Download OU (Osmania University) B.Sc Fifth Year (5th Year) B.Sc Maths Question Bank Question Paper
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LinearAlgebra
Unit-I
I
'2?l
l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=
l4.
.Letv;=
.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li
v7=
iii
Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that
m
v=
{lS]=1
3
0.51
z
o}
a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch
l?2
?--2andv;=l7
Determineif{vbv?isabasicforR?.ls{vbv?a
3?9
basislorlll2'.?
.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof
p(l)=l+~lf+7|?relativetoB.
Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.
l_?
v;=[__3I1spmtlllzbutdonotfomiabasis.Find
.l...twodifferentwaystoexpressiasalinearcombinationofv?.v;.v;.
.Findthe(liineiisioiioftliesubspaceofallwctorsin1R3whose?rstandthirdentriesareequal.
...21?2?3
.FindthediineiisonofthesubspaceHofIRspannedby_5,w.l5.
.LetHbeanll-(llIllCllSlOllIllsubspaceo!?anll-tlllll0ll?0ll?lvectorqmceV.ShowthatH=V.
.ExplainwhythespacePol?allpolynomialsismiin?nitediiiietisioiialspace.
Unit-II
.Ifa-lx7iiiatrixAhasrank3.?iuldiiii.\'ulA.dimRowA.iiiidniiikAT.
.lla7x5niatrixAhasrank2.?nddiiii.\'iilA.diinRowA.niidmiikAT.
.lftheiiullspaeeofan8x5inatrixAisZl-diiueiiiional.whatisthedimensionoftherowspace
ofA?
IfAisa3x7matrix.whatisthesniallmtpossiblediniensioiiof.\'iilA?
2.12.1PracticalsQuestionBank
LinearAlgebra
Unit-I
I
'2?l
l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=
l4.
.Letv;=
.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li
v7=
iii
Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that
m
v=
{lS]=1
3
0.51
z
o}
a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch
l?2
?--2andv;=l7
Determineif{vbv?isabasicforR?.ls{vbv?a
3?9
basislorlll2'.?
.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof
p(l)=l+~lf+7|?relativetoB.
Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.
l_?
v;=[__3I1spmtlllzbutdonotfomiabasis.Find
.l...twodifferentwaystoexpressiasalinearcombinationofv?.v;.v;.
.Findthe(liineiisioiioftliesubspaceofallwctorsin1R3whose?rstandthirdentriesareequal.
...21?2?3
.FindthediineiisonofthesubspaceHofIRspannedby_5,w.l5.
.LetHbeanll-(llIllCllSlOllIllsubspaceo!?anll-tlllll0ll?0ll?lvectorqmceV.ShowthatH=V.
.ExplainwhythespacePol?allpolynomialsismiin?nitediiiietisioiialspace.
Unit-II
.Ifa-lx7iiiatrixAhasrank3.?iuldiiii.\'ulA.dimRowA.iiiidniiikAT.
.lla7x5niatrixAhasrank2.?nddiiii.\'iilA.diinRowA.niidmiikAT.
.lftheiiullspaeeofan8x5inatrixAisZl-diiueiiiional.whatisthedimensionoftherowspace
ofA?
IfAisa3x7matrix.whatisthesniallmtpossiblediniensioiiof.\'iilA?
20.
2L
22.
23.
24.
25.
.Letu=l?
FindvinR?suchthatl
l?3~l
=T
2438]uv.
.lfAisa7X5matrix.whatisthelargestpossiblerankofA?lfAisa5x7matrix.whatis
thelargestpossiblerankofA?Explain_vouranswers.
.Withoutcalculations.listrankAanddiin.\'ulA
26?6636
A=?2?36-30?6
-l9?l293l2'
i-23633?(ii
.UseapropertyofdcteriiiinantstoshowthatAandAThavethesamecharacteristicpolynomial.
.Findthecharacteristicequationof
if;?Z?0
Azlo054]
000i
Findthecharacteristicpolynomialandtherealeigenvaluesofl
iii-FH-
$3111i
._.-QIOQNI
b?
Unit-lll
5712
2ain=i23l
LetB={bhbmba}andD={dhdg}bebasesforvectorspacesVandW.respectively.
LetT:V->ll"benlineartransformationwiththepropertythatT(h|)=3di?5th.
T(b;)=-d|+6d-,-.T(b;)=Jdq.FindthematrixforTrelativetoBandD.
LetA=PDP"andcomputeA?.whereI?=[
Let?D={dbd?andB={bpbg}bebasesforvectorqiiicesVandll".respectively.Let
T:V?>WbealineartransfonnationwiththepropertythatT(d|)=3b;?13b1,T(d;)=
-2b|+5b,.FindthematrixforTrelativetoBand?D.
