alxi 93-113
DESCRIPTION
mathematicsTRANSCRIPT
ALGEBRA - clasa a XI - a (simbol AL - XI)
102Culegere de probleme
Algebr XI 103
MATEMATIC , clasa a XI a
ALGEBR SUPERIOAR
(simbol AL - XI)
AL - XI. 001 Se dau matricele ;
S se calculeze matricea C = A + B.
a) ;
b)
c)
d)
e)
f)
AL - XI. 002 Se dau matricele ptratice de ordinul al doilea i .
S se calculeze matricea
A = 2E 3F
a)
b)
c)
d)
e)
f)
AL - XI. 003 Fie .
Dac s se calculeze .
a) b) c)
d)
e)
f)
AL - XI. 004 S se calculeze produsul de matrice A(B, unde
,
a)
b)
c)
d)
e)
f)
AL - XI. 005 S se rezolve ecuaia matriceal:
a)
b)
c)
d)
e)
f)
AL - XI. 006 S se rezolve ecuaia matriceal:
a)
b)
c)
d)
e)
f)
AL - XI. 007 S se rezolve ecuaia matriceal
a)
b)
c)
d)
e)
f)
AL - XI. 008 Aflai astfel ca matricea diagonal constant
s fie soluia comun a ecuaiilor matriceale
i
a)
b)
c)
d)
e)
f)
AL - XI. 009 S se determine toate matricile X, cu proprietatea c ,
unde A =.
a)
; (,((R
b)
c)
; ((Rd)
; ((R
e)
; (,((R f)
; (,((RAL - XI. 010 S se determine matricea X care verific relaia: .
a) X =
b) X =
c) X =
d) X =
e) X =
f) X =
AL - XI. 011 Care este valoarea parametrului a(R pentru care exist x,y,z,t (R , nu toi nuli, astfel nct
?
a)
b)
c)
d)
e)
f)
AL - XI. 012 S se determine constantele reale p i q pentru care matricea
A =
satisface relaia A3=pA2+qA .
a)
b)
c)
d)
e)
f)
AL - XI. 013 S se rezolve ecuaia matriceal X
.
a) X =
b) X =
c) X =
d) X =
e) X =
f) X =
AL - XI. 014 S se determine matricea X care verific ecuaia
.
a) X =
b) X =
c) X =
d) X =
e) X =
f) X =
AL XI. 015 S se rezolve ecuaia matricial
a) ;
b)
c) ;
d)
e) ;
f)
AL XI. 016 S se determine toate matricile formate cu elemente din codul binar
B= care s transforme prin nmulire matricea coloan n matricea coloan
a) i
b)
c) i
d)
e) i
f)
AL - XI. 017 S se rezolve ecuaia: , X(M2(Z).
a) X =
b) X =
c) X =
i X =
d) X =
e) X =
f) X =
AL - XI. 018 S se determine toate matricile X (M2( Z ) astfel ca: X 2 =
.
a)
b)
i
c)
d)
i
e)
i
f)
i
AL - XI. 019 Se dau matricele A cu m(R.. S se determine valorile lui m (R astfel nct s existe trei constante nu toate nule, a,b,c(R cu condiia aA+bB+cC = 0, 0 - matricea nul.
a)
b)
c) orice
d)
e) f)
AL - XI. 020 S se calculeze suma:
.
a) b)
c) d)
e)
f)
AL XI. 021 Dac iar , s se determine numrul
an( R astfel nct s avem
N .
a)
b)
c)
d)
e)
f) .
