QCM corrigé Analyse numérique licence master 0 analyse numérique, arithmétique, continuité, denombrement, dénombrement, mathematique, nombres complexes, probabilité, QCM licence math, QCM Master math, series, suites 06 h 58 QCM Analyse numérique licence master Q u e s t i o n 1 On désigne par x la partie entière de x . lim x → 1 x 2 − 1 x 2 − 1 = □ 2 □ 0 □ 1 □ n ' e x i s t e pas Question 2 On considére la suite x n pour tout n ∊ ℕ définie par ; x n = x n − 1 + x n − 2 + .... + x 0 , ∀ n ∊ ℕ Quelle est la nature de x n ? □ x n est une suite géomtrique □ x n est une suite arithmétique □ x n est une suite constante □ On ne peut pas en conclure Question 3 L ' e q u a t i o n x 2 − 2 2 − 5 2 = 1 admet dans ℝ : □ 4 solutions □ 5 solutions □ 6 solutions □ 7 solutions □ 8 solutions Question 4 Soit f d é rivable en a , lim h → 0 f 2 a + 3 h − f 2 a − h h = □ 8 f a f ′ a □ 8 f 2 a f ′ a □ 4 f a f ′ a □ 2 f a f ′ a Question 5 On pose f x = 1 x 2 ∫ 3 x 2 u − 3 f ′ u du . f ′ 3 = □ 1 / 2 □ 3 / 2 □ 1 / 4 □ − 1 / 3 Question 6 Siut u n la suite réelle définit par 1 − u n u n + 1 > 1 / 4 et 0 ≤ u n ≤ 1 ∀ n ∊ ℕ , lim u n = n → + ∞ □ 0 □ 1 / 2 □ 1 □ + ∞ □ Aucune de ces réponses Question 7 Pour quelle valeur de α la série dont le terme générale est − 1 n − 1 n + n α converge pour n ≥ 2 ? □ α ≥ 3 / 2 □ α ≤ 2 / 3 □ α ≤ 1 / 2 □ α > 2 / 3 □ α > 3 / 2 Question 8 Soit l ' a p p l i c a t i o n f dans ℂ définie par f ( z ) = z + i z − i pour tout z different de i et on considére E = z ∊ ℂ , ∣ z ∣ = 1 et i ℝ = z ∊ ℂ , Rel z = 0 □ f réalise une bijection de ℂ ∖ i dans ℂ . □ f i ℝ = ℝ □ f i ℝ = E □ f E = i ℝ □ Aucune réponses n ' e s t c o r r e c t e . Question 9 On désigne P E l ' e n s e m b l e des parties de E . Soit E une ensemble de cardinal fini n . le nombre de couples A , B ∊ P E × P E tel que A ⊂ B v a u t : □ 2 n □ 3 n − 1 □ 3 n □ 2 n 2 □ n 2 n − 1 Question 10 Une urne contient 2 boules blanches