加拿大安大略省十年级数学试卷-英文
加拿大国际袋鼠数学竞赛试题及答案-2016年ParentsQuestions
加拿⼤国际袋⿏数学竞赛试题及答案-2016年ParentsQuestionsCanadian Math Kangaroo ContestPart A: Each correct answer is worth 3 points1.Which letter on the board is not in the word "KOALA"?(A) R (B) L (C) K (D) N (E) O2.In a cave, there were only two seahorses, one starfish and three turtles. Later, five seahorses, three starfishand four turtles joined them. How many sea animals gathered in the cave?(A) 6 (B) 9 (C) 12 (D) 15 (E) 183.Matt had to deliver flyers about recycling to all houses numbered from 25 to 57. How many houses got theflyers?(A) 31 (B) 32 (C) 33 (D) 34 (E) 354.Kanga is 1 year and 3 months old now. In how many months will Kanga be 2 years old?(A) 3 (B) 5 (C) 7 (D) 8 (E) 95.(A) 24 (B) 28 (C) 36 (D) 56 (E) 806. A thread of length 10 cm is folded into equal parts as shown in the figure.The thread is cut at the two marked places. What are the lengths of the three parts?(A) 2 cm, 3 cm, 5 cm (B) 2 cm, 2 cm, 6 cm (C) 1 cm, 4 cm, 5 cm(D) 1 cm, 3 cm, 6 cm (E) 3 cm, 3 cm, 4 cm7.Which of the following traffic signs has the largest number of lines of symmetry?(A) (B) (C) (D) (E)8.Kanga combines 555 groups of 9 stones into a single pile. She then splits the resulting pile into groups of 5 stones. How many groups does she get?(A) 999 (B) 900 (C) 555 (D) 111 (E) 459.What is the shaded area?(A) 50 (B) 80 (C) 100 (D) 120 (E) 15010.In a coordinate system four of the following points are the vertices of a square. Which point is not a vertexof this square?(A) (?1;3)(B) (0;?4)(C) (?2;?1)(D) (1;1)(E) (3;?2)Part B: Each correct answer is worth 4 points11.There are twelve rooms in a building and each room has two windows and one light. Last evening, eighteen windows were lighted. In how many rooms was the light off?(A) 2 (B) 3 (C) 4 (D) 5 (E) 612.Which three of the five jigsaw pieces shown can be joined together to form a square?(A) 1, 3 and 5 (B) 1, 2 and 5 (C) 1, 4 and 5 (D) 3, 4 and 5 (E) 2, 3 and 513.John has a board with 11 squares. He puts a coin in each of eight neighbouring squareswithout leaving any empty squares between the coins. What is the maximum numberof squares in which one can be sure that there is a coin?(A) 1 (B) 3 (C) 4 (D) 5 (E) 614.Which of the following figures cannot be formed by gluing these two identical squares of paper together?(A) (B) (C) (D) (E)15.Each letter in BENJAMIN represents one of the digits 1, 2, 3, 4, 5, 6 or 7. Different letters represent different digits. The number BENJAMIN is odd and divisible by 3. Which digit corresponds to N?(A) 1 (B) 2 (C) 3 (D) 5 (E) 716.Seven standard dice are glued together to make the solid shown. The faces of the dice thatare glued together have the same number of dots on them. How many dots are on the surfaceof the solid?(A) 24 (B) 90 (C) 95 (D) 105 (E) 12617.Jill is making a magic multiplication square using the numbers 1, 2, 4, 5, 10, 20, 25, 50 and 100. The productsof the numbers in each row, in each column and in the two diagonals should all be the same. In the figure you can see how she has started. Which number should Jill place in the cell with the question mark?(A) 2 (B) 4 (C) 5 (D) 10 (E) 2518.What is the smallest number of planes that are needed to enclose a bounded part in three-dimensional space?(A) 3 (B) 4 (C) 5 (D) 6 (E) 719.Each of ten points in the figure is marked with either 0 or 1 or 2. It is known thatthe sum of numbers in the vertices of any white triangle is divisible by 3, while thesum of numbers in the vertices of any black triangle is not divisible by 3. Three ofthe points are marked as shown in the figure. What numbers can be used to markthe central point?(A) Only 0. (B) Only 1. (C) Only 2. (D) Only 0 and 1. (E) Either 0 or 1 or 2.20.Betina draws five points AA,BB,CC,DD and EE on a circle as well as the tangent tothe circle at AA, such that all five angles marked with xx are equal. (Note thatthe drawing is not to scale.) How large is the angle ∠AABBDD ?(A) 66°(B) 70.5°(C) 72°(D) 75°(E) 77.5°Part C: Each correct answer is worth 5 points21.Which pattern can we make using all five cards given below?(A) (B) (C) (D) (E)22.The numbers 1, 5, 8, 9, 10, 12 and 15 are distributed into groups with one or more numbers. The sum of thenumbers in each group is the same. What is the largest number of groups?(A) 2 (B) 3 (C) 4 (D) 5 (E) 623.My dogs have 18 more legs than noses. How many dogs do I have?(A) 4 (B) 5 (C) 6 (D) 8 (E) 924.In the picture you see 5 ladybirds.Each one sits on its flower. Their places are defined as follows: the difference of the dots on their wings is the number of the leaves and the sum of the dots on their wings is the number of the petals. Which of the following flowers has no ladybird?(A) (B) (C) (D) (E)25.On each of six faces of a cube there is one of the following six symbols: ?, ?, ?, ?, ? and Ο. On each face there is a different symbol. In the picture we can see this cube shown in two different positions.Which symbol is opposite the ??(A) Ο(B)?(C) ?(D) ?(E) ?26.What is the greatest number of shapes of the form that can be cut out from a5 × 5 square?(A) 2 (B) 4 (C) 5 (D) 6 (E) 727.Kirsten wrote numbers in 5 of the 10 circles as shown in the figure. She wants to writea number in each of the remaining 5 circles such that the sums of the 3 numbers alongeach side of the pentagon are equal. Which number will she have to write in the circlemarked by XX?(A) 7 (B) 8 (C) 11 (D) 13 (E) 1528. A 3×3×3 cube is built from 15 black cubes and 12 white cubes. Five faces of the larger cube are shown.Which of the following is the sixth face of the large cube?(A) (B) (C) (D) (E)29.Jakob wrote down four consecutive positive integers. He then calculated the four possible totals made bytaking three of the integers at a time. None of these totals was a prime. What is the smallest integer Jakob could have written?(A) 12 (B) 10 (C) 7 (D) 6 (E) 330.Four sportsmen and sportswomen - a skier, a speed skater, a hockey player and a snowboarder - had dinnerat a round table. The skier sat at Andrea's left hand. The speed skater sat opposite Ben. Eva and Filip sat next to each other.A woman sat at the hockey player`s left hand. Which sport did Eva do?(A) speed skating (B) skiing (C) ice hockey (D) snowboarding(E) It`s not possible to find out with the given information.International Contest-Game Math Kangaroo Canada, 2016Answer KeyParents Contest。
AMC 10A 2017 英语题目及答案
