ffiIIEWI(c,S TAR.GET IIT JEE 2OI3

auEsTroN BANK ON DETERMINANT

& MATRICES

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a lsl

lu' \,/

The value of the deterrninant

lcos{nx1

lsin(nx)

(c)

(B) a

ttR= (A)

1

[r

rl liLr il^l anddet.

(D)

x

a,

n and x

1-In,neNhenthevalueof l,is (c) 3

Ais an iirvotutary mabix given ry

o=

Io

r'l

I

(c)tA

A-r

(B)

A

-3 4 | tnen the inverse of ; Ll -3 4)

z

lf a, 0 & y are real numbers, then

I t O

=

(D)A2

cos(p-cr)

lcos(cr-P)

I

cos(y cos(T

lcos(o-1) cos(P-y) (A) -1 (C) cosry + cosP + cosl

(B)

matric*

[;

lj[l?][;

then the value of n is (A) 26

I

16.

Ji

(c)-qr 3l U

Il

F

i Lx y zj

iAldet{8)

(c) 377

(D) 378

=

-

and supposethatdet.(A)=2thenthedet.(B)equals, where

Zar

2a -pl 2b -ql 2c -rl

2

(D) det(B) = $

\/l

z-^dl-'l

2ap I Zaq I (A) A., = A,

g= [1,. | 4Y 14,

(B)det1a1=-g

2

detiB) = - 16

fr 37sl [o tJ

l-l rl is "qr"tto the matrix

io qh .;l r

letn=l

=

cosc cosp

lt|.lj$ ^

4r

lf the product of n

-a) -P)

zeto

,,o=[-3;$ ;l'Jr-],A-1 sgvenby (A)-A

will be

P

KJ Lr

1

(B) A, =

2At

\=2d--.

lf an idempotent matrix is also skew symmetric then it must be (B) an identi$ matrix (A) an involutiary matrix a nullmatrix. (C) an orthogonal matrix

dr

(D)4 +24=0

flo PflI

'o

Vibri

Let the matri

(A)2

Let f(x) =

Ir |

I

()2

+

wino st gtenlents is not conect

-g

+Ar

ftfo ,.6

(D) (kA)r = kAr (k is a scalar)

,'oll yqf

of a 3 x 3 matrixA= (a,,) is

defin follo

"9 fil

)l-1

?r+

dzz+ aee (i.e. T,(A) is sum of the main

ways hold?

1n; t, (kA)= kT.(A)(k is a scalar) (c) T,(t.)= 3

,

is not coi'rect

(D)

nalelements). Which of the

rr(A+ts)=T..(A)+Tn(B) T,(A2) = (T,(A) )2

Put )t=o (A)

16.

'r]

(B)(Ar)r=A

lf A and B are invertible matrices, which one of the

lr ftt tu_

|

(A)Adj.4=lAlA-1 .r$XA + B)-1 = B-1 + A-1

=lJ -t -_ -\ n g-{

' fl

33

(B) 0

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 =

(A)1,=5,p=13

p

has no solution if

(B)1,=5 (D)p=13

tfll.=S,p+i3

I

a.t 0 clfl'4' r.- r -l p=

(B) a2 + 621 sz (D) ab+bc+ca ect of two square matrices A and B conformable for sum and product.

(B)tr(cA)=ot(A),aeR

(A) 0 (C) independentofQ Vibrant Academy (I) Pvt Ltd.'A-14(A)"

i

ofO

of

0&Q

both

Road tlo.1.Indraprastha IndustrialArea, Kota (Raj.) Ph.O744-242340G

t3l

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

| + 0)

fr

(/

dr2 ert

I

lutt Let Ao = art azz dzt | lut. atz att

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

4

A^ denote the determinant formed by the cofactor at at An-., then the determinant value of An is

and so on An denotes the determinant formed by the

c6factors

tn

(A)

(D)

(c)

A2o'

^i;

^1

^1

o sin a cos cr I t l- cos2 p sin B cos p I lf A= ,ino.oro sin2cr | I ' = lsinp cosp sin2p i [

.or'

integral ,r"ln that, AB is a null matrix, then which of the following should necessarily be an odd

"r"

(D)c+P

(B) p

cos

The deterrninant I cos I

sin

gq a"n cL

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

cos (Y + z)

(x+y)

sin (Y+z)

(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 ragivenmatrixA=l sine cosu I I

wnicn of the following statement holds good?