et n − 2 boules r o u g e s . On effectue des tirages successifs sans remise d ' u n e boule et on note X le rang pour lequel la boule blanche apparait pour la premiere f o i s . E X = □ n + 1 2 □ 2 n + 1 2 □ 3 n + 1 2 □ n + 1 3 Question 11 Soit S l ' ensemble des entiers x tel que x 2 − x + 1 est divisible par 7 . S est alors : □ S = 7 k + 2 , 7 k + 5 , k ∊ ℤ □ S = 7 k + 1 , 7 k + 5 , k ∊ ℤ □ S = 7 k + 3 , 7 k + 5 , k ∊ ℤ □ S = 7 k + 2 , 7 k + 3 , k ∊ ℤ Question 12 lim n → + ∞ ∑ k = 0 n sin 1 n + k = □ 0 □ sin 1 □ ln 2 □ 1 □ + ∞ Question 13 Soit une fonction f d é rivable en x 0 = 0 d é finie de ℝ → ℝ tel que f 0 = 0 et f '( 0 ) = 1 ∀ n ∊ ℕ * lim x → 0 f x × f 2 x × f 3 x × ..... × f n − 1 x x n − 1 = □ n n − 1 / 2 □ n ! □ n n + 1 / 2 □ n − 1 ! □ + ∞ MMF.7h^7D000cA_KL]^hm*_l3niCZ8jR402_jL))k5ECCf(k]KCIC5MmH6CJHB=CGTY:|lWhOn^o:022i068*hTN)gFbl*8h^9`KcPgdo)C|[nLGYn?IbOWI`ON3nO75I3`k_g1)TlfWdn*jFabn:eJ7_fKihIMXi=;1`Gaf(W|kLBJoKi==]|YO(SkdOW8aiN|MOgPH30mM?_;cN3IaX^4Q9ObOBcR0N:m9m9XA3YalN?MfO7;V30k^3^KCbFafL_I6W3fMWXe?9lh_jk*hW:JKCIIOlmW[SN()Cdn?dgbc;OhhBomCW|27RGfH6|)d76K=l?WblTfAYYm[V6N3ZNelHc]B3POf`l?Q;;]9ehMlk?1RMI?T=B*J7ZnfALIYPo=SM3k1Z4K89PSMY8]`dTDi*DPW2)d49ikHZBLhnM*TGae)(JTSi=(^lVTGnA*QWb;TDialJRNOh^*cA?X(8InIi(_aKocZd25QII|(bM1Ghgadb?bbcGRKN:[SlDjX9_T2*1DP403E3|5)TIPDZ4k()i5ZL`DV`g;e[B?JK]TDJ13EYV2:*((OAFG7Zo6kUFS4ZRf`8:XML][LDCD[TSfZe|OP48V8GgD8`8*:kZRf)=iSZR)i(58k4h50e*kPaX8?Q5Dm`HVPj|*JL]B5A0(DZ)10G7DHX8ij|))39A056SJhdDPo4o:0***H0*aQDPIEQmD2i1d?B|OG=VM1Lbh;0N4|PW_7X)=153b1*Z*fl2PPeF=`VPLk9*YN=E6Pl;86:VfXm2L2Q7Pa8(Ag`F4nA((GJ:Se?QB5k`TmZMC1UmNRX|X?X0kh0P^ZfQ5XahfV1BiX0eD8J(?B038QT(MG7KoI=*S0CT88[(8SR2C5EBm^CPWQI*RQ8XCPo52L[bi3j4W)QZ[W`bF2n)[(4:QR:(kgE5_ZXD8i4|Lc]EL4Shm:?Jaj38:P8TJU5UBmX24]4PRl[)e1121aPe[|PSIPOdc1RC5[)1]k*;eS7nQ57(0U8HA4|1)36dILEk(];]6kE3=L35PNeh(M_a4*LCFSi8IJ;h;|8Ri*A49L3JJIAT:1eBGB)[j|;AdY[fGMmN7207K2XMMd15=NeU]ZG946|YHnTBJBEQd:)`cJKlPBJB=YgH)b8M9(AWD_PW^0bd4Hm5?BC=H6QS1`?