2017 AMC 10A ProblemsContents[hide]▪1 Problem 1▪2 Problem 2▪3 Problem 3▪4 Problem 4▪5 Problem 5▪6 Problem 6▪7 Problem 7▪8 Problem 8▪9 Problem 9▪10 Problem 10▪11 Problem 11▪12 Problem 12▪13 Problem 13▪14 Problem 14▪15 Problem 15▪16 Problem 16▪17 Problem 17▪18 Problem 18▪19 Problem 19▪20 Problem 20▪21 Problem 21▪22 Problem 22▪23 Problem 23▪24 Problem 24▪25 Problem 25▪26 See alsoProblem 1What is the value of ?SolutionProblem 2Pablo buys popsicles for his friends. The store sells single popsicles for each, -popsicle boxes for each, and -popsicle boxes for . What is the greatest number of popsicles that Pablo can buy with ?SolutionProblem 3Tamara has three rows of two -feet by -feet flower beds in her garden. The beds are separated and also surrounded by -foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?Problem 4Mia is “helping” her mom pick up toys that are strewn on the floor. Mia’s mom manages to put toys into the toy box every seconds, but each time immediately after those seconds have elapsed, Mia takes toys out of the box. How much time, in minutes, will it take Mia and her mom to put all toysinto the box for the first time?Problem 5The sum of two nonzero real numbers is times their product. What is the sum ofthe reciprocals of the two numbers?SolutionMs. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which of of these statements necessarily follows logically?SolutionProblem 7Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?SolutionProblem 8At a gathering of people, there are people who all know each other and people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?SolutionProblem 9Minnie rides on a flat road at kilometers per hour (kph), downhill at kph, and uphill at kph. Penny rides on a flat road at kph, downhill at kph, and uphill at kph. Minnie goes from town to town , a distance of km all uphill, then from town to town , a distance of km all downhill, and then back to town , a distance of km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the -km ride than it takes Penny?SolutionJoy has thin rods, one each of every integer length from cm through cm. She places the rods with lengths cm, cm, and cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?SolutionProblem 11The region consisting of all points in three-dimensional space within units of line segment has volume . What is the length ?SolutionProblem 12Let be a set of points in the coordinate plane such that two of the three quantities and are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description forSolutionProblem 13Define a sequence recursively by and the remainder when is divided by for all Thus the sequence startsWhat isSolutionProblem 14Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was dollars. The cost of his movie ticket was of the difference between and the cost of his soda, while the cost of his soda was of the difference between and the cost of his movie ticket. To the nearest whole percent, what fraction of did Roger pay for his movie ticket and soda?SolutionProblem 15Chloé chooses a real number uniformly at random from the interval . Independently, Laurent chooses a real number uniformly at random from the interval . What is the probability that Laurent's number is greater than Chloé's number?SolutionProblem 16There are 10 horses, named Horse 1, Horse 2, , Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse runs one lap in exactly minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time , in minutes, at which all 10 horses will again simultaneously be at the starting point is . Let be the least time, in minutes, such that at least 5 of the horses are again at the starting point. What is the sum of the digits of ?SolutionProblem 17Distinct points , , , lie on the circle and have integer coordinates. The distances and are irrational numbers. What is the greatest possible value of the ratio ?SolutionProblem 18Amelia has a coin that lands heads with probability , and Blaine has a coin thatlands on heads with probability . Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is , where and are relatively prime positive integers. What is ?SolutionProblem 19Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of 5 chairs under these conditions?SolutionProblem 20Let equal the sum of the digits of positive integer . For example,. For a particular positive integer , . Which of the following could be the value of ?SolutionProblem 21A square with side length is inscribed in a right triangle with sides of length , , and so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length is inscribed in another right triangle with sides of length , , and so that one side of the square lies on the hypotenuseof the triangle. What is ?SolutionProblem 22Sides and of equilateral triangle are tangent to a circle at points and respectively. What fraction of the area of lies outside the circle?SolutionProblem 23How many triangles with positive area have all their vertices at points in the coordinate plane, where and are integers between and , inclusive?SolutionProblem 24For certain real numbers , , and , the polynomialhas three distinct roots, and each root of is also a root of the polynomial What isSolutionProblem 25How many integers between and , inclusive, have the property that some permutation of its digits is a multiple of between and For example, both and have this property.Solution2017 AMC 10A Answer Key1. C2. D3. B4. B5. C6. B7. A8. B9. C10. B11. D12. E13. D14. D15. C16. B17. D18. D19. C20. D21. D22. E23. B24. C25. A2017 AMC 10A Problems/Problem 1Contents[hide]▪1 Problem▪2 Solution 1▪3 Solution 2▪4 Solution 3▪5 Solution 4▪6 See AlsoProblemWhat is the value of ?Solution 1Notice this is the term in a recursive sequence, defined recursively asThus:Solution 2Starting to compute the inner expressions, we see the results are . This is always less than a power of . The only admissible answer choice by this rule is thus .Solution 3Working our way from the innermost parenthesis outwards and directly computing, we have .Solution 4If you distribute this you get a sum of the powers of . The largest power of in the series is , so the sum is .2017 AMC 10A Problems/Problem 2 ProblemPablo buys popsicles for his friends. The store sells single popsicles for each, -popsicle boxes for each, and -popsicle boxes for . What is the greatest number of popsicles that Pablo can buy with ?Solutionboxes give us the most popsicles/dollar, so we want to buy as many of those as possible. After buying , we have left. We cannot buy a third box, so we opt for the box instead (since it has a higher popsicles/dollar ratio than the pack). We're now out of money. We bought popsicles, so theanswer is .2017 AMC 10A Problems/Problem 3 ProblemTamara has three rows of two -feet by -feet flower beds in her garden. The beds are separated and also surrounded by -foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?Finding the area of the shaded walkway can be achieved by computing the total area of Tamara's garden and then subtracting the combined area of her six flower beds.Since the width of Tamara's garden contains three margins, the total width isfeet.Similarly, the height of Tamara's garden is feet.Therefore, the total area of the garden is square feet.Finally, since the six flower beds each have an area of square feet, the area we seek is , and our answer is2017 AMC 10A Problems/Problem 4Contents[hide]▪1 Problem▪2 Solution▪3 Solution 