AL - XI. 022 Dac
este o rdcin a ecuaiei x2+x+1 = 0 i n = 3p, p(N*, s se
calculeze suma:
.
a)
b)
c)
d)
e)
f)
AL XI. 023 Fie ;
, unde ( este o rdcin cubic complex a unitii i fie ecuaia matriceal AX = B. Fie S suma modulelor elementelor matricei X. Atunci :
a) S = 4;
b) S = 16;
c) S = 3;
d) S = ;
e) S = ;
f) S =
AL XI. 024 Fie M mulimea tuturor matricelor cu 4 linii i 5 coloane n care toate elementele sunt numerele +1 i - 1 i astfel nct produsul numerelor din fiecare linie i din fiecare coloan este -1 . S se calculeze numrul elementelor mulimii M.
a) 2
b) 7
c) 6
d) 4
e) 0
f) 1
AL - XI. 025 Se consider matricea M =
, a,b,c,d(R. S se determine
condiiile n care exist p,q(R , unici astfel ca M 2-pM-qI = 0, I fiind matricea unitate, 0 matricea nul. S se determine n acest caz valorile lui p i q.
a) b = c, a = d, p = a, q = b2-a2 b) b,c(R, a = d, p = 2a, q = bc-a2c) b = c, a,d(R, p = a+d, q = b2-a2 d) b ( 0 sau c ( 0 sau a ( d, p=a+d, q = bc-ade) b = 0, c = 0, a = d, p = a+d, q = bc-ad f) b ( 0, a ( d, c(R, p = a+d, q = -ad
AL - XI. 026 Fie A,B,C ( Mn ( C ) cu proprietile A+B = AB, B+C = BC,
C+A = CA. Pentru ce valoare m(R are loc egalitatea A+B+C = mABC ?
a)
b)
c)
d)
e)
f)
AL - XI. 027 Fie A =
o matrice nenul cu ad = bc , a,b,c,d(R. S se determine (n funcie de elementele matricii A) numrul real r asfel nct s aib loc egalitatea An = rn-1A pentru orice n(N, n ( 2.
a) r = a-d
b) r = a+d
c) r = b+cd) r = b-c
e) r = a+c
f) r = b+d
AL - XI. 028 S se determine puterea a matricei .
a)
b)
c)
EMBED Equation.3
d)
EMBED Equation.3 e)
EMBED Equation.3
f)
EMBED Equation.3 AL - XI. 029 Fie matricea A = . Calculai det P(A), unde P(x) = x100 - 1.
a) 0 b) 1 c) -1 d) 99 e) 100 f) -100
AL - XI .030 Fie A = . S se arate c An este de forma: An = i s se
determine apoi an , n ( N.
a) b) c)
d) e) f)
AL - XI. 031 S se determine An, n(N*, unde A(M3(Z) este o matrice care verific
relaia: (1 1+x 1+x2) = (1 x x2)A pentru orice x(R .
a) A
b) A
c) A
d) A
e) A
f) A
AL - XI. 032 Fie matricea A =
. S se calculeze A
, (n ( 1).
a) A
b) A
c) A
d) A
e) A
f) A
AL - XI. 033 S se calculeze
.
a)
b)
c)
d)
e)
f)
AL - XI. 034 Fiind dat matricea A =, s se calculeze matricea An, n(N*.
a) An =
b) An = c) An =
d) An =
e) An = f) An =
AL - XI. 035 Fie matricea A =. S se arate c An, n ( 1 are forma
i s se determine an i bn.
a)
,
b)
,
c)
,
d)
,
e)
,
f)
,
AL - XI. 036 Fie matricea A =. S se calculeze An, n(N, n ( 2.
a) b) c)
d)
e)
f)
AL - XI. 037 S se calculeze An, n(N* unde A =.
a) An = b) An = c) An =
d) An = e) An = f) An =
AL - XI. 038 Care sunt valorile parametrului a(R pentru care matricea
A =
este inversabil.
a) orice a(R\
b) orice a([-7,2]
c) orice a(R
d) orice a(
e) orice a(
f) orice a(R\
AL - XI. 039 S se calculeze inversa matricei
a)
b)
c)
d)
e)
f)
AL - XI. 040 S se determine parametrul astfel nct matricea
s fie inversabil i apoi s se afle inversa sa.
a)
b)
c)
d)
e)
f)
AL - XI. 041 Matricea are rangul doi pentru:
a)
b)
c)
d)
e)
f)
AL - XI. 042 S se determine valorile parametrilor reali
i
pentru care matricea:
A =
are rangul 2.a)
b)
c)
d)
e)
f)
AL - XI. 043 Se d matricea
. S se determine parametrul
real a pentru care rangul matricei este egal cu 2.
