Grade 10 / Colinearity and parallelism 106 exercises (100% corrected)

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ChingQuizz : 9 exercises available for Quizz assessment : 1. Opposite vectors and subtractions E.8530 Definition: Let −→ u be a vector. The opposite vector of vector −→ u is the vector denoted by − −→ u , defined as : the same direction as vector −→ u the opposite direction to vector −→ u the same length as −→ u In the plane, consider the 7 vectors below : 1 Name the vector(s) opposite to vector −→ u . 2 Draw a vector −→ e opposite to vector −→ d . E.524 Definition: Let −→ u and −→ v be two vectors. We define the subtraction of vector −→ u by vector −→ v by the relation: −→ u − −→ v = −→ u + − −→ v In other words, we never subtract a vector, we add its op-posite Let −→ u and −→ v be two vectors in the plane. 1 What can we say about the difference : −→ u − −→ u ? 2 For each of the three cases shown below, draw a repre-sentation of the subtraction : −→ u − −→ v E.495 Determine a representative of each of the sums below : 1 −→ EI − −−→ GF 2 −−→ HE + −→ BI − −→ JF 3 −−→ FG − −→ IF − −−→ GE E.9342 Let ABCD be a parallelogram. Note: I the middle of segment [ AB ] ; J the middle of segment [ DC ] . Determine in each case a representative of the resulting vec-tor: a −→ AC − −→ AJ b −−→ AB + −→ IJ − −→ DJ c −−→ DJ − −→ JB E.8531 In the plane, consider the two vec-tors −→ u and −→ w shown below. Draw the vector −→ v realizing the relation: −→ u − −→ v = −→ w E.6488 The plane is given a reference frame ( O ; −→ i ; −→ j ) and the three points : A 1+ 2 ; 2 − 2 ; B 2 − 1 ; 2+1 ; C 2+3 ; 3 − 3 2 1 Show that the vectors −−→ AB and −→ AC are opposite. 2 What can be said about the point A relative to the seg-ment [ BC ] . 2. Multiplication by an integer https://chingmath.fr chapExoCorrec/8530 sacados/8530 −u−w−v−a−b−c−d chapExoCorrec/524 sacados/524 uvuvuv chapExoCorrec/495 sacados/495 ABCDJHIGEF chapExoCorrec/9342 sacados/9342 chapExoCorrec/8531 sacados/8531 −u−w chapExoCorrec/6488 sacados/6488
E.9713 Proposition: in the plane, consider a vector −→ u and an integer n ∈ N ∗ . We define the vector n · −→ u by: n · −→ u = −→ u + −→ u + · · · + −→ u  n fois In the plane, consider the three vectors and three points shown below : 1 Place the point B such that : −−→ AB = 3 · −→ u 2 Place the point C such that : −−→ CD = 2 · −→ v 3 Place the point F such that : −→ w = 4 · −−→ EF E.515 On a graduated line, are placed the points A , B , C , D , E : For each question, complete the dotted lines correctly: a −−→ BC = : : : : : : ×−→ AC b −−→ ED = : : : : : : ×−→ AC c −→ AC = : : : : : : ×−→ CA d −−→ ED = : : : : : : ×−→ CA e −→ EA = : : : : : : ×−−→ AB f −−→ BA = : : : : : : ×−−→ BE E.9716 The drawing below shows a straight line with a regular scale. Complete the blanks with the missing number: a −−→ DG = : : : : : : −−→ DE b −−→ CE = : : : : : : −→ GI c −−→ DB = : : : : : : −−→ DF d −→ EI = : : : : : : −→ AC E.5287 On a graduated line, we place the points A , B , C , D , E : For each question, determine the value of the number k veri-fying the equality: a −−→ BC = k · −→ AC b −−→ ED = k · −→ AC c −→ AC = k · −→ CA d −−→ ED = k · −→ CA e −→ EA = k · −−→ AB f −→ AC = k · −−→ BA E.9811 Let A , B , C be three non-aligned plane points : Simplify the expression : 2 · −−→ AB − −−→ BA E.2077 Consider, in the plane, the two vec-tors −→ u and −→ v below : Draw in the grid a representative of the vector −→ w defined by: −→ w = 2 −→ u + 3 −→ v . E.2106 Consider the plane below, with a regular grid. Let −→ u and −→ v be two vectors of the plane : 1 Draw a representative −→ w of the vector −→ v −−→ u . 2 Draw a representative −→ x of the vector 4 −→ u +3 −→ v . E.4813 Consider the parallelogram ABCD shown below où the points I and J are the respective middles of the segments [ AB ] and [ CD ] . For each question, give without justification a vector equal to the proposed expression : a 2 ×−→ DJ + −−→ BD b 3 · −→ DJ + 2 · −→ IA c 2 · −→ AJ − −−→ BC 3. Multiplications by a real https://chingmath.fr chapExoCorrec/9713 sacados/9713 −u−v−wADE chapExoCorrec/515 sacados/515 ABCDE chapExoCorrec/9716 sacados/9716 ABCDEFGHIJK chapExoCorrec/5287 sacados/5287 ABCDE chapExoCorrec/9811 sacados/9811 chapExoCorrec/2077 sacados/2077 −u−v chapExoCorrec/2106 sacados/2106 −u−v chapExoCorrec/4813 sacados/4813 ABCDIJ
E.523 Definition: In the plane, consider a vector −→ u and a num-ber k ∈ R . The vectors −→ u and k · −→ u are collinear, and : −→ u et k · −→ u −→ v k< 0 opposite direction − k × −→ u k =0 × 0 k> 0 same direction k × −→ u The figure below shows a line with evenly spaced markings. Fill in the dotted line with the missing number: a −−→ EF = : : : : : : −→ GJ b −−→ BD = : : : : : : −−→ CF c −→ JI = : : : : : : −→ AC d −−→ EF = : : : : : : −→ JG e −→ AE = : : : : : : −−→ FC f −−→ CG = : : : : : : −−→ KA E.484 Let A and B be two points of the plane, and let I be the midpoint of the segment [ AB ] 1 Complete the dotted lines to verify the following vector relationship : −→ AI + −→ AI = −−−→ A:::::: 2 Copy and complete with the words ˇ double ı and ˇ moitié ı the following sentences : a −→ AI est . . . de −−→ AB b −−→ AB est . . . de −→ AI 3 In relation to the previous question, complete the dotted lines with the appropriate number: a −→ AI = : : : : : : −−→ AB b −−→ AB = : : : : : : −→ AI E.485 Let ABC be any triangle. Place the points D and E verifying the following vector relations : −−→ AD = 2 · −−→ AB ; −→ AE = 2 · −→ AC Compare −−→ BC and −−→ DE . Justify. E.6544 Consider the two concentric circles of center O and whose radius is twice that of the other : 1 Justify vector equality: −→ LJ = 2 · −−→ DB 2 Without justification, complete the equalities: a −−→ ED = −−−→ : : : : : : = 1 2 · −−−→ : : : : : : = 1 2 · −−−→ : : : : : : b −−→ FB = 2 · −−−→ : : : : : : = 2 · −−−→ : : : : : : = 1 2 · −−−→ : : : : : : E.509 Consider below the two vectors −→ u and −→ v : 1 Draw a vector −→ w representing from : 3 · −→ u +1.5 · −→ v 2 Draw a vector −→ y representing from : −→ v − 1.5 · −→ u 4. Simplification and algebraic manipulation E.9809 In the plane, consider the three points A , B , C and define the two vectors −→ r and −→ s by: −→ r = 5 · −→ AC − 2 · −−→ BC ; −→ s = 2 · −−→ AB + 3 · −→ AC Show that the two vectors −→ r and −→ s are equal. E.8532 Consider the parallelogram ABCD shown below où the points I and J are the respective middles of the segments [ AB ] and [ CD ] . Using the points in the figure, give a representative of the https://chingmath.fr chapExoCorrec/523 sacados/523 ABCDEFGHIJK chapExoCorrec/484 sacados/484 chapExoCorrec/485 sacados/485 chapExoCorrec/6544 sacados/6544 OABCDEFGHIJKLMNPQ chapExoCorrec/509 sacados/509 −u−v chapExoCorrec/9809 sacados/9809 chapExoCorrec/8532 sacados/8532 ABCDIJ
sum : 2 · −→ AJ + 2 · −−→ CB E.8533 In the plane, consider A , B , C three non-aligned points of the plane. For each question, determine the value of the real k verifying the equality: a 3 · −−→ AB − −−→ CB + −→ CA = k · −−→ AB b 3 −−→ AB − −−→ BC + −→ AC + 2 · −−→ BA = k · −−→ AB 5. Vectors and distributivity E.2047 Proposition: in the plane, consider the two vectors −→ u and −→ v and a real k . We have the two identities below : k · −→ u + −→ v = k · −→ u + k · −→ v k · −→ u − −→ v = k · −→ u − k · −→ v Let −→ u and −→ v be two vectors. Simplify each of the following vector sums : a 3 −→ u − 2 −→ v + 2 −→ u − −→ v b 2 · −→ u + −→ v − −→ u d − −→ u + −→ v + 2 · −→ u − −→ v e 2 3 · 2 · −→ u − 3 2 · −→ v − 1 6 −→ u E.2947 Let A , B , C be three non-aligned points of the plane : For each question, determine the value of the real k verifying the proposed relationship : a 2 · −−→ AB +2 −−→ BC + −→ AC = k · −→ AC b 3 · −−→ AB − 3 · −−→ CB = k · −→ AC E.2048 1 a Place three points A , B and C not aligned in the plane. b Draw a representative of the sum : −→ u = −−−→ AB − 2 · −−→ BC + 2 · −→ AC c What conjecture can we make? 2 Establish that : −→ u = −−→ AB Hint: We’ll use the Chasles relationship : −−→ BC = −−→ BA + −→ AC E.9720 In the plane, consider A , B , C three non-aligned points of the plane. For each question, determine the value of the real k verifying the equality: a 2 · −−→ AB + 2 · −−→ BC + −→ AC = k · −→ AC b −−→ AB + 2 · −→ AC + 4 −−→ BC = k · −→ AC + −−→ BC 6. Decomposition in a vector basis E.937 Definition: vectors −→ i and −→ j are said to form a vector basis if they have different directions. The graph below shows two vectors −→ i and −→ j with differ-ent directions. The aim of this exercise is to decompose any vector in the plane as a function of the vectors −→ i and −→ j . 1 a Draw a representative of the vector −→ y defined by: −→ y = 4 ×−→ i b Place the point B such that : −−→ AB = −→ y 2 a Place the point D such that : −−→ CD = −−→ i . b Place the point F such that : −−→ EF = − 3 ·−→ j 3 a Place the point H such that : −−→ GH =2 ·−→ i . b Place the point K such that : −−→ HK =4 ·−→ j . c Complete the following equality: −−→ GK = : : : : : : · −→ i + : : : : : : · −→ j 4 Complete the following dotted lines : −→ u = : : : : : : · −→ i ; −→ v = : : : : : : · −→ j −−→ LM = : : : : : : · −→ i + : : : : : : · −→ j 5 Complete the following dotted lines : a −→ w = : : : · −→ i + : : : · −→ j b −→ z = : : : · −→ i + : : : · −→ j c −→ r = : : : · −→ i + : : : · −→ j d −→ s = : : : · −→ i + : : : · −→ j https://chingmath.fr chapExoCorrec/8533 sacados/8533 chapExoCorrec/2047 sacados/2047 chapExoCorrec/2947 sacados/2947 chapExoCorrec/2048 sacados/2048 chapExoCorrec/9720 sacados/9720 chapExoCorrec/937 sacados/937 jisuLvMrwzGACE
E.486 1 Place the point W such that : −−→ AW = −−→ i +2 ·−→ j . Place the point Z such that : −−→ BZ = − 2 −→ i + 5 3 ·−→ j The vectors −→ i and −→ j are two vectors of different directions. For any vector −→ u of the plane, there exist two real numbers ¸ and ˛ achieving equality: −→ u = ¸ · −→ i + ˛ · −→ j This decomposition is called ˇ linear combination ı. 2 a Determine the linear combination of each of the vec-tors shown in the graph as a function of the vectors −→ i and −→ j . b Deduce the equality of the vectors −−→ QR and −→ a ? 3 a Draw a representative of the vector −→ u + −→ v . b Graphically, express the vector −→ u + −→ v using the vec-tors −→ i and −→ j . c Find this decomposition using those of the vectors −→ u and −→ v obtained in question 2 . E.5290 In the plane, consider any triangle ABC . Note respectively I and J the respective symmetries of B and C with respect to A : In the reference frame A ; −−→ AB ; −→ AC , give the coordinates of the following vectors : a −→ IA b −→ AJ c −−→ BC d −−→ CB e −→ IJ f −→ IC E.9808 In the plane, consider the three points A , B , C . Consider the vector −→ v defined by: −→ v = 5 6 · −→ AC − 1 2 · −−→ AB − 1 3 · −−→ BC + 1 3 · −→ CA 1 Express the vector −→ v as a function of −−→ AB and −→ AC : 2 In the reference frame A ; −−→ AB ; −→ AC , give the coordi-nates of the vector −→ u E.7215 In the plane, consider the reference frame O ; −→ i ; −→ j and the two non-collinear vectors −−→ AB and −−→ CD colinear vectors shown below : Without justification, give the decompositions of the vectors −−→ AB and −−→ CD in the base −→ i ; −→ j . 