`PgU6hcDNYWQJ6bU7hmEI))=9He8jFeK7XeBmj(gg6Gj*hUfT?6oeofbS(5U(RN3^Eo5EAZTHC3oa(5IGaEY(HSmC)BUTm0?KhSDa19YN`c1OGUB)^_PPHB2`b[|8J64QYZd:QcKMc=SAg(8T8TEZ`_c|mWSS_RmlTM1O`O4*f^I7a8nXFK:b?m:(N8IHbfPlTB`2bC;I5W2A1AikPXEVZN23J?|S`YoRSk8NnOYMO99Rgk4NmOY4^IS9DS(AmiWaAIlW6IDXF]866bFJRN*7kbME(TZTo1R[4JHf3|B8eiH)aHSOUPk6Le5]PH5]X68h`;9;Haf(FVbj`0L4eV0i1](Pf0O9)Q_lhhj[Li9h?mVW(de3U7XcKWJ=cVW8cV3Lk9B=kPW175Zd6KnS1DOiQ_VajPdd)MLb`b)(MRPg)Nfn:LAmZLlbSTW(MdcWUNVg)NgnJL5k*iihD6ii8U_cDRM)D:4|Won1CEhFeG]VF?2C)F5=YdC`a;Nb06?3GC4i_`bLM_;kP1)5|E=lWbL9IngBRKF?hU8eMJEC5;^9F;=dL_SiKKE9UMMaSh?_?Qa?)ginJ6g=C8?iH6GeH_d_JgCXE[S_h^(NN]jBnWSY39bMW(4BQ(ogKnjm5h)Q70)k7ifFab=Q(5V5?QL`KOidOOilNo3HOLQ8Xoh]ao)OoHYV|QL3:h(n0LN9h?iU?^6`K6n)Gm)[_)Dc_`=RTZ`4cTZJf3nIb_o;CgH|Ik9n1=fcK;1=enTb57lh^*oKM0P9O8n=MABOZ(FaLK[|_|a^2=bh)_TX[im;MbFSGT^FC*W|RIFa5I(BH`SW:i5EIkLW66;BRAD2XgZ|kT1Uh)a]ZIJYADE;E5d9c5YE1_a^BHT8Ta92CdN3RmicIKR9`kRK14LL[C:=ND2^f0^Ad`dP7;GjAO|oD6eM]ej`kXMlBb[Ya0nebNaBYOIiOgV9YRN[oNIQSVYLjH6UFB;8ASd;WJ5_Im=Z_]aPhAY?2d]98YHaGP33(*EeV)gE9P87C0Gfa|g4ELZDQm5_4U^WYFRR(fHf7d29]AM;(Aoc?XOHkK691SIchD6i2hSi*|S)EJ_EaRQWB=:0:VWGW2*i2nQ[NGf_kDLK5gjTPO4[NHHn^hPMOgZi]=TOfnCIlcUSbdfWbj^MllMcb5QM`TnJH;BLbjXU59R]TfHONAJ095Kk4Df_i0Kl5|]o7]Re*m_EG[FR5DAcAUllm3H8S|9X[kHRI2((o^h7M|h?OJ`6:T2J:UUcLY8Z[;9=MUeIRh2o^:eadZi26:^EY^QAcF7D_mQbl=NRee`XNO5)fe];onNVZQ(eeUSJ2LZjIi)IQ_NIS7A;_8_XQl[P62JnDT]CmCbUj7h23|MSh9_eI6e]0CjT7d|8U[?bVV=)P3h5G[|P2PE5KcWP5h0R9n=M*hLSSH7_YT2KII)mPFc3F?J2kO4m9Q?FAoY)4l]B4BiP^m?2kAKo3S66WFI)b8V^ag)O6^JbJfMLfI=RlNjBc_jLRPSkieoe_^mlUYKUM[a)mNkJ`2e1*Y9f7;SNU`3cmW`ALTW3=AQG=aCeZW|m_1W]T|Ak`FP[V;:A(C_^ec|E1AH|64H)b9N9|Og)UXNm:;KU]9BKU9RaX[:O;8U]Pj(_9GGKTljhBBEih6]FM^[mY(f5?90a_gYQTF:Vjk(d6aXYFg?22O;nkac:c([=5*MWOYSB1Lk:C]CU?3OUbYE5[E`Rb9[dZmoM2cbMlNVb(L:7G4KlFJ)jRc;jY[BDR^KCRl7_DA4O9deA_kJelCI9MZZ5|MaG^;)6AHIIFA)DQYF1UIY2ZW9Bj?o1)c;4miI|O9?naLEBVJB_HooJ4hbN?]0|gU|*[?8TEBjn;nURLPjH=CT=1V]mnQQLgO)dYMGi9URZe[j1;FbCh7]Nog1EH0_EcUF7T=FK59Ra^dm9CO5lTNQK`j[YZ9Un9f98*W124H9C^coR[40KFRZCe;bV4leiRKF0;mD7TM]hZ)CKof50QGhJ95_AJ[o4]JG6?e9ECiI5FFa2f_AngCOc827ZUg]_EVi5A=]CPbFTE8^jNJdEXee*mKV6:kFZKF^?JojY4eA5]Q8M3bAG9k^l`FFRT=5ROdVhPGU8ieVfR;`_Ihga076jkOnCHO3;ZBWE9__kD33;^2IjJcacO*HbnhPCeikjGWn)_A=hB9fMEEFZC05Z:)a:Aaf1D3FYEYJ7]hLbI04(*N;`_TIFPWC4=H1KlF?JTNIVPDfaQ*SV)5D1h(Z^?5(A72O8cRc5X)OEc2m(DGjK8SaoQVCNaKhF:;J3bhA)i_LIl|(jcdP8H[7k=oY`_4j;BS6IdSSTNl3ZIV=COgTfRg0K9VjAQoD6WH]0)ESckYBHoS]fo_4nCLGBYU?aTkXB^P[N=C^dMoPCig;EI5`GD[7CgDgLF?lQf)[?)lja2BbDj[9ddi2fhnXYD8S3gU5cX84;?`TcK7)_glY)NMgdD;8|jT^(cbI6V7RQP0dA0[=MSYnN[VHmo7llEZNk]|V8`:NS`O7QVb=UamDkIhV)XhG^0m]^i]DXCUh^GJ2QQ;flaJ=^48boZfVjjg*HZGVV5Nlf=CL=J5E=MG;Ll8cAo1)abeC[5f?bkl(;agdI1g4h6Z`Id(AAkTU8Q[(fNoH;IfFf0*7X9ceh1E?i2(n)=Z^lCLbLMUTRlnHM0D53l*HmCa1ML^8mF5Ek7JGZM[ZaO0W7W:|j751[=_Z;OMI4EbS*7GflDRGJnc:`a^QYl=0NT=6UQO_|13Jl6CGL;H;eg;EmSkb0Leg(?55TUnSD*[J4:kC4GM484QEKl)nYFZfX79;HnJ4_AM1lLAndYJb2o5PXZ[EFKGcDW7COV`jmT?[fFHaAm*0=b_A46jBaA?HWGkTd)O:Cf?CBMkYWC^X|Ml9Whd=FbfUGlj_2Z0TnYG;Xc_Z:I=LbUn_1S8SjJ(m4Ec2?8G7m8237a]VS2Fe^9Whg|:oE(o^H=R0oQBCo_*dkW(_VC[C4=;:gk:;7mDThW|TRaG47DS;9O;ZU8F:0S8_Cl396UMZF[3O5FFDIFHJYjLHW^foNL3LB3?00Of373XTPEk60jm[iNbG|PgQC;^:anoh*|[n;R*aNZAC:gh?6PmRJ|jU]]|NSMojgRnn_;6FFNil*D_l|ERY`D2PimeG^bfBeJ7[L_dZMk]Zoe=9QbKCg9?M)kB?=HJdSoAhOnW3`*NMU)HgA7189n7S?T2Ne;Kog]OoEL;e(NlcCFbKZRo])7EnH_jAU|0)gfQFFmfTIZeU6jZOoK(OOCLZH7PMJoA0g3]bXn9)WnU2W[U?d)KI3=UQTgZo:E4(=Se8CEQ8koJ3)82T^FjY=1NI7TLd?O|oB5d_KHY=OHhT5ZFo70Tf0m1HZAn_N`Q:6_ZKdQJnoS?n6kGlR^M^X(akG3WgZmDn?0XESPG?okd1)KMBU8M0ch5BGRf|bmW7i(?^0)j)kPkn1lJ(i:C.mmf Question 14 Soit n un entier n a t u r e l . On d é f i n i t l ' e n t i e r A n = 1 + 5 n + 5 2 n + 5 3 n . A n ≡ 0 13 si est seulement si : □ n est un multiple de 5 □ n est un multiple de 4 □ n n ' e s t pas un multiple de 5 □ n n ' e s t pas un multiple de 4 