a) a = 4 b) a = -2 c) a = 3 d) a = 8 e) a = -1 f) a = 0
AL - XI. 044 Pentru ce valori ale parametrilor , matricele
i au ambele rangul 2.
a)
b)
c)
d)
e)
f)
AL - XI. 045 Fie matricea , ; dac rangul matricii este 2, atunci suma elementelor sale este soluie a ecuaiei:
a)
b)
c)
d)
e)
f)
AL - XI. 046 S se determine valorile parametrilor pentru care matricea
are rangul minim.
a)
b)
c)
d)
e)
f)
AL - XI. 047 Se d matricea:
. S se determine parametrii reali
pentru care rangul matricei s fie doi.
a)
b)
c)
;
d)
e)
f)
RAL - XI. 048 Pe care din urmtoarele mulimi de variaie ale parametrilor reali
i
matricea
are rangul 3?
a)
b)
c)
d)
e)
f)
AL XI. 049 Se consider matricea
.
S se precizeze valoarea parametrului (, pentru care rangul matricei este doi.
a) ( = 3; b) ( = 1; c) ( = -5; d) ( = 5; e) ( = -3; f) ( = 4 .
AL XI. 050 Fie matricea
Pentru ce valori reale ale lui a i x matricea A are rangul 2?
a) a = 0; x = 1b) x = 1; a ( R c) a = 0; x ( Rd) a = 0; x ((-1,2)
e) pentru nici o valoare real a lui a i x.
f) a = 0; x = 0
AL - XI. 051 S se rezolve sistemul
unde A=
, B=
.
a)
b)
c)
d)
e)
f)
AL - XI. 052 S se precizeze care dintre perechile de matrice (X,Y), date mai jos, reprezint o soluie a sistemului:
.
a)
b)
c)
d)
e)
f)
AL - XI. 053 S se calculeze determinantul:
a) 8
b) 6
c) 16
d) 17
e) 18
f) 0
AL - XI. 054 S se calculeze determinantul:
a) 0
b) 2a2
c) 4a2d) 6a2
e) 1
f) -1
AL - XI. 055 S se calculeze det dac
a) 1
b)
c)
d)
e)
f)
AL - XI. 056 Fie matricea ,
S se determine mulimea matricelor
a) sau
b) sau
c) sau
d) sau
e) sau
f) sau
AL - XI. 057 Calculai determinantul
.
a)
b)
c)
d)
e)
f)
_1043051471.unknown
_1086165682.unknown
_1086169829.unknown
_1093416543.unknown
_1094976116.unknown
_1094976209.unknown
_1094978541.unknown
_1098856442.unknown
_1098856474.unknown
_1098856490.unknown
_1098856498.unknown
_1098856481.unknown
_1098856451.unknown
_1094979010.unknown
_1094978502.unknown
_1094978514.unknown
_1094978427.unknown
_1094976171.unknown
_1094976192.unknown
_1094976141.unknown
_1093416640.unknown
_1094976047.unknown
_1094976092.unknown
_1093416850.unknown
_1093417873.unknown
_1094976021.unknown
_1093417710.unknown
_1093416759.unknown
_1093416602.unknown
_1093416622.unknown
_1093416580.unknown
_1086171036.unknown
_1086172506.unknown
_1086416063.unknown
_1086416444.unknown
_1086416477.unknown
_1086416504.unknown
_1093412822.unknown
_1086416491.unknown
_1086416465.unknown
_1086416322.unknown
_1086416387.unknown
_1086416416.unknown
_1086416361.unknown
_1086416273.unknown
_1086172727.unknown
_1086415826.unknown
_1086415991.unknown
_1086415780.unknown
_1086172591.unknown
_1086172623.unknown
_1086172532.unknown
_1086171145.unknown
_1086172392.unknown
_1086172453.unknown
_1086171819.unknown
_1086171089.unknown
_1086171121.unknown
_1086171059.unknown
_1086170530.unknown
_1086170763.unknown
_1086170937.unknown
_1086171014.unknown
_1086170844.unknown
_1086170695.unknown
_1086170732.unknown
_1086170560.unknown
_1086170028.unknown
_1086170412.unknown
_1086170495.unknown
_1086170326.unknown
_1086169969.unknown
_1086169998.unknown
_1086169867.unknown
_1086167877.unknown
_1086168787.unknown
_1086169552.unknown
_1086169699.unknown
_1086169783.unknown
_1086169649.unknown
_1086168869.unknown
_1086168917.unknown
_1086168827.unknown
_1086168416.unknown
_1086168712.unknown
_1086168751.unknown
_1086168686.unknown
_1086168135.unknown
_1086168295.unknown
_1086168016.unknown
_1086167061.unknown
_1086167461.unknown
_1086167659.unknown
_1086167753.unknown
_1086167605.unknown
_1086167261.unknown
_1086167349.unknown
_1086167168.unknown
_1086166068.unknown
_1086166909.unknown
_1086166968.unknown
_1086166099.unknown
_1086165930.unknown
_1086165975.unknown
_1086165829.unknown
_1086091754.unknown
_1086155070.unknown
_1086155803.unknown
_1086155898.unknown
_1086165199.unknown
_1086165242.unknown
_1086155920.unknown
_1086155853.unknown
_1086155874.unknown
_1086155825.unknown
_1086155299.unknown
_1086155426.unknown
_1086155522.unknown
_1086155343.unknown
_1086155202.unknown
_1086155243.unknown
_1086155173.unknown
_1086092577.unknown
_1086092827.unknown
_1086153998.unknown
_1086154035.unknown
_1086153897.unknown
_1086092741.unknown
_1086092826.unknown
_1086092668.unknown
_1086092381.unknown
_1086092533.unknown
_1086092554.unknown
_1086092411.unknown
_1086092337.unknown
_1086092362.unknown
_1086092291.unknown
_1044253154.unknown
_1044255837.unknown
_1086090683.unknown
_1086091450.unknown
_1086091561.unknown
_1086091640.unknown
_1086091753.unknown
_1086091495.unknown
_1086091395.unknown
_1086091426.unknown
_1086091330.unknown
_1086090463.unknown
_1086090595.unknown
_1086090623.unknown
_1086090548.unknown
_1086090337.unknown
_1086090431.unknown
_1086090138.unknown
_1044254561.unknown
_1044255013.unknown
_1044255275.unknown
_1044255409.unknown
_1044255536.unknown
_1044255669.unknown
_1044255473.unknown
_1044255338.unknown
_1044255137.unknown
_1044255216.unknown
_1044255087.unknown
_1044254860.unknown
_1044254924.unknown
_1044254793.unknown
_1044253888.unknown
_1044254069.unknown
_1044254279.unknown
_1044254370.unknown
_1044254125.unknown
_1044253966.unknown
_1044253390.unknown
_1044253827.unknown
_1044253208.unknown
_1043059485.unknown
_1043060789.unknown
_1044252173.unknown
_1044252622.unknown
_1044252825.unknown
_1044252861.unknown
_1044252269.unknown
_1043061027.unknown
_1043063968.unknown
_1043064365.unknown
_1043065035.unknown
_1043065514.unknown
_1043064402.unknown
_1043064332.unknown
_1043061052.unknown
_1043061457.unknown
_1043060976.unknown
_1043060999.unknown
_1043060926.unknown
_1043060130.unknown
_1043060619.unknown
_1043060679.unknown
_1043060178.unknown
_1043059736.unknown
_1043059814.unknown
_1043060066.unknown
_1043059659.unknown
_1043059092.unknown
_1043059281.unknown
_1043059363.unknown
_1043059134.unknown
_1043058606.unknown
_1043058965.unknown
_1043059015.unknown
_1043058646.unknown
_1043058900.unknown
_1043057239.unknown
_1043058500.unknown
_1043056886.unknown
_919325586.unknown
_940510940.unknown
_940670760.unknown
_940673470.unknown
_940673880.unknown
_942492455.unknown
_943204587.unknown
_943205141.unknown
_942492530.unknown
_942492541.unknown
_942492548.unknown
_942492535.unknown
_942492521.unknown
_942492523.unknown
_940674030.unknown
_942492431.unknown
_942492445.unknown
_940674031.unknown
_940673954.unknown
_940674028.unknown
_940674029.unknown
_940673955.unknown
_940673881.unknown
_940673953.unknown
_940673743.unknown
_940673878.unknown
_940673879.unknown
_940673876.unknown
_940673877.unknown
_940673744.unknown
_940673875.unknown
_940673577.unknown
_940673578.unknown
_940673471.unknown
_940673576.unknown
_940672711.unknown
_940672988.unknown
_940673466.unknown
_940673468.unknown
_940673469.unknown
_940673467.unknown
_940673284.unknown
_940673463.unknown
_940673465.unknown
_940673286.unknown
_940673462.unknown
_940672989.unknown
_940673283.unknown
_940672858.unknown
_940672859.unknown
_940672987.unknown
_940672712.unknown
_940672857.unknown
_940670919.unknown
_940672709.unknown
_940672710.unknown
_940672539.unknown
_940672708.unknown
_940670920.unknown
_940670917.unknown
_940670918.unknown
_940670915.unknown
_940670916.unknown
_940670913.unknown
_940670914.unknown
_940670761.unknown
_940669335.unknown
_940670191.unknown
_940670283.unknown
_940670757.unknown
_940670758.unknown
_940670284.unknown
_940670192.unknown
_940670282.unknown
_940670097.unknown
_940670190.unknown
_940669336.unknown
_940667846.unknown
_940668702.unknown
_940669111.unknown
_940668700.unknown
_940668701.unknown
_940510987.unknown
_940513964.unknown
_940514020.unknown
_940512101.unknown
_940510959.unknown
_919334262.unknown
_919340410.unknown
_940508475.unknown
_940509072.unknown
_940509133.unknown
_940509161.unknown
_940509101.unknown
_940508987.unknown
_940509029.unknown
_940508510.unknown
_919447655.unknown
_940508353.unknown
_940508444.unknown
_940508459.unknown
_940508421.unknown
_919451978.unknown
_925212945.unknown
_925214586.unknown
_919452142.unknown
_919452204.unknown
_919452237.unknown
_919452060.unknown
_919447765.unknown
_919447807.unknown
_919447711.unknown
_919447290.unknown
_919447473.unknown
_919447614.unknown
_919447332.unknown
_919340537.unknown
_919447251.unknown
_919340477.unknown
_919337885.unknown
_919338206.unknown
_919339729.unknown
_919340233.unknown
_919338324.unknown
_919338004.unknown
_919338051.unknown
_919337934.unknown
_919335471.unknown
_919337736.unknown
_919337809.unknown
_919335907.unknown
_919335391.unknown
_919335410.unknown
_919335449.unknown
_919334301.unknown
_919328075.unknown
_919333934.unknown
_919334157.unknown
_919334228.unknown
_919334011.unknown
_919328699.unknown
_919328767.unknown
_919328109.unknown
_919326063.unknown
_919327118.unknown
_919327201.unknown
_919327069.unknown
_919325723.unknown
_919325882.unknown
_919325655.unknown
_919238967.unknown
_919323466.unknown
_919325186.unknown
_919325389.unknown
_919325421.unknown
_919325260.unknown
_919324994.unknown
_919325072.unknown
_919324862.unknown
_919321877.unknown
_919323348.unknown
_919323395.unknown
_919322285.unknown
_919321615.unknown
_919321822.unknown
_919239017.unknown
_919237492.unknown
_919237930.unknown
_919238618.unknown
_919238884.unknown
_919238336.unknown
_919237611.unknown
_919237683.unknown
_919237572.unknown
_919236957.unknown
_919237212.unknown
_919237444.unknown
_919237146.unknown
_919236877.unknown
_919236908.unknown
_919236639.unknown