2▪4 See alsoProblemMia is helping her mom pick up toys that are strewn on the floor. Mia’s mom manages to put toys into the toy box every seconds, but each time immediately after those seconds have elapsed, Mia takes toys out of the box. How much time, in minutes, will it take Mia and her mom to put all toys into the box for the first time?SolutionEvery seconds toys are put in the box, so after seconds there will be toys in the box. Mia's mom will then put toys into to the box and we have our total amount of time to be seconds, whichequals minutes.Solution 2Though Mia's mom places toys every seconds, Mia takes out toys right after. Therefore, after seconds, the two have collectively placed toy into the box. Therefore by minutes, the two would have placed toys into the box. Therefore, at minutes, the two would have placed toys into the box. Though Mia may take toys out right after, the number of toys in the box firstreaches by minutes.2017 AMC 10A Problems/Problem 5Contents[hide]▪1 Problem▪2 Solution▪3 Solution 2▪4 Solution 3▪5 See AlsoProblemThe sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?SolutionLet the two real numbers be . We are given that and dividing both sides by ,Note: we can easily verify that this is the correct answer; for example, 1/2 and 1/2 work, and the sum of their reciprocals is 4.Solution 2Instead of using algebra, another approach at this problem would be to notice the fact that one of the nonzero numbers has to be a fraction. See for yourself. Andby looking into fractions, we immediately see that and would fit the rule.Solution 3Notice that from the information given above,Because the sum of the reciprocals of two numbers is just the sum of the two numbers over the product of the two numbers orWe can solve this by substituting .Our answer is simply .Therefore, the answer is .2017 AMC 10A Problems/Problem 6 ProblemMs. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows logically?SolutionRewriting the given statement: "if someone got all the multiple choice questions right on the upcoming exam then he or she would receive an A on the exam." If that someone is Lewis the statement becomes: "if Lewis got all the multiple choice questions right, then he got an A on the exam." The contrapositive: "If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong (did not get all of them right)" must also be true leaving B as the correct answer. B is also equivalent to the contrapositive of the original statement, which implies that it must be true, so the answer is.2017 AMC 10A Problems/Problem 7 ProblemJerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which ofthe following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?SolutionLet represent how far Jerry walked, and represent how far Sylvia walked. Since the field is a square, and Jerry walked two sides of it, while Silvia walked the diagonal, we can simply define the side of the square field to be one, and find the distances they walked. Since Jerry walked two sides, Since Silvia walked the diagonal, she walked the hypotenuse of a 45, 45, 90 triangle with leglength 1. Thus, We can then take2017 AMC 10A Problems/Problem 8 Contents[hide]▪1 Problem▪2 Solution 1▪3 Solution 2▪4 Solution 3▪5 See AlsoProblemAt a gathering of people, there are people who all know each other and people who know no one. People who know each other a hug, and people who do not know each other shake hands. How many handshakes occur?Solution 1Each one of the ten people has to shake hands with all the other people they don’t know. So . From there, we calculate how many handshakes occurred between the people who don’t know each other. This is simply countinghow many ways to choose two people to shake hands, or . Thus the answer is .Solution 2We can also use complementary counting. First of all, handshakes or hugs occur. Then, if we can find the number of hugs, then we can subtract it from to find the handshakes. Hugs only happen between the 20 people whoknow each other, so there are hugs. . Solution 3We can focus on how many handshakes the 10 people get.The 1st person gets 29 handshakes.2nd gets 28......And the 10th receives 20 handshakes.We can write this as the sum of an arithmetic sequence.Therefore, the answer is2017 AMC 10A Problems/Problem 9 ProblemMinnie rides on a flat road at kilometers per hour (kph), downhill at kph, and uphill at kph. Penny rides on a flat road at kph, downhill at kph, and uphill at kph. Minnie goes from town to town , a distance of km all uphill, then from town to town , a distance of km all downhill, and then back to town , a distance of km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the -km ride than it takes Penny?SolutionThe distance from town to town is km uphill, and since Minnie rides uphill at a speed of kph, it will take her hours. Next, she will ride from town to town , a distance of km all downhill. Since Minnie rides downhill at a speed of kph, it will take her half an hour. Finally, she rides from town back to town , a flat distance of km. Minnie rides on a flat road at kph, so this will take her hour. Her entire trip takes her hours. Secondly, Penny will go from town to town , a flat distance of km. Since Penny rides on a flat road at kph,it will take her of an hour. Next Penny will go from town to town , which is uphill for Penny. Since Penny rides at a speed of kph uphill, and town and are km apart, it will take her hours. Finally, Penny goes from Townback to town , a distance of km downhill. Since Penny rides downhill atkph, it will only take her of an hour. In total, it takes her hours, which simplifies to hours and minutes. Finally, Penny's Hour Minute trip was minutes less than Minnie's Hour Minute Trip2017 AMC 10A Problems/Problem 10 ProblemJoy has thin rods, one each of every integer length from cm through cm. She places the rods with lengths cm, cm, and cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?SolutionThe triangle inequality generalizes to all polygons, so andto get . Now, we know that there are numbers between and exclusive, but we must subtract to account for the 2 lengthsalready used that are between those numbers, which gives2017 AMC 10A Problems/Problem 11Contents[hide]▪1 Problem▪2 Solution▪3 Diagram▪4 See AlsoProblemThe region consisting of all point in three-dimensional space within 3 units of line segment has volume 216. What is the length ?SolutionIn order to solve this problem, we must first visualize what the region contained looks like. We know that, in a three dimensional plane, the region consisting of all points within units of a point would be a sphere with radius . However, we need to find the region containing all points within 3 units of a segment. It can be seen that our region is a cylinder with two hemispheres on either end. We know the volume of our region, so we set up the following equation (the volume of our cylinder + the volume of