classify the quadrilateral, showing full justification. Perpendicular Bisector Definition/Properties: What steps do I take to find it? Find the equation of the perpendicular bisector to the line segment with endpoints (0, 5) and (6, 7). Draw as you go to confirm your solution.  Triangles: Median Definition/Properties: What steps do I take to find it? Find the equation of the median of triangle ABC (A(-2, 5), B(6, 9), C(7, -3)) that passes through vertex C. Triangles: Altitude Definition/Properties: What steps do I take to find it? Given the triangle ABC with A(-2, 5), B(9, 2) and C(3, -7), determine the equation of the altitude from A onto BC.  Shortest distance from a point to a line The shortest distance from a point to a line is always the perpendicular path! What steps do I take to find it? Find the shortest distance from the point A(-6, 0) to the line Did not do this ---- Circles Formula if the centre is at the origin: Formula if the centre is not at the origin: Determine the equation of a circle with a centre at that passes through the point. State its domain and range.  Application Problems In her amazing high wire act, Laura is holding a long pole at its middle to help her balance. If she is positioned at (1, 1) and one end of her balancing pole is positioned at coordinates (-2, -3), find the coordinates of the other end of the balancing pole and its length. Spencer, Gillian and Emily run a small delivery company. For their business, they use licensed two-way radios with a 20-km range. Spencer is at their office, which they have marked as the origin on their map of the town. The grid lines on the map are scaled at 1 km apart. Gillian is dropping off a package at (-8, 16) while Emily is making a pick-up at (4, 20). Are Gillian and Emily both within range of the radio at the office? Explain your answer. In a rural neighbourhood, a gas line runs between two homes that are located at points and. A new home, located at , is to be connected to the existing gas line (measurements are in metres.) Determine the length of pipe that the gas company will need to connect the new home. No Calculator Problems (Paper 1) 1. Find the midpoint between the given points. a) and b) and c) and Calculator Problems (Paper 2) 1. Find the distance between the given points. a) and b) and c) and 2. Given the points , and , find: a) The length of the median from C to AB b) The equation of the median from C to AB c) The equation of the altitude from C to AB 3. Given the points , and , find: a) The length of the perpendicular bisector from D to EF b) The equation of the perpendicular bisector from D to EF 4. Find the shortest distance from the given point and the line. a) origin to the line b) to the line c) to the line 5. Classify the triangle with vertices at , and. 6. The pilot of a training mission calculates that there is just enough fuel to fly 15 km. The plane’s location is at and the closest airport is at. Does the plane have enough fuel to reach the airport or should the pilot return to its base at ? Give reasons for your answer. 7. A cargo ship is heading in a straight path that can be described as. If another ship is currently at the position (4, -1) and wants to rendezvous with that cargo ship, at what point on the straight path should they meet to minimize the distance the second ship has to travel? What is that distance? (measurements are in kilometres) 8. Determine the area of the parallelogram that has vertices , and.  3 - Trigonometry Summary Pythagorean Theorem Similar Triangles (Classifying & Solving) Formula/Example/things to remember Types of Similarity/Setting up ratios to solve Primary Trig (SOH CAH TOA) Applications (angles of elevation and depression) How to set up/things to remember Sine Law Two versions of the formula: When to use it: In triangle ABC, 𝑎 = 7𝑚, 𝑏 = 5𝑚, ∠𝐴𝐵𝐶 = 40º. Find the missing sides and angles in the triangle.  Cosine Law Two versions of the formula: When to use it: In triangle ABC, 𝑎 = 8𝑚, 𝑏 = 11𝑚, ∠𝐴𝐶𝐵 = 37º. Find the missing sides and angles in the triangle. In triangle ABC, 𝑎 = 8𝑚, 𝑏 = 11𝑚, 𝑐 = 15𝑚. Find the largest angle in the triangle. Area of a Triangle Bearings Formula / where the angle needs to be: How do bearings work? Draw 200º Find the area of ABC with AB = 10cm, BC = 9cm and angle ABC = 44°. Draw 130º.  