Grade 9 / Equations 88 exercises (100% corrected)

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2kg5kgx4kgxx ABC3x−2x4DEFG2x−13x−5 ChingQuizz : 10 exercises available for Quizz assessment : 1. Reminders E.841 Give the value of the following ex-pressions for x =4 : a 3 x + 4 b − 2 x + 1 + 3 × (2 x − 1) c x 2 − 2 x + 1 d x 2 − 4 x E.2505 Students were asked the fol-lowing question : ˇ Is it true that, for any value of the number x , we have : 5 x 2 − 10 x + 2 = 7 x − 4 ı Lea replied : "Yes, it’s true. Indeed, if we replace x with 3, we have : 5 × 3 2 − 10 × 3 + 2 = 17 and 7 × 3 − 4 = 17 ‘’ Myriam replied : "No,it’s not true. In fact, if we replace x with 0, we get : 5 × 0 2 − 10 × 0 + 2 = 2 et 7 × 0 − 4 = − 4 ." One of these students gave an argument that allows us to answer the question in the exercise correctly. Indicate which one and explain why. E.5250 We consider the balance for which three weights are placed on each of the two plates : The mass of each weight is noted on the front of the weight. All the weights noted ˇ x ı are identical in mass, but their mass will change over the course of the questions : 1 Which way does the scale tip when the weights noted ˇ x ı have mass 2 kg ? 2 Which way does the scale tip when the weights noted ˇ x ı have mass 10 kg ? 3 What mass should the weights ˇ x ı be assigned in order for the balance to be balanced? E.5256 Consider the equation ( E ) defined by: ( E ) : 3 x + 2 = 6 − x 1 Tell whether or not the numbers 1 and 2 are solutions of the equation ( E ) . 2 a Write the equation ( E ) by removing 4 from each member of the equation ( E ) . b Tell whether or not the numbers 1 and 2 are solutions of the equation ( E ) . 3 a Write the equation ( E  ) by multiplying each mem-ber of the equation ( E ) by 3 . b Tell whether or not the numbers 1 and 2 are solutions of the equation ( E  ) . E.811 Determine whether the following equations have x =2 as a solution : a 3 x + 1 = 2 x − 1 b 3( x + 1) − 3(2 − x ) = x + 1 c 2 x + 1 3 x + 4 = 1 2 d 3 x 2 + 4 = 4 2. Posing an equation E.4075 The chocolatier sold 315 boxes within a week. Each box contains 19 chocolates. An empty box costs 200 F . 1 Assuming a chocolate costs 100 F . a Calculate the price of a box of chocolates? b Deduce how much the sale of 315 boxes brings in dur-ing the week? 2 What should the price of a chocolate be if the chocolatier wanted to sell his 2 290 F box? E.5248 We consider the two geometric fig-ures below : Write the equation, in terms of x , characterizing the following situation : ˇLe triangle ABC and rectangle DEFG https://chingmath.fr chapExoCorrec/841 sacados/841 chapExoCorrec/2505 sacados/2505 France - Septembre 2008 chapExoCorrec/5250 sacados/5250 2kg5kgx4kgxx chapExoCorrec/5256 sacados/5256 chapExoCorrec/811 sacados/811 chapExoCorrec/4075 sacados/4075 chapExoCorrec/5248 sacados/5248 ABC3x−2x4DEFG2x−13x−5