our two hemispheres will equal ):, where is equal to the length of our line segment. Solving, we find that .Diagram/cwNt293.png2017 AMC 10A Problems/Problem 12ProblemLet be a set of points in the coordinate plane such that two of the three quantities and are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description forSolutionIf the two equal values are and , then . Also, because 3 is the common value. Solving for , we get . Therefore the portion of the line where is part of . This is a ray with an endpoint of .Similar to the process above, we assume that the two equal values are and . Solving the equation then . Also, because 3 is the common value. Solving for , we get . Therefore the portion of the line where is also part of . This is another ray with the sameendpoint as the above ray: .If and are the two equal values, then . Solving the equation for , we get . Also because is one way toexpress the common value. Solving for , we get . Therefore the portion of the line where is part of like the other two rays. Thelowest possible value that can be achieved is also .Since is made up of three rays with common endpoint , the answer is2017 AMC 10A Problems/Problem 13 ProblemDefine a sequence recursively by and the remainder when is divided by for all Thus the sequence startsWhat isSolutionA pattern starts to emerge as the function is continued. The repeating pattern isThe problem asks for the sum of eight consecutive terms in the sequence. Because there are eight numbers in the repeating sequence, we just need to find the sum of the numbers in the sequence, which is2017 AMC 10A Problems/Problem 14 ProblemEvery week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was dollars. The cost of his movie ticket was of the difference between and the cost of his soda, while the cost of his soda was of the difference between and the cost of his movie ticket. To the nearest whole percent, what fraction of did Roger pay for his movie ticket and soda?SolutionLet = cost of movie ticketLet = cost of sodaWe can create two equations:Substituting we get:which yields:Now we can find s and we get:Since we want to find what fraction of did Roger pay for his movie ticket and soda, we add and to get:2017 AMC 10A Problems/Problem 15 Contents[hide]▪ 1 Problem▪ 2 Solution 1▪ 3 Solution 2▪ 4 Solution 3:▪ 5 See AlsoProblemChloé chooses a real number uniformly at random from the interval . Independently, Laurent chooses a real number uniformly at random from the interval . What is the probability that Laurent's number is greater than Chloé's number?Solution 1Denote "winning" to mean "picking a greater number". There is a chance that Laurent chooses a number in the interval . In this case, Chloécannot possibly win, since the maximum number she can pick is . Otherwise, if Laurent picks a number in the interval , with probability , then the two people are symmetric, and each has a chance of winning. Then,the total probability isSolution 2We can use geometric probability to solve this. Suppose a point lies in the-plane. Let be Chloe's number and be Laurent's number. Then obviously we want , which basically gives us a region above a line. We know thatChloe's number is in the interval and Laurent's number is in the interval , so we can create a rectangle in the plane, whose length is andwhose width is . Drawing it out, we see that it is easier to find the probability that Chloe's number is greater than Laurent's number and subtract this probability from . The probability that Chloe's number is larger than Laurent's number issimply the area of the region under the line , which is . Instead of bashing this out we know that the rectangle has area . Sothe probability that Laurent has a smaller number is . Simplifying the expression yields and so .Solution 3:Scale down by to get that Chloe picks from and Laurent picks from . There are an infinite number of cases for the number that Chloe picks, butthey are all centered around the average of . Therefore, Laurent has a range of to to pick from, on average, which is a length of out of a total length of . Therefore, the probability is2017 AMC 10A Problems/Problem 16 Contents[hide]▪ 1 Problem▪ 2 Solution 1▪ 3 Solution 2▪ 4 See AlsoProblemThere are 10 horses, named Horse , Horse , . . . , Horse . They get their names from how many minutes it takes them to run one lap around a circular race track: Horse runs one lap in exactly minutes. At time all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time , in minutes, at which all horses will again simultaneously be at the starting point is . Let be the least time, in minutes, such that at least 5 of the horses are again at the starting point. What is the sum of the digits ofSolution 1If we have horses, , then any number that is a multiple of the all those numbers is a time when all horses will meet at the starting point. The least of these numbers is the LCM. To minimize the LCM, we need the smallest primes, and we need to repeat them a lot. By inspection, we find that. Finally, .Solution 2We are trying to find the smallest number that has one-digit divisors. Therefore we try to find the LCM for smaller digits, such as ,, , or . We quickly consider since it is the smallest number that is the LCM of , , and . Since has single-digit divisors, namely , , , , and , our answer is2017 AMC 10A Problems/Problem 17 ProblemDistinct points , , , lie on the circle and have integer coordinates. The distances and are irrational numbers. What is thegreatest possible value of the ratio ?SolutionBecause , , , and are integers there are only a few coordinates that actually satisfy the equation. The coordinates areand We want to maximize and minimize They also have to bethe square root of something, because they are both irrational. The greatest value of happens when it and are almost directly across from each other and are in different quadrants. For example, the endpoints of the segmentcould be and because the two points are almost across from each other. The least value of is when the two endpoints are in the samequadrant and are very close to each other. This can occur when, for example,is and is They are in the same quadrant and no other point on the circle with integer coordinates is closer to the point than Using the distance formula, we get that is and that is2017 AMC 10A Problems/Problem 18 Contents[hide]▪ 1 Problem▪ 2 Solution▪ 3 Solution 2▪ 4 See AlsoProblemAmelia has a coin that lands heads with probability , and Blaine has a coin that lands on heads with probability . Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is , whereand are relatively prime positive integers. What is ?SolutionLet be the probability Amelia wins. Note that, as if she gets to her turn again, she is back where she started with probability of winning . The chance she wins on her first turn is . The chance she makes itto her turn again is a combination of her failing to win the first turn - and Blaine failing to win - . Multiplying gives us . Thus, Therefore, , so the answer is .Solution 2Let be the probability Amelia wins. Note thatThis can be represented by an infinite geometric series,. Therefore, , so the answer isSolution by ktong2017 AMC 10A Problems/Problem 19 Contents[hide]▪ 1 Problem▪ 2 Solution 1▪ 3 Solution 2▪ 4 See AlsoProblemAlice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of 5 chairs under these conditions?Solution 1For notation purposes, let Alice be A, Bob be B, Carla be C, Derek be D, and Eric be E. We can split this problem up into two cases:A sits on an edge seat.Then, since B and C can't sit next to A, that must mean either D or E sits next to A. After we pick either D or E, then either B or C must sit next to D/E. Then, we can arrange the two remaining people in two ways. Since there are two different edge seats that A can sit in, there are a total of .A does not sit in an edge seat.In this case, then only two people that can sit next to A are D and E, and there are two ways to permute them, and this also handles the restriction that D can't sit next to E. Then, there are two ways to arrange B and C, the remaining people. However, there are three initial seats that A can sit in, so there areseatings in this case.Adding up all the cases, we have .Solution 2Label the seats through . The number of ways to seat Derek and Eric in the five seats with no restrictions is . The number of ways to seat Derek and Eric such that they sit next to each other is (which can be figure out quickly), so the number of ways such that Derek and Eric don't sit next to each other is . Note that once Derek and Eric are seated, there are three cases.The first case is that they sit at each end. There are two ways to seat Derek and Eric. But this is impossible because then Alice, Bob, and Carla would have to sit in some order in the middle three seats which would lead to Alice sitting next to Bob or Carla, a contradiction. So this case gives us ways.Another possible case is if Derek and Eric seat in seats and in some order. There are 2 possible ways to seat Derek and Eric like this. This leaves Alice, Bob, and Carla to sit in any order in the remaining three seats. Since no two of these three seats are consecutive, there are ways to do this. So the second case gives us total ways for the second case.The last case is if once Derek and Eric are seated, exactly one pair of consecutive seats is available. There are ways to seat Derek and Eric like this. Once they are seated like this, Alice cannot not sit in one of the two consecutive available seats without sitting next to Bob and Carla. So Alice has to sit in the other remaining chair. Then, there are two ways to seat Bob and Carla in the remaining two seats (which are consecutive). So this case gives us ways.So in total there are . So our answer is .。
国际学校数学十年级试卷
1. The prime factorisation form of 156
A) 2 ´ 3´ 13
B) 22 ´ 32 ´ 13
2. The logarithemic form of 8x = 2
Chapter Real numbers
2.
Sets
3.
Polynomials
Weightage % 48% 20% 32% 100%
Marks 12 5 8 25
Table : 3 Weightage to difficulty level
S. No.
Difficulty level
Weightage % Marks
A) logx 8 = 2
B) log 2 = x 8
p 3. The q form of the decimal 0.875
7
A) 23
7
B) 16
C) 22 ´ 3 ´ 13 C) log 2 = 8
x
3
C) 8
() D) 2 ´ 32 ´ 13
() D) log x = 8
2
7
D) 22
()
3
1.
Easy
24%
06
2.
Average
3.
Difficulty
52%
13
24%
06
100%
25
Table : 4 Weightage to type of questions
S. No. Type of Questions
Weightage % No. of Questions Marks
amc10英语真题及答案新课标
amc10英语真题及答案新课标AMC10是美国数学竞赛（American Mathematics Competitions）的10年级级别，面向10年级及以下的学生。
以下是一套模拟的AMC10英语真题及答案，请注意，这只是一个示例，并非真实的AMC10题目。
AMC10 英语真题及答案Part 1: Multiple Choice1. The word "innovate" is most closely related to which of the following?A. InnovatorB. InventionC. InventionistD. InnovatoryAnswer: D. Innovatory2. Which of the following is the correct spelling of the word meaning "to make something new"?A. InnovateB. InnovateC. InnovateD. InnovateAnswer: A. Innovate3. The phrase "a leap of faith" is used to describe:A. A large jumpB. A risky decisionC. A new religionD. A sudden increaseAnswer: B. A risky decision4. In the sentence "The company is looking to streamline its operations," the word "streamline" means:A. To make more expensiveB. To make more efficientC. To make more complicatedD. To make more visibleAnswer: B. To make more efficient5. The word "altruistic" is an antonym for:A. SelfishB. AltruismC. AltruisticallyD. AltruistAnswer: A. SelfishPart 2: Fill in the Blanks6. The scientist was awarded the Nobel Prize for his _______ contributions to the field of physics.Answer: innovative7. The _______ of the old building was a significantachievement for the preservation society.Answer: renovation8. The _______ of the new policy was met with mixed reactions from the public.Answer: implementation9. The _______ of the company's profits was due to a series of successful marketing campaigns.Answer: increase10. The _______ of the ancient ruins provided valuable insights into the history of the civilization.Answer: excavationPart 3: Reading ComprehensionRead the following passage and answer the questions that follow.Passage:In recent years, there has been a significant increase in the number of people who are interested in sustainable living. This trend has led to the development of various eco-friendly products and practices. One such practice is the use of solar panels to generate electricity. Solar panels are becoming more popular due to their ability to harness the power of the sun and convert it into usable energy.Questions:11. What is the main topic of the passage?Answer: Sustainable living and the use of solar panels.12. Why are solar panels becoming more popular?Answer: Because they can harness the power of the sun and convert it into usable energy.13. What is the trend mentioned in the passage?Answer: An increase in the number of people interested in sustainable living.Part 4: Vocabulary in Context14. The _______ of the old factory was a major concern for the environmentalists.Answer: pollution15. The _______ of the new technology was celebrated by the scientific community.Answer: advancement16. The _______ of the endangered species was a top priority for the conservation organization.Answer: preservation17. The _______ of the ancient artifact provided evidence ofa previously unknown civilization.Answer: discovery18. The _______ of the new policy was met with skepticism by some members of the community.Answer: enforcement请注意，AMC10是一个数学竞赛，通常不包含英语题目。
加拿大国际袋鼠数学竞赛试题-2013年
(A) 12
(B) 8
(C) 4
(D) 2
(E) 0
3. Which of the dresses has less than seven dots, but more than five dots?
(A)
(B)
(C)
(D)
(E)
Grade 1-2
Year 2013
4. A lot of babies were born in the zoo last year: two baby lions, three baby dolphins and four baby eagles.
tulips. They started at 9:00 in the morning. At what time will they finish planting all 20 tulips?
