BREVET BLANC N 1 JANUARY 2012

Transcription

1 Exercise. (5 pts) duration: h Presentation and explanations (4 points) Numerical Activities Consider the figure opposite, which is made up of two squares.. a) Calculate the area A of the white part. b) For which value of x is the value of A zero?. a) Calculate the area A of the grey part and expand and simplify your answer. b) Expand the expression ( x+ 5) ( x+ 7) and thus deduce the factorised form of A. c) Solve the equation( x )( x ) BREVET BLANC N JANUARY = 0. Exercise. (5 pts) Questions and are independent.. Consider the function f defined by f : x x. a. Calculate the images of the numbers and then under f. b. Find the arguments of under f.. Consider the function g whose graph is drawn below : a) Using the graph, find the images of and. b) Using the graph, find the arguments of 0 and then of. Brevet Blanc n Page sur 8

2 Exercise. (6 pts) This exercise is Multiple Choice. No justification is required. For each of the questions, three answers are proposed, of which only one is right. No points will be taken off if a wrong answer is given. For each of the questions, circle the right answer. Proposed answers 5 7. =, The standard form of, is, , 005, ( ) = 0 = The equation x = 4x The inequation x has the 6. numbers x for solutions where Has no solutions Has 0 for solution x x Has 0 and 0.5 for solutions x Brevet Blanc n Page sur 8

3 GÉOMÉTRICAL ACTIVITIÉS (0 pts) In the figure opposite, (DE) // (BC). AD = 5 cm, AE = 4 cm and BC = 7 cm. The distance DE is denoted x. ST PART : In this part, we take x = 5.6 cm. - Calculate AC and prove that AB = 6.5 cm. Carefully justify your calculations and give the name of the properties you use. - Calculate the perimeter of the triangle ABC. - Draw the figure its actual size. ND PART : In this part, we calculate as a function of x. - Explain why AC = 8 and AB = x 5. x Carefully justify your calculations and give the name of the property you use. - From the above, deduce that the perimeter p(x) of the triangle ABC in terms of x is p ( x ) = x - Copy and fill in the following table of values of the function p : x p ( x ) Brevet Blanc n Page sur 8

4 4- On the set of axes below, draw the graph of the function p for x between and 6. Use the following units : cm for 0.5 units on the x-axis, cm for 5 units on the y-axis. 5- From the graph, find the values of x (rounded to decimal place) for which the perimeter of the triangle ABC is equal to : a- 50 cm, b- 5 cm. Write the answers carefully on your sheet and make the corresponding dotted lines appear on the graph. Brevet Blanc n Page 4 sur 8

5 Problems Exercise. (.5 pts) You are not asked to reproduce the figure. The unit is cm. ABCD is a rectangle. Given AE = AD = and EB = x,. Calculate the perimeter of ABCD as a function of x. (justify). Solve the equation x + = 0.. Find the value of x for which the perimeter of ABCD is 0. Exercise. ( pts) Pieces of jewellery are made from triangles which all have the same shape. Some are made out of metal (represented in grey), others out of glass (represented in white). Three examples of these pieces of jewellery are shown below. All the metal triangles cost the same price. All the glass triangles cost the same price. Piece of jewellery n is worth ; Piece of jewellery n is worth 9.0. Piece of jewellery n Piece of jewellery n Piece of jewellery n What is the price of piece of jewellery n? Exercise. Even if your work is not finished, leave proof of your research: it will be taken into account in the mark. (.5 pts) The human heart beats at about 5000 beats per hour.. Give the standard form of Calculate the number of heartbeats in a life that lasts 80 years (Consider that there are 65 days in a year) Give the result in standard form and show details of your calculations. Brevet Blanc n Page 5 sur 8

6 Exercise. (5 pts). a) The area of the white part is A ( x ) b) A ( x ) Numerical Activities = + 4. = + 4 = 0 if, and only if x + 4 = 0,so for 4 x = 4. a) The area of the grey part is A A ( x 4) Exercise. ( )( ) ( ) = = +. So A = x + x + = x + x + x + = x x = x x b) ( x + 5)( x + 7) = 9x 45x + x + 05 = 9x 4x + 05 = A. ( x 5)( x 7) thus the factorised form of A. x 5 x 7 0 c) The equation( + )( + ) = is a zero-product equation : ( x )( x ) ( x + 5) = 0 or ( x ) (5 pts). Consider the function f defined by + 7 = 0. The two only solutions are thus f : x x. + + is = 0 only if 5 x = = 5 and 7 7 x = = a. The image of is f ( ) = = and the image of - is f ( ) = ( ) = 9 = 5. BREVET BLANC N CORRECTION b. The arguments of are the solutions of the equation x = 0. So 0 is the only argument of - under f. x =,which is equivalent to. a) The image of is -. The image of is 0. (in red) b) Graphically, the arguments of 0 are -, and 5 (in blue) Those of are -.4 and.7 (in green) Exercise. 5 (6 pts) = = = The standard form of, = = = ( ), is = = = The equation x = 4x is equivalent to x 4x = 0 or x ( x) = 0 which has two solutions (nullproduct) : 0 and 0.5 The inequation x is equivalent to x = (multiplying both sides by /) Brevet Blanc n Page 6 sur 8

7 GEOMETRICAL ACTIVITIES (0 pts) ST PART : In this part, we take x = 5.6 cm. - Since (DE) // (BC), ADE and ABC form a Thales configuration and thus, according to AD AE DE Thales theorem = =. So AB AC BC AB = AC = 7. We thus have 4 7 AC = = 5. Now AB = = The perimeter of the triangle ABC is AC + CB + BA = = 8.5 cm. ND PART : In this part, we calculate in terms of x Since (DE) // (BC), ADE and ABC form a Thales configuration and thus, according to AD AE DE 5 4 x Thales theorem = =. So = =. We thus have AC = =. AB AC BC AB AC 7 x x Now x AB = = 5 4 x 5 =. 4 x - The perimeter p(x) of the triangle ABC in terms of x is p( x) = AC + CB + BA = = + 7. x x x - x p ( x ) Graphically the values of x for which the perimeter of the triangle ABC is equal to 50 cm is.5 cm Graphically the values of x for which the perimeter of the triangle ABC is equal to 5 cm is.5 cm Brevet Blanc n Page 7 sur 8

8 Problems Exercise. (.5 pts) ABCD is a rectangle. We have : AE = AD = and EB = x. AE + EB + AD = + x + 6 = x +. The perimeter of ABCD with respect to x is ( ) ( ). x + = 0 x = 8 x = 4.. The value of x for which the perimeter of ABCD is 0 is 4 (from the two previous questions). Exercise. ( pts) To make pieces of jewellery n, you need 6 metal triangles and 0 glass ones, exactly the same as to make piece n and piece n. The price of pieces of jewellery n is thus 9.+=0. The price of piece of jewellery n is thus Exercise. (.5 pts) The human heart beats about 5000 times per hour.. The standard form of is The number of heart beats in a life of 80 years is = Brevet Blanc n Page 8 sur 8