LetB={b|.bg.b;)beabassforavectorspaceVandletT:V?>R7bealinear
transformationwiththepropertythat
_2:1?3r;+1,
T(.r|b|+13b;+.'r,b;)-_2rl+51.?
FindthematrixforTrelativetoBandthestandardbasisforR2.
LetT:P;?>P;bethetmnsformationthatmapsapolynomialp(t)intothepolynomial
(f+3)P(')-
a.Findtheiniageofp(t)=3-2t+l?.
2.12.1PracticalsQuestionBank
LinearAlgebra
Unit-I
I
'2?l
l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=
l4.
.Letv;=
.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li
v7=
iii
Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that
m
v=
{lS]=1
3
0.51
z
o}
a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch
l?2
?--2andv;=l7
Determineif{vbv?isabasicforR?.ls{vbv?a
3?9
basislorlll2'.?
.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof
p(l)=l+~lf+7|?relativetoB.
Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.
l_?
v;=[__3I1spmtlllzbutdonotfomiabasis.Find
.l...twodifferentwaystoexpressiasalinearcombinationofv?.v;.v;.
.Findthe(liineiisioiioftliesubspaceofallwctorsin1R3whose?rstandthirdentriesareequal.
...21?2?3
.FindthediineiisonofthesubspaceHofIRspannedby_5,w.l5.
.LetHbeanll-(llIllCllSlOllIllsubspaceo!?anll-tlllll0ll?0ll?lvectorqmceV.ShowthatH=V.
.ExplainwhythespacePol?allpolynomialsismiin?nitediiiietisioiialspace.
Unit-II
.Ifa-lx7iiiatrixAhasrank3.?iuldiiii.\'ulA.dimRowA.iiiidniiikAT.
.lla7x5niatrixAhasrank2.?nddiiii.\'iilA.diinRowA.niidmiikAT.
.lftheiiullspaeeofan8x5inatrixAisZl-diiueiiiional.whatisthedimensionoftherowspace
ofA?
IfAisa3x7matrix.whatisthesniallmtpossiblediniensioiiof.\'iilA?
20.
2L
22.
23.
24.
25.
.Letu=l?
FindvinR?suchthatl
l?3~l
=T
2438]uv.
.lfAisa7X5matrix.whatisthelargestpossiblerankofA?lfAisa5x7matrix.whatis
thelargestpossiblerankofA?Explain_vouranswers.
.Withoutcalculations.listrankAanddiin.\'ulA
26?6636
A=?2?36-30?6
-l9?l293l2'
i-23633?(ii
.UseapropertyofdcteriiiinantstoshowthatAandAThavethesamecharacteristicpolynomial.
.Findthecharacteristicequationof
if;?Z?0
Azlo054]
000i
Findthecharacteristicpolynomialandtherealeigenvaluesofl
iii-FH-
$3111i
._.-QIOQNI
b?
Unit-lll
5712
2ain=i23l
LetB={bhbmba}andD={dhdg}bebasesforvectorspacesVandW.respectively.
LetT:V->ll"benlineartransformationwiththepropertythatT(h|)=3di?5th.
T(b;)=-d|+6d-,-.T(b;)=Jdq.FindthematrixforTrelativetoBandD.
LetA=PDP"andcomputeA?.whereI?=[
Let?D={dbd?andB={bpbg}bebasesforvectorqiiicesVandll".respectively.Let
T:V?>WbealineartransfonnationwiththepropertythatT(d|)=3b;?13b1,T(d;)=
-2b|+5b,.FindthematrixforTrelativetoBand?D.
LetB={b|.bg.b;)beabassforavectorspaceVandletT:V?>R7bealinear
transformationwiththepropertythat
_2:1?3r;+1,
T(.r|b|+13b;+.'r,b;)-_2rl+51.?
FindthematrixforTrelativetoBandthestandardbasisforR2.
LetT:P;?>P;bethetmnsformationthatmapsapolynomialp(t)intothepolynomial
(f+3)P(')-
a.Findtheiniageofp(t)=3-2t+l?.
26.
27.
28.
29.
30.
b.ShowthatTisalineartransformation.
c.FindthematrixforTrelativetothebases{LL17}and{LLFJJ}.
AssumethemappingT:1?;?>ll?;de?nedbyT(?Q+?|l+l|;lz)=3ao+(5no?2zii)l+(~lrii+01)!?
islinear.FindthematrixrepresentationofTrdativetothebasisB=(l.Li?).
P(-1)
9(3)
PU)J
9(0)
a.ShowthatTisalineartransformation.
b.FindthematrixforTrelativetothebases(l,t.t?.t?)forP;andthestandardbasisfor
R4.