7. Vector base and introduction to coordinate operations E.8534 Definition: in the plane equipped with a coordinate sys-tem O ; I ; J , we call the unit vector with abscissa (or ordinate) , noted −→ i (or −→ j ) , the vector −→ OI (resp. −→ OJ ) . In the plane equipped with a coordinate system O ; I ; J , consider the two points A and B shown below : 1 Decompose the vector −−→ AB into the vector basis −→ i ; −→ j . That is, complete the dotted lines in the equation : https://chingmath.fr chapExoCorrec/486 sacados/486 −j−i−u−v−aCDEFGHKLMNOPQRSTBA chapExoCorrec/5290 sacados/5290 ABCIJ chapExoCorrec/9808 sacados/9808 chapExoCorrec/7215 sacados/7215 O−i−jABCD chapExoCorrec/8534 sacados/8534 x-6-5-4-3-2-12Iy-12JOijAB
−−→ AB = : : : : : : ×−→ i + : : : : : : ×−→ j 2 a Give the coordinates of points A and B . b Determine the coordinates of the vector −−→ AB . 3 What comparison can be made between the coordinates of a vector and its decomposition in the vector basis of the unit vectors of the coordinate system? E.527 We equip the plane with an orthonor-mal coordinate system O ; −→ i ; −→ j . 1 Give, without proof, the coordinates of the vectors −→ u and −→ v . 2 a Draw a line segment representing the vector −→ w de-fined by: −→ w = 3 · −→ u . b Graphically, find the coordinates of −→ w . c Compare the coordinates of vectors −→ u and −→ w . 3 a Draw a line segment representing vector −→ z defined by: −→ z = −→ u + −→ v b Graphically, find the coordinates of −→ z . c Compare the coordinates of vector −→ z relative to those of vectors −→ u and −→ v . 8. Coordinates of points E.9814 Consider the rectangle ABCD shown below Consider the plane provided with the reference frame A ; −−→ AB ; −−→ AD . 1 a Complete the blanks in the equality: −→ AC = : : : · −−→ AB + : : : · −−→ AD Deduce the coordinates of point C . b Give the coordinates of the points A , B , D . 2 The point E is the midpoint of the segment [ CD ] . De-termine the coordinates of point E . 3 The F is defined by the equality: −−→ CF = 1 3 · −→ CA Give the coordinates of the point F . 9. Operations on coordinates E.8535 Proposition: Let −→ u ( x ; y ) and −→ v ( x ; y ) be two vectors and k a real number ( k ∈ R ) . The vector −−→ u , opposite to the vector −→ u , has the following coordinates : −−→ u =( − x ; − y ) The sum −→ u + −→ v has the following coordinates : −→ u + −→ v ( x + x ; y + y ) The vector k ×−→ u has the following coordinates : k ×−→ u ( k × x ; k × y ) https://chingmath.fr chapExoCorrec/527 sacados/527 -4-3-2-101234-3-2-1123ijuv chapExoCorrec/9814 sacados/9814 ABCEDF chapExoCorrec/8535 sacados/8535
In the plane with the coordinate system O ; −→ i ; −→ j , consider the two vectors −→ u and −→ v shown below : 1 Give the coordinates of the vectors −→ u and −→ v . 2 Let −→ w be the vector defined by: −→ w = −→ u +2 · −→ v a Give the coordinates of the vector −→ w . b Draw the vector −→ w . 3 Let −→ z be the vector defined by: −→ z = −→ u − −→ v a Give the coordinates of the vector −→ z . b Draw the vector −→ z E.8536 We consider equipped with the ref-erence frame O ; −→ i ; −→ j orthonormal and the three points A , B , C shown below : 1 a Give, without justification, the coordinates of the vectors : −−→ AB ; −−→ BC ; −→ AC b Determine the coordinates of the vector −→ u defined by: −→ u = 3 · −−→ AB − −−→ CB + −→ CA 2 Determine the unique real number k ( k ∈ R ) verifying: −→ u = k ×−−→ AB E.8537 In the plane provided with a refer-ence frame O ; −→ i ; −→ j , consider the three points A , B , C defined by: A (2 ; − 3) ; B ( − 4 ; 2) ; C (0 ; − 1) 1 Determine the coordinates of the vector −→ u defined by: −→ u = 2 ×−−→ AB + 2 ×−−→ BC + −→ AC 2 What simplified expression does the vector −→ u admit? 10. Operations and search for the coordinates of a point E.516 Consider a plane equipped with an arbitrary coordinate system O ; −→ i ; −→ j and the following three points, determined by their coordinates : A (2 ; 1) ; B (3 ; 2) 1 Determine the coordinates of the vector 3 · −−→ AB . 2 Find the coordinates of the point D such that : −−→ AD = 3 · −−→ AB . E.307 In the plane with the origin O ; I ; J , consider the three points A , B , and C with coordinates : A (2 ; 1) ; B ( − 1 ; 3) ; C (0 ; − 2) Determine the coordinates of point M that satisfy the follow-ing vector relationship : −−→ CM = 2 · −−→ AB E.8539 Consider the plane provided with a O ; −→ i ; −→ j any coordinate system and the following three points determined by their coordinates : A (2 ; 1) ; B (3 ; 2) ; C ( − 1 ; − 1) 1 Determine the coordinates of the vector defined by the expression : 2 · −−→ AB − 4 · −→ AC 2 Determine the coordinates of the point E verifying the relation: −→ AE =2 · −−→ AB − 4 · −→ AC E.8201 Consider the plane provided with a reference frame O ; −→ i ; −→ j . Consider the three points A , B , C verifying the following relationships : 2 −→ OA (4 ; 6) ; 3 −−→ AB (9 ; 3) ; 2 −−→ BC − −−→ OB = −→ u where the vector −→ u has coordinates : −→ u (3 ; 3) Determine the coordinates of the points A , B and C . https://chingmath.fr -5-4-3-2-101234-1123ij−u−v chapExoCorrec/8536 sacados/8536 -6-5-4-3-2-10123456-4-3-2-112ijABC chapExoCorrec/8537 sacados/8537 chapExoCorrec/516 sacados/516 chapExoCorrec/307 sacados/307 chapExoCorrec/8539 sacados/8539 chapExoCorrec/8201 sacados/8201
