# MATEMATIKA ANGOL NYELVEN

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1 Matematika angol nyelven középszint 03 ÉRETTSÉGI VIZSGA 03. május 7. MATEMATIKA ANGOL NYELVEN KÖZÉPSZINTŰ ÍRÁSBELI ÉRETTSÉGI VIZSGA JAVÍTÁSI-ÉRTÉKELÉSI ÚTMUTATÓ EMBERI ERŐFORRÁSOK MINISZTÉRIUMA

2 Instructions to examiners Formal requirements:. Mark the paper in ink, different in colour from the one used by the candidate. Indicate the errors, incomplete solutions, etc. in the conventional way.. The first one of the rectangles under each problem shows the maximum attainable score on that problem. The points given by the examiner are to be entered in the rectangle next to that. 3. If the solution is perfect, it is enough to enter the maximum scores in the appropriate rectangles. 4. If the solution is incomplete or incorrect, please indicate the individual partial scores in the body of the paper, too. 5. Except diagrams, do not assess anything that is written in pencil. Assessment of content:. The markscheme may contain more than one solution for some of the problems. If the solution by the candidate is different, allocate the points by identifying the parts of the solution equivalent to those of the one given in the markscheme.. The subtotals in the markscheme can be further divided, but the scores awarded should always be whole numbers. 3. If it is clear that the reasoning and the final answer are both correct, you may award the maximum score even if the solution is less detailed than the one in the markscheme. 4. If there is a calculation error or inaccuracy in the solution, only take off the points for that part where the error occurs. If the reasoning remains correct and the error is carried forward, the points for the rest of the solution should be awarded. 5. In the case of a principal error, no points should be awarded at all for that section of the solution, not even for steps that are formally correct. (These logical sections of the solutions are separated by double lines in the markscheme.) However, if the wrong information obtained owing to the principal error is carried forward to the next section or in the next part of the problem and is used correctly, the maximum score is due for the next part, provided that the error has not changed the nature of the task to be completed. 6. Where the markscheme shows a unit or a remark in brackets, the solution should be considered complete without that unit or remark as well. 7. If there are more than one different approaches to a problem, assess only the one indicated by the candidate. 8. Do not give extra points (i.e. more than the score due for the problem or part of problem). 9. Do not take off points for steps or calculations that contain errors but are not actually used by the candidate in the solution of the problem. 0. Assess only two out of the three problems in part B of Paper II. The candidate was requested to indicate in the appropriate square the number of the problem not to be assessed and counted in their total score. Should there be a solution to that problem, it does not need to be marked. However, if it is still not clear which problem the candidate does not want to be assessed, assume automatically that it is the last one in the question paper, and do not assess that problem. írásbeli vizsga 03 / 03. május 7.

3 . The simplified fraction:. A right circular cylinder is obtained. Base radius is 5 cm, height is cm. I. a b. points 3 Total: points V = 5π (cm3 ). The volume of the right circular cylinder is 300π cm 3. The points cannot be divided. The point is due if these ideas appear in a diagram. The answer is also accepted in decimal form. 3. The number of real roots is. points Total: points if the answer is x = 5, no points if the answer is x = 4, x = 4 points Total: points for each correct answer. 5. r r r a + b The position vector of the midpoint is f =. r r r Rearranged: b = f a. Total: points 6. The smallest positive angle is 30º. points Total: points for stating that 750º = 360º+30º. 7. ( ) x + 8 x + 8 = x + 9 The square of a number is a minimum when 0 is squared. The smallest value of the function occurs at x = 9. Total: points írásbeli vizsga 03 3 / 03. május 7.

4 8. There are 4 =6 positive five-digit numbers. points Total: points 9. Group I: 80 people, Group II: 40 people, Group III: 300 people. each 0. Rearranged: x 7y = 0. A normal vector of a perpendicular line e is n ( 7 ; ), Thus the equation of line e is 7x + y = 33. The equations of the lines are accepted in any correct form.. A: true; B: false; C: true; D: true. each. Total: 4 points. The terms of the sequence are a =, a =, a = 0, a =, a =, a = points Therefore S = 6 4. írásbeli vizsga 03 4 / 03. május 7.

5 II. A 3. a) The side of the square is a, the sides of the rectangles are a and 3 a. The perimeter of one rectangle is a + a = 4, 3 points hence a = 9 cm. The area of the square is 8 cm. Total: 5 points The point is also due if the notations only appear in a diagram. 3. b) Solution From the Pythagorean theorem: 3 = x (or: 3,, 5 are a Pythagorean triple), the other leg of the right-angled triangle (BP) is 5 cm. The area T of the right-angled triangle can be expressed in two ways: a b c m T c points = =, where m c is the height drawn to hypotenuse c). Hence a b = c mc, a b 60 that is, m c = =. c 3 The height drawn to the hypotenuse is 4.6 cm long. Total: 7 points 3. b) Solution From the Pythagorean theorem: 3 = x (or: 3,, 5 are a Pythagorean triple), the other leg of the right-angled triangle (BP) is 5 cm. It is known that the leg is the geometric mean of its projection and the hypotenuse: 5 = 3 p, points 5 p = From the Pythagorean theorem the square of the height m c drawn to the hypotenuse is 5 m c = 5 3 The height drawn to the hypotenuse is 4.6 cm long. Total: 7 points írásbeli vizsga 03 5 / 03. május 7.

