Grade 12 / Exercises and competitions 31 exercises (100% corrected)

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-+0-0135∞∞−3−1−2VariationdefSignedefx 1. Advance Competition E.5867 Concours Advance - Session 2013 - Calculator prohibited - Circle the correct answer(s) : The following reasonings are correct: a All the students are called Bob. However, some Bobs are not gifted. So some students are gifted. b All the gifted students are called Bob. But Bob is not gifted. So Bob is not a student. c Most Bobs are not gifted. But all the students are gifted. So no student is named Bob. d Most of the students are called Bob. But all Bobs are gifted. So some students are gifted. e Bob is gifted. Now all students are gifted. So Bob is a student. E.5873 Concours Advance - Session 2012 - Calculator prohibited - Circle the correct answer(s) : Let P the statement : ˇ For a pawn to be white, it must be in bois ı Then P means : a ˇ Any white pawn is wooden ı b ˇ Any wooden pawn is white ı c ˇ If a pawn is white then it is wooden ı d ˇ If a pawn is wooden then it is white ı e ˇ For a pawn to be made of wood it only needs to be white ı E.5875 Concours Advance - Session 2012 - Calculator prohibited - Circle the correct answer(s) : Let f be the function defined on R \{ 1 } by: f ( x ) = x 2 − x + 1 x − 1 Note C the representative curve of f in an orthonormal ref-erence frame. a The straight line x =1 is a vertical asymptote of C . b lim x ↦→ + ∞ f ( x ) = −∞ c f is increasing on 1 ; + ∞ . d f is decreasing on 0 ; 1 . E.5877 Concours Advance - Session 2012 - Calculator prohibited - Circle the correct answer(s) : Let f be the function defined and derivable on R by: f ( x ) = ( x + 1) · e − 2 x We have : a For all x ∈ R : f ( x )= x +2 · e − 2 x b f is increasing on −∞ ; − 1 2 c The tangent to the curve of f at the point of abscissa 0 has equation y = x +1 d lim x ↦→ + ∞ f ( x ) = −∞ e lim x ↦→−∞ f ( x ) = −∞ E.5868 Concours Advance - Session 2013 - Calculator prohibited - Circle the correct answer(s) : Let f be the function defined and derivable on R : f ( x ) = x 3 1 + x 2 a For any x ∈ R : f ( − x ) = − f ( x ) b For any x ∈ R : f ( x )= f ( − x ) c There is a single a ∈ R : f ( a )=0 d f is strictly increasing on R . e For any x ∈ 0 ; + ∞ : f ( x ) x . E.5876 Concours Advance - Session 2012 - Calculator prohibited - Circle the correct answer(s) : Let f be defined on R by: f ( x ) = 1 − x 2 si x < 0 f ( x ) = x 2 + 2 si x 0 Then : a f admits no limit in 0 . b f is derivable in 0 . c f is increasing on R . d The equation f ( x )=0 admits one and only one solution in R . e f admits a reciprocal function defined in R . E.5874 Concours Advance - Session 2012 - Calculator prohibited - Circle the correct answer(s) : Let f be a function defined and derivable on 1 ; + ∞ whose table of variations is : Note C its representative curve in a reference frame O ; −→ i ; −→ j . Then : a f (4) < 0 b C admits a vertical asymptote. c C admits a horizontal asymptote. d The equation f ( x )=0 admits no solution in 3 ; + ∞ . e The equation f ( x )= − 1 admits two solutions in 1 ; + ∞ . https://chingmath.fr chapExoCorrec/5867 sacados/5867 Concours Advance Session 2013 chapExoCorrec/5873 sacados/5873 Concours Advance Session 2012 chapExoCorrec/5875 sacados/5875 Concours Advance Session 2012 chapExoCorrec/5877 sacados/5877 Concours Advance Session 2012 chapExoCorrec/5868 sacados/5868 chapExoCorrec/5876 sacados/5876 Concours Advance Session 2012 chapExoCorrec/5874 sacados/5874 Concours Advance Session 2012 -+0-0135∞∞−3−1−2VariationdefSignedefx
E.5870 Concours Advance - Session 2013 - Calculator prohibited - Circle the correct answer(s) : Let f be the function defined on R by: f ( x ) = sin x 2 + cos x . Then : a For any x ∈ R : f ( − x )= f ( x ) b f is periodic ı . c f is decreasing on ı 3 ; ı d f is increasing on 0 ; ı 3 e For any x ∈ 0 ; ı : f ( x ) 5 4 E.5879 Concours Advance - Session 2012 - Calculator prohibited - Circle the correct answer(s) : The real sequence u n is convergent : a u n = n · n + 1 n 2 + 1 b u n = 1 + ( − 1) n · n n + 1 c u n = cos n n + 1 d u n = ( − 1) n · n n + 1 e u n = ln e n +1 n + 1 E.5878 Concours Advance - Session 2012 - Calculator prohibited - Circle the correct answer(s) : Let u n be a real