Grade 10 / Algorithms 15 exercises (including 0 corrected)

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On choisitAetBdeux nombresrest di∑érent de 0?Le PGCD estBˆAprend pour valeurBˆBprend pour valeurrOn e∑ectue la division euclidienne deAparB.Le reste est a∑ecté àrOne∑ectueladivisioneuclidi-ennedeAparB.Le reste est a∑ecté àrNonOuiABCD1D2E Donner une valeur deAA∑ecter àXla valeurA×2−5A`cher la valeur deXA Donner une valeur deAA∑ecter àXla valeurA×2−5Xest supérieur ou égal à 10?A`cherAXA`cherA−XOuiNon Donner une valeur deAA∑ecter àXla valeurA−3Tant queXest inférieurou égal à10A∑ecter àXla valeurA×XA`cherXSortiedeboucle 1. General study of algorithms E.3043 The diagram below shows Euclid’s algorithm for determining the greatest common factor of two integers : Here is the execution of this algorithm with the values : A =254 ; B =16 : A A =254 and B =16 B r =14 C Oui D 1 A = 16 et B = 14 D 2 r =2 C Oui D 1 A = 14 et B = 2 B r =0 C Non D 2 GCF (254 ; 16)=2 1 By analogously reproducing the table below, determine the GCF of the following integers : a A =1542 ; B =36 b A = 18 ; B = 543 2 For the values in question b , what does the algorithm perform at the start of this algorithm? E.3042 Consider the algorithm shown in the graph below : Complete the table below : Value of A 0 3 12 5 2 − 4 Value affichée E.3045 Consider the algorithm shown in the graph below : Complete the table below : Value of A 5 8 − 2 0 21 Value affichée E.3046 Consider the algorithm shown in the graph below : 1 Justify that by assigning the value 5 to the variable A , the algorithm displays the value 50? 2 Determine the value displayed by the algorithm in the following cases : a A = 10 b A = 4 c A = − 2 3 a What happens when we assign the value 0 to the variable A ? b Find another example where the algorithm never ter-minates. 2. First use of algoBox https://chingmath.fr sacados/3043 On choisitAetBdeux nombresrest di∑érent de 0?Le PGCD estBˆAprend pour valeurBˆBprend pour valeurrOn e∑ectue la division euclidienne deAparB.Le reste est a∑ecté àrOne∑ectueladivisioneuclidi-ennedeAparB.Le reste est a∑ecté àrNonOuiABCD1D2E sacados/3042 Donner une valeur deAA∑ecter àXla valeurA×2−5A`cher la valeur deXA sacados/3045 Donner une valeur deAA∑ecter àXla valeurA×2−5Xest supérieur ou égal à 10?A`cherAXA`cherA−XOuiNon sacados/3046 Donner une valeur deAA∑ecter àXla valeurA−3Tant queXest inférieurou égal à10A∑ecter àXla valeurA×XA`cherXSortiedeboucle
Lire les nombresAetBTant que Le reste de la di-vision deAparBvaut 0?Le PGCD estBˆAprend pour valeurBˆBprend pour valeurrOn e∑ectue la division euclidienne deAparBSortiedeboucle Lire le nombrexA`cher 1Pouriallant de 2 àxLe reste de la division euclidiennedexpariest vaut 0?A`che " - "A`cheiFin de l’algorithmeSortiedebouclesOuiNon E.3069 1 The command floor is used to obtain the integer part of a number; suppose the variable a has the value 3.1415926535 a Give the value of floor(a*10) . b Deduce the command to obtain the default value of a to the nearest tenth ; to the nearest hundredth. 2 Determine the remainders of the following Euclidean di-visions : a 10 par 3 b 33 par 5 c 27 par 4 d 69 par 8 The command a % b returns the remainder of the Eu-clidean division of a by b : e What can be the values of a %2 ? of 2 % a ? f Using a conditional structure, write an algorithm that asks for the input of a value and then displays the sen-tences ˇthis number is pairı or ˇ this number is impair ı as appropriate. 3 The command sqrt(2) returns the square root of the number 2: a Write an algorithm asking the user for four numbers representing the coordinates of two points, and return-ing the distance separating these two points. b Modify the algorithm so that it displays the default value of this distance to the nearest tenth. E.3044 1 Enter the algorithm below in the programming language of your choice : For a ranging from 1 to 25 x ← a × a End For 2 By a step-by-step execution of this algorithm, give the set of values that will be assigned to the variable x . E.3068 1 Enter the algorithm below in the programming language of your choice : a ← 0 As long as a<100 x ← A%2 If x=0 Then y ← a End If a ← a+1 End As long as 2 When executing the step-by-step algorithm, what are dif-ferent values assigned to the variable y . 