MMF.7h|A6000gESKK]]64?d2oh?N8P6(|?N;gfA7MHeJDRXaBH6Z3fc:Y49UbI3XG6[XPo|GWEd^bN5531[TXHP5fg?KfITc|k=[[foW?bbF|dUl^iQO?5f|[iOCBKaH3VM9m^L|NKmi)gQif0oNK7J33gZ|AaO[n3JnV`ho0OEj^Uc1ZZ6(1RXJ492lV(CCXHT634B(D?Q9mBDgUdB0L_[;bk_9kG`h^SQM[5OC);jMgkPMEj_iI3HM_SZVQl4ZcK;=kSeH7k(QRFJcN7)O7W?W8:4]2J|T;9M`T2B3Vd)JoUEJR*jI[5KBG:9JgWGTnL4loCQHk^nCGJTadOGnlK21T45GBVdd)FbBKAE_A`[]76Pk2MZE1Ne:PkKcX)e4j?U(J7LZe4HgjBkmT5A*TmI)[:(PkFAHEc:|:aWFCXJeTf7WTf7MbK1F([aM6Mi)Q[)6j6mXBQIAUM(dXY4(LY16G)Hd1iZZX130V41;Yf31BP6S0jfA9h=GFk*J^PWfl(c3d=4dd1CA3=U`R9F(NLjh)5Q*b7;[1am6|ESS1LHc;9QI_9oKg1I(QL:3JfGH9M2lQ)C1mBiH1M[3D30:(`h7FXA?7A:ZH2b)SKTbQ4G*Q:09=4)A(Ek]b*CJQ[T02R)l?m(HBE?K4*?0(OXL0l0IPYUcU03g1*PKLHUmLe`3[_4RPgfk46C`8708PZ:d1*i1L6`V(9)7D;QC5EHR]4716ZBb5Bea119785d49]0LhB^5)cg5WS;_Pf:A`[FG;PHJJ8=XFm6:81ZO0LFZ:1GNGoWm2b|7*)QEYI0[E`1NE42IZUND[FQ=D6DdAKGE35UaA0|?Z`jLa)]MkZc*j2YjSK3G_P=3`(I];`X6Mj39bem`7:]`2iYJoHfZ0SD^P)M5n*d2`:3bFh9XF]UHU;kU5JaFX;jf4RTD2]5ZE2i[D;cFX[JVa)f^RnPYXGFFeJHEVP)Dh454d22PA=DFjAYW(5RDX2jTU=Ad5651oCbdYHYkH4]UOQa;5PM35FId92[6PO:lM5U3QN6VY8bRHLc`I6JX;BW3R?SAb4Y)eOcYlS9bW(4nd(VPOSZFS0_RN*TTAdNC`W`TUJ5_c_;^2=eIV]J:hfMTRJ(OTQEG8W8cN*WG98Ulf167gcVGknVde7=e;Gb2eSNAen?O=6Pi[391K|9?hjNNd`[`b8?6InUi7[CBBeZOX5DnRW=AjE:[_EKG]:Ig[Ne7h`]PnG^f(n[UHQ4?bAR)4aT[n:J)P2H3TG4E_7mG?NIcU]IImlkbCl6LiGF]Z6_M8g5b))`oiZak85i]M|WQLli[h)OYnbA;Lmh0_dbgBKKIkg:91LW[15kZ_fmC5^9alDfc]h5chDdoIHLTl0b]V0*IAk:[81=8MQeT4|UN19UZX:4K_:UWC6d3;U;GngLhP|*o`S4VoPV)*O7?kcXZC;IQlJo_4QJVjk0`dhJ5fCH|o[7MP(Fo]Q^`E(o]`3OjP3LJPL^6GSGd^Ph;=`eH^6g08TP;5T7K|0R6HA6l3X|*KER4K((RE1|FXA^`95_XMcM[XNI`iL:mkmiP)*)dm7*^Mi(Y)MC(QA?kHnX4`|fM8;EPO7fgQ7(igao^TndPCSmUHECU7cXVO]QM;nhFCC|hf?j[Ph1U|)SZ9nlLZ=F[fM31MS^?QlkMj|O5Vj_9J^ZD9nMm7ToW|O]7`(a=jm7Cn^YYOOe[5(7`RF3TSbELO]5_`ilOdj)_bWX50gCDID;5j7AVmFZobGYFmWUmk5)V^`cnf)ccW6B?Qg*k?Q_IX|om7onlfnaj8mlnnf88Tm4JX5MWTMf=CZOeb|W[J1[MSO:X`iQ[dfT|oMkj_nkm]NkH=oK7_Kl)aOUjOSGJgA4Bi`iN3fOmD*jQ_0knR7lgiXVeI0X(ccK9LM?GiLNn5SbVSm_d7]Z`cjK?oFD)IQ|D2GmAO]fQkHnhmdSO?fjccL(fkC^FOD[iGFDS_Wdf^fOmnC`TannXM_oSK??JWTi`3ol;B3l^J`.mmf Voir corrigé de QCM QCM corrigé Analyse numérique licence master QCM Analyse numérique licence master Q u ... 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QCM Suites numériques 0 analyse numérique, arithmétique, corrigé, mathematique, QCM, QCM analyse numérique, QCM calcul de somme, QCM convergence, QCM développement limité, qcm suites numeriques, séries, suites 17 h 44 MathMagic Equation - www.mathmagic.com QCM Suites n u m é r i q u e s Q u e s t i o n 1 : la suite de terme g é n é r a l u n n ∊ N * = cos 1 n n 2 tend vers : □ + ∞ □ e □ 1 2 e ☑ e − 1 2 □ n ' a d m e t pas de limite Q u e s t i o n 2 : Soit la suite u n n ∊ N * = 1 + 1 2 + 1 3 + ... + 1 n □ la suite u n est convergente □ la suite u n est positive □ la suite u n est m a j o r é e ☑ la suite u n est croissante Q u e s t i o n 3 : La somme S = ∑ n = 1 + ∞ n 5 n vaut : □ 1 4 □ 4 5 ☑ 5 16 □ 1 16 Q u e s t i o n 4 : Soit la suite de terme g é n é r a l a n = a 2 n − 1 + a 2 n + 1 2 , ∀ n ≥ 1 ☑ la suite a n est constante ou divergente □ la suite a n est arithmetique □ la suite a n est g é o m e t r i q u e □ On peut pas en conclure Q u e s t i o n 5 : La suite x n n ∊ N definit par x n = π − n − π 10 est : □ b o r n é e □ croissante □ convergente ☑ d é c r o i s s a n t e Q u e s t i o n 6 On note S n = v 0 + v 1 + ... + v n pour tout n ∊ N avec v n = ln n + 2 n + 3 Alors : □ lim n → + ∞ S n = 0 □ lim n → + ∞ S n = 1 □ lim n → + ∞ S n = 2 □ lim n → + ∞ S n = 3 ☑ lim n → + ∞ S n = − ∞ Q u e s t i o n 7 Soit la suite α n d é f i n i e par α n = n − n + a n + b avec a , b ∊ R On a alors : □ lim n → + ∞ α n = a − b □ lim n → + ∞ α n = − a + b ☑ lim n → + ∞ α n = − a + b 2 □ lim n → + ∞ α n = a + b Q u e s t i o n 8 ∑ k = 1 n − 2 n − k C n k vaut : □ 1 − − 1 n □ 1 − − 2 n □ 0 ☑ − 1 n − − 2 n □ − 1 n + − 2 n 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