geometric sequence with first term u 0 =36 and reason q . Then a If u 3 = 4 3 then q = 1 3 b If u 2 u 4 =4 then q = 2 c If q< 1 3 then u 4 < 1 d If there exists n ∈ N such that u n =1 then q = 1 6 e If lim n ↦→ + ∞ u 0 + u 1 + u 2 + ··· + u n =30 then q = − 1 5 E.5869 Concours Advance - Session 2013 - Calculator prohibited - Circle the correct answer(s) : For any real sequence u n : a If u n is not increased then : lim n ↦→ + ∞ u n =+ ∞ b If u n is increasing and majoring by 1 then : lim n ↦→ + ∞ u n = 1 c If lim n ↦→ + ∞ u n =+ ∞ then u n is increasing from a certain rank. d If lim n ↦→ + ∞ u n =1 then u n is positive from a certain rank. e If lim n ↦→ + ∞ u n + u n +1 2 =1 then lim n ↦→ + ∞ u n =1 . E.5871 Concours Advance - Session 2013 - Calculator prohibited - Circle the correct answer(s) : Two laboratories each offer their own flu vaccine. We know that a quarter of the population used the 1 vaccine and a sixth the 2 vaccine. It is not possible for an individual to be vaccinated twice. The epidemic having taken place, we find that 1 % of sufferers used the vaccine 1 and 0.6 % the vaccine 2 . An individual is chosen at random from the population, noting : M : ˇ the individual is malade ı ; I : ˇ the individual has received the vaccine 1 ı ; II : ˇ the individual has received the vaccine 2 ı. We have : a The probability that the individual is vaccinated is : P ( I ) + P ( II ) . b Data do not allow calculation P I ( M ) . c P ( I ) = 1 100 d P M II = 0.94 e P II ( M ) P II ( M ) = P M ( II ) ·P II P M ( II ) ·P II E.5880 Concours Advance - Session 2012 - Calculator prohibited - Circle the correct answer(s) : A recruitment department receives 15 dossiers of which 6 in-clude a favorable opinion and the remaining 9 an unfavorable opinion. The 15 files are ranked randomly. The probability of the event : a ˇ the first file is favorable and the second unfavorable ı is 9 35 b ˇ the first two files are favorable ı is 1 7 c ˇ the first two files are unfavorable ı is 6 7 d ˇ at least one of the first two files is unfavorable ı is 6 7 e ˇ the second file is favorable knowing that the first is un-favorable ı is 3 7 . https://chingmath.fr chapExoCorrec/5870 sacados/5870 Concours Advance Session 2013 chapExoCorrec/5879 sacados/5879 Concours Advance Session 2012 chapExoCorrec/5878 sacados/5878 Concours Advance Session 2012 chapExoCorrec/5869 sacados/5869 Concours Advance Session 2013 chapExoCorrec/5871 sacados/5871 Concours Advance Session 2013 chapExoCorrec/5880 sacados/5880 Concours Advance Session 2012
E.5881 Concours Advance - Session 2012 - Calculator prohibited - Circle the correct answer(s) : Two dice are thrown whose faces are numbered from 1 to 6 . For each die, the probabilities of getting one of the six faces are equal. We note S the sum of the points of the top faces : If 2 S 3 , we gain 20 points ; if 3 <S 5 , we gain 10 points if 5 <S < 10 , we gain 5 points if 10 S 12 , we gain 1 point. We note X the random variable gives the number of points per throw. a P X =20 = P X =1 b P X =5 = 5 9 c P X 5 = 13 18 d P X 10 = 5 18 e The expectation of X is 64 9 . E.5872 Concours Advance - Session 2013 - Calculator prohibited - Circle the correct answer(s) : Let I , J and K be three points in the plane such that : IJ = 3 ; IK = 2 ; ∠ JIK = ı 3 Let L and M be the points of the plane defined by: −→ IL = 2 · −→ IJ − 3 · −→ IK ; −−→ IM = −−→ IJ + 4 · −→ IK Then : a −→ IL · −→ IK = 3 b −→ IL · −→ IL = 30 c −→ IL · −−→ IM = − 33 d cos ∠ LIM = − 11 14 e A measure of ∠ LIM belongs to ı 2 ; ı . 2. Alpha competition E.5884 Concours Alpha - Session 2011 - Calculator prohibited - Choose the answer among the pro-posed assertions : Let the equation be : ( E ): 6 x 3 +7 x 2 =1 . a ( E ) has a unique negative solution. b ( E ) has a unique integer solution. c ( E ) has a unique real solution. E.5883 Concours Alpha - Session 2011 - Calculator prohibited - Choose the answer among the pro-posed assertions : lim x ↦→ 0 sin(5 x ) tan( x ) = a 5 b 0 c + ∞ E.5889 Concours Alpha - Session 2011 - Calculator prohibited - Choose the answer among the pro-posed assertions : tan(2 x ) = : : : a 2 · tan x tan x 2 − 1 b 2 · tan x 1 − tan x 2 c 2 · tan x 1 + tan x 2 E.5890 Concours Alpha - Session 2011 - Calculator prohibited - Choose