3 Modify this algorithm so that the variable y is succes-sively assigned all multiples of 13 less than 100 . E.3070 1 Enter the algorithm below in the programming language of your choice : Function f(a) a ← a As long as a>=1 a ← a − 1 End While Return a 2 a Make a call to the function f with each of the fol-lowing values : 1 ; 2 ; 3 ; 4 ; 9 ; 10 b What is the role of the function f ? 3. Algorithm creation E.3071 The diagram below shows Euclid’s algorithm. Build this algorithm using algoBox. E.3090 Consider the algorithm below where the variables x and i are of type number: https://chingmath.fr sacados/3069 sacados/3044 sacados/3068 sacados/3070 sacados/3071 Lire les nombresAetBTant que Le reste de la di-vision deAparBvaut 0?Le PGCD estBˆAprend pour valeurBˆBprend pour valeurrOn e∑ectue la division euclidienne deAparBSortiedeboucle sacados/3090 Lire le nombrexA`cher 1Pouriallant de 2 àxLe reste de la division euclidiennedexpariest vaut 0?A`che " - "A`cheiFin de l’algorithmeSortiedebouclesOuiNon
VARIABLESxEST_DU_TYPENOMBREyEST_DU_TYPENOMBREiEST_DU_TYPENOMBREDEBUT_ALGORITHMEPOURiALLANTDE-5A5DEBUT_POURxPREND_LA_VALEURiyPREND_LA_VALEURx*xTRACER_POINT(x,y)FIN_POURFIN_ALGORITHME VARIABLESx1EST_DU_TYPENOMBREy1EST_DU_TYPENOMBREx2EST_DU_TYPENOMBREy2EST_DU_TYPENOMBREiEST_DU_TYPENOMBREDEBUT_ALGORITHMEPOURiALLANTDE10A100DEBUT_POURx2PREND_LA_VALEURi/10y2PREND_LA_VALEURsqrt(x2-1)TRACER_POINT(x2,y2)FIN_POURFIN_ALGORITHME 1 Build this algorithm using Algobox. 2 Mathematically, what is the purpose of this algorithm? 4. Drawing of curves E.3091 1 a In AlgoBox and in the tab ˇ Draw in a marker ı, check the box ˇ Use a repère ı ; enter the following values for the axis terminals : Xmin:-5 ; Xmax:5 ; GraduationsX:1 Ymin:0 ; Ymax:25 ; GraduationsY:1 b Enter the following algorithm into AlgoBox: c Run the algorithm to observe its display. d What does this algorithm appear to display? 2 We want to plot more points representing this curve, to do this we want to modify the iterative loop so that the abscissas of the points are 0.1 apart in 0.1 : a Edit line x PREND_LA_VALEUR i to : x PREND_LA_VALUE i/10 b Run the algorithm to observe the effect of these modi-fications. c What modification needs to be made to the algorithm so that the representative curve is displayed over the interval − 5 ; 5 ? d Apply these changes and run this algorithm again. E.3092 Consider the function f defined by: f ( x ) = x − 1 In a reference frame O ; I ; J orthogonal, note C f the repre-sentative curve of the function f . 1 a Determine the definition set of the function f . b Determine the coordinates of the point on the curve C f having abscissa 1 . 2 a In AlgoBox and in the ˇ tab Draw in a marker ı, check the ˇ box Use a repère ı ; enter the following values defin-ing the axis bounds : Xmin:0 ; Xmax:9 ; GraduationsX:1 Ymin:0 ; Ymax:3 ; GraduationsY:1 b Enter the following algorithm into AlgoBox: c Run the algorithm and observe the resulting graph. 3 The aim of this question is to draw the curve C f by seg-ments connecting each of the previous points. a Delete the command ˇ TRACER_POINT (x2,y2) ı to re-place it with the command TRACER_SEGMENT connect-ing the points with coordinates (x1,y1) and (x2,y2) . b Run the algorithm to observe the changes. The plot is made up of segments only. What is the common origin of all these segments? Why or why not? c Before defining the loop for and in relation to the question 1 b , correctly initialize the values of x1 and y1 in order to improve the curve plot C f . d To plot the curve C f segment by segment, the algo-rithm must connect the current point with the previ-ous point. Just before the end of the loop POUR , make the point with coordinates (x1,y1) represent the coordinate point (x2,y2) for the next loop execution. 