Applications (double right triangle, bearings) A 5m ladder rests against a building wall and makes an angle of 20° with the wall. What is the distance between the wall and the foot of the ladder to one decimal place? The angle of elevation to the top of the Rialto Tower in Australia is 47° from a point on the road below. 90m closer on the road, the angle of elevation is 60°. What is the height of the Rialto Tower to the nearest metre? Ms. Muir watches a hot air balloon being released into the air at a summer festival. She is standing 75m from the launch point and the angle of elevation to the hot air balloon changes from 25° to 51° as the balloon rises. What is the increase in altitude, to the nearest metre? Ms. Johannsen is planning a triangular course for a sailing regatta. The boats are to travel 4.8km on a bearing of 65°T, followed by 7.3km on bearing of 172°T. What distance are the boats from the starting point? What bearing is required to return to the starting point? No Calculator Problems (Paper 1) Q1. Solve for the unknown. a) b) Calculator Problems (Paper 2) 1. Solve for the unknown. Use appropriate formulas for the triangles given. Trigonometry Word Problems 1. A pole casts a shadow which is 5.6m long. The sun is at an angle of 70° to the ground. Find the height of the pole. 2. Town B is 8km directly North of town A while town C is 10km on a bearing of 120°T from A. Find the distance of town B from town C. 3. The angle of depression from the top of a 150m cliff to a ship in the sea is 5°. How far is the ship from the foot of the cliff? 4. Find the smallest angle in a triangle with sides 5cm, 6cm and 8cm. 5. A triangle has angles of 110°, 40° and 30°. The longest side is 20cm. Calculate the perimeter of the triangle. 6. A vertical radio tower is held in position by a wire which is 48m long. The wire is attached to a point 32m up the tower. Find the angle that the wire makes with the tower. 7. A speed boat leaves buoy A and travels in a direction of 060°T for a distance of 9.6km to reach buoy B. It then heads due North for 7.8km to buoy C. How far is buoy C from buoy A, to two decimal places? 8. From the window of her apartment, Brooke can see the top of a nearby building at an angle of elevation of 50° and the street level entrance to the same building at an angle of depression of 15°. If the buildings are 60m apart, how tall is the building that Brooke is looking at? 9. Nikki and Kris travel 8.2km on a bearing of 040°T and then 6.3km on a bearing of 160°T. What distance should they travel to return directly to their starting point? 10. In his sailboat, Ryan travels 9.2km on a bearing of 147°T and then changes direction to travel 5.1km on a bearing of 265°T. How far is the sailboat from its starting point? 11. When preparing to lay underground cables, a worker uses the 250m straight edge of the property as a reference line. The point from which the cable is to enter the building is at an angle of 64° from one end and 71° from the other end. What is the shortest possible cable length the worker can use to reach the building from any point off the property, to the nearest metre? 12. A golfer hits her ball a distance of 127m so that it finishes 31m from the hole. If the direct distance from the tee to the hole is 150m, calculate the angle between the line of her shot and the direct line to the hole. 13. Two scouts travel 3.7km on a bearing of 140°T, and then 5.2km on a bearing of 260°T. What distance must they travel to return directly to the starting point? 14. A destroyer and a cruiser leave port at the same time. The destroyer sails 25km on a bearing of 040°T and the cruiser sails 30km on a bearing of 320°T. How far are the ships from each other, to the nearest metre? 15. A triathlon requires competitors to run 230m along a shore line from point A directly North to point B. From point B, they swim across to point C on the edge of shore. They cycle 260m from point C directly to point A. The bearing of C from A has been recorded as 074°T. a) Write down the value of internal angle A. b) Find the distance the competitors had to swim, to the nearest metre. c) Find the bearing of C from B, to the nearest degree.  