question bank on determinants & matrix

7/29/2019 Question Bank on Determinants & Matrix

1/12

ffiIIEWI(c,STAR.GET IIT JEE 2OI3

auEsTroN BANK ONDETERMINANT&MATRICES

UBRANT ACADEFTY (India) private LimitedAehern Jn a,cel,h,nceA-1 4(A), Road No. 1, Indraprastha Industrial Area, Kota-324005 (Raj.)

Tel. (07 44) 2428664, 2428666, 24234;06, 320526 1, 320 5262 F ax: 242340 5Email: [email protected] Website : www.vibrantacademy.com


2/12

lsllu'The value of the deterrninant lcos{nx1

lsin(nx)(B) a

A-r(B) z

[r rlttR= li il anddet. 1-In,neNhenthevalueof l,isLr ^l(A) 1 (c) 3IoAis an iirvotutary mabix given ry o= Ll

r'l A4 | tnen the inverse of ; will be4)I-3-3

(c) x

A(c)t

(B) cosc cospzeto

(c)-qr3lU

(c) 377

(D) a, n and x

I t cos(p-cr)lf a, 0 & y are real numbers, then O = lcos(cr-P) Ilcos(o-1) cos(P-y)

(D)A2

cos(y -a)cos(T -P) =

I

,,o=[-3;$ ;l'Jr-],A-1 sgvenby ^ lt|.lj$ Ji(A) -1(C) cosry + cosP + cosl

(A)-A 4r

2a -pl2b -ql2c -rl2P\/lKJ

Lr drz-^dl-'l2ap IZaq IZar 1

lf the product of n matric* [;then the value of n is(A) 26

l-l is "qr"tto the matrixrl(D) 378

Ilj[l?][; fr 37sl[o tJ

(B) A, = 2At \=2d--. (D)4 +24=0lf an idempotent matrix is also skew symmetric then it must be flo PflI 'o(A) an involutiary matrix(C) an orthogonal matrix

io h .;l [1,.letn=l F q r i and supposethatdet.(A)=2thenthedet.(B)equals, where g= | 4YLx y zj 14,iAldet{8) = - 2

detiB) = - 16

(A) A., = A,

(B)det1a1=-g(D) det(B) = $

(B) an identi$ matrixa nullmatrix.


3/12

Let the matri(A)2 Iret f(x) = |

I()2wino st-g+Ar

gtenlents is not conect' fl | (B)(Ar)r=A(D) (kA)r = kAr (k is a scalar)

n -\ftt ,"9 fil

lf A and B are invertible matrices, which one of the ,'oll is not coi'rect(A)Adj.4=lAlA-1 yqf )l-1.r$XA + B)-1 = B-1 + A-1 (D)of a 3 x 3 matrixA= (a,,) is defin ?r+ dzz+ aee (i.e. T,(A) is sum of the manalelements). Which of the follo ways hold?rr(A+ts)=T..(A)+Tn(B)

T,(A2) = (T,(A) )21n;t, (kA)= kT.(A)(k is a scalar)

(c) T,(t.)= 3

(A) 0(C) independentofQ

Put )t=o(A) 33 (B) 0

16. lf Aand B are difiere'rt matrices satisfying As =(A) det (A2 + 82) must be zero.(B) det (A - B) must be zerc.(C) det (A2 + 82) as well as det (A - B) must be(D) At least one of det (A2 + 82) or det (A- B) must be zero.The system of equations x + v + z = 5, x + 2y + 32 = 9, x + 3y + ).2 = p has no solution if(A)1,=5,p=13 (B)1,=5tfll.=S,p+i3 (D)p=13

Ia.t 0 clfl'4'.- r -lp=(B) a2 + 621 sz(D) ab+bc+ca

ect of two square matrices A and B conformable for sum and produ(B)tr(cA)=ot(A),aeR

i ofOof 0&Q bothVibrant Academy (I) Pvt Ltd.'A-14(A)" Road tlo.1.Indraprastha IndustrialArea, Kota (Raj.) Ph.O744-242340G


4/12

Which of the following statements is incorrect for a square matrix A' ( | A | + 0)(A) lf A is a diagonal matrix, A-1 will also be.a diagonal T_?J'I_^-,..

,B,- R,rRr+ /\

The deterrninant II

I

luttdr2 ert

Let Ao = | art azz dztlut. atz att

[ .or' o sin a cos cr I l- cos2 p sin B cos p Ilf A= | ,ino.oro sin2cr I ' t = lsinp cosp sin2p i"r" ,r"ln that, AB is a null matrix, then which of the following should necessarily be an odd integral

and let A" denote the determinant formed by the cofactors of elements of Ao and

A^ denote the determinant formed by the cofactor at 4 and so on An denotes the determinant formed by thec6factors at An-., then the determinant value of An is(A) A2o' (c) ^i; (D) ^1tn^1

(B) p

cos (x-y) cos (Y-z)cos (x+y) cos (Y + z)sin (x+y) sin (Y+z)

(D)c+Pgq a"n cL

(A) 2 sin (x - Y) sin (Y - z) sin (z - x)(C) 2 cos (x - Y) cos (Y - z) cos (z- x)')'' 2 sin (x - y) sin (y - z) sin (z- x) *-'-> :2 cos (x - Y) cos (Y- z) cos (z- x)

I coso -ti1e I wnicn of the following statement holds good?ragivenmatrixA=l sine cosu I(A)A=A-1 V0eR

(A) (1 +a+b+c) (1 +x+V+z) - 3(ax+by+cz)tei i tx+ y) + b (y + z) + c(z+ x) - (xy +yz+ zx\iCi x(a * 6) * y(b+ c) + z(c+ a) - (ab+ bc+ca)(D) none oftheEelf A is matrix such th + A+ 2l = O) tnen which of the following is INCORRECT ?1 I'(A)Ais non-singular (B)A I O \J%n-'= ; {n+ l) (D)A-1 (A+ l)1 ere t is unit matrix of order 2 and O is null matrix of order 2 )

1l(B)A is symmetric, for 0 = (2n + 1) , ' nel(D)Ais a skew symmetric, for0 = nr ; n e I

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5/12

lz -r sl,o-'= [B lt t)(A) 0 (B) 1

a +bl.*91 and D, =a-ol

2

(l+b2)xl+b2x(l + b2)x

(D) none

and

bythe

(l+c2)x(l + c2)xl+c2x

t4l -. Mbrant Academy (I) Pvt. L6. "A-14(A)" Road No.1, Indraprastha lndustrialApa, Kota (Raj.) Ph.o744-242 C 2

(B) | is true but not ll(D) both land ll are fialse.

Let A+2'.= [_L + f] .,,0then Tr (A) - Tr(B) has the value equalto

ad * bc, is (B) 0 (c)-2bThe number of solutions of the matrix equation X2 =l other than I, is(A) 0 (B) 1(where I is the 2 x 2 unit matrix ) (c)2

aca+clD,b d bld I tn"n the value of I wnere b * 0 anda c a+b+cl D2

Det (A) e 12,41 (D) DetA e [2, o)h=oDrt4 n-+0 ,vt^r21Vas on vLvia tion

;ST { gTfr&t9 HBcn

(A) 0 (B) 1o,u"no= [l t] ;l = [l ?] 'to-r.risasinsurarmatrixthen(A)i'e0 ,%"-3X-4=0 (C)1,2+32,,+4=0 (D)).,2-3i,-6=0Thevaluesof0,l.forwhichthefollowingequations n - AsinOx-cos0y+(I+1)z=0; cos0x+sin0Y- :=0; l.x+(l+1)y+@s0z=0 U " L'have non trivial solution, is(A)0= Dn,l. e R-t0) G)g=2nn,L isanyrationalnumbern(G)e=(2n+1)ll,1. eR*,nel =(2n+1);,1eR,nel

I t sino 1lLet A= l-sine I sinol,whereoso


6/12

m of equations,ax+ (2a - 1)y = as - 2 possess no solution is(A)o (B)l -t|sfz

1o{ e_"= o0-' L\ D- =-O ^Dl =-o(D)infinite Dl u

I t tan*lO = [_ * * ^i ^.] then let us define a function f (x) = 691. (ArAt ) then which of the following can not be

where n is order of matrices. lxThe number of real values of x satisfyin tltr|^-t,(A) 3 (B) o

[r -r 1l l+ 2 21Let A= l? I -,, I and 1oB = l-'t o ct IL^ J L' -2 3-l(A)-2 (B)-1 (c)2x-lv -145. lf D(x)= ^x

(A) 5 (B)-2 (c)6 (D)o

tn) J (o) T *", 3 \-x, 3 Fr^U_wz( "A' A= 3 od f,"=-9 \/41. Thefollowingsystemofequations 3x-7y+52=3; 3x+Y+52=7and2x+3y+52=5are : 3(A) consistent wltn trivial solution (B) consistentwith unique non trivial solution(Ci consistent with infinite solution (D) inconsistent with no solutionlf every element of a square non singular matrix A is mulA-11 and I B-11 are related as ..I(A) lArl = k l B-11 (B) lA-11 = pl Brl \.19f1

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7/12

not be

9 (C) 12 (D)none

(B) has 4 real solutions(C) has two realand two no(D) has infinite number of sA isa2x2matrixsuchth(A)-1 (B(^ t rlh=leL b \ a-f,= -lt_.c cL), C-d_ = I

,/\ y' MatrixAsatisfies42 =2A- lwhere listhe identity matrixthen for n >2, An is equalto (n e N)(A)nA-l (B)2n-1A-(n-1)l (C)nA-(n-1)l (D)2n-1A-l*\=AhL- ATrhesetorEi"l'lll.r,'=o I{ :?,'LAA'L) -&{3x+y+22 =Q ,, }n-g,t-(cosO)x+Y+22=00 < e < 2n , has non- trivial solution(s)119tfor no value of l. and e (B) for all values of I and 0(C) for all values of l. and only two values of 0 (D) for only one value of l. and all values of 0[r 2 3'l to'lletA= li 0 5 | and o= l-3l. wnirnof thefoltowing istrue? D f OLo 21.1 Ll.l

(B) Ax = b has exactly three solutions.(D) Ax = b is inconsistent.

(A)a+b+c=0 (B)a+b+c=1 (C)a+b+c=-1 a=b=c-0lf A, B and C are n x n matrices and det(A) = 2, de. is equalto612(A); (B);))

VA) Ax = b has a unique solution.(C) Ax = b has infinitely many solutions./

ele

-LLl&=

lf a, b, c are real then the value of determinant

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8/12

"Or!3he number of positive integral solutionsl-iL 2 I-3 l. -22 -2 l+1. =Q is

Ir 4eso-l(B)Lo 1 J

For a matrixo= [l t'it], the varue " il tl "f t] is equartor=lIr tool(A) Lo r .J

22 then det. A is equal tol+z(B) z2\"(D) z")'A) (1 + xy + Yz * zx)'(C) (xV + yz + zx)3

considera matrixA(o) = l-ttge !1,i3] *.'(A) A(0) is symmetrictlZ{x'Pl=A(n-o)

(B) A(0) is skew sYmmetric(* \(D) A2(o) = ^[; - ,t,J

*, o =f o f+i"'l and B =[';'L to*'' Iand degree'n'then (m + n) is equalto

,..,0,.'l =orrtion tr(AB) = * * a difierential equation of order'm'

l-r sosol 25ool(c)10 t J I I

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9/12

lf f(x) satisfies the equation \x,{at-

(A) -r0 (c) A


10/12

2if B = 0 then tl'a value of 'a'for which AX = B have non trivial solution, is(B) 3 (c) 1Faragraph f,orquestion nos.69 to 7tlfAisasymmetricandBskerssymrnetricmatrixandA+BisnonsinEularandC=(A+B)-1(A-B)then

(c)A (D) B

(c)A

(D) none

(D) B

(D) B

(D) - 80CI0(D)491

Paragraph forquestion nos.72 to74I z -z -4f l-+ -3 -3.llfA=l-l 3 4-landBo=11 g ll" lr -2 -31 14 4 3J

Bn = adj(Bn_ r), n e N and I is an identi$ matrix of order 3 then 3nsv/er ihe following questions.22. det. (\. n3 g3 + Afl + nf, efi + ....... 10 terms) is equalto(A) 1000 (B)- 800?3. 8., u B, + ......, + B*n is equal to(A) Eo (Bi 7Bo

(A) unique solutian(C) firiieiy many soiuiion

(c) 0

(c)498c

(B) infinite solution(D) no solution;4. For a vatlable matiix X the equation AoX = Bo will have

A. is(A) non singular"ct;l - -Qt(B) symmetric

(A)-1 (BiI

\9rlA' is orthogonai A-: is orthogonal (79/ ThesetofequationsN-y 3z=2,2x-y +(A) unique soiuton oniy for = S(C) inflnite number of solutions for a = I

Paragraph for question nos. 75 &76Iu,, isnotequaito \f 'l' Il


11/12

(where bc + 0) satisfies the equat(B) k = -lAl

l^' a2 - 1b-c;t b.lant lu' b2 - (c-a)z ."1 it oiuil.t c2 - 1a-b;2 ubl

c2

IA-r = 5 (A-4ls)VPln'is invertible

and 0 ,Rr

value of 0 lying betweenl+sin2 A cos'A 2sin

sin2 A 1+ cos2 A 2sinsin2A costA l+2si are :

0,)-:_){,il--0nsri

*r,

I I r -rl 1[l o'l(A)x=iL-, ;l 1a1v=5[+ oj86.,/ lfthesystemof equations' ?2\-by=a2-b &\,/ then the possible values of 'a'and'b'are\J$f a=1, b=-1 (B) a=l'b=-2(C) a=-1, b=-1 (D) a= -1 ' b=-2nd B are two 3 x 3 matices such that their productAB is a null matrix thenet. A* O = B must be a null matrix'

-Zt*t^=[i i ?] .n".'' then the conect statement isA-51.=9

(C) Aa is not invertiblex 3 diagonal matrix. Which of the follo is not true?D '-v - or every matrix A of order 3 x 3if exists is a scalar matrix ese

fr -ll [r -tlss.,,/civenthematricesAandeas A= l+ -iJ anot= LZ -21\/ The two matrices X and y are such that XA = B and AY = B then which of the following hold(s) true?

tra C

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12/12

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