Grade 12 - Exp. / Other past papers 2 exercises corrected

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1. Unclassified financial years E.3152 For each question, only one of the four proposed answers is correct. The candidate will indicate on the copy the number of the question and the letter corre-sponding to the chosen answer. Each correct answer gains 1 point. Each wrong answer deducts 0.5 point. An absence of response is counted as 0 points. If the total is negative, the mark is reduced to zero. No justification is required. 1 In the set of relative integers, consider the equation : x 2 − x + 4 ≡ 0 ( mod. 6) a toutes solutions are even integers b il no solution c les solutions verify x ≡ 2 ( mod. 6) d les solutions verify: x ≡ 2 ( mod. 6) ou x ≡ 5 ( mod. 6) 2 We propose to solve the equation ( E ) : 24 x + 34 y = 2 , where x and y are relative integers. a Les solutions of ( E ) are all of the form : ( x ; y ) = (34 k − 7 ; 5 − 24 k ) , k ∈ Z b L the equation ( E ) has no solution c Les solutions of ( E ) are all of the form : ( x ; y ) = (17 k − 7 ; 5 − 12 k ) , k ∈ Z d Les solutions of ( E ) are all of the form ( x ; y ) = ( − 7 k ; 5 k ) , k ∈ Z 3 Consider the two integers n =1789 and p =1789 2005 . Then we have : a n ≡ 4 ( mod. 17) et p ≡ 0 ( mod. 17) b p is a prime integer c p ≡ 4 ( mod. 17) d p ≡ 1 ( mod. 17) 4 Consider, in the complex plane referred to an orthonor-mal reference frame, the points A and B of affixes a and b respectively. The triangle MAB is a direct isosceles rectangle of hypotenuse [ AB ] if, and only if, the point M of affix z is such that : a z = b − i a 1 − i b z − a = e i π 4 · ( b − a ) c a − z = i · ( b − z ) d b − z = ı 2 · ( a − z ) 5 Consider in the oriented plane two distinct points A and B ; note I the midpoint of the segment [ AB ] . Let f be the direct similitude of center A , ratio 2 and angle 2 ı 3 ; let g be the direct similitude of center A , ratio 1 2 and an-gle 2 ı 3 ; either g the direct similitude of center A , ratio 1 2 and angle ı 3 ; or h the central symmetry of center I . a h ◦ g ◦ f transforms A into b and this is a rotation. b h ◦ g ◦ f is the reflection having axis the bisector of the segment [ AB ] c h ◦ g ◦ f is not a similarity. d h ◦ g ◦ f is the vector translation −−→ AB E.3165 The complex plane P is referred to a direct orthonormal reference O ; −→ u ; −→ v . The graphic unit will be 4 cm . Consider the points A , B , C and D with respective affixes a , b , c and d such that : a = 1 ; b = 1 + 2i ; c = 2e i π 4 ; d = 3 + 2i Consider the direct similitude s which transforms A into b and C into D . Let M be a point of affix z and M , of affix z , its image by s . 1 Express z as a function of z . Determine the characteristic elements of s . Let U n be the numerical sequence defined by: U 0 = 0 U n +1 = 2 U n + 1 pour tout n ∈ N . 2 Show that, for any natural number n , U n +1 and U n are prime to each other. 3 Interpret geometrically, using the similarity s , the terms of the sequence U n . 4 Show that for any natural number n : U n =2 n − 1 . 5 Show that, for all non-zero natural numbers n and p such that n p : U n = U p · U n − p + 1 + U n − p The notation pgcd ( a ; b ) is used, in the sequel, to denote the greatest common divisor of two natural numbers a and b . Show for n p the equality: pgcd ( U n ; U p ) = pgcd ( U p ; U n − p ) 6 Let n and p be two non-zero natural numbers, show that : pgcd ( U n ; U p ) = U pgcd ( n ; p ) Determine integer: pgcd ( U 2005 ; U 15 ) https://chingmath.fr sacados/3152 sacados/3165