E.8538 In the plane provided with a refer-ence frame O ; −→ i ; −→ j , consider the three points A , B and C with coordinates : A (2 ; 1) ; B ( − 1 ; 3) ; C (0 ; − 2) Determine the coordinates of the point N verifying the fol-lowing vector relation: 4 · −−→ AN − −−→ BN − 2 · −−→ CN = −→ 0 E.518 Consider a plane equipped with an or-thonormal coordinate system ( O ; −→ i ; −→ j ) and the three points A , B , and C with coordinates ( − 2 ; 1) , (0 ; 3) , and (3 ; 0) , re-spectively. 1 a Find the coordinates of the vectors −−→ AB and −→ AC . b Find the coordinates of the vector −−→ AB + −→ AC . 2 Consider the point D that satisfies the relation: −−→ AB + −→ AC = −−→ AD a Let ( x D ; y D ) denote the coordinates of the point D . Justify that the following two equalities hold : x D + 2 = 7 y D − 1 = 1 b Use this to find the coordinates of point D . E.11715 Dans le plan muni d’un repère O ; I ; J , on considère les trois points A , B , C : A (5 ; − 2) ; B ( − 3 ; 1) ; C (4 ; 4) Déterminer les coordonnées du point M tel que : −−→ CM = 4 · −−→ AB 11. Vector collinearity E.8543 Definition: Let −→ u and −→ v be two non-zero vectors in the plane. Two vectors are said to be collinear if there exists a real number k such that : −→ u = k · −→ v The real number k is called the coefficient of collinearity of −→ u with respect to −→ v . 1 Let −→ u and −→ v be two vectors such that : 2 −→ u = 3 −→ v Justify that vectors −→ u and −→ v are collinear and that their collinearity coefficient is 3 2 . 2 Let −→ u and −→ v be two vectors such that : −→ u + −→ v = −→ 0 . Justify that these two vectors are collinear. 3 For each of the questions below, vectors −→ u and −→ v are collinear. Determine the value of the collinearity coeffi-cient of −→ u with respect to −→ v : a 1 2 · −→ u = 3 4 · −→ v b 3 · −→ u − 2 · −→ v = −→ 0 E.6998 In the plane, consider the three points O , A , B below and the vector −−→ AB : 1 a Draw the vector −−−→ A B image of the vector −−→ AB by the homothety of center O and ratio 3 . b Draw the vector −−−→ A  B  image of the vector −−→ AB by the homothety of center O and ratio − 1 2 . 2 What can we say about the vectors −−→ AB , −−−→ A B and −−−→ A  B  ? E.520 Consider the plane equipped with a coordinate system O ; −→ i ; −→ j . For each question, determine whether the two vectors −→ u and −→ v are collinear. If they are, give the associated collinearity coefficient of −→ u with respect to −→ v : a −→ u − 1 2 ; −→ v 4 − 8 b −→ u 3 2 ; −→ v 9 4 c −→ u 2 3 ; −→ v 4 ; 2 6 ; 3 d −→ u 0 ; 7 4 ; 1 ; −→ v − 2 ; 8 16 ; 4 E.5295 For each question, specify whether the vectors −→ u and −→ v are collinear and, if so, give the coeffi-cient of collinearity of the vector −→ u with respect to the vector −→ v : a −→ u ( − 2 ; − 10) et −→ v (4 ; 20) b −→ u (0 ; 5) et −→ v ( − 5 ; 0) https://chingmath.fr chapExoCorrec/8538 sacados/8538 chapExoCorrec/518 sacados/518 chapExoCorrec/11715 sacados/11715 chapExoCorrec/8543 sacados/8543 chapExoCorrec/6998 sacados/6998 ABO chapExoCorrec/520 sacados/520 chapExoCorrec/5295 sacados/5295
E.8202 In the plane provided with a refer-ence frame O ; −→ i ; −→ j , consider the five points : A (2 ; − 2) ; B (11 ; − 14) ; C ( − 3 ; 1) ; D (5 ; 3) ; E (12 ; − 19) Of the four vectors above, only one is collinear with the vector −−→ AB . Which is it? Justify your answer. −−→ BC ; −−→ CD ; −−→ DE ; −−→ CE E.9813 For each question, specify whether the vectors −→ u and −→ v are collinear and, if so, give the coeffi-cient of collinearity of the vector −→ u with respect to the vector −→ v : a −→ u ( − 6 ; 9) et −→ v 1 4 ; − 1 2 b −→ u − 4 3 ; 4 et −→ v (3 ; − 9) c −→ u 1 3 ; 2 5 et −→ v (5 ; 6) d −→ u (6 ; − 5) et −→ v 14 5 ; − 2 E.6624 The plane is given a reference frame O ; I ; J and the points A , B and C below are considered : 1 a Give the coordinates of points A , B and C . b Determine the coordinates of the vectors −−→ AB and −−→ BC . c Deduce the coordinates of the vector −→ v defined by: −→ v = −−→ AB + 2 · −−→ BC 2 Justify that the vectors −→ u and −→ v are collinear. E.500 In the plane, consider the points A , B , C , D and E such that : C is the middle of [ AF ] ; B is the middle of [ AD ] . The quadrilateral ABEC is a parallelogram. Let’s equip the plane with the reference frame A ; B ; C any. 1 In the reference frame A ; B ; C , give, without justifica-tion, the coordinates of the six points in this plane. 2 Justify that the points E , F and D are aligned. 12. Colinearity and algebraic manipulation E.9718 Let A , B , C and D be four points of the plane verifying the relation: −−→ AB + −−→ AD = −→ AC Show that the vectors −−→ AB and −−→ CD are collinear. E.9812 For each of the questions below, the vectors −→ u and −→ v are collinear. Determine the value of the collinearity coefficient of −→ u with respect to −→ v : a 3 · −→ u − 2 · −→ v = −→ 0 b − 2 · −→ u + −→ v = 2 · −→ u + 3 · −→ v E.4812 Consider the three points A , B and C shown in the grid below : 1 a Place the point M verifying the vector relation: −−→ AM = 2 · −→ CA https://chingmath.fr chapExoCorrec/8202 sacados/8202 chapExoCorrec/9813 sacados/9813 chapExoCorrec/6624 sacados/6624 -3-2-12345I-2-123JOABCu chapExoCorrec/500 sacados/500 ABCDEF chapExoCorrec/9718 sacados/9718 chapExoCorrec/9812 sacados/9812 chapExoCorrec/4812 sacados/4812 ABC