De?iieT2P;?>R?byT(p)=l
LetAbea2x2matrixwitheigenvalues-3and-landcorrespondingeigenvcctorsvi=
l_:1andv;=li
Letx(t)bethepositionofaparticleattimet.Solvetheinitialvalue
problemx'=Ax.x(0)=[g
Constructthegeneralsolutionofx?=Ax.A=i
_?
1g1
ComputetheorthogonalprojectionoflQ1ontothelinethroughl_;1andtheorigin.
2.12.1PracticalsQuestionBank
LinearAlgebra
Unit-I
I
'2?l
l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=
l4.
.Letv;=
.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li
v7=
iii
Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that
m
v=
{lS]=1
3
0.51
z
o}
a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch
l?2
?--2andv;=l7
Determineif{vbv?isabasicforR?.ls{vbv?a
3?9
basislorlll2'.?
.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof
p(l)=l+~lf+7|?relativetoB.
Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.
l_?
v;=[__3I1spmtlllzbutdonotfomiabasis.Find
.l...twodifferentwaystoexpressiasalinearcombinationofv?.v;.v;.
.Findthe(liineiisioiioftliesubspaceofallwctorsin1R3whose?rstandthirdentriesareequal.
...21?2?3
.FindthediineiisonofthesubspaceHofIRspannedby_5,w.l5.
.LetHbeanll-(llIllCllSlOllIllsubspaceo!?anll-tlllll0ll?0ll?lvectorqmceV.ShowthatH=V.
.ExplainwhythespacePol?allpolynomialsismiin?nitediiiietisioiialspace.
Unit-II
.Ifa-lx7iiiatrixAhasrank3.?iuldiiii.\'ulA.dimRowA.iiiidniiikAT.
.lla7x5niatrixAhasrank2.?nddiiii.\'iilA.diinRowA.niidmiikAT.
.lftheiiullspaeeofan8x5inatrixAisZl-diiueiiiional.whatisthedimensionoftherowspace
ofA?
IfAisa3x7matrix.whatisthesniallmtpossiblediniensioiiof.\'iilA?
20.
2L
22.
23.
24.
25.
.Letu=l?
FindvinR?suchthatl
l?3~l
=T
2438]uv.
.lfAisa7X5matrix.whatisthelargestpossiblerankofA?lfAisa5x7matrix.whatis
thelargestpossiblerankofA?Explain_vouranswers.
.Withoutcalculations.listrankAanddiin.\'ulA
26?6636
A=?2?36-30?6
-l9?l293l2'
i-23633?(ii
.UseapropertyofdcteriiiinantstoshowthatAandAThavethesamecharacteristicpolynomial.
.Findthecharacteristicequationof
if;?Z?0
Azlo054]
000i
Findthecharacteristicpolynomialandtherealeigenvaluesofl
iii-FH-
$3111i
._.-QIOQNI
b?
Unit-lll
5712
2ain=i23l
LetB={bhbmba}andD={dhdg}bebasesforvectorspacesVandW.respectively.
LetT:V->ll"benlineartransformationwiththepropertythatT(h|)=3di?5th.
T(b;)=-d|+6d-,-.T(b;)=Jdq.FindthematrixforTrelativetoBandD.
LetA=PDP"andcomputeA?.whereI?=[
Let?D={dbd?andB={bpbg}bebasesforvectorqiiicesVandll".respectively.Let
T:V?>WbealineartransfonnationwiththepropertythatT(d|)=3b;?13b1,T(d;)=
-2b|+5b,.FindthematrixforTrelativetoBand?D.
LetB={b|.bg.b;)beabassforavectorspaceVandletT:V?>R7bealinear
transformationwiththepropertythat
_2:1?3r;+1,
T(.r|b|+13b;+.'r,b;)-_2rl+51.?
FindthematrixforTrelativetoBandthestandardbasisforR2.
LetT:P;?>P;bethetmnsformationthatmapsapolynomialp(t)intothepolynomial
(f+3)P(')-
a.Findtheiniageofp(t)=3-2t+l?.
26.
27.
28.
29.
30.
b.ShowthatTisalineartransformation.
c.FindthematrixforTrelativetothebases{LL17}and{LLFJJ}.
AssumethemappingT:1?;?>ll?;de?nedbyT(?Q+?|l+l|;lz)=3ao+(5no?2zii)l+(~lrii+01)!?
islinear.FindthematrixrepresentationofTrdativetothebasisB=(l.Li?).
P(-1)
9(3)
PU)J
9(0)
a.ShowthatTisalineartransformation.
b.FindthematrixforTrelativetothebases(l,t.t?.t?)forP;andthestandardbasisfor
R4.