4 - Quadratics Summary Expanding What is it/my method/things to remember/important tips Examples: 2 (𝑥 + 3)(2𝑥 − 4) (𝑥 − 5) 1 2 − 2(𝑥 + 2)(𝑥 + 3) 2 (𝑥 − 5) + 1 Factoring What it does/why it's useful/my method/how to check it/tips/look for and factor out the GCF Examples: 2 2 2 𝑥 − 16𝑥 + 63 − 𝑥 + 2𝑥 + 15 𝑥 − 25 2 2 2 3𝑥 − 9𝑥 + 6 2𝑥 + 5𝑥 + 2 4𝑥 − 15𝑥 − 25  Forms of a Quadratic Form ‘Formula’/what it usually looks like What it tells me & where in the form Factored Form or Vertex Form Standard Form How do I move between forms? Factored to Standard: Standard to Factored: (not always possible) Vertex to Standard: Standard to Vertex: Factored to Vertex: (two methods) Vertex to factored: (uncommon, n.a.p)  Transformations & Tips for Graphing 2 (vertex form is sometimes called transformations form because we see how it changed from 𝑦 = 𝑥 ) Describe the transformations required to take to Finding the Minimum/Maximum Value (aka the Vertex) How would you find the vertex of a parabola? To locate the vertex from standard form… To locate the vertex from factored form… Find the vertex of 2 𝑦 = 2𝑥 + 20𝑥 + 46  Solving: Finding zeros/roots/x-intercepts/solutions Tips/method/how to choose your approach By Factoring 2 2 Find the x-intercepts of 𝑦 = 𝑥 + 5𝑥 + 6 Solve by factoring 6𝑥 + 𝑥 − 2 = 0 Using the Quadratic Formula 2 2 2 Find the x-intercepts of 𝑦 = − 𝑥 + 7𝑥 + 2 Solve 2𝑥 − 8𝑥 − 4 = 3𝑥 − 𝑥 Applications General tips: - Draw a diagram! - Decide if you are finding the max/min (vertex) or if you are solving for x (by factoring or quad formula). - There is sometimes more than one way to solve, but sometimes there is a more efficient way! - Check to see if your answer makes sense. - You can use your graphing calculator to check your work but still need to show algebraic reasoning. Ms Kunkel looks at one of her Australia photos, measuring 12cm by 8cm, which is to be surrounded by a mat for framing. The width of the mat is to be equal on all sides of the photo. The area of the photo and mat backing is 210 square cm. Find the width of the border. The following function gives the height, h in metres, of a batted ball (hit baseball) as a function of time, t in seconds, since the ball was hit. a. What is the maximum height of the ball? b. What was the height of the ball when it was hit? c. How many seconds after the ball was hit did it hit the ground, to the nearest second? The city bus company carries, on average, 3500 passengers daily. Each passenger pays $2.25 to ride the bus. Market research has shown that for every $0.25 increase in bus fare, the company loses 50 customers. What price would maximize revenue? A farmer wants to make a rectangular corral along the side of the barn and has enough materials for 60m of fencing. Only three sides must be fenced, since the barn wall will form the fourth side. What width of rectangle should the farmer use so that the maximum area is enclosed? No Calculator (Paper 1) 1. State the number of roots that exist using the discriminant. a) b) c) 2. Solve the following equations (use the method that is most appropriate – factoring or quadratic formula). a) 𝑥2 − 3𝑥 − 54 = 0 b) c) 3. Rewrite the equations in vertex form. Describe the transformation required to take to the given parabola. a) b) 4. For the quadratic function , find: direction of opening y-intercept zeros/roots (if they exist) axis of symmetry coordinates of the vertex. sketch the relation showing all the important features 5. Find the equation of the quadratic relation, in vertex form, with a vertex at and passing through. 6. Find the equation of the quadratic relation, in vertex form, with a vertex at and passing through. 7. Determine the equations of the parabolas in the graphs below. Give your final answer in standard form. Quadratics Word Problems - No calculator 7. Two consecutive integers are added. The sum of their squares is 113. What are the integers? 