b Place the point N verifying the vector relation: −−→ AN = −−→ AB + 2 · −−→ CB c What conjecture can be made about the vectors −−→ AB and −−→ MN ? 2 Demonstrate, using vector calculus, establish the equal-ity: −−→ MN = 3 · −−→ AB E.9717 Let A , B , C and D be four points of the plane such that : −−→ AD + −−→ BD + 2 · −−→ CB = −→ 0 1 Establish equality: −−→ AB = − 2 · −−→ CD 2 What can be said about the vectors −−→ AB and −−→ CD ? E.510 Let A , B , C and D be four points of the plane verifying the relation: −→ AC − 3 · −−→ BD + 2 · −−→ BC = −→ 0 Show that the vectors −−→ AB and −−→ CD are collinear. E.501 In each case, consider three points A , B , C of the plane verifying a vector relationship. Show that in each case, the points A , B and C are aligned : a 3 · −−→ AB + −−→ BC = − 2 · −→ AC b − 2 · −−→ AB = 3 × −−→ CB + −→ CA E.2917 In the plane, shown below fitted with a grid, consider the points A , B , C , M : Give a representative of the vector −→ u defined by the relation: −→ u = 2 · −−→ AB + −−→ CB − −→ AC 1 Place the point N such that : −−→ MN = −→ u . 2 We define the vector −→ v defined by: −→ v = −−→ CB + 1 3 · −→ AC Show that the vectors −→ u and −→ v are collinear. E.5293 Let A , B , C and D be four points in the plane such that : 5 · −−→ AD = 2 · −→ AC + 3 · −−→ BD Show that the vectors −−→ AB and −−→ CD are collinear. E.9719 Let A , B , C and D be four points in the plane such that : 3 · −−→ AD + 4 · −−→ BC = 7 · −→ AC Show that the vectors −−→ AB and −−→ CD are collinear. E.8122 Consider the four points A , B , C and D verifying the vector relation: 2 · −−→ DC + 5 · −−→ CB + 5 · −−→ AD − 3 · −−→ AB = −→ 0 Demonstrate that the vectors −−→ AB and −−→ CD are collinear. E.2055 Let A , B , C be three points in the plane verifying the relation: − 1 2 · −−→ AB + 5 2 · −−→ BC − −−→ BA + −−→ CB = −→ 0 1 Show that these three points verify: −−→ AB = 3 2 · −→ AC 2 What can be said about the points A , B , C ? E.5343 In the plane, consider a ABC non-aplatized triangle. Consider the three points M , N and P defined by: −−→ BM = 1 3 · −−→ BA ; −−→ BN = 1 2 · −−→ BC ; −→ AP = 2 · −→ AC Show that the points M , N and P are aligned. E.2903 Let A , B , C be three points in the plane. Show that the vector −→ u defined below is collinear with the vector −→ AC by: −→ u = 3 · −−→ AB + 2 3 · −−→ BC − 5 3 · −→ CA + 7 3 · −−→ BA E.5294 Consider a triangle ABC and M a point belonging to side [ AB ] verifying the relation: AM = 2 3 · AB P is the point of intersection of the line ( BC ) and the par-allel to ( AC ) passing through the point M . N is the point of intersection of the straight lines ( AC ) and the parallel to ( AB ) passing through the point P 1 Make a representation of this configuration. 2 Show that : AN = 1 3 · AC ; CP = 2 3 · CB . 3 Decompose the vectors below in terms of the vectors −−→ AB and −→ AC : a −→ AP b −−→ MC 4 Decompose the vectors below in terms of the vectors −→ CA and −−→ CB : a −→ AP b −−→ NM 13. Introduction to the collinearity criterion E.8364 In the plane provided with a ref-erence frame O ; I ; J , consider two vectors −→ u ( x ; y ) and −→ v ( x ; y ) such that : 0 <x <x ; 0 <y <y https://chingmath.fr chapExoCorrec/9717 sacados/9717 chapExoCorrec/510 sacados/510 chapExoCorrec/501 sacados/501 chapExoCorrec/2917 sacados/2917 ABCM chapExoCorrec/5293 sacados/5293 chapExoCorrec/9719 sacados/9719 chapExoCorrec/8122 sacados/8122 chapExoCorrec/2055 sacados/2055 chapExoCorrec/5343 sacados/5343 chapExoCorrec/2903 sacados/2903 chapExoCorrec/5294 sacados/5294 chapExoCorrec/8364 sacados/8364
Consider the two points A and B such that : −→ OA = −→ u ; −−→ OB = −→ v 1 a Express the areas of the following figures in terms of x , x , y and y : OBB ; OAA ; AA B B b Deduce the expression for the area of triangle OAB as a function of x , x , y and y . 2 a What can be said about the points O , A , B when : x × y − x × y =0 ? b Is the reciprocal true? 14. Criteria of collinearities E.8541 Definition: Let −→ u ( x ; y ) and −→ v ( x ; y ) be vectors. We call the determinant of vectors −→ u and −→ v , noted det( −→ u ; −→ v ) , defined by: det( −→ u ; −→ v ) = x × y − x × y For each of the pairs of vectors −→ u and −→ v defined below, de-termine the value of det( −→ u ; −→ v ) : a −→ u (2 ; − 1) ; −→ v (3 ; 4) b −→ u ( − 5 ; 1) ; −→ v (2 ; − 2) E.11689 Dans le plan muni d’un repère O ; −→ i ; −→ j , on considère les points : A ( − 3 ; 2) ; B (1 ; 4) ; C (5 ; − 1) Déterminer la valeur de det −−→ AB ; −→ AC . E.5288 Proposition: In the plane provided with a reference frame, consider the two vectors −→ u and −→ v . The two vectors −→ u and −→ v are collinear with each other if, and only if, their determinant is zero. Consider the plane provided with a reference frame ( O ; −→ i ; −→ j ) and the four points : A (3 ; − 5) ; B (1 ; − 1) ; C (13 ; 2) ; D (18 ; − 