(B)A is symmetric, for 0 = (2n + 1)

(A)A=A-1 V0eR

1l

, ' nel

(D)Ais a skew symmetric, for0 = nr ; n e

B,(A) (1 +a+b+c) (1 +x+V+z)

R,rRr+

,{

I

/\

- 3(ax+by+cz)

+yz+ zx\ tx+ y) + b (y + z) + c(z+ x) - (xy (ab+ bc+ca) iCi x(a * 6) * y(b+ c) + z(c+ a) -

tei i

(D) none oftheEe lf A is matrix such

th

+

A+ 2l = O) tnen which of the following is INCORRECT ?

(A)Ais non-singular (B)A

1

ere

t

IO

is unit matrix of order

Vibrant Academy (I) hrt. Ltd.

'A

2

1 \J%n-'= ; {n+ l)

and O is null matrix of order 2

(D)A-1

)

A+

o.'1, Indraprastha IndustrialArea, Kota (Raj.) Ph'

It

I'

(A+ l)

&ft-l

0744-242uo6

=Q t4l

Mbra

Let A+2'.= then Tr (A) (A) 0

-

(B) a

*

[B

lt t) (D) none

2

1

+bl

.*91 a-ol

ad

,o-'=

sl

Tr(B) has the value equalto

rts of Ao and

rmed bythe

+ f] .,,0

[_L

lz -r

and D, =

bc, is

aca+clD, b d bld I tn"n the value of ID2 a c a+b+cl

wnere b

*

0 and

(c)-2b

(B) 0

The number of solutions of the matrix equation X2 =l other than I, is (c)2 (B) 1 (A) (where I is the 2 x 2 unit matrix )

0

(l+b2)x (l+c2)x l+b2x (l + c2)x (l + b2)x l+c2x

dd integral

(A)

0

(B)

o,u"no=

(A)i'e0

[l t] ;l = [l

1

?]

'to-r.risasinsurarmatrixthen

(C)1,2+32,,+4=0 (D)).,2-3i,-6=0

,%"-3X-4=0 Thevaluesof0,l.forwhichthefollowingequations n A sinOx-cos0y+(I+1)z=0; cos0x+sin0Y- :=0; l.x+(l+1)y+@s0z=0 U -" L' have non trivial solution, is

R-t0) (G)e=(2n+1)ll,1. eR*,nel (A)0= Dn,l. e

I t

el Let A=

=l

G)g=2nn,L isanyrationalnumber n

=(2n+1);,1eR,nel

sino 1l

I sinol,whereoso<2r,then l-sine L -l -sinO I J Det (A) e

12,41

(D) DetA e [2, o)

h=o v

Drt4 n-+0 vt^r21V nec Br= Ar=BCD; forthe matrix S =ABCD, LetA, B, C, D be (not

I ll

t4l -.

Mbrant Academy (I) Pvt. L6. "A-14(A)"

;ST

{

gT

fr&t9 HBcnA

S3=S 52=Sa

(A) ll is true but not I oth land llaretrue

=Q

as on vL via tion ch that r t\T ^ \fi') \!-'

,

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

Road No.1, Indraprastha lndustrialApa, Kota (Raj.)

Ph.o744-242

C

2

tsl

m of equations,

ax+ (2a

(A)o O

-

1)y = as

-

2

(B)l

possess no solution is

-t|sfz

0-'

L\

1o{

e_"=

o

D- =-O ^

Dl =-o

(D)infinite

Dl u

I t

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

=

0

\,1/((

t

$1

v

(

tn)

41.

(o)

J

3

*",

T

Fr^U_wz(

A= 3

(B) consistentwith unique non trivial solution (D) inconsistent with no solution

(A) consistent wltn trivial

"A

od f,"=-9

3x+Y+52=7and2x+3y+52=5are

Thefollowingsystemofequations 3x-7y+52=3;

solution (Ci consistent with infinite solution

\-x, 3 '

:

\/

sG

3

lf every element of a square non singular matrix A is mu lA-11 and I B-11 are related

(A)

lArl = k l B-11

as

.. I

(B) lA-11 =

pl Brl \.19f1

where n is order of matrices.

\

lx The number of real values of x satisfyin (A)

3

tltr|^-t,

(B) o

[r -r 1l Let A= l? I -,, I and J L^ (A)-2

1oB

l+ 2 o = l-'t -2 L'

21 ct 3-l I

(B)-1

(c)2

(B)-2

(c)6

x-l 45.

v

lf

D(x)= ^ -1 x

(A) 5

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Mbn

\ y'

L

,/

MatrixAsatisfies42

(A)nA-l

=2A- lwhere listhe

identity matrixthen for n >2, An is equalto (n e N)

(B)2n-1A-(n-1)l (C)nA-(n-1)l

(D)2n-1A-l

*\=AhL- AT rhesetorEi"l'lll.r,'=o I{ :?,'LAA'L) -&{ 3x+y+22 =Q

g can not be

,,

(cosO)x+Y+22=0

0 < e < 2n , has non- trivial solution(s) 119tfor no value of l. and (C) for all values of l. and only two values of

e

letA=

}n-g,t-

(B) for all values of I and 0 (D) for only one value of l. and all values of 0

0

2 3'l to'l [r 5| o= l-3l. wnirnof thefoltowing istrue? D f O li 0 21.1 Lo Ll.l and

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

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

VA)

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

(A)a+b+c=0

.3

.

(B)a+b+c=1

(C)a+b+c=-1

a=b=c-0

lf A, B and C are n x n matrices and det(A) = 2, de is equalto

612 (A); ))

(B);

9

(C)

12

(D)none

(B) has 4 real solutions

(C) has two realand two no (D) has infinite number of s

A isa2x2matrixsuchth

(A)-1

(B

(^

t rl

t_.c

cL),

h=leL b \ 16l -.

ele

Mbrant Academy (I) Pvt Ltd. 'A-14(A)"

a-f,= C-d_ = Road

-l

I

No.l, Indraprastha IndustrialArea, Kota (Raj.) Fh.

-LLl&=O

0744-242UOG

t4

lf f(x)

|"Or!3

The number of positive integral solutions

l-iL 2

I

itix -

6

-3 l. -2 2 -2 l+1.

=Q

1l

is

Thet

w

L€I

Sr SE

,,r-{

For a

(A)

matrixo=

[l

t'it],

Ir tool r Lo

.J

(^l

t]

equarto " il tl "f l-r sosol Ir 4eso-l (c)10 (B)Lo tJ 1 the varue

(q

is

G

r=l

J

25ool

I

I

2

2

then det. A is equal to

l+z (B)

(A) (1 + xy + Yz * zx)' (C) (xV + yz + zx)3

considera matrixA(o) =

l-ttge

!1,i3]

(A) A(0) is symmetric

to*'' I L f+i"'l

z")'

*.' (B) A(0) is skew sYmmetric

(* (D) A2(o) =

tlZ{x'Pl=A(n-o)

*, o =f o

z2\"

(D)

and B

=[';'

,..,0,.'l =orrtion

^[;

tr(AB) =

\ - ,t,J

*

*

a difierential equation of order'm'

and degree'n'then (m + n) is equalto

Vibnnt Academy (I) pvg Ltd. "A-14(A)"

Road N0.1, lndraprasthe IndustrialArea' Kota (Raj.) Ph'0744-242y,06

g

\x,

lf f(x) satisfies the equation

{at-

t(x-2) f(x+5) r(x-z)l

5 4 -5 l=0vt.R 10 12 3ol

(c)(11) he equation the elements Let g(x)

=

)

./

lf(x+c1

f(c) | f'(c) I

f(x+2c) f(x+3c)l f(2c) f(3c) f'(2c) f'(3c) I,

where c is constant tnen

,l!f Y

&:t\tt) ,.

"ou!,,o

|

Vr,fo

k))ft cotumn relueof det.

(D)(17)

(B)

1

(c)

-

1

(D) f(c)

Letcr+1,F+l,bethedistinctnonzerorealrootsoftheequationax2+bx+c=0,wherea,b,ceR,a+0.

It+s6 1+s1

rfl rlrag oots are

LetSn=cr,n+ 9n,

n>0and A=11+51 1+52

l1+52 1+53 (B) A>0

(A) -r<0

1+s2l 1+S3l,then 1+Sal

(c) A<0

(D) A=0

LetAand B be 3 ' 3 matrices such thatAB +fi+ $ = Q Staternent-l AB = BA

ry

Stagnent-2 Pr1=

br every mabix P wttich is invertible. (A) ffirnent-1 b Uue. ffirnent-2 b fue and StsEment-2 is conect explanation for Statement-1 (B) ftnent-1 bfue. Sdernentr2 btLeand Stabment-2 is NOTconectexplanation brStatementl I=

F1

P

(C) $atement-1 is tn-'e, Statement-2 is false (D)

Srnent-1

isfalse, $atement-2 istrue

LetAbeany3x2matrix

Statement-l Statement-2

l6fsatement-1

Inverse of Mr does not exist.

AAr is a singular matrix is true, Statement-2 is true and Statement-2 is correct explanation for Statement-1

(B) $atement-1 is fue, Statement-2 is fue and Statement-2 is NOTconect explanation for Statement

l

(C) $atement-1 is true, Statement-2 is false

(D) Statement-1 is false, Statement-2 is true

Paragraph forquestion nos.67 & 6a Gonsider the system of equations

l\

by AX = B then for a = 1, the value of

n

A-1

-312|'

L-1 1 -l_l

[8]

'

Mbrant Academy (I) Pvt. Ltd.'A-14(A)"

"\

\a, 0\ \ \ -\

tlzI Irlz o F)l-Il2 | -3121 L-l I 1 I o rlzf Irlz p)l-rl2 |

of order'm'

\

\ \

t\

I

q

ot

t

Road No.1,Indraprastha IndustrialArea, Kota (Raj.)Ph.0744-2423406

teI

if B = 0 then tl'a value of 'a'for which AX = B have non trivial solution, is (c) 1 (B) 3 2

(D) none

Faragraph f,orquestion nos.69 to 7t lfAisasymmetricandBskerssymrnetricmatrixandA+BisnonsinEularandC=(A+B)-1(A-B)then

(c)A

(D) B

(c)A

(D) B

(D) B

Paragraph forquestion nos.72 to74

I z -z -4f l-+ -3 -3.l 4-landBo=11 g ll 3 lfA=l-l " lr -2 -31 14 4 3J Bn = adj(Bn_

22.

det.

(\.

r), n e N and I is an identi$

n3 g3 + Afl + nf, efi

8., u B, +

(A)

;4.

+ ....... 10 ) is equalto

(B)- 800

(A) 1000

?3.

matrix of order 3 then 3nsv/er ihe following questions.

Eo

(c) 0

(D)

(c)498c

(D)491

-

80CI0

......, + B*n is equal to (Bi 7Bo

For a vatlable matiix X the equation AoX = Bo will have (B) infinite solution (A) unique solutian (D) no solution (C) firiieiy many soiuiion

Paragraph for question nos. 75 &76

Iu,, isnotequaito

\f 'l'

l
(A) t.' +

i.

+

1,

(B)

zero

I

(c) (det(A))2

ct;l - -Qt

A. is

(B) symmetric

(A) non singular"

(A)-1

\9rlA'

79/ \./ ^

is

(BiI

orthogonai

A-: is orthogonal

ThesetofequationsN-y 3z=2,2x-y (A) unique soiuton oniy

(

+

for

=S (C) inflnite number of solutions for a =

I

( pYLl

V,b,

(where bc + 0) satisfies the equat

-

(B) k = -lAl B) then

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

-Zt*t^=[i i

?]

.n".'' then the conect statement is

A-r

A-51.=9 (C) Aa is not invertible

VPln'is

-

x 3 diagonal matrix. Which of the '-v

D

if exists is a scalar

The value of

A cos'A 1+ cos2 A sin2 A sin2A costA

brnents

invertible

follo

is not true? or every matrix A of order 3 x 3

matrix

ese

and 0 <

0 lying between

l+sin2

I = 5 (A-4ls)

;and

g

2sin

are

2sin

:

satisfying the equation

tra C*?

,-, R r-

F rel="nofollow">,Rr

l+2si

I

=J

t iIfriFo

' -Qli

ss.,,/civenthematricesAandeas A=

0g

\/

Ais

I I r -rl (A)x=iL-, ;l

0t )-: f_t t=l {, _)

il--0 \sn nsri + (fr

\,/

l+ -iJ

LZ

the following hold(s) true? The two matrices X and y are such that XA = B and AY = B then which of

he det,

t- D ) =G 0,

fr -ll anot= [r -tl -21

86.,/

1[l

lfthesystemof equations'

?2\-by=a2-b

then the possible values of 'a'and'b'are

\J$f a=1, b=-1 (C)

o'l

1a1v=5[+ oj

a=-1, b=-1

&

(B) a=l'b=-2 (D) a= -1 ' b=-2

matrix then nd B are two 3 x 3 matices such that their productAB is a null B must be a null matrix' et. A*

O

=

h ++)

,l'?,, *r,

Mbnnt Acadenry (I) p111 L6. "A-t4(A)"

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t\ (-,

ANSWERKEY lsrNGLE CORRECTI

Q.1 Q.6

A

Q.1'1

C

Q16

D

B

Q.21 C Q.26 A Q.31 C Q.36 C Q.41 B Q.46 C Q.51 A

0.56

B

Q.61 Q.66 Q.71

A

Q.76

C

C

C

Q.2BQ.3AQ.4 Q.7CQ.8DQ.9 o.12AQ.13CQ.14 Q.17CQ.18AQ.19 Q.23 C Q.22 B Q.27CQ.28CQ.29 Q.328Q.33DQ.34 Q.37CQ.38DQ.39 Q.43 C Q.42 C Q.47AQ.48AQ.49 Q.52DQ.53DQ.54 Q.578Q.58DQ.59 Q.62AQ.63AQ.64 Q.67DQ.68AQ.69 Q.73 C Q.72 C

Q.24

Q.4

Q.74

D

Q.5

B

D

Q.10

D

D

Q.15

c

D

Q.20

B

B

Q.25

c

A

Q.30

D

c

Q.35

B

B

Q.40

D

D

Q.45

A

D

Q.50

B

D

Q.55

D

B

Q.60

c

B

Q.65

A

A

Q.70

B

D

Q.75

D

Q.89

A, C

IMULTTPLE CORREGT ANSWERS TYPEI

j, ,I

Q.77

A, B, C

Q.78 A. B

Q.79 B, D

Q.8O

A,C

Q.81

A, C, D

Q.82 A, B, D

Q.83 B, C

Q.84

A,B,C,D

Q.85

C, D

Q.86 A, B, C, D

Q.87

Q.88

A,B,C

Vrbrant Academy (I)Pvt. Ltd. "A-14(A)"

A, B, C,

D

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