6 3. b) Solution 3 Let α denote the angle at vertex A in the right-angled triangle ABP, and let Q be the foot of the altitude drawn to the hypotenuse. AB In the right-angled triangle ABP, cos α = =. AP 3 points α. 6. BQ BQ In the right-angled triangle AQB, sin α = =. AB points BQ = sinα The height drawn to the hypotenuse is 4.6 cm long. Total: 7 points 4. a) Domain of definition: 5 ( since x 5 > 0 and x > 0), x >. (It follows from the identities of logarithms that) x points x 5 =. 3 Hence x = 3. This root is in the domain of definition, therefore it is the solution of the equation. Total: 5 points This point is also due if the value of sin α is calculated correctly later on. This point is also due if the candidate uses the equation obtained by cancelling the logarithms, but checks the root by substitution. írásbeli vizsga 03 6 / 03. május 7.

7 4. b) 0 3 x, so x 6, x = x 5, hence 5 x. Thus the equation may only have a solution 5 x If both sides are squared, the left-hand side will be 3 x. The square of the right-hand side: x 0x + 5. The quadratic equation obtained: 0 = x 8x +. Hence x = 6 or x =. The only solution on the domain of the equation is 6. Total: 7 points 5. a) There are 0 employees who have both kinds of diploma, since the total number of diplomas is 4+8=70, which is 0 more than the number of diploma holders. Thus there are with technician s certificates only. 5. b) If the number of those under 30 is x, then the mean is x ( 50 x) = x =6 There are 6 employees under 30 in the lab. Total: 4 points These points are also due if the candidate uses the equation obtained by squaring, checks the results by substitution and rejects the extraneous root. Award this point for a correct set diagram, too. The point is also due if this idea is only reflected by the solution. írásbeli vizsga 03 7 / 03. május 7.

8 5. c) 5 participants will receive financial support. 5 The number of all cases is, 5 7 The number of favourable cases is. 5 (According to the classical model, the probability is) = The probability of selecting 5 women is 0. (that is,.65%). Total: 5 points írásbeli vizsga 03 8 / 03. május 7.

9 6. a) II. B If c is the length of the third side of the triangle, (it follows from the triangle inequality that) 0 + c > and c < 0+. Therefore < c < 4. If the third side is also an integer then the smallest possible value of c is 3, and its largest possible value is 4. That means 39 possible triangles. Total: 5 points 6. b) (If γ denotes the angle enclosed by the two given sides,) 0 sin γ 88 =. Hence sin γ = o γ 3. 6 o γ Total: 4 points 6. c) o For γ 3. 6, the length (c ) of the third side is obtained by using the cosine rule. c o cos3. 6 c Hence c 8 8 units long.. o For γ 56 4 and the length (c ) of the third side: c o cos c ( ), that is c and hence c units long. The length of the third side of the triangle may be 8.8 units or 4. units long. Total: 8 points The point is also due if this idea is only reflected by the solution Award 4 points at most for this part if only one of the two cases is investigated. írásbeli vizsga 03 9 / 03. május 7.

10 7. a) The rent to be paid by Gábor increases as a geometric progression: a 00 and a 4 = = 3 = = 00 q 00, q (where the common ratio is p q = + ) 00 3 q = = 3 (.0306) p = 3.06, that is, Gábor pays 3.06% more each month. Total: 5 points 7. b) Péter s rent follows an arithmetic progression: b =00 and b 4 = 00, 00 = d 00 d = 4. 35, that is, the monthly increment is dollars. 7. c) The sums of the first 4 terms of the individual sequences: 4 3 ( ) The point is also due if this idea is only reflected by the solution. S Gábor = points S Péter = 4 = points Péter pays 3 dollars more rent than Gábor during the 4 months. Total: 6 points If the monthly rents are represented graphically, it shows that Péter would pay more rent than Gábor in each month (except the st and 4th months) Therefore Péter would clearly pay more during the 4 months than Gábor. 3 points may be awarded for a reasoning based on clear graphs. írásbeli vizsga 03 0 / 03. május 7.

11 7. d) During the first months, Péter pays a total rent of S 3 = 487 dollars, and he pays 3 dollars in the second months. 3. 4, thus Péter pays 4.% more rent in 487 the second year than in the first year. 8. a) Solution If we disregard the restriction that the two products cannot pe placed next to each other then the number of orders is 6! If the two products are placed next to each other but their order is not considered, the number of possible arrangements is 5!. If the order of the two product is also considered, there are 5! arrangements of the six products. The number of orders in question is obtained by subtracting these from the number of all cases: 6! 5!. Thus the number of possible orders of the six products is 480. Total: points 6 points The points are also due for a reasoning less detailed. Award at most if the restriction on two kinds of product not being next to each other is disregarded. 8. a) Solution Semolina and bread crumbs alone can be placed on the shelf in 6 5, that is, 30 ways if they are allowed to be next to each other as well. If their order is not considered, they may be next to each other in 5 places. Since order also matters, that makes 0 cases. Thus there are ( 30 0 = ) 0 ways to place these two product on the shelf so that they are not next to each other. In each of the 0 cases, the remaining four products may be arranged in 4! different ways. Therefore the total number of possible orders is 0 4! = 480. Total: 6 points írásbeli vizsga 03 / 03. május 7.

12 8. b) The number of bread loaves was 35 (= ), and 4 were sent back. This is.9% of the quantity ordered. The total number of rolls and croissants was 695 (=34+38), and 34 were sent back. This is 4.9% of the quantity ordered. Total: 4 points 8. c) The numbers of rolls and croissants together, sold on the individual days are 4; 33; 3; ; 50. The two days can be selected in ways. (There were 3 days when at least 30 pieces were sold.) The two days can be selected in ways, Thus the probability in question is 0.3 Total: 4 points 8. d) The quantities ordered were: -kg loaves of white bread: ( ) 3, /-kg loaves of white bread: ( ) 9, loaves of rye bread: ( 6.6) 7, points bread rolls: 58, croissants: 74. for two correct answers, no points for only one correct answer. írásbeli vizsga 03 / 03. május 7.

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