8) Establish that the vectors −−→ AB and −−→ CD are collinear. E.504 Consider the plane equipped with a coordinate system O ; −→ i ; −→ j and the two vectors : −→ u 1 − 2 √ 3 3+ 2 ; −→ v 6 − 3 − 3 − 6 Prove that the vectors −→ u and −→ v are collinear. E.512 In the plane marked with the refer-ence point O ; −→ i ; −→ j , consider the following two vectors : −→ u 2 + 3 2 1 − 10 ; −→ v 5 2 + 4 3 2 − 2 5 Show that the vectors −→ u and −→ v are collinear. E.8542 In the plane provided with a refer-ence frame O ; −→ i ; −→ j , consider the two collinear vectors : −→ u x + y 2 ; 4 ; −→ v 2 2 − 1 ; − 2 where x and y are two relative integers. Determine the values of x and y . E.7888 The plane is given a reference frame O ; −→ i ; −→ j orthonormal and we consider the points : A 1 4 ; 1 3 ; B 1 ; 5 6 ; C − 1 2 ; 7 6 Show that the vectors −−→ AB and −→ AC are not two collinear vec-tors. E.11716 Dans le plan muni d’un repère O ; I ; J , on considère les trois points : A (5 ; 1) ; B (9 ; 3) ; C ( − 9 ; − 5) 1 Déterminer les coordonnées des vecteurs −−→ AB et −→ AC . 2 Est ce que les vecteurs −−→ AB et −→ AC sont colinéaires? Jus-tifier votre réponse. 15. Parallelism and collinearity E.499 In the plane, consider the coordinate system ( O ; −→ i ; −→ j ) and the points : O (49 ; − 100) ; P (14 ; 5) ; Q (1 ; − 85) ; R ( − 58 ; 92) Determine whether the lines ( OP ) and ( QR ) are parallel. E.5289 In the plane provided with a reference frame ( O ; −→ i ; −→ j ) , consider the three points : A ( − 3 ; − 1) ; B (1 ; 5) ; C ( − 1 ; 2) Show that the points A , B , C are aligned. https://chingmath.fr OIJAABBuvuv chapExoCorrec/8541 sacados/8541 chapExoCorrec/11689 sacados/11689 chapExoCorrec/5288 sacados/5288 chapExoCorrec/504 sacados/504 chapExoCorrec/512 sacados/512 chapExoCorrec/8542 sacados/8542 chapExoCorrec/7888 sacados/7888 chapExoCorrec/11716 sacados/11716 chapExoCorrec/499 sacados/499 chapExoCorrec/5289 sacados/5289
E.6507 In the plane provided with a refer-ence frame O ; −→ i ; −→ j , consider the four points : A (2 ; − 5) ; B ( − 2 ; 2) ; C ( − 4 ; 5) ; D 2 ; − 11 2 Justify that the straight lines ( AB ) and ( CD ) are parallel. E.5296 In a plane with an orthonormal co-ordinate system O ; −→ i ; −→ j . Consider the following four points in the plane : A − 2 ; − 6 ; B 2 2 ; 0 ; C ( − 2 √ 3 ; 3 ) ; D (0 ; 5) Show that the straight lines ( AB ) and ( CD ) are parallel. E.8540 The plane is given a reference frame ( O ; −→ i ; −→ j ) and the three points : A 2 2+1;3+2 √ 2 ; B 2 − 1; 2+1 ; C 2+5; 2+7 Show that the points A , B and C are aligned. E.5313 Consider the plane provided with the reference frame O ; −→ i ; −→ j shown below : Consider the four vectors below : −→ u 9 4 ; − 3 4 ; −→ v 7 2 ; − 3 2 ; −→ w − 15 4 ; 5 4 1 Represent the three vectors −→ u , −→ v and −→ w with the point O as their origin. 2 a Conjecture the collinearity of vectors −→ u , −→ v and −→ w with each other. b Establish your conjecture. E.1144 In a plane with an orthonormal co-ordinate system O ; −→ i ; −→ j , consider the points : D (5 ; − 2) ; E ( − 3 ; 10) ; F ( − 3 ; − 2) ; G (3 ; − 11) Show that the lines ( DE ) and ( FG ) are parallel. Hint: To show that the lines ( DE ) and ( FG ) are parallel, it suffices to show that the vectors −−→ DE and −−→ FG are collinear. 16. Modeling and collinearity E.9806 In the plane, consider the rectan-gle ABCD such that AB =4 cm and AD =2 cm and the two points E and F verifying: F ∈ AB and AF =9 cm E ∈ AD and AE =3.6 cm The plane is given the reference A ; −−→ AB ; −−→ AD . 1 a Complete the following equalities: −→ AE = : : : −−→ AD ; −→ AF = : : : −−→ AB b In the reference frame A ; −−→ AB ; −−→ AD , give the coordi-nates of the six points of this plane. 2 Determine the coordinates of the vectors −−→ CE and −−→ CF . 3 Deduce that the points C , E , F are aligned. E.5824 In the plane, consider the parallelo-gram ABCD . Let I be the midpoint of segment [ AB ] and J the point on segment [ AC ] verifying the relation: AJ = 1 3 · AC The plane is given the reference frame A ; −−→ AB ; −−→ AD . 1 Determine the coordinates of points D , I and J . 2 Demonstrate that the points D , I and J are aligned. https://chingmath.fr chapExoCorrec/6507 sacados/6507 chapExoCorrec/5296 sacados/5296 chapExoCorrec/8540 sacados/8540 chapExoCorrec/5313 sacados/5313 -4-3-2-101234-2-112ij chapExoCorrec/1144 sacados/1144 chapExoCorrec/9806 sacados/9806 ABCDEF chapExoCorrec/5824 sacados/5824 ABCDIJ
E.7214 Consider the figure below consisting of the two squares ABEF and BCDE : Note G the point of intersection of the straight lines ( AD ) and ( BF ) and H the point in the plane verifying the vector relation −→ AG = −−→ GH 1 Placing yourself in the reference frame A ; −−→ AB ; −→ AF , de-termine the coordinates of the point G . 2 Establish that the points E , H and C are aligned. E.5394 Consider the figure above composed of a square ABCD and two equilateral triangles DIC and BJC : In this question any trace of research, however incomplete, or initiative however unsuccessful, will be taken into account in the assessment. Show that the points A , I , J are aligned. (In an equilateral triangle of side a , we admit that all its heights have length a 3 2 ) . E.5393 Consider the triangle oppo-site où I and G are the re-spective middles of the seg-ments [ AB ] and [ CI ] , the point J is defined by the re-lation: −→ CJ = 1 3 · −→ CA Consider the vector basis −−→ AB ; −→ AC . 