ABCx5x have the same périmètreı E.5252 Consider the triangle ABC shown below whose side measures depend on an undetermined x : Of the three propositions below, of which equation must x be a solution in order for the triangle ABC to be rectangular in B a 25= x 2 +( x +3) 2 b ( x +3) 2 =25+ x 2 c x +3= x +5 3. First degree equation E.10966 Solve the following equations, de-tailing your approach : a 3 x + 2 = x + 6 b 5 x + 2 = 3 x + 9 E.11346 Solve the following equations : a 3 x − 5 = 3 + 2 x b 3 x + 1 = 5 x − 1 E.2373 Solve the following equations, detail-ing your approach : a 2 x − 4 = 5 x + 3 b 6 x + 7 = x − 13 E.5257 Solve the following equations, detail-ing your approach : a 3 x + 3 = 5 − 5 x b 2 − x = x + 5 E.9210 Solve the following equations : a − 5 x + 15 = − 17 x + 6 b 3 x + 2 = − 6 x + 5 E.10965 Solve the following equations, de-tailing your approach : b 7 x + 2 = − 3 x + 1 b 1 + x = − 2 x + 4 4. First-degree equation with expansion E.10967 Solve the following equations, de-tailing your approach : a 3( x − 2) + 4 = 2 − x b 5( x + 1) = 3(3 − x ) E.812 Solve the following equations, detail-ing your approach : a 2( x + 5) = 3(2 x − 2) b 2( x − 2) + 4( x − 1) = 4 E.10968 Solve the following equations : a 3( x − 4) = 4( x + 4) b 5 2(3 − x ) − 2 = 5 x + 1 E.4923 Solve the following equations : a − 2( x + 1) = 3(3 − 2 x ) b 5(3 − 2 x ) − 4( x − 2) = 5 E.9214 Solve the following equations : a 2 × ( x + 4) − 3 × (4 − x ) = 0 b 3 × (2 x + 4) − 4 = 5 x E.818 Solve the following equations : a 2 x − (3 x − 5) = 4(2 − x ) b 2( x + 1) − 3( x − 7) = 1 E.5255 We consider the two calculation pro-grams below : Program A : Choose a number; Multiply it by 3 ; Subtract 4 ; Write the final result. Program B: Choose a number; Add to it 3 ; Multiply it by − 2 ; Write the final result. 1 Let x be the number to choose so that these two cal-culation programs display the same result. Write the equation verified by the number x . 2 Solve the previous equation. 5. First-degree equation with double distributivity E.5258 Solve the following equations : a x 2 − 3 x + 5 = x 2 + 4 x + 19 b (2 x − 1)( x + 1) + ( x − 4)(3 − 2 x ) = 5 https://chingmath.fr chapExoCorrec/5252 sacados/5252 ABCx5x chapExoCorrec/10966 sacados/10966 chapExoCorrec/11346 sacados/11346 chapExoCorrec/2373 sacados/2373 chapExoCorrec/5257 sacados/5257 chapExoCorrec/9210 sacados/9210 chapExoCorrec/10965 sacados/10965 chapExoCorrec/10967 sacados/10967 chapExoCorrec/812 sacados/812 chapExoCorrec/10968 sacados/10968 chapExoCorrec/4923 sacados/4923 chapExoCorrec/9214 sacados/9214 chapExoCorrec/818 sacados/818 chapExoCorrec/5255 sacados/5255 chapExoCorrec/5258 sacados/5258
xx3cm4cmABCDEFGHIJxx E.820 After expanding and simplifying, solve the following equations : a x + 1 3 x + 5 = 3 x 2 − 5 b ( x +1) × x = x 2 +2 x +5 E.5261 Consider the two polygons shown below : où x is an indeterminate measure measured in centimeters and où : polygon ABCDEF consists of a square of side x and a rectangle of dimensions 4 cm and 3 cm . polygon GHIJ is a square of side x +2 . 1 Express the areas of the polygons ABCDEF and GHIJ in terms of x . 2 Determine the value of x so that the polygons ABCDEF and GHIJ have the same area. 6. First-degree equation with development of remarkable identities E.3756 We pose : H =( x − 4) 2 − x · ( x − 10) 1 Expand and reduce H . 