5. Observation of the law of large numbers E.3108 1 a Enter the following algorithm in the algorithm of your choice : c ← 0 For i ranging from 0 to 100 x ← random value belonging to 0 ; 1 x ← integer part of 3 × x End For b Running this algorithm step by step, what values are assigned to the variable ˇ x ı? https://chingmath.fr sacados/3091 VARIABLESxEST_DU_TYPENOMBREyEST_DU_TYPENOMBREiEST_DU_TYPENOMBREDEBUT_ALGORITHMEPOURiALLANTDE-5A5DEBUT_POURxPREND_LA_VALEURiyPREND_LA_VALEURx*xTRACER_POINT(x,y)FIN_POURFIN_ALGORITHME sacados/3092 VARIABLESx1EST_DU_TYPENOMBREy1EST_DU_TYPENOMBREx2EST_DU_TYPENOMBREy2EST_DU_TYPENOMBREiEST_DU_TYPENOMBREDEBUT_ALGORITHMEPOURiALLANTDE10A100DEBUT_POURx2PREND_LA_VALEURi/10y2PREND_LA_VALEURsqrt(x2-1)TRACER_POINT(x2,y2)FIN_POURFIN_ALGORITHME sacados/3108
VARIABLESxEST_DU_TYPENOMBREiEST_DU_TYPENOMBREmaxEST_DU_TYPENOMBREDEBUT_ALGORITHMEmaxPREND_LA_VALEUR100POURiALLANTDE1A100DEBUT_POURxPREND_LA_VALEURrandom()xPREND_LA_VALEURfloor(x*3)TRACER_POINT(10*i/max,x)FIN_POURFIN_ALGORITHME VARIABLESborneMinEST_DU_TYPENOMBREborneMinEST_DU_TYPENOMBRExEST_DU_TYPENOMBREDEBUT_ALGORITHMETANT_QUE(borneMax-borneMin>pow(10,-3))FAIREDEBUT_TANT_QUESI(x<(borneMin+borneMax)/2)ALORSDEBUT_SIborneMaxPREND_LA_VALEUR(borneMin+borneMax)/2FIN_SISINONDEBUT_SINONborneMinPREND_LA_VALEUR(borneMax+borneMin)/2FIN_SINONAFFICHERborneMinAFFICHER"-"AFFICHERborneMaxFIN_TANT_QUEFIN_ALGORITHME 2 a Add a conditional structure inside the POUR loop so that the : c ← c+1 are executed whenever the variable ˇ x ı is assigned the value 2 . b Run the algorithm several times and observe the value of the variable c . Can we explain the variations in the values of the vari-able c ? 3 a Modify the algorithm so that the loop performs 500 iterations and add the instruction below at the end of the algorithm: f ← c 500 b Run this algorithm several times and observe the vari-ations in the value of the variable f at the end of the algorithm. 4 What can be done to make the variations in the variable f stabilize? E.3109 1 a Enable the use of a marker in AlgoBox by taking the following parameters : Xmin : 0 Xmax: 10 Graduations X: 1 Ymin : 0 Ymax: 10 Graduations Y: 1 b Enter the following algorithm in AlgoBox: c Execute this algorithm. What does it do? 2 a Using the previous exercise, modify the present al-gorithm so that it displays the frequency of occurrence of the number 2 (in the variable x ) . b Modify the algorithm so that it displays the straight line with equation y=1/3 . c Increase the number of runs of this algorithm. What observation can be made when running the algorithm? 6. Dichotomy E.3146 1 Enter the algorithm below : 2 a Execute this algorithm with the following values : xMin=1 ; borneMax=3 ; x=1.9384 b By observing the successive values taken by borneMin and borneMax , to what value do the numbers borneMin and borneMax point? 3 Modify this algorithm so that these two values approach 2 . https://chingmath.fr sacados/3109 VARIABLESxEST_DU_TYPENOMBREiEST_DU_TYPENOMBREmaxEST_DU_TYPENOMBREDEBUT_ALGORITHMEmaxPREND_LA_VALEUR100POURiALLANTDE1A100DEBUT_POURxPREND_LA_VALEURrandom()xPREND_LA_VALEURfloor(x*3)TRACER_POINT(10*i/max,x)FIN_POURFIN_ALGORITHME sacados/3146 VARIABLESborneMinEST_DU_TYPENOMBREborneMinEST_DU_TYPENOMBRExEST_DU_TYPENOMBREDEBUT_ALGORITHMETANT_QUE(borneMax-borneMin>pow(10,-3))FAIREDEBUT_TANT_QUESI(x<(borneMin+borneMax)/2)ALORSDEBUT_SIborneMaxPREND_LA_VALEUR(borneMin+borneMax)/2FIN_SISINONDEBUT_SINONborneMinPREND_LA_VALEUR(borneMax+borneMin)/2FIN_SINONAFFICHERborneMinAFFICHER"-"AFFICHERborneMaxFIN_TANT_QUEFIN_ALGORITHME