(A) At 9:10
(B) At 9:20 (C) At 9:40
(D) At 9:50
(E) At 10:00
(C) 2 and 0
(D) 6 and 9
(E) 7 and 1
2. There are twelve books on a shelf and four children in a room. How
many books will be left on the shelf if each child takes one book?
(A) 16
(B) 30
(C) 50
(D) 52
(E) 70
18. In a park there are babies in four-wheel strollers and children on two-wheel bikes. Paula counted wheels and the total was 12. When she added the number of strollers to the number of bikes, the total was 4. How many two-wheel bikes are there in the park?
国际学校十年级数学期末测试卷(英语)
2. If cos q = 0.6691306, then q = a) 48° b) 42° c) 33.8° d) 22.9°
3. If DABC ~ DPQR, then a) AB = PQ b) A = P c) a = b d) c = r
Final Practice Exam
4. The cost of 4 L of milk is $7.35. The unit cost per litre is a) $3.68 b) $1.84 c) $2.45 d) $0.735
5. A 10% discount off an item priced at $30.00 would save you a) $3.00 b) $27.00 c) $10.00 d) $1.00
8. A rotation of a shape is a) a slide b) a translation c) a turn d) a mirror image
9. Which statement is true about angles with parallel lines with a transversal? a) All angles are 60° b) Interior alternate angles are complementary c) Corresponding angles are supplementary d) Exterior alternate angles are congruent
Final Practice Exam
3
4
Grade 10 Essential Mathematics
加拿大数学十年级(高一)省考模拟试题A
PART A: MULTIPLE-CHOICE QUESTIONS (calculator not permitted)
Value: 12 marks
Suggested Time: 30 minutes Allowable Time: 40 minutes
Page 1
Use the following graph to answer question 2.
y
10
5
–10
–5
–5
x
5
10
–10
2. Which of the following equations describes the linear relation graphed above?
Do not access your calculator until directed by the supervisor. At the end of the 40 minutes, the supervisor will give you permission to access your calculator.
where n is the number of students attending the prom. Calculate the cost of renting the hall if 70 students attend. A. $108 B. $500 C. $780 D. $970
6. Which of the following statements are true?
Contents: 31 pages 54 multiple-choice questions (maximum of 54 marks) 6 numeric-response questions (maximum of 6 marks)
加拿大安大略地区十学年数学试卷英文
GRADE 10 PRINCIPLES OF MATHEMATICS (ACADEMIC)MPM 2DTotal Marks:INSTRUCTIONS:1. Calculators may be used.2. Read all instructions carefully in order to maximize your mark.A/C[K] Part A – Multiple Choice 25 Marks (25 questions * 1 mark each)For each of the following questions in this section, circle the letterrepresenting the correct answer.1. A linear system of two equations that has one solution represents twolines that are:a) parallel b) coincident c) intersecting d) none of these2. The midpoint of RS is M(8, -1). If point S has coordinates (11, 4) what are the coordinates of point R ? a) (3, -6)b) (15, -6)c) (5, -6)d) (3, 9)3. The midpoint of the line segment with end points A(-8, 8) and B(6, 4) is: a) (0, 10)b) (1, 2)c) (7, 2)d) (-1, 6)4. The equation of a horizontal line passing through the point (4, 2) is: a) 2=xb) 4=yc) 2=yd) 4=x5. The equation of a line with a slope of 5=m and a y intercept of 8 is: a) 85+=x yb) 85+-=x yc) 85--=x yd) 58+=x y6. The slopes of 2 lines are -7 and 71. These lines are said to be:a) parallelb) perpendicularc) coincidentd) none of these7. The slope of a line segment passing through 2 points (10,- 4) and (-2, -16) is: a) 1b) 2c) -1d) -28. The length of a line segment with end points (-6, 7) and (-1, -5) is: a) 12b) 5c) 13d) 1699. The diameter of a circle whose equation is 28922=+y x is:a) 15b) 16c)17d) none of these10. T he equation of a circle with a centre of (0, 0) that also passes through the point (-8, -6) is: a) 1022=+y xb) 10022=+y xc) 1422=+y xd) 4822=-y x11. T he y-intercept of the line 01052=+-y x is: a) 2b) -2c) 10d) 512. T he slope of the line 0124=-+y x is: a) 2b) -2c) 1d) 013. I f (-3, y) is a solution to the equation 132=+y x , what is the value of y ? a) 3b) 6c) 5d) 814. T he product ()()z y x z y x 323243-- is equal to:a) 2612z xyb) 26412z y xc) 2612z xy -d) 00412z y x15. A simplified expression for ()()n m n m ----52 is: a) m 7b) n m 27+c) m 3-d) n m 27-16. A simplified expression for 242927abcbc a -- is:a) ac 3 b) abc 3 c) 23acd) 223c a17. T he slope of the line, which is perpendicula r to the line, 084=+-y x is: a) -4 b) 4 c) 1 d) -118. T he shortest distance from the point (2, -3) to the line 4-=x is: a) 5 b) 3 c) 2 d) 619. T he value of the polynomial 8542+-a a when 3-=a is: a) 59 b) 44 c) 13 d) 2920. W hich of the following is not a function : a)()()(){}7,6,5,4,3,2b) 22x y =c) 22y x =d) ()()(){}3,8,3,7,2,621. T he range of the relation whose equation is 52--=x y is: a) 5-≤y b) 5≤y c) 5-≥y d) 5≥y22. T he vertex of the parabola ()642--=x y is:a) ()6,4- b) ()6,4- c) ()4,6- d) ()4,6-23. T he equation of the axis of symmetry of the parabola ()5242+--=x y is:a) 5=x b) 5-=x c) 2=xd) 2-=x 24. A parabola with a vertex of ()3,2 and a stretch factor of 41- (relative to2x y =) would have an equation of: a) ()32412+--=x y b) ()32412++-=x y c)()23412-+-=x y d) ()23412++-=x y25 The parabola k x y +-=24 passes through the point ()3,2-. T he value of k is: a) -19b) 11c) 13d) 19A/CPart B – Short AnswersFor each of the questions in this section, write your answers in the spaces provided . Use the foolscap provided for any rough work. Show details of calculations wherever requested.1. In the accompanying diagram, state each of the following: (4 Marks) [K] a) domain: __________ (1 Mark)[K]b)range: __________ (1 Mark)[C]c)Is the relation a function? Justify your answer. (2 marks)[A] 2. The x-intercepts of the parabola 2892-y are: __________ and=x__________.