2 Solve the equation : H =16 . E.10969 Solve the following equations : a ( x +1) 2 = x 2 − 3 x + 5 b x 2 − 25 = ( x + 5) 2 Hint: First, expand each side of the equation. E.821 Löse die folgenden Gleichungen : a (2 x + 1)(8 x − 1) = (4 x − 1) 2 b ( x + 1) 2 = ( x − 1) 2 E.11082 After development and reduction, solve the following equations : a 3 x − 2 2 = x + 1 9 x − 1) b x 4 x + 7 − 2 x + 1 2 x − 1 = 0 E.10970 After expanding and simplifying, solve the following equations : a (3 x + 1) 2 = (3 x − 6) 2 b (2 x + 1) 2 = (2 x − 5) 2 E.11347 After expanding and simplifying, solve the following equations : a 3 × ( x + 1) 2 = 3 x 2 − 2 b 4( x + 1) 2 = (2 x − 5) 2 7. Rational numbers and first degree equations E.1997 Solve the following equations : a 1 3 x + 3 10 = − 4 3 x − 1 5 b 3 2 x + 4 = 1 7 x − 1 14 E.1117 Solve the following equation : a − 1 3 x − 1 2 = 1 3 x + 1 4 b − 5 2 x − 2 3 = 3 4 x − 5 E.1114 Solve the following equations using cross multiplication: a 1 4 x + 1 2 = 1 3 x + 1 6 b 1 5 x + 1 3 = 1 15 x + 1 9 E.11022 Solve the following equations using the cross product : a 2 3 x − 2 = − 4 5 x − 1 2 b 3 x + 1 2 = 3 4 − 5 8 x E.1120 Solve the following equations : a 1 7 + 2 14 x = − 4 7 b 1 2 x + 3 = 1 3 x − 1 8 E.1119 Solve the following equations : a 2 3 ( x + 4) = 4 3 x + 4 b 2 3 6 x − 3 4 = x + 1 E.9211 Solve the following equations : a 3 x +2 3 − 5 2 = − 2 5 x − 3 b 3 − 2 x 4 + 3 10 = x − 3 5 − 7 2 8. Product equation https://chingmath.fr chapExoCorrec/820 sacados/820 chapExoCorrec/5261 sacados/5261 xx3cm4cmABCDEFGHIJxx chapExoCorrec/3756 sacados/3756 chapExoCorrec/10969 sacados/10969 chapExoCorrec/821 sacados/821 chapExoCorrec/11082 sacados/11082 chapExoCorrec/10970 sacados/10970 chapExoCorrec/11347 sacados/11347 chapExoCorrec/1997 sacados/1997 chapExoCorrec/1117 sacados/1117 chapExoCorrec/1114 sacados/1114 chapExoCorrec/11022 sacados/11022 chapExoCorrec/1120 sacados/1120 chapExoCorrec/1119 sacados/1119 chapExoCorrec/9211 sacados/9211
E.5266 1 Which pairs of numbers have a product equal to 0 (we say a zero product) ? (5 ; − 5) ; (2 ; 0) ; 3 ; 1 3 ; 2 ; − 1 2 0 ; 1 2 ; (0 ; − 3) ; (3 ; − 3) 2 What condition must two numbers a and b satisfy in or-der for their product to be zero? That is, so that they verify: a × b = 0 E.5262 Solve the following equations : a (2 x − 1)(3 x + 1) = 0 b ( x − 2)(2 x + 4) = 0 E.822 Solve the following product equa-tions : a x (1 − x ) = 0 b x + 1 3 x − 2 E.11083 Solve the following product equa-tions : a 3 x + 2 5 x − 8) = 0 b 8 − 3 x 2 x + 7 = 0 E.10972 Solve the following product equa-tions : a (3 x + 6)(2 x + 1) = 0 b ( x + 1)(2 − x ) = 0 E.10971 Solve the following equations : a (3 − 2 x ) x = 0 b (5 x + 1)(5 + x ) = 0 E.823 Consider the expressions : E = (4 x + 5)( x − 2) − x ( x + 4) ; F = (3 x − 10)( x + 1) 1 By expanding and reducing the expressions E and F , show that : E = F . 2 Deduce the solutions of the equation : E =0 . E.2333 We give the expression : E = ( x − 5) 2 +( x − 5)(2 x +1) 1 To calculate the exact value of E when x = 3 , Marc chose to expand E . a What expression does he get? b Calculate the exact value of E when x = 3 . c Was Marc right to develop E ? Why? 2 a Léa has mentally found a solution to the equation E = 0 . Which one do you think it is? b To find the other solution, Léa chooses to factor E . Show that : E =( x − 5)(3 x − 4) . c Give, then the second solution of the equation E =0 . 3 When x = 1 9 , choose whichever form of E you think is best for calculating the exact value of E as an irreducible fraction. Do this calculation. 9. Equation: equality of two squares E.7999 Solve the following equations : a x 2 = 10 2 b x 2 = 9 c x 2 = 5 d x 2 = − 2 E.794 Give the solutions of the following equations, giving reasons : a ( x + 1) 2 = 4 b ( x + 2) 2 = 4 E.778 1 a Give all the numbers that have a square equal to 9. What are the solutions of the equation x 2 =9 ? b Deduce the solutions of the equation : ( x +1) 2 =9 2 By similar reasoning, solve the equation : ( x − 2) 2 = 2 E.9217 We pose : I =(7 x − 3) 2 − 5 2 . 1 Factor I . 2 Solve the equation I =0 . E.9215 Solve the following equations : a ( x + 3) 2 = ( x − 2) 2 b 9 x 2 − ( x + 1) 2 = 0 10. Product equations with factorization E.5354 Solve the following equations : a 2 x 2 − 5 x = 0 b x x + 3 + 2 x + 3 = 0 E.5330 Solve the following equations : a (3 x − 2)( x + 1) + (3 x − 2)(2 − 3 x ) = 0 b ( x + 1)(2 − x ) − ( x + 1)(2 x + 5) = 0 Hint: we factor the left-hand member to get a null product equation. https://chingmath.fr chapExoCorrec/5266 sacados/5266 chapExoCorrec/5262 sacados/5262 chapExoCorrec/822 sacados/822 chapExoCorrec/11083 sacados/11083 chapExoCorrec/10972 sacados/10972 chapExoCorrec/10971 sacados/10971 chapExoCorrec/823 sacados/823 chapExoCorrec/2333 sacados/2333 Brevet 2008 chapExoCorrec/7999 sacados/7999 chapExoCorrec/794 sacados/794 chapExoCorrec/778 sacados/778 chapExoCorrec/9217 sacados/9217 chapExoCorrec/9215 sacados/9215 chapExoCorrec/5354 sacados/5354 chapExoCorrec/5330 sacados/5330
E.9865 Solve the following equations : a (3 − 2 x )( x + 1) = 3(3 − 2 x ) b (2 − 3 x )( x + 4) − (2 − 3 x )( x + 2) = 0 Hint: we factor the left-hand member to get a null product equation. E.11064 Solve the equation : 3 x + 2 5 − x − 2 x + 4 3 x + 2 = 0 Note: First, factor the left side of the equation. E.11630 On considère les fonctions f et g définies par : f ( x ) = ( x + 2) 2 − x ; g ( x ) = 7 x + 4 1 Démontrer que l’équation f ( x )= g ( x ) peut se ramener à l’équation x 2 − 4 x =0 . 2 Factoriser l’expression x 2 − 4 x . 3 En déduire les solutions de l’équation f ( x )= g ( x ) . E.9216 Modify the proposed equations to obtain zero products, then solve them : a (3 x + 1)( x − 1) − ( x − 1) 2 = 0 b (2 x − 1) 2 = (2 x − 1)(4 x + 7) E.9866 Modify the proposed equations to obtain zero products, then solve them : a (5 x + 1)( x − 2) = (5 x + 1) 2 b ( x − 2)(2 x + 1) = ( x − 2) 2 11. Develop, factor, solve E.837 The expression given is : A = ( x − 3)( x +3) − 2( x − 3) 1 Factor A . 2 Expand and simplify A . 3 By choosing the most appropriate expression for A from those found in questions 1 and 2, determine the value of A for x = − 1 and for x =0 . 4 Solve the equation : ( x − 3)( x +1)=0 E.814 Let the expression : E =( x +1) 2 + ( x +1)(2 x − 3) 1 Expand then reduce the expression E . 2 Factor the expression E . 3 Solve the equation : ( x +1)(3 x − 2)=0 E.817 Consider the expression : C =(2 x +5) 2 − ( x +3)(2 x +5) 1 Expand and simplify C . 2 Factor C . 3 Solve the equation : (2 x +5)( x +2)=0 4 Evaluate the expression C for x = − 2 3 . E.3761 1 We pose : A =( x − 1) 2 + x 2 +( x +1) 2 a Expand and reduce A . b Determine three consecutive positive integers, ( x − 1) , x , and ( x +1) whose sum of squares is 1 325 . 2 We pose : B =9 x 2 − 64 . a Factor B . b Determine the two relative numbers whose square of the triple is equal to 64 . E.4054 1 Consider the expression : A =9 x 2 − 1+(3 x − 1)(2 x +1) a Determine the expanded and reduced form of the ex-pression A . b Factor the expression 9 x 2 − 1 . Derive the factorized form of the expression A . c Solve the equation : (3 x − 1)(5 x +2)=0 d Evaluate the expression A for the following two values of x : x = − 3 ; x = 3 2 Consider the expression B defined by: B = (3 x − 2)(5 x + 3) + 2 x + 4 Justify that the two expressions A and B are equal. 12. Problems E.6313 We consider these two calcu-lation programs : Program A : Choose a number. Subtract 0.5 . Multiply the result by twice the number chosen at the beginning. Program B: Choose a number. Calculate its square. Multiply the result by 2 . Subtract from this new re-sult the number chosen at the beginning. https://chingmath.fr chapExoCorrec/9865 sacados/9865 chapExoCorrec/11064 sacados/11064 chapExoCorrec/11630 sacados/11630 chapExoCorrec/9216 sacados/9216 chapExoCorrec/9866 sacados/9866 chapExoCorrec/837 sacados/837 chapExoCorrec/814 sacados/814 chapExoCorrec/817 sacados/817 Groupe Est - Juin 2003 5,5 points chapExoCorrec/3761 sacados/3761 chapExoCorrec/4054 sacados/4054 chapExoCorrec/6313 sacados/6313
1234567ABCNombrechoisi123456ProgrammeA1615284566ProgrammeB1615284566 Programme 1Choisir un nombreLe multiplier par3Ajouter1Programme 2Choisir un nombreSoustraire 1Ajouter 2Multiplier les deux nombres 1 a Show that if we apply A to the number 10 , the result is 190 . b Apply the program B to the number 10 . 2 We used a spreadsheet to calculate results from these two programs. Here is what we got: a What formula was entered in cell C2 and then copied down? b What conjecture can be made from this table? c Prove this conjecture. 3 What two numbers should be chosen at the start to get 0 from these programs? E.6055 The following calculation pro-gram is given : Choose a number Add 1 to it. Compute the square of this sum. Remove 16 from the result. 1 a Verify that when the starting number is 4 , the result is 9 . b When the starting number is ( − 1) , what result do we get? We call P this expression. c Check that : P = x 2 +2 x − 15 2 a Check that : ( x − 3)( x +5)= P . b What numbers can be chosen at the start so that the final result is 0 ? Justify your answer. E.8683 Here are two calculation programs : 1 Verify that if we choose 5 as the starting number: a the result of the program 1 is 16 . b the result of the program 2 is 28 . We call A ( x ) the result of the program 1 as a function of the number x chosen at the beginning. The function B : x −→ ( x − 1)( x +2) gives the result of the pro-gram 2 as a function of the number x chosen at the start. 2 a Express A ( x ) in terms of x . b Determine the number that should be chosen at the start to get 0 as the result of the program 1 . 3 Expand and reduce the expression : B ( x )=( x − 1)( x +2) 4 a Show that : B ( x ) − A ( x )=( x +1)( x − 3) b What numbers must be chosen at the start for the pro-gram 1 and the program 2 to give the same result? Explain the process. E.3273 Two calculation programs are proposed : Program A Choose a number. Add 5. Calculate the square of the result obtained. Program B Choose a number. Subtract 7. Calculate the square of the result obtained. 1 We choose 5 as the starting number. Show that the result of the program B is 4. 2 We choose − 2 as the starting number. What is the result with the program A ? 3 a What number must be chosen so that the result of the program A is 0 ? b What numbers must be chosen so that the result of the program B is 9? 4 Which number should be chosen to get the same result with both programs? E.829 Subtracting the same number from the numerator and denominator of the fraction 4 5 gives the fraction 5 4 . What is this number? Leave the steps of your reasoning. E.5699 The following calculation pro-gram is proposed : Choose a number. Subtract 6 . Calculate the square of the result ob-tained. What number could be chosen so that the result of the pro-gram is the number 144 ? Justify the answer. Hint: for this question, any trace of research, even if in-complete, will be considered for evaluation 13. Problem and geometry E.5264 Consider the figure below composed of the rectangle ABCD and the triangle CDE isosceles rect-angle at D : https://chingmath.fr 1234567ABCNombrechoisi123456ProgrammeA1615284566ProgrammeB1615284566 chapExoCorrec/6055 sacados/6055 chapExoCorrec/8683 sacados/8683 Programme 1Choisir un nombreLe multiplier par3Ajouter1Programme 2Choisir un nombreSoustraire 1Ajouter 2Multiplier les deux nombres chapExoCorrec/3273 sacados/3273 Brevet juin 2010 - 4 points chapExoCorrec/829 sacados/829 chapExoCorrec/5699 sacados/5699 chapExoCorrec/5264 sacados/5264
ABCDE2x3 ABCDEFG 4x−2xABCD ABCMNx2x3cm2cm The dimensions are shown in the figure où x is a positive number. 1 a Express the area A 1 of the triangle CDE in terms of x . b Express the area A 2 of the rectangle ABCD in terms of x . 2 a Show that : 2 ×A 1 − 2 ×A 2 = x + 3 x − 1 b Determine the possible values of x so that the area of the rectangle ABCD is equal to the area of the triangle CDE . E.5697 The drawing below shows a figure composed of a square ABCD and a rectangle DEFG . E is a point on segment [ AD ] . C is a point on segment [ DG ] . In this figure the length AB can vary but we always have : AE = 15 cm ; CG = 25 cm Can we find the length AB so that the area of the square ABCD is equal to the area of the rectangle DEFG ? If yes, calculate AB . If no, explain why. Even if the exercise is not completed, any trace of research will be considered in scoring. E.5265 Consider the rectangle ABCD shown below : whose dimensions, de-pending on an indeter-minate value x , are x +1 and 4 x − 2 expressed in centimetres. The number x must be greater than 1 2 . Determine the possible values of x so that the area of ABCD , expressed in cm 2 , is equal to the perimeter of ABDC , ex-pressed in cm . 14. Problem, geometry and theorems E.4908 Consider a triangle ABC où M and N belong to the segments [ AB ] and [ AC ] , respectively, such that the lines ( MN ) and ( BC ) are parallel. The measurements are plotted in the figure où x is an un-known number. 1 Determine the length of segment [ AC ] as a function of x . 2 Show that the number x checks the equality: x + 2 2 x + 2 = 3 5 3 Determine the measure of segment [ AC ] . Hint: we will solve the equation obtained in question 2 using a cross product. https://chingmath.fr ABCDE2x3 chapExoCorrec/5697 sacados/5697 ABCDEFG chapExoCorrec/5265 sacados/5265 4x−2xABCD chapExoCorrec/4908 sacados/4908 ABCMNx2x3cm2cm
2;5x;5xxyABCMN E.9218 We consider the following configura-tion where : which verify the following properties : the triangle AMN rectangular in A ; point B belongs to side [ AM ] such that : AB =2.5 point C belongs to segment [ AB ] such that ( BC ) is par-allel to ( MN ) ; Noting x the measure of the segment [ AC ] , we have : BM = x ; BC = x +0.5 ; AC = x 1 Determine the measure of [ AC ] . 2 Determine the measure of [ CN ] . 15. Share E.11287 1 Solve the equation : 3 x − 4 3 x − 5 = 3 x + 2 2 Expand and simplify the expressions : a 3 x + 2 4 x − 5 b 3 x − 4 2 16. Unclassified exercises E.813 1 Consider the equation ( E ) defined by: ( E ) : 1 2 x + 4 3 = x − 2 3 a Give the simplified expression of the equation ( E ) ob-tained by multiplying each member of the equation ( E ) by 6 . b Solve the equation ( E ) . c What property allows us to state that the solutions of the equation ( E ) are the same as those of the equation ( E ) ? 2 Consider the equation ( F ) defined by: ( E ) : 1 5 x + 1 3 = 1 6 ( x + 1) a Give the simplified expression of the equation ( F ) ob-tained by multiplying each member of the equation ( F ) by 30 . b Deduce the solutions of the equation ( F ) . E.5260 Solve the following equations : a 1 2 x + 1 3 = 4 3 x − 5 2 b 3 5 (2 − x ) = 1 10 (2 x − 1) E.4400 Which of the following equations admit the number 2 as a solution? a 3 x + 1 = 2 x − 1 b 3( x + 1) − 3(2 − x ) = x + 1 c 2 x + 1 3 x + 4 = 1 2 d 3 x 2 + 4 = 4 E.739 Consider a disk with an area equal to 235 cm 2 . Determine the radius of this disk, rounded to the nearest mil-limeter. Recall : let r be the radius of the disk. Its area A is : A = ır 2 Indication : : we will use : ı ≈ 3.1416 E.732 We have a rectangular piece of fabric that is 15 m long and 3 m wide. Consider a square piece of fabric with the same area as the first piece of fabric. What is the length of the side of this square, to the nearest decimeter? E.764 Here is the formula that gives the dis-tance d , in meters, traveled by a skydiver in free fall during a time t expressed in seconds (, ignoring air resistance ) : d = 9 ; 81 2 t 2 . 1 Calculate the time it takes for the skydiver to complete a jump of 50 m 2 Same question for a jump of 4 000 m https://chingmath.fr chapExoCorrec/9218 sacados/9218 2;5x;5xxyABCMN chapExoCorrec/11287 sacados/11287 chapExoCorrec/813 sacados/813 chapExoCorrec/5260 sacados/5260 chapExoCorrec/4400 sacados/4400 chapExoCorrec/739 sacados/739 chapExoCorrec/732 sacados/732 chapExoCorrec/764 sacados/764
LLA0LLA1 ODD3cmx fx123ABCDEFGHNombrededépart-3-2-10123RésultatduprogrammeA3625169410RésultatduprogrammeB7557111725B2B1− E.763 Leonardo da Vinci wanted to create a sequence of rectangles with the following properties : The first rectangle has an area of 1 m 2 . The second rectangle is obtained by folding the first rect-angle in half along its longest side. It must have the same proportions as the first rectangle. That is, the ratio L ongueur L argeur must be identical for the first two rectangles. And so on ; we can create the third, fourth, and subse-quent types of sheets.. . . Let’s label these rectangles A 0 , A 1 , A 2 . . . 1 a Based on L and L , , express the ratio L ongueur L argeur in rectangle A 1 . b Given that the proportions of rectangles A 0 and A 1 are equal, deduce the following equality: L = 2 × L . c Rectangle A 0 has an area of 1 m 2 ; show that : L ≈ 1 ; 19 m et L ≈ 0 ; 84 m . Note: We have thus recreated the European paper sizes : sizes A 0 , A 2 , . . . , A 13 2 Given that most papers used have a ˇbasis weightı of 80 g = m 2 , find the weight of a ream of 500 sheets in size A 4 . E.768 Hint: We will use : ı ≈ 3 ; 1416 Consider two disks D and D with center O . Disk D has radius 3 cm . 1 Find the exact area of disk D , then round it to the nearest tenth of cm 2 . 2 What must be the ra-dius of disk D so that the area of disk D is twice that of disk D ? 3 Provide a set of construction steps to draw disk D start-ing from disk D using an unmarked ruler and a compass. E.2470 Solve the following equations and in-equations : a x (2 x − 7) = 0 b 4 x 2 = 100 E.9264 Program A. Choose a number; Subtract 3 ; Calculate the square of the result obtained. Program B. Choose a number; Calculate the square of this number; Add three times the orig-inal number; Add 7 . 1 Corinne chooses the number 1 and applies the program A . Explain in detail the calculations that show that the re-sult of the calculation program is 4 . 2 Tidjane chooses the number − 5 and applies the program B . What result does he get? 3 Lina wants to group the results of each program using a spreadsheet. She creates the spreadsheet below. Which formula, copied to the right in cells C3 à H3 , , did she enter in cell B3 ? 4 Zoé is trying to find a starting number for which the two calculation programs give the same result. To do this, she calls x the number chosen at the start and expresses the result of each calculation program as a function of x . a Show that the result of the program A as a function of x can be written in expanded and reduced form : x 2 − 6 x +9 b Write the result of program B . c Is there a starting number for which both programs give the same result? If so, which one? E.476 Find three consecutive integers whose sum equals 2 010 . E.11023 Solve the following equations : a 1 4 x = 1 6 x − 5 2 b x + 2 2 + 3 = 2 x 3 + x − 2 E.11066 1 By increasing the length of the sides of a square by 3 cm , its area becomes 144 cm 2 . What was the original area? 2 By increasing the length of the sides of a square by 5 cm , its area increases by 144 cm 2 . What was the length of its sides before the increase? https://chingmath.fr chapExoCorrec/763 sacados/763 LLA0LLA1 chapExoCorrec/768 sacados/768 ODD3cmx chapExoCorrec/2470 sacados/2470 chapExoCorrec/9264 sacados/9264 fx123ABCDEFGHNombrededépart-3-2-10123RésultatduprogrammeA3625169410RésultatduprogrammeB7557111725B2B1− chapExoCorrec/476 sacados/476 chapExoCorrec/11023 sacados/11023 chapExoCorrec/11066 sacados/11066
x4cm2cmF22x−5x−2F1 E.11633 On considère les deux figures ci-dessous : la figure F 1 est un rectangle de longueur 2 x − 5 et de largeur x − 2 . On note son aire A 1 . la figure F 2 est composé d’un carré auquel a été enlévé un rectangle de dimension 2 cm × 4 cm . On note son aire A 2 . 1 Déterminer les expressions de A 1 et A 2 en fonction de x . 2 a Etablir l’égalité des expressions : A 1 − A 2 = ( x − 6)( x − 3) b En déduire les valeurs de x pour lesquelles les figures F 1 et F 2 ont la même mesure. https://chingmath.fr chapExoCorrec/11633 sacados/11633 x4cm2cmF22x−5x−2F1