De?iieT2P;?>R?byT(p)=l
LetAbea2x2matrixwitheigenvalues-3and-landcorrespondingeigenvcctorsvi=
l_:1andv;=li
Letx(t)bethepositionofaparticleattimet.Solvetheinitialvalue
problemx'=Ax.x(0)=[g
Constructthegeneralsolutionofx?=Ax.A=i
_?
1g1
ComputetheorthogonalprojectionoflQ1ontothelinethroughl_;1andtheorigin.
2.13.1PracticalsQuestionBank
r~
ll.
SolidGeometry
Unit-l
.FindthecquationofthespherethroughthefourpointsH.?l.2).(0.?2.3).(1.?5.?1).(2.0.l).
.Findtheequationofthespherethroughthe[ourpoints(0.0.0).(?a.b.c).(a.?b.c).(a.b.?r).
.Findthecentreandtheradiusofthecircle1+2y+2=15.124-yz+22-2y?4:=ll.
.Showthatthefollowingpointsaremncyclie:
(a)(s.o.2).(2.-c.o).(n-rus).(4~_9.6).
(u)(-s.s.2).(-s.2.2)(-'r.o.o).(-4.a.s).
.Findthecentresofthetwoqihenswhichtouchtheplane41+3y+=47ntthepoints(8,5.4)
andwhichto\|chthespherer?+y?+z?=1.
.Showthatthespheres
1?+y?+=?=25
1?+y?+=?-24:-40y-1s=+22s=o
touchexternallyand?ndthepointofthecontact.
.Findtheequationofthespherethatpassesthroughthetwopoints(0,3.0)_(?2,?l,?1l)and
cutsorthogonallythetwospheres
.r?+y?+:'+.r-3:-2=0.2(:?+y'+:?)+:+3y+4=0.
.Findthelimitingpointsoftheco-axalsystemofspheres
;?+y?+=?-2o?+aoy-4o=+29+,\(21-sy+4=)=0.
.Findthetquationtothetwospheresoftheco-axalsystems
:'+y?+;'-s+,\(2:+y+a=-s)=0.
whichtouchtheplane
3r+-ly=l5.
.Showthattheradicalplanesofthesphereofaco-axalsystemandofanygivenspherepass
throughaline.
Unit-H
Findtheequationoftheconewhosevertmcisthepoint(l.I.0)andwhomguidingcurveis
2.12.1PracticalsQuestionBank
LinearAlgebra
Unit-I
I
'2?l
l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=
l4.
.Letv;=
.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li
v7=
iii
Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that
m
v=
{lS]=1
3
0.51
z
o}
a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch
l?2
?--2andv;=l7
Determineif{vbv?isabasicforR?.ls{vbv?a
3?9
basislorlll2'.?
.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof
p(l)=l+~lf+7|?relativetoB.
Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.
l_?
v;=[__3I1spmtlllzbutdonotfomiabasis.Find
.l...twodifferentwaystoexpressiasalinearcombinationofv?.v;.v;.
.Findthe(liineiisioiioftliesubspaceofallwctorsin1R3whose?rstandthirdentriesareequal.
...21?2?3
.FindthediineiisonofthesubspaceHofIRspannedby_5,w.l5.
.LetHbeanll-(llIllCllSlOllIllsubspaceo!?anll-tlllll0ll?0ll?lvectorqmceV.ShowthatH=V.
.ExplainwhythespacePol?allpolynomialsismiin?nitediiiietisioiialspace.
Unit-II
.Ifa-lx7iiiatrixAhasrank3.?iuldiiii.\'ulA.dimRowA.iiiidniiikAT.
.lla7x5niatrixAhasrank2.?nddiiii.\'iilA.diinRowA.niidmiikAT.
.lftheiiullspaeeofan8x5inatrixAisZl-diiueiiiional.whatisthedimensionoftherowspace
ofA?
IfAisa3x7matrix.whatisthesniallmtpossiblediniensioiiof.\'iilA?
20.
2L
22.
23.
24.
25.
.Letu=l?
FindvinR?suchthatl
l?3~l
=T
2438]uv.
.lfAisa7X5matrix.whatisthelargestpossiblerankofA?lfAisa5x7matrix.whatis
thelargestpossiblerankofA?Explain_vouranswers.
.Withoutcalculations.listrankAanddiin.\'ulA
26?6636
A=?2?36-30?6
-l9?l293l2'
i-23633?(ii
.UseapropertyofdcteriiiinantstoshowthatAandAThavethesamecharacteristicpolynomial.
.Findthecharacteristicequationof
if;?Z?0
Azlo054]
000i
Findthecharacteristicpolynomialandtherealeigenvaluesofl
iii-FH-
$3111i
._.-QIOQNI
b?
Unit-lll
5712
2ain=i23l
LetB={bhbmba}andD={dhdg}bebasesforvectorspacesVandW.respectively.
LetT:V->ll"benlineartransformationwiththepropertythