Grade 12 - Exp. / Graphs and matrices 21 exercises (100% corrected)

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ABCDE ABCD AFBCDE 1. Order and degree E.6226 Consider the graph below : 1 Give the order of the graph. 2 Complete the table below : Point A B C D E Degré du point 3 In the double entry table below, put a cross in a box if the two corresponding vertices are adjacent : A B C D E A B C D E E.6227 Consider the graph below : 1 Give the order of the graph. 2 Complete the table below : Point A B C D Degré du point 3 In the double-entry table below, put a cross in a box if the two corresponding vertices are adjacent : A B C D A B C D 2. Simple graph E.6224 Consider the graph below : 1 What is the order of this graph? 2 Complete the table below : Sommet A B C D E F Degré du summit 3 Is the sum of the degrees of all vertices equal to twice the number of edges? 4 Is this graph simple? E.6225 Justify that each of the graphs below is not a simple graph : a b c c https://chingmath.fr chapExoCorrec/6226 sacados/6226 ABCDE chapExoCorrec/6227 sacados/6227 ABCD chapExoCorrec/6224 sacados/6224 AFBCDE chapExoCorrec/6225 sacados/6225
A1A2A3A4A5A6A7A8 A1A2A3A4A5A6A7 A1A2A3A4A5A6A7 ABCDEFG CDEFG ABCE ABCD ABCDEFG ABCDEF ABCDEF E.6209 For each of the graphs below : 1 Name the order of the graph ; 2 Name the degree of vertex A 2 ; 3 Give the vertices adjacent to vertex A 5 ; 4 Say whether the graph is simple or not. Fig 1: Fig 2: Fig 3: 3. Subgraph E.6229 Consider the graph below : Which of the graphs below are sub-graphs of the main graph? a b c 4. Stable subgraph E.6272 A dog-sitting company currently accom-modates seven dogs, represented in the graph below by the vertices of the graph ; the edges represent dogs that cannot be locked in the same cage. 1 What is the largest stable subgraph containing a maxi-mum number of vertices? 2 What is the minimum number of cages needed to keep these seven dogs without risking confrontations between them? 5. Complex and related E.6245 Consider the two graphs below : https://chingmath.fr chapExoCorrec/6209 sacados/6209 A1A2A3A4A5A6A7A8 A1A2A3A4A5A6A7 A1A2A3A4A5A6A7 chapExoCorrec/6229 sacados/6229 ABCDEFG CDEFG ABCE ABCD chapExoCorrec/6272 sacados/6272 ABCDEFG chapExoCorrec/6245 sacados/6245 ABCDEF ABCDEF
ABCDEFG Specify whether these graphs are related or not. 6. Chain and cycle E.6242 Consider the graph below : Which of the ordered lists below form a string of this graph? 1 A − E − F − E − G − D 2 E − F − A − B − C 3 D − A − C − D 4 D − C − E − A − F − E 7. Introduction to matrices E.6142 We call ˇ magic matrix ı any matrix où the sums of the coefficients per row, per column , by the main diagonal and by the second diagonal all give the same result. Which of the following matrices are magic matrices? a 9 4 5 2 6 10 7 8 3 b 2 7 6 9 5 1 4 3 8 c 8 1 3 0 5 7 4 6 2 E.6141 Consider the matrix A defined by: A = 2 3 − 1 5 3 0 4 − 2 1 3 − 4 7 To highlight the coefficients of the matrix, we note : A = a ij 1 Give the value of the following matrix coefficients : a a 23 b a 34 c a 11 d a 31 2 Determine the value of the following calculations : a a 11 + a 21 b a 32 × a 14 c a 24 2 3 Name the set of mutually opposite pairs of coefficients of the matrix. E.6123 A company lists the prices in euros of the products it uses in conjunction with its various suppliers. It obtains the following double-entry table : Product 1 Product 2 Product 3 Fournisseur 1 3.50 2.80 1.75 Fournisseur 2 3.25 3 1.70 He notes all these values in the matrix M : M = 3.50 2.80 1.75 3.25 3 1.70 The various numbers placed in the matrix are called ˇ the coefficients of the matrix ı. Generally speaking, m ij are the coefficients of the matrix où i and j are two integers used to locate the place of the coefficient in the matrix. More precisely, coefficient m ij is the price of ˇ Product j ı at ˇ Supplier i ı. 1 a Give the values of the following coefficients : m 12 ; m 22 ; m 13 b Complete the following sentence : ˇ The coefficient m ij lies in the matrix at the intersec-tion of row . . . and column . . . ı 2 Write the new matrix N = n ij where the coefficient n ij has the value of the product price i at supplier i . 8. Adjacent matrix of a non-oriented graph E.6171 Definition: Let G be a graph with n vertices ( n ∈ N ∗ ) . We call the ad-jacency matrix of the graph G the matrix M = m ij square of order n whose coefficients have the value : m ij = 1 si the vertex A i est connected to the vertex A j by a arête 0 sinon pour i;j ∈ [[1; n ]] https://chingmath.fr chapExoCorrec/6242 sacados/6242 ABCDEFG chapExoCorrec/6142 sacados/6142 fichierPlus/6142/ chapExoCorrec/6141 sacados/6141 chapExoCorrec/6123 sacados/6123 chapExoCorrec/6171 sacados/6171
A1A2A3A4A5A6A7 A1A2A3A4A5A6A7A8 ABCDEFG ABCDEF ABCDEF ABCDEF ABCDEFG A1A2A3A4A5A6A7 The graph below represents 7 cities connected by roads : Give the adjacency matrix of this graph. E.6196 During an election campaign, a politician must tour the eight cities rated from A 1 to A 8 , using the motorway network. The graph G below, represents the various cities on the tour and the freeway sections connect-ing these cities (a city is represented by a vertex, a freeway section by an edge) : Give the expression for the M adjacency matrix of the graph G . E.6228 In the graph below, the ver-tices represent different zones of residence or activity in a mu-nicipality. An edge connecting two of these vertices indicates the existence of a main access route between two correspond-ing locations. 1 Give, without justification, the degree of each of the ver-tices (the answer may be presented in tabular form où the vertices will be put in alphabetical order) . 2 Give the matrix M associated with the graph (vertices will be put in alphabetical order) . E.6330 Consider a graph G admitting the matrix M as adjacency ma-trix. Of the representations below, only one corresponds to the graph G . M = 0 1 0 0 0 1 1 0 1 0 1 0 0 1 0 0 1 0 0 0 0 0 1 1 0 1 1 1 0 0 1 0 0 1 0 0 a b c Specify the correct representation and justify your choice, ex-plaining why the other two representations were discarded. 9. Adjacent matrix of a directed graph E.6230 Give the adjacency matrix of the graph below : E.6170 Give the M adjacency matrix of the directed graph given below : 10. Introduction to matrix operations E.6712 Indication : during the fiscal year, a matrix M = m ij will have its coefficient m ij which will express the ˇ price of the product j at the supplier i . ı https://chingmath.fr A1A2A3A4A5A6A7 chapExoCorrec/6196 sacados/6196 A1A2A3A4A5A6A7A8 chapExoCorrec/6228 sacados/6228 ABCDEFG chapExoCorrec/6330 sacados/6330 ABCDEF ABCDEF ABCDEF chapExoCorrec/6230 sacados/6230 ABCDEFG chapExoCorrec/6170 sacados/6170 A1A2A3A4A5A6A7 chapExoCorrec/6712 sacados/6712
A company lists the price in euros of the products it uses in connection with its various suppliers. It obtains the following double-entry table : Product 1 Product 2 Product 3 Fournisseur 1 3.50 2.80 1.75 Fournisseur 2 3.25 3 1.70 1 Write the matrix A related to these prices... 2 The suppliers decide on an increase on their products, the amounts of which are given in the table below : Product 1 Product 2 Product 3 Fournisseur 1 0.50 0.20 0.25 Fournisseur 2 0.25 0 0.3 Write the B matrix corresponding to these increases. 3 Write the matrix C corresponding to the new prices of these products at each of the suppliers. E.6153 A company sources its raw materials from two different countries. Consider the following two tables : Alu M1 Bois M2 Fer M3 Pays 1 25 14 12 Pays 2 23 11 11 Material prices premières Alu M1 Bois M2 Fer M3 Pays 1 3 2 1 Pays 2 2 3 1 Price from transport 1 a Write the matrix A = a ij ij où the coefficient a ij represents ˇ the price of the material j in the country i ı. b Write the matrix B = b ij ij où the coefficient b ij rep-resents ˇ the transport price of the material j in the country i ı. c Write the matrix C = c ij ij où the coefficient c ij rep-resents ˇ the purchase price of the material j in the country i ı. 2 To start production, the company needs 2 units of Aluminium, 1 unit of wood and 3 units of Iron. We represent these data in the matrix D = d ij given below : D = 2 1 3 a Give an interpretation of the number: c 11 × d 11 + c 12 × d 21 + c 31 × d 31 b Determine the total price of its purchases when the company sources from the country 2 ? E.6143 Three tri-athletes take part in one race. The course times for each event are given in the table below, expressed in minutes : Jacques 1 er athlète Henry 2 e athlète Paul 3 e athlète Course à pied (épreuve 1) 45 62 34 Natation (épreuve 2) 35 24 28 Cyclisme (épreuve 3) 92 95 104 We wish to represent the data of this double entry table by a matrix A = a ij où the coefficient a ij is the running time, expressed in minutes, of the athlete i during the event j . 1 a Give the values of the following coefficients : a 13 ; a 32 b Write the matrix A . 2 Consider the matrix B = b ij o where the coefficient b ij is the duration, expressed in hours and rounded to the near-est hundredth, of the course time taken by the athlete i during the event j . a Write the matrix B b What simple mechanism allows you to go from matrix A to matrix B ? https://chingmath.fr chapExoCorrec/6153 sacados/6153 chapExoCorrec/6143 sacados/6143