(Show your work) (2 Marks)[A] 3. The roots of the quadratic equation 032=1710x are: __________ and+-x__________.(Show your work) (3 Marks)[A] 4. Write the equation of the parabola with a vertex of (4, 23) if it passesthrough the point (-1, -2): (Show your work) (3 Marks)____________________[T] 5. A line passes through 2 points (1, 4) and (2,-4). Calculate the slope of the line. Also show the equation of the line in the form 0ByAx. (Show+=+Cyour work) (4 Marks)____________________ ____________________Slope Equation[K] 6. The Tangent of 45 is: __________ (1 Mark)[A] 7. a) In the accompanying diagram, the two triangles are similar. What is thevalue of x?(Show your work) (2 Marks) Array=x__________[T] b) If the area of the smaller triangle is 8 cm 2, what is the area of the larger triangle?(Show your work) ( 2 Marks)Area = __________[K]8 Given that sin A = 21, find A ∠ (to the nearest degree) __________ (1Mark)[A] 9. In the accompanying right triangle , find the value of x to one decimal place.(Show your work) (2 Marks)=x ________[A] 10. U se the SINE LAW to find the value of side x to one decimal place.(Show your work) (2 Marks)3028︒x56︒42︒x30x = ________[A] 11. U se the COSINE LAW to find the value of side x to one decimal place.(Show your work) (2 Marks)x = ________[T] 12. F actor each of the following to the fullest extent possible: (4 Questions * 2 marks each)a) y x my mx 22--+________________________b) 31142--x x________________________c) 2416916y x -________________________56︒2030xd) 2225rs-________________________ r+9s30A/C Part C – Full Solutions RequiredFor each of the questions in this section, full solutions are required.Record your answers in the spaces provided. Use the foolscap providedfor any rough work.[A] 1. Solve the linear system using the elimination method. Remember tofind values for both x and y. (5 Marks)22+yx5=-yx=32-21[C] Explain what the solution above represents geometrically. How do youknow that the solution you arrived at is the correct answer? (2 Marks)[A]2. Expand and simplify the polynomial ()()()21432+-+-x x x . (4 Marks)[T]3. Find the equation of the line perpendicular to the line 088=-+y x and passing through the point (-4, 1). (4 Marks)[T] 4. From the window of one building, a man finds that the angle ofelevation to the top of a second building is 47︒ and the angle ofdepression to the bottom of the same building is 58︒. The buildings are 60 m apart. Find the height of the 2nd building to the nearest metre.A diagram is required. (6 Marks)[T] 5. ABC has vertices A(1, 7), B(-5, 3) and C(3, -1). Determine the equation for AE, the altitude from vertex A to the opposite side BC.(5 Marks)6. The hypotenuse of a right triangle is 26 cm. The sum of the other twosides is 34 cm. (9 Marks)[T] a) Find the length of the other two sides of the triangle. (3 Marks)[T] b) Find the measure of the other two angles. Round to the nearest degree. (3 Marks)[C] c) Describe a situation where you would be able to use knowledge ofthe Pythagorean theorem in a practical, real life situation. (3 Marks)[T] 7. A rectangular skating rink measures 20m by 20m. It has been decided to increase the area of the rink by a factor of 4. Determine how mucheach side should be extended. Assume that each side is extendedby the same amount. (6 Marks)[C]What is the significance of keeping the skating rink in the shape of a square? Justify your answer. (3 Marks)[A]8. a) Solve 35122+=d d using the quadratic formula. (2 Marks)[A] b) Solve 03122=-x by factoring. Check your solutions. (2 Marks)。
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加拿大安大略省十年级数学试卷-英文2 of 11GRADE 10 PRINCIPLES OF MATHEMATICS (ACADEMIC)MPM 2DTotal Marks:INSTRUCTIONS:1. Calculators may be used.2. Read all instructions carefully in order to maximize your mark.A/C[K]Part A – Multiple Choice 25 Marks (25 questions * 1 mark each)For each of the following questions in this section, circle the letter representing the correct answer.1. A linear system of two equations that has one solution represents two lines that are:a)parallel b)coincident c)intersecting d)none of these2.The midpoint of RS is M(8, -1). If point S has coordinates (11, 4) what are the coordinates of point R?a)(3, -6)b)(15, -6)c)(5, -6)d)(3, 9)3.The midpoint of the line segment with end points A(-8, 8) and B(6, 4) is:a)(0, 10)b)(1, 2)c)(7, 2)d)(-1, 6)4.The equation of a horizontal line passing through the point (4, 2) is:a)b)c)d)5.The equation of a line with a slope of and a y intercept of 8 is:a)b)c)d)2=x 4=y 2=y 4=x 5=m 85+=x y 85+-=x y 85--=x y 58+=xy3 of 116.The slopes of 2 lines are -7 and. These lines are said to be:a)parallel b)perpendicular c)coincident d)none of these7.The slope of a line segment passing through 2 points (10,- 4) and (-2, -16) is:a)1b)2c)-1d)-28.The length of a line segment with end points (-6, 7) and (-1, -5) is:a)12b)5c)13d)1699.The diameter of a circle whose equation is is:a)15b)16c)17d)none of these10.The equation of a circle with a centre of (0, 0) that also passes through the point (-8, -6) is:a)b)c)d)11.The y-intercept of the line is:a)2b)-2c)10d)512.The slope of the line is:a)2b)-2c)1d)013.If (-3, y) is a solution to the equation , what is the value of y?a)3b)6c)5d)814.The product is equal to:a)b)c)d)15.A simplified expression for is:a)b)c)d)7128922=+y x 1022=+y x 10022=+y x 1422=+y x 4822=-y x 01052=+-y x 0124=-+y x 132=+y x ()()z y x z y x 323243--2612z xy 26412z y x 2612z xy -00412z y x ()()n m n m ----52m 7n m 27+m 3-nm 27-4 of 1116.A simplified expression for is:a)b)c)d)17.The slope of the line, which is perpendicular to the line, is:a)-4b)4c)1d)-118.The shortest distance from the point (2, -3) to the line is:a)5b)3c)2d)619.The value of the polynomial when is:a)59b)44c)13d)2920.Which of the following is not a function:a)b)c)d)21.The range of the relation whose equation is is:a)b)c)d)22.The vertex of the parabola is:a)b)c)d)23.The equation of the axis of symmetry of the parabola is:a)b)c)d)24.A parabola with a vertex of and a stretch factor of (relative to ) would have an equation of:a)b)c)d)25The parabola passes through the point . The value of is:242927abc bc a --ac 3abc 323ac 223c a 084=+-y x 4-=x 8542+-a a 3-=a ()()(){}7,6,5,4,3,222x y =22y x =()()(){}3,8,3,7,2,652--=x y 5-≤y 5≤y 5-≥y 5≥y ()642--=x y ()6,4-()6,4-()4,6-()4,6-()5242+--=x y 5=x 5-=x 2=x 2-=x ()3,241-2x y =()32412+--=x y ()32412++-=x y ()23412-+-=x y ()23412++-=x y k x y +-=24()3,2-k5 of 11a)-19b)11c)13d)19A/C Part B – Short AnswersFor each of the questions in this section, write your answers in the spaces provided.Use the foolscap provided for any rough work. Show details of calculationswherever requested.1.In the accompanying diagram, state each of the following: (4 Marks)[K]a)domain: __________ (1 Mark)[K]b)range: __________ (1 Mark)[C]c)Is the relation a function? Justifyyour answer. (2 marks)[A] 2.The x-intercepts of the parabola are: __________ and__________.(Show your work) (2 Marks)[A] 3.The roots of the quadratic equation are: __________ and__________.(Show your work) (3 Marks)2892-=x y 0101732=+-x x 24-2-46 of 11x[A] 4.Write the equation of the parabola with a vertex of (4, 23) if it passes through thepoint (-1, -2): (Show your work) (3 Marks)____________________[T] 5. A line passes through 2 points (1, 4) and (2,-4). Calculate the slope of the line. Alsoshow the equation of the line in the form . (Show your work) (4 Marks)________________________________________Slope Equation[K] 6.The Tangent of is: __________ (1 Mark)[A]7.a) In the accompanying diagram, the two triangles are similar. What is the value of?(Show your work) (2 Marks)__________[T]b) If the area of the smaller triangle is 8 cm 2, what is the area of the largertriangle?(Show your work) ( 2 Marks)Area = __________[K]8Given that sin A = , find (to the nearest degree) __________ (1 Mark)0=++C By Ax 45x =x 21A ∠1127 of 11[A]9.In the accompanying right triangle, find the value of x to one decimal place.(Show your work) (2 Marks)________[A]e the SINE LAW to find the value of side x to one decimal place.(Show your work) (2 Marks)x = ________[A]e the COSINE LAW to find the value of side x to one decimal place.(Show your work) (2 Marks)x = ________[T]12.Factor each of the following to the fullest extent possible: (4 Questions * 2 marks each)a)________________________b)________________________=x y x my mx 22--+31142--x x 328︒x56︒42︒x3056︒2030x8 of 11c)________________________d)________________________A/C Part C – Full Solutions RequiredFor each of the questions in this section, full solutions are required. Record your answers in the spaces provided. Use the foolscap provided for any rough work.[A] 1.Solve the linear system using the elimination method. Remember to find valuesfor both x and y. (5 Marks)[C]Explain what the solution above represents geometrically. How do you knowthat the solution you arrived at is the correct answer? (2 Marks)[A] 2.Expand and simplify the polynomial . (4 Marks)2416916y x -2225309s rs r +-225=+y x 2132-=-y x ()()()21432+-+-x x x9 of 11[T] 3.Find the equation of the line perpendicular to the line and passingthrough the point (-4, 1). (4 Marks)[T] 4.From the window of one building, a man finds that the angle of elevation to thetop of a second building is 47︒ and the angle of depression to the bottom of thesame building is 58︒. The buildings are 60 m apart. Find the height of the 2ndbuilding to the nearest metre. A diagram is required. (6 Marks)[T] 5.∆ABC has vertices A(1, 7), B(-5, 3) and C(3, -1). Determine the equation for AE,the altitude from vertex A to the opposite side BC. (5 Marks)088=-+y x6.The hypotenuse of a right triangle is 26 cm. The sum of the other two sides is 34cm. (9 Marks)[T]a)Find the length of the other two sides of the triangle. (3 Marks)[T]b)Find the measure of the other two angles. Round to the nearest degree. (3 Marks)[C]c)Describe a situation where you would be able to use knowledge of thePythagorean theorem in a practical, real life situation. (3 Marks)10 of 1111 of 11[T]7. A rectangular skating rink measures 20m by 20m. It has been decided toincrease the area of the rink by a factor of 4. Determine how much each side should be extended. Assume that each side is extended by the same amount. (6 Marks)[C]What is the significance of keeping the skating rink in the shape of a square?Justify your answer. (3 Marks)[A]8.a)Solve using the quadratic formula. (2 Marks)[A]b)Solve by factoring. Check your solutions. (2 Marks)35122+=d d 03122=-x。