8. The following function gives the height, h in metres, of a tennis shot as a function of time, t in seconds, since the ball was hit. a) What is the maximum height of the tennis ball? b) What was the height of the tennis ball when it was hit? 9. A rectangular area in my backyard has been grassed for my dog. If the area of a grassed section is 16m² and it is surrounded by 20m of fence, what are the dimensions of the grassed area in my backyard? Solve algebraically 10. A stone is thrown up in the air and its distance (d) above the ground at any time (t) is given 2 by the formula 𝑑 = 45𝑡 − 5𝑡. At what time will it reach a height of 70m? 11. A collectors book of wildlife photography sits on the coffee table. The length of the book is 6 inches less than twice its width and the area of the cover is 80 square inches. Find the dimensions of the book. Calculator Problems (Paper 2) 1. Solve the following equations (use the method that is most appropriate – factoring or quadratic formula). a) b) c) 2. Describe the transformation required to take to the parabola. 3. For the quadratic function , find: direction of opening y-intercept zeros/roots (if they exist) axis of symmetry coordinates of the vertex. sketch the relation showing all the important features 4. The profit on a new design of sunglasses is determined by the relation P = −3n2 + 78n − 315, where n is the number of sunglasses sold and P is the profit in dollars. What is the maximum profit that the company can earn? What are the break-even points? Which makes more sense? 5. Hannah is planting impatiens in the backyard of her new house this summer. A rectangular flowerbed is 40 m long and 30 m wide. To present a pleasing look, a solid block of impatiens is planted around the bed so that the area of the block of flowers is exactly one-half the area of the entire bed. The distance from the edge of the flowerbed to the block of impatiens is always constant. Find the dimensions of the block of impatiens. 6. Burger Shack, a fast-food restaurant, determines that each 10¢ increase in the price of a hamburger results in 25 fewer hamburgers sold. The usual price for a hamburger is $2.00 and the restaurant sells 300 hamburgers each day. Find the optimum price for a hamburger. 7. A rocket is fired from the floor of the Grand Canyon. The height of the rocket, h, in meters, above the floor of the canyon after t seconds is modeled by. A person sitting at the top of the canyon, 1734 m above its floor, can only see the rocket when it is above that height. For how many seconds will the rocket be visible? 8. The underpass of a bridge is shaped like a parabola. The width of the road passing under the bridge is 10 m, which includes soft shoulders 2 m wide on each side. The maximum height of a vehicle that can pass under the bridge and not drive on the shoulder is 3.5 m. The bridge designer has allowed at least 0.5 m clearance for all allowed vehicles at the centre of the bridge. Find an algebraic expression in vertex form that models the shape of the underpass. 9. A theatre company has 300 season ticket subscribers. The board of directors has decided to raise the price of season tickets from the current price of $400. A survey of the subscribers has determined that for every $20 increase in the price, 10 subscribers would not renew their season tickets. What is the maximum revenue possible from the season tickets? 10. A rectangular photo of my dog has a border around it that is even on all four sides. The total length of the picture and border is 60cm and the total width is 40cm. If the area of the photograph is 800cm², find the width of the border. 11. Find the points where the graphs of the relations and intersect.  5 - Calculator Steps Graphing Y = to graph equations 2nd - CALC - Intersect 2nd - CALC - Zero 2nd - CALC - 2nd - CALC - Value Adjusting Your Window Maximum/Minimum (ZoomFit - ZOOM 0) Calculating ALPHA - Y= - n/d for fraction Using ANS button sin (vs. sin-1) notation (or any other trig ratio) Converting fractions/decimals DEL vs CLEAR vs INS BRACKETS MATH -> frac  GRADE 10 FORMULA SHEET Quadratics Solutions of a quadratic equation in the form Forms: 2 𝑦 = 𝑎𝑥 + 𝑏𝑥 + 𝑐 2 𝑦 = 𝑎(𝑥 − ℎ) + 𝑘 𝑦 = 𝑎(𝑥 − 𝑝)(𝑥 − 𝑞) Discriminant Axis of Symmetry of a parabola in the form 𝑏 𝑥 =− 2𝑎 Analytic Geometry Slope formula: Distance formula: Midpoint formula: Trigonometry Sine law: Cosine law: SOH CAH TOA: 𝑜𝑝𝑝 𝑠𝑖𝑛θ = ℎ𝑦𝑝 𝑎𝑑𝑗 𝑐𝑜𝑠θ = ℎ𝑦𝑝 𝑠𝑖𝑛 𝐴 𝑠𝑖𝑛 𝐵 𝑠𝑖𝑛 𝐶 𝑜𝑝𝑝 𝑡𝑎𝑛θ = 𝑎 = 𝑏 = 𝑐 𝑎𝑑𝑗 Area Triangle: Pythagorean Theorem: 1 𝐴𝑟𝑒𝑎 = 2 𝑎𝑏𝑠𝑖𝑛𝐶