1 Express the vectors −→ AI and −→ AJ in the vector basis −−→ AB ; −→ AC . 2 Establish that the vector decomposition of the vector −→ AG : −→ AG = 1 4 · −−→ AB + 1 2 · −→ AC 3 Deduce the alignment of points B , G , J . 17. Colinearity and coordinate search E.5291 Consider the plane provided with a reference frame O ; −→ i ; −→ j . Let A , B , C and D be four points in the plane with coordi-nates : A ( − 5 ; 1) ; B (2 ; 4) ; C ( − 1 ; − 2) ; D (3 ; y D ) Determine the coordinates of the point D such that the straight lines ( AB ) and ( CD ) are parallel and the point D has 3 as its abscissa. E.8200 Consider the plane provided with a reference frame O ; −→ i ; −→ j . Let A , B and C be three points in the plane with coordinates : (4 ; − 1) ; (1 ; 3) ; (1 ; − 2) Determine the coordinates of the point D such that the straight lines ( AB ) and ( CD ) are parallel and the point D has 3 as its abscissa. E.5314 In a plane with a reference frame O ; −→ i ; −→ j , consider the three points A , B , C with coordi-nates : A (1 ; 2) ; B − 2 ; 5 2 ; C ( − 1 ; 4) Determine the value of x so that the point D of coordinates ( x ; 3) is such that the straight lines −−→ AB and −−→ CD are collinear. E.5822 In the plane provided with a refer-ence frame O ; −→ i ; −→ j orthonormal, consider the following three points : A ( − 1 ; 1) ; B ( − 3 ; − 1) ; C (2 ; 3) 1 Are the points A , B and C aligned? Justify your answer. 2 Determine the coordinates of the single point D with ab-scissa − 2 such that the straight lines ( AB ) and ( CD ) are parallel. E.5746 Consider the plane with a reference frame O ; −→ i ; −→ j and the four points : A ( − 3 ; 2) ; B (2 ; − 1) ; C (1 ; 5) ; D (7 ; 2) 1 Are the straight lines ( AB ) and ( CD ) parallel? 2 Determine the coordinates of the point E with abscissa 7 so that the vectors −−→ AB and −−→ CE are collinear. https://chingmath.fr chapExoCorrec/7214 sacados/7214 ABCDEFGH chapExoCorrec/5394 sacados/5394 ABCDIJ chapExoCorrec/5393 sacados/5393 ABCIJG chapExoCorrec/5291 sacados/5291 chapExoCorrec/8200 sacados/8200 chapExoCorrec/5314 sacados/5314 chapExoCorrec/5822 sacados/5822 chapExoCorrec/5746 sacados/5746
E.2080 Consider the plane with any refer-ence point ( O ; −→ i ; −→ j ) and the following three points in the plane : A (3 ; 2) ; B ( − 1 ; 3) ; D ( − 4 ; − 1) 1 Determine the coordinates of the point C such that ABCD is a parallelogram. 2 Determine the coordinates of the point E belonging to the x-axis such that : ( BD ) == ( AE ) . E.507 Consider a plane with a coordinate system O ; −→ i ; −→ j and the three points : A 4+4 2 ; 3+3 2 ; B 2 − 2 ; 2 2+1 ; C 2+1 ; y C where y C is a real number. Given that the points A , B , and C lie on a straight line, deter-mine the y-coordinate of point C in the form a + b 2 , where a and b are two real numbers. E.6625 Consider the plane provided with a reference frame O ; −→ i ; −→ j . Let A , B , C , D be four points in the plane with re-spective coordinates : A 2 ; 8 3 ; B 2 3 ; 2 ; C 4 5 ; 0 ; D x D ; − 1 2 where x D is a real number. Knowing that the straight lines ( AB ) and ( CD ) are parallel, determine the coordinates of the point D . 18. Vector geometry E.2107 Consider the plane provided with a O ; −→ i ; −→ j orthonormal datum : 1 Place the three points A , B , C in the frame below : A (3 ; − 3) ; B ( − 4 ; 3) ; C ( − 5 ; − 1) 2 Determine the coordinates of the middle M of segment [ AB ] . 3 a Determine lengths AB and MC b Establish that the triangle ABC is right-angled at C . 4 Note N the point of intersection of the ordinate axis with the line parallel to ( CM ) passing through the point B . a Place the point N in the marker. b Determine the coordinates of point N . E.2918 In the plane provided with the refer-ence frame O ; I ; J , consider the straight lines ( d ) and ( d ) below : 1 Graphically determine the slope-intercept forms of the straight lines ( d ) and ( d ) . 2 a Give graphically the coordinates of the points M and N . b Justify that the line ( MN ) is parallel to the line ( d ) . 3 Let Q be a point on the line ( d ) such that the line ( MQ ) is parallel to ( d ) . We note x the abscissa of point Q . a Justify that the vectors −−→ MQ and −→ u 1 ; − 1 3 are collinear. b Justify that the vector −−→ MQ has coordinate as a func-tion of x : −−→ MQ x + 3 ; 5 4 · x + 3 4 c Solve the equation : x +3= − 3 · 5 4 · x + 3 4 . d Deduce the coordinates of point Q . https://chingmath.fr chapExoCorrec/2080 sacados/2080 chapExoCorrec/507 sacados/507 chapExoCorrec/6625 sacados/6625 chapExoCorrec/2107 sacados/2107 -6-5-4-3-2-1234I-3-2-1234JO chapExoCorrec/2918 sacados/2918 -4-3-2-1234I-2-123JO(d(dMN
E.2902 In the plane provided with a O ; I ; J orthonormal, consider the line ( d ) shown below and the point M with coordinate ( − 2 ; 1) : 1 Determine the reduced equation of the line ( d ) . 2 a Draw a representative of the vector −→ u (2 ; 1) . b Determine the coordinates of the point P belonging to the line ( d ) such that the vectors −−→ MP and −→ u are collinear. 3 Solve the following equation : (1 − 2 x ) (10 x + 13) = ( x + 2) (15 x + 24) 4 Let N be the point with coordinates − 13 10 ; 1 5 . Let x be a real number, consider the two points R and S be-longing respectively to the lines ( d ) and (Δ) each having as abscissa the value x a We admit that the reduced equation of the line (Δ) is : y = 1 2 · x + 1 Express as a function of x the coordinates of the two vectors −−→ MR and −−→ NS We wish to determine a value of x for which the vectors −−→ MR and −−→ NS are collinear. Let us now assume that x verifies this constraint : b Justify that x verifies the following condition : x + 13 10 − 2 3 x + 1 3 = x + 2 1 2 x + 4 5 c Deduce the coordinates of points R and S . 19. Deepening: any benchmarks E.4968 The plane is provided with a refer-ence frame O ; −→ i ; −→ j any represented below : 1 a In the marker below, place the two points : A ( − 1 ; 2) ; B (4 ; 1) b Justify graphically that the vector −−→ AB has coordinates (5 ; − 1) . 2 Consider the following two vectors : −→ u (3 ; 2) ; −→ v ( − 2 ; − 2) Give a representative of your choice of each of these two vectors in the above reference frame. E.5339 The plane is provided with a refer-ence frame O ; −→ i ; −→ j any represented below : 1 Draw a representative of each of the two vectors : −→ u (5 ; 2) ; −→ v ( − 3 ; − 2) 2 a Draw a representative of the vector −→ w defined by: −→ w = −→ u + −→ v b Graphically, determine the coordinates of the vector −→ w . c Compare the coordinates of the vector −→ w relative to those of the vectors −→ u and −→ v . https://chingmath.fr chapExoCorrec/2902 sacados/2902 -4-3-2-1234I-2-123JO(dM chapExoCorrec/4968 sacados/4968 O−i−j-1123456789-112 chapExoCorrec/5339 sacados/5339 O−i−j-1123456789-112345
E.5744 In the plane, consider the two non-colinear vectors −→ i and −→ j shown below : The vectors −→ u and −→ v are also shown above. 1 In the vector basis of −→ i ; −→ j , give the coordinates of the vectors −→ u and −→ v . 2 By the method of your choice, determine the coordinates of the sum vector: −→ w = −→ u + −→ v . 3 By the method of your choice, determine the coordinates of the vector −→ t achieving the following equality: −→ v = −→ u + −→ t 20. Deepening: any base and affine function E.4592 Let ABCD be a parallelogram. Let E be the point belonging to segment [ AC ] verifying: AE = 2 3 · AC The straight lines ( BE ) and ( CD ) intersect at the point G . The plane is provided with the reference frame A ; −−→ AB ; −→ AC . 1 Give, without justification, the coordinates of the points : A ; B ; C ; D ; E 2 a Justify that the straight line ( BE ) has the equation : y = − 2 3 · x + 2 3 b Deduce the coordinates of point G . 3 What does the point G represent for the segment [ CD ] ? Justify. E.2074 Consider a square ABCD of side 1 cm . The point I is the only point such that the triangle AIB is equilateral and is contained in the square ABCD ; The point J is the only point such that the triangle BCJ is equilateral and is outside the square ABCD . We note : K is the foot of the height from the vertex I in the tri-angle AIB . L is the foot of the height from J in the triangle BCJ . Here is the representation of this configuration : The purpose of the exercise is to show that the points D , I , and J are aligned. Consider the plane munite of the reference frame ( A ; −−→ AB ; −−→ AD ) . 1 What is the nature of the benchmark ( A ; −−→ AB ; −−→ AD ) ? 2 a Determine the length of segment [ KI ] . b Give the coordinates of the point I in the considered coordinate system. 3 Assuming the length LJ measures √ 3 2 , give the coordi-nates of the point J . 4 Consider the line ( d ) of equation : y = 3 − 2 x + 1 Show that the three points D , I , and J belong to the line ( d ) . 21. Deepening: decomposition of vectors in any basis E.8112 Consider a rectangle ABCD and the three points E , F , G defined by: −→ AE = 1 2 · −−→ AB ; −−→ DF = 2 3 · −−→ DC ; −−→ DG = 4 · −−→ DA https://chingmath.fr chapExoCorrec/5744 sacados/5744 −i−juv chapExoCorrec/4592 sacados/4592 ABCDEG chapExoCorrec/2074 sacados/2074 Utilisation de la simplification de radicaux KLABCDIJ chapExoCorrec/8112 sacados/8112
Consider the two vectors −−→ AB and −−→ AD non-colinear forming a vector basis. 1 Determine the decomposition of the vector −−→ EF in the base −−→ AB ; −−→ AD . 2 Establish the decomposition of the vector −−→ EG in the base −−→ AB ; −−→ AD : −−→ EG = − 1 2 · −−→ AB − 3 · −−→ AD 3 Demonstrate that the points E , F and G are aligned. E.6664 In the plane, consider the triangle ABC shown below : The points M , N and P are defined by the relations : −−→ AM = 4 5 · −−→ AB ; −−→ BN = 1 2 · −−→ BC ; −→ AP = 4 3 · −→ AC The study will be carried out in the reference frame B ; −−→ BA ; −−→ BC . 1 Give the coordinates of points M and N . 2 a Determine the coordinates of the vector −→ AC . b Deduce the coordinates of the point P . 3 Justify that the points M , N and P are aligned. E.5342 In the plane, consider the triangle ABC : Consider the points M and N defined by: −−→ BM = 1 4 · −−→ BA ; −−→ BN = 1 2 · −−→ BC We define the point P by the vector relation: −→ AP = ¸ · −→ AC où ¸ ∈ R 1 Express −→ AC in terms of the vectors −−→ BA and −−→ BC . 2 The plane is given the reference frame B ; −−→ BA ; −−→ BC : a Determine the coordinates of the vector −−→ MN and the vector −−→ MP as a function of the real ¸ . b Determine the value of ¸ so that the points M , N and P are aligned. https://chingmath.fr ABCDEFG chapExoCorrec/6664 sacados/6664 ABCMNP chapExoCorrec/5342 sacados/5342 ABCMNP