the answer among the pro-posed assertions : Let the equation be : ( E ): sin x 2 +2 · sin x − 3=0 a ( E ) has two real solutions b ( E ) has a single solution in the interval 0 ; 2 ı c ( E ) has four solutions in the interval 0 ; 2 ı E.5882 Concours Alpha - Session 2011 - Calculator prohibited - Choose the answer among the pro-posed assertions : Let u n and v n be two numerical sequences. Then a If the sequence u n + v n converges then u n and v n converge. b If the suite u n + v n diverges then u n or v n diverge. c If u n and v n diverge then the sequence u n + v n di-verges. E.5887 Concours Alpha - Session 2011 - Calculator prohibited - Choose the answer among the pro-posed assertions : Let a , b and c be three consecutive terms of an arithmetic sequence such that : a + b + c = 27 ; a + 2 c = 25 Then, . . . a the sequence is increasing. b the next term of the sequence is equal to 5 c the next term in the sequence is equal to 2 . E.5886 Concours Alpha - Session 2011 - Calculator prohibited - Choose the answer among the pro-posed assertions : Let the sequence u n be defined by u 0 =0 and for any integer n : u n +1 = u n + 2 n + 1 Then :. . . a The sequence u n is arithmetic. b u 99 = 10000 c u 100 = 10000 https://chingmath.fr chapExoCorrec/5881 sacados/5881 Concours Advance Session 2012 chapExoCorrec/5872 sacados/5872 Concours Advance Session 2013 chapExoCorrec/5884 sacados/5884 Concours Alpha Session 2011 chapExoCorrec/5883 sacados/5883 Concours Alpha Session 2011 chapExoCorrec/5889 sacados/5889 Concours Alpha Session 2011 chapExoCorrec/5890 sacados/5890 Concours Alpha Session 2011 chapExoCorrec/5882 sacados/5882 Concours Alpha Session 2011 chapExoCorrec/5887 sacados/5887 Concours Alpha Session 2011 chapExoCorrec/5886 sacados/5886 Concours Alpha Session 2011
E.5885 Concours Alpha - Session 2011 - Calculator prohibited - Choose the answer among the pro-posed assertions : A factory manufactures plasma screens. After manufacture, each screen is tested. If the test is positive, the screen is deliv-ered to the customer ; if not, the screen is repaired and then tested a second time. If the second test is positive, the screen is delivered, otherwise it is destroyed. 80 % of screens are positive in the first test ; of screens nega-tive in the first test, 60 % are positive in the second test. The cost of manufacturing a screen is 1 000 euros, plus 100 euros if a second test is required. To make a profit, the selling price must be at least . . . a 1020 euros b 1095 euros c 1075 euros E.5888 Concours Alpha - Session 2011 - Calculator prohibited - Choose the answer among the pro-posed assertions : There are three urns noted A , B , C . The urn A contains one ball numbered 1 ; the urn B contains two balls : one numbered 1 and one numbered 2 ; the urn C contains three balls : one numbered 1 , one numbered 2 and one numbered 3 . An urn is chosen at random and a ball is drawn from the chosen urn. a The probability of obtaining a numbered ball 1 is equal to 1 2 . b The probability of choosing the urn A knowing that a numbered ball 1 was obtained is equal to 6 11 . 3. ECE competition E.5892 Concours ECE - Session 2007 - Calculator prohibited - For each question, specify whether the assertion is True or False Let f be the function defined on R by f ( x )= x 3 +3 x 2 − 9 x of representative curve C f in an orthonormal frame O ; −→ i ; −→ j . a The function f admits at the point a local minimim in x = − 3 . b The function f is increasing on the interval − 3 ; 1 . c The curve C f admits a horizontal tangent at the point of abscissa x =1 . d The area of the plane domain bounded by the represen-tative curve of f and the straight lines of equations y =0 , x = − 1 , x =1 is equal to 2 area units. E.5894 Concours ECE - Session 2007 - Calculator prohibited - For each question, specify whether the assertion is True or False Let f be the function defined on R by ( f ( x )=e x · x 2 + x +1 of representative curve C f in a reference framel O ; −→ i ; −→ j orthonorma. a On R : f ( x )=e x · x 2 +3 x +2 . b The function f admits a local maximum at x = − 1 . c The curve C f admits a horizontal tangent at the point of abscissa x = − 2 . d C f admits a horizontal asymptote of equation y =0 . E.5896 Concours ECE - Session 2007 - Calculator prohibited - For each question, specify whether the assertion is True or False Let f be the function defined on R by f ( x ) = e − x · 3 x 2 + 10 x + 11 representative curve C f in a repère O ; −→ i ; −→ j or-thonormé a On R : f ( x )= − e − x · 3 x 2 +4 x +1 . b The function f admits a local minimum at x = − 1 3 . c The equation f ( x )=4 e has a single real solution. d C f admits a horizontal asymptote of equation y =0 . https://chingmath.fr chapExoCorrec/5885 sacados/5885 Concours Alpha Session 2011 chapExoCorrec/5888 sacados/5888 Concours Alpha Session 2011 chapExoCorrec/5892 sacados/5892 Concours ECE Session 2007 chapExoCorrec/5894 sacados/5894 Concours ECE Session 2007 chapExoCorrec/5896 sacados/5896 Concours ECE Session 2007
E.5891 Concours ECE - Session 2007 - Calculator prohibited - For each question, specify whether the assertion is True or False In R × R , the system of equations : 2 x − 3 y = 1 5 x − y = 9 has for solution (2 ; 1) . Say whether each of the statements below is true or false : a In R × R , the system of equations : 2 x 2 − 3 y 2 = 1 5 x 2 − y 2 = 9 has for solution ( √ 2 ; 1 ) . b In R × R , the system of equations : 2 x − 3 y = 1 5 x − 1 y =9 has for solution 1 2 ; 1 . c In R × R , the system of equations : 2 ln x − 3 ln y = 1 5 ln x − ln y = 9 has as solution (e 2 ; 0) . d In R × R , the system of equations : 2 sin x − 3 cos y = 1 5 sin x − cos y = 9 has an infinite number of solutions. E.5895 Concours ECE - Session 2007 - Calculator prohibited - For each question, specify whether the assertion is True or False Let f be the function defined on R by f ( x )=( x − 3)( x +4) and g the function defined on R \{− 1 } by: g ( x )= 1 x +1 . a At points x où the function g ◦ f is defined : g ◦ f ( x ) = 1 x 2 + x − 11 b At points x où the function g ◦ f is derivable : g ◦ f ( x ) = − 2 x − 1 x 2 + x − 11 2 c At points x où the function g ◦ g is defined : g ◦ g ( x ) = x + 2 x + 1 d On R \{− 1 } : g ◦ g ( x )= − 1 x +1 2 E.5893 Concours ECE - Session 2007 - Calculator prohibited - For each question, specify whether the assertion is True or False a The limit of the general term sequence : u n = ( − 1) n n 2 + 2 n + 1 n 2 + 1 does not exist. b The limit of the general term sequence : u n = 1 n · sin n + ln n 2 does not exist. c The general term sequence u n = e n + n 3 e 2 n + n 2 converges to 1 . d The general term sequence u n = − 2 3 n + 2 3 n con-verges to 0 . 4. Unclassified financial years E.6907 For each of the following statements, say whether it is true or false, justifying the answer One point is awarded for each justified correct answer. An unjustified answer will not be taken into account, and the absence of an answer is not penalized. Let f be the function defined on R by: f ( x ) = 3 4 + 6 · e − 2 x Assertion 1: The equation f ( x )=0.5 has a unique solution on R . Consider the algorithm: X ← 0 Y ← 3 10 While Y<0.5 X ← X+0.01 Y ← 3 4+6 · e − 2X Fin As long as Assertion 2: At the end of the execution of this al-gorithm, the variable X has the value 0.54 : Zoé walks or drives to work. Whereù lives, it rains one day in four. When it rains, Zoé drives to work 80 % of the time. When it’s not raining, she walks to work with a proba-bility equal to 0.6 . Assertion 3: Zoë uses the car every other day. In the set E of outcomes of a random experiment, con-sider two events A and B . Assertion 4: If A and B are independent, then A and B are also independent. An institute conducts a survey to find out, in a given population, the proportion of people who are in favor of a land development project. To do this, a random sam-ple of 700 people from this population is interviewed, and each person is asked a question. It is assumed that whether a person agrees to answer or not is independent of the other people questioned, and that the probability of that person agreeing to answer the question is equal to 0.6 . Assertion 5: The probability of at least 400 peo-ple answering the question has a probability of 0.93 rounded to the nearest hundredth. https://chingmath.fr chapExoCorrec/5891 sacados/5891 Concours ECE Session 2007 chapExoCorrec/5895 sacados/5895 Concours ECE Session 2007 chapExoCorrec/5893 sacados/5893 Concours ECE Session 2007 chapExoCorrec/6907 sacados/6907