1QUESTTUTORIALSHead Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 653954391.111222a a bcb b a cc c a b =(A)0 (B)a3+ b3 + c3 3 abc(C)3abc (D)(a+b+c)32. Thefollowingsystemofequations,3x 2y + z = 0, x 14y + 15z = 0,x+2y-3z=0hasasolutionotherthan,x=y=z=0for equalto:(A)1 (B)2(C)3 (D)53. Therootsoftheequation,1 4 201 2 51 2 52x x = 0are :(A)- 1, - 2 (B)- 1,2(C)1,- 2 (D)1,24. If 000x a x bx a x cx b x c + + +=0,thenthevalueof x is:(A)0 (B)1(C)2 (D)35. Ifisthecuberootofunity,then111222 =(A)1 (B)0(C) (D)26. Ifxxx+++1 3 52 2 52 3 4 = 0, then x =(A)1,9 (B)- 1,9(C)- 1, - 9 (D)1,- 97. Thevalueofthedeterminant,7 9 794 1 415 5 55is :(A)- 7 (B)0(C)15 (D)278.a b c a ab b c a bc c c a b 2 22 22 2 =(A)(a + b + c)2(B)(a + b + c)3(C)(a + b + c) (ab + bc + ca)(D)Noneofthese9.a b a b a ba b a b a ba b a b a b+ + ++ + ++ + +2 32 3 44 5 6 =(A)3 (a + b) (B)3ab(C)3a+5b (D)010.b c a ab c a bc c a b+++ =(A)abc (B)2abc(C)3abc (D)4abc11. One of the roots of the given equation2QUESTTUTORIALSHead Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439x a b cb x c ac a a b+++ = 0, is :(A)- (a + b) (B)- (b + c)(C)- a (D)- (a + b + c)12. If2x+3y-5z=7,x+y+z=6,3x-4y+2z=1,thenx=(A) 2 5 71 1 63 2 17 3 56 1 11 4 2+(B) +7 3 56 1 11 4 22 3 51 1 13 4 2(C)7 3 56 1 11 4 22 3 51 1 13 4 2+13. x+ky-z=0,3x-ky-z=0andx-3y+z=0hasnon-zerosolutionfor k =(A)- 1 (B)0(C)1 (D)214. If = a b ca b ca b c1 1 12 2 23 3 3andA1,B1,C1denotetheco-factorsofa1, b1, c1respectivel y,thenthevalueofthedeterminant, A B CA B CA B C1 1 12 2 23 3 3is:(A) (B)2(C)3(D)015. Thenumberofsolutionsofequationsx+y-z=0,3x-y-z=0andx - 3y + z = 0is :(A)0 (B)1(C)2 (D)Infinite16. Thenumberofsol ut i onsoft heequations,x+4y-z=0,3x 4y z = 0 and x 3y + z = 0 is(A)0 (B)1(C)2 (D)Infinite17.b c a b ac a b c ba b c a c+ + + =(A)a3+b3+c3-3 abc(B)3 abc-a3-b3-c3(C)a3+b3+c3-a2b-b2c-c2a(D)(a + b + c) (a2 + b2 + c2 + ab + bc + ca)18. Ifx = cy + bz, y = az + cx, z = bx + ay,wherex,y,zarenotallzero,then:(A)a2 + b2 + c2 - 2 abc=0(B)a2 + b2 + c2 + 2 abc = 0(C)a2 + b2 + c2 + 2 abc = 1(D)a2 + b2 + c2 - 2 abc=119. Ifisacuberootofunity,thenarootofthefollowingequation,xxx+++111222 = 0, is :3QUESTTUTORIALSHead Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439(A)1 (B)(C)2(D)020. Inaskew-symmet ri cmat ri x, t hediagonalelementsareall:(A)Differentfromeachother(B)Zero (C)One(D)Noneofthese21. If A,B,Carethree nnmartices,then(ABC)=(A)A B C (B)C B A (C)B C A (D)B A C 22. If M = 1 22 3 and M2 M I2 = O,then =(A)- 2 (B)2(C)- 4 (D)423. IfA = 1 0 00 1 01 a b , then A2 =(A)Unitmatrix (B)Nullmatrix(C) A (D)- A24. If A = 1 10 1 , then An =(A) 10 1n (B)n nn 0(C)nn10(D) 1 10 n25. IfA = 1 11 1 , then A2 =(A) A (B)2 A(C)- A (D)- 2 A26. AB=O,ifandonlyif:(A)A O, B = O(B)A = O, B O(C)A = O or B = O(D)Noneofthese27. Inverseofthematrix 3 2 14 1 12 0 0 is :(A) 1 2 33 3 72 4 5 (B) 1 3 57 4 64 2 7 (C) 1 2 32 5 72 4 5 (D) 1 2 48 4 53 5 2 28. A =4 6 13 0 21 2 5 , B = 2 40 11 2 andC= 312,thentheexpressionwhichisnotdefinedis:(A)A2 + 2B 2A (B)CC (C)B C (D)AB29. If the matrix 1 3 22 4 83 5 10+ is singular,then =(A)- 2 (B)4(C)2 (D)- 430. Out oft hefol l owi ngaskew-symmetricmatrixis:4QUESTTUTORIALSHead Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439(A) 0 4 54 0 65 6 0 (B) 1 4 54 1 65 6 1 (C) 1 4 54 2 65 6 3 (D) iii+ 1 4 54 65 631. If AB = C, then matrices A, B, C are(A)A2 3 ,B3 2 ,C2 3(B)A3 2 , B2 3 ,C3 2(C)A3 3 , B2 3 ,C3 3(D)A3 2 ,B2 3 ,C3 332. If A= 4 13 2and I= 1 00 1,thenA2-6A=(A)3 I (B)5 I(C)- 5 I (D)Noneofthese33. If AandBarenon-singularmatrices,then:(A)(AB) -1 = A -1 B-1(B)AB = BA (C)(AB) = A B (D)(AB) -1 = B -1 A -134. If A = 11 , then for what valueof , A2 = O(A)0 (B) 1(C)- 1 (D)135. If A = 0 1 21 0 52 5 0 , then :(A)A = A (B)A = - A(C)A = 2 A (D)Noneofthese36. If A,B,Carethreesquarematricessuch that AB = AC imples B = C, thenthematrix Aisalways:(A) Asingularmatrix(B)Anon-singularmatrix(C)Aorthogonalmatrix(D)Adiagonalmatrix37. If A= 1 3 20 2 3/x ,B= 3 60 1 and AB=I,thenx=(A)- 1 (B)1(C)0 (D)238. The matrix A = 12121212 is :(A)Unitary (B)Orthogonal(C)Nilpotent (D)Involutary39. Fromthefollowingfindthecorrectrelation.(A)AB = A B (B)AB = B A(C)A -1 = adj AA(D)(AB) -1 = A -1 B-140. If A is a square matrix of order 3, thenthetruestatementis(whereIisunitmatrix).(A)det (- A) = - det A(B)det A = 0(C)det (A + I) = 1 + det A(D)det2A=2det A41. IfkisascalarandIisaunitmatrixoforder3,thenadj.(k I)=(A)k3 I (B)k2 I(C)-k3 I (D)-k2 I42. IfA = (1, 2, 3) & B = 5 4 00 2 11 3 25QUESTTUTORIALSHead Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439then AB =(A) 5 4 00 4 23 9 6 (B) 311(C)[ ] 2 1 4 (D) 5 8 00 4 31 6 643. If A = ii00 2 / ( )i = 1 , then A 1 =(A) ii00 2 /(B)ii00 2(C) ii00 2(D)i ii 2 044. IfA = 123 ,then AA =(A)14 (B)143(C)1 2 32 4 63 6 9(D)Noneofthese45. If A = 1 0 00 1 00 0 1 , then A2 =(A)Nullmatrix (B)Unitmatrix(C) A (D)2 A46. Adjointofthematrix,N = 4 3 31 0 14 4 3is :(A)N (B)2 N(C)- N (D)Noneofthese47. IfAi sasymmet ri cmat ri x, t henmatrix M AM is :(A)Symmetric(B)Slew-symmetric(C)Hermitian(D)Skew-Hermitial48. IfA = 2 0 00 2 00 0 2 ,then A2 =(A)5 A (B)10 A(C)16 A (D)32 A49. If A = 0 10 0 and AB = O, then B =(A) 1 11 1(B)0 11 0 (C)0 11 0 (D) 1 00 050. IfAandBaresquarematricesoforder2,then(A+B)2=(A)A2 + 2 AB + B2(B)A2 + AB + BA + B2(C)A2 + 2 BA + B2(D)Noneofthese51. If A = a cd b , then A -1 =6QUESTTUTORIALSHead Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439(A)1a b cd b cd a(B)1a d bc b cd a(C)1a b cd b dc a(D)Noneofthese52. IfR (t) = cos sinsin cost tt t , thenR (s) R (t) =(A)R (s) + R (t) (B)R (st)(C)R (s + t) (D)None of these53. Theelementofsecondrowandthirdcolumn in the inverse of 1 2 12 1 01 0 1 is(A)- 2 (B)- 1(C)1 (D)254. Thesolutionoftheequation,1 0 11 1 00 1 1 xyz = 112is (x, y, z) =(A)(1,1,1) (B)(0,- 1,2)(C)(- 1,2,2) (D)(- 1,0,2)55. IfAdenotesthevalueofthedeterminantofthesquaremartix Aoforder3,then 2 A=(A)- 8A (B)8 A(C)- 2 A (D)Noneofthese56. The inverse of the matrix 1 0 00 1 00 0 1is(A) 0 0 10 1 01 0 0(B) 1 0 00 1 00 0 1(C) 0 0 00 1 11 0 0(D) 1 0 00 0 10 1 057. Ift hemat ri x 0 1 21 0 33 0 i ssingular,then=(A)- 2 (B)- 1(C)1 (D)258. If AisasquarematrixofordernandA = kB, where k is a scalar, than A=(A)B (B)k B(C)knB (D)n B59. Ifp 4 + q 3 + r 2 + s + t = 23 1 31 2 43 4 3+ ++ + ,the value oft is :(A)16 (B)18(C)17 (D)1960. If A = cos sinsin cos and7QUESTTUTORIALSHead Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439Aadj. A= kk00thenkisequalto(A)0 (B)1(C)sin cos (D)cos2 61. If A = 10 1a , then A4 is equal to :(A) 10 14a(B) 4 40 4a (C) 40 44a(D) 1 40 1a 62. Theinverseof 2 34 2is:(A) 18 2 34 2(B) 18 3 22 4(C)18 2 34 2(D) 18 3 22 463. Thevalueofthedeterminant,a anx n x n xnx n x n x211 21 2cos( ) cos( ) cos( )sin( ) sin( ) sin( )+ ++ +isindependentof:(A)n (B)a(C)x (D)Noneofthese64. Thedeterminant,a b a bb c b ca b b c +++ + 0 = 0,ifa,b,carein:(A) A.P. (B)G.P.(C)H.P. (D)Noneofthese65. Let A= 4 6 13 0 21 2 5,B= 2 40 11 2 andC= [ ]3 1 2 .Theexpressionwhichisnotdefinedis:(A)B B (B)CAB(C)A + B (D)A2 + A66. LetA= 1 0 05 2 01 6 1 ,thentheadjointof Ais:(A) 2 5 320 1 60 0 2 (B) 1 0 05 2 01 6 1(C) 1 0 05 2 01 6 1(D)Noneofthese67. If 1= x b ba x ba a xand1= x ba xaregivendeterminants,then:(A)1 = 3 (2)2(B)ddx (1) = 3 2(C)ddx(1) = 2 (2)2(D) 1 = 3 23 2 /8QUESTTUTORIALSHead Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 6539543968. If A = 3 21 4 , thenA (adj.) A =(A) 10 00 10(B)0 1010 0(C)10 11 10(D)Noneofthese69. If a1x + b1y + c1z = 0, a2x + b2y + c2z = 0,a3x + b3y + c3z = 0 and a b ca b ca b c1 1 12 2 23 3 3 = 0,thenthegivensystemhas:(A)Onetrivialandonenon-trivialsolution(B)Nosolution(C)Onesolution(D)Infinitesolution70. IfA = [ ]a b,B = [ ] b aandC = aa , then the correct statementis :(A)A = - B (B)A + B = A - B(C)AC = BC (D)CA = CB71. Forpositivenumbersx,yandz,thenumericalvalueofthedeterminant,111log loglog loglog logx xy yz zy zx zx yis :(A)0 (B)1(C)loge xyz (D)Noneofthese72. If= a b cx y zp q r ,then ka kb kckx ky kzkp kq krisequalto:(A) (B)k (C)3k (D)k3 73. Theinverseofthematrix 3 21 4 is(A) 414214114314(B) 314214114414(C) 414214114314(D) 31421411431474. Thevalueof nN=1Un,ifUn = nn N Nn N N1 52 1 2 13 323 2+ + is :(A)0 (B)1(C)- 1 (D)Noneofthese75. MatrixAissuchthat A2=2A-I,whereIistheidentitymatrix.Thenfor n 2, An =(A)nA-(n-1)I(B)nA - I(C)2n - 1 A - (n - 1) I(D)2n-1 A-I76. Inathirdorderdeterminant,eachelementofthefirstcolumnconsistsofsumoftwoterms,eachelementofthesecondcolumnconsistsofsumofthreetermsandeachelementofthethirdcolumnconsistsofsumoffour9QUESTTUTORIALSHead Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439terms.Thenitcanbedecomposedintondeterminants,wherenhasthevalue:(A)1 (B)9(C)16 (D)2477. Ifvalueofathirdorderdeterminantis11,thenthevalueofthesquareoft hedet ermi nant formedbyt hecofactorswillbe:(A)11 (B)121(C)1331 (D)1464178. IfAandBaretwomatricesand(A+B)(A-B)= A2 - B2,then:(A)AB = BA(B)A2 + B2 = A2 - B2(C)A B = AB(D)Noneoftheabove79. A = 5 32 4 and B = 6 43 6 (A) 11 75 10 (B) 1 11 2(C) 11 75 10 (D) 12 75 1080. If A = 2 0 00 2 00 0 2 and B = 1 2 30 1 30 0 2,then ABisequalto:(A)4 (B)8(C)16 (D)3281. If A and B are square matrices of sameorder,then:(A)A + B = B + A(B)A + B = A - B(C)A - B = B - A(D)AB = BA82. If AandBaresquarematricesofthesameorder,then:(A)(AB) = A B (B)(AB) = B A (C)AB = O ; if A = 0 or B = 0(D)AB = O ; if A = 1 or B = 183. IfA= 1122tantanandAB=1,thenB=(A)cos2 2. A (B)cos2 2 . AT(C)cos2 2.I (D)Noneofthese84. IfAandBaresquarematricesoforder 3 such that A = 1, B = 3,then 3 AB =(A)- 9 (B)-81(C)-27 (D)8185. If 2X + 2Y = I and 2X - Y = O, whereIandOareunitandnullmatricesoforder3respectively,then:(A)X = 17 , Y = 27(B)X = 27 , Y = 17(C)X = 17 I , Y = 27 I(D)X = 27 I , Y = 17 I86. Theequations, x+2y+3z=1,10QUESTTUTORIALSHead Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 653954392x+y+3z=2,5x+5y+9z=4have:(A)Uniquesolutions(B)Infinitelymanysolutions(C)Inconsistent(D)Noneofthese87. The order of [xyz] a h gh b fg f c xyzis :(A)3 1 (B)1 1(C)1 3 (D)3 388. If AandBaretwomatricessuchthatAB=BandBA= A,then A2+B2=(A)2 AB (B)2BA(C)A + B (D) AB89.1 33 101 =(A) 10 33 1(B)10 33 1(C)1 33 10(D) 1 33 1090. IfA = 0 11 0 , then A4 =(A)1 00 1(B)1 10 0(C)0 01 1(D) 0 11 091. If A = 3 11 2 , then A2 =(A) 8 55 3(B)8 55 3 (C)8 55 3 (D) 8 55 3 92. If X = 3 41 1 , then the value of Xnis :(A) 3 4 n nn n(B) 2 5 + n nn n(C) 3 41 1n nn n( )( )(D)Noneofthese93. If A = 5 23 1 , then A -1 =(A) 1 23 5(B) 1 23 5(C) 1 23 5(D) 1 23 594. Theinverseofmatrix A= 0 1 01 0 00 0 1is :(A) A (B)AT(C) 1 0 00 1 00 0 1(D) 1 0 01 0 00 1 095. If A = 0 10 0 , I is the unit matrix of11QUESTTUTORIALSHead Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439order2&a,barearbitraryconstants,then(aI+bA)2isequalto:(A)a2I + ab A (B)a2I + 2 ab A(C)a2I+b2A (D)Noneofthese96. Whichofthefollowingisnottrue?(A)Everyskew-symmetricmatrixofoddorderisnon-singular(B)Ifdeterminantofasquarematrixis non-zero, then it is non-singular(C) Adjointofasymmetricmatrixis symmetric(D)Adjointofadiagonalmatrixisdiagonal97. Whi choneoft hefol l owi ngstatementsistrue?(A)Non-singularsquarematrixdoesnothaveauniqueinverse(B)Determinantofanon-singularmatrixiszero(C)If A= A,then Aisasquare matrix(D)IfA0,then A . adj A = A(n - 1)whereA = [aij]n nANSWERS1. A 2.D 3.B 4.A 5.B 6.D7.B 8.B 9.D 10.D 11.D 12.C13.C 14.B 15.D 16.B 17.B 18.C19.D 20.B 21.B 22.D 23.A 24.A25.B 26.D 27.C 28.A 29.B 30.A31.D 32.C 33.D 34.B 35.B 36.B37.B 38.C 39.B 40.A 41.B 42.C43.B 44.C 45.B 46.A 47.A 48.C49.D 50.B 51.A 52.C 53.B 54.D55.A 56.B 57.D 58.C 59.B 60.B61.D 62.A 63.A 64.B 65.C 66.D67.B 68.A 69.D 70.C 71.A 72.D73.A 74.A 75.A 76.D 77.D 78.A79.B 80.C 81.A 82.B 83.B 84.B85.C 86.A 87.B 88.C 89.B 90.A91.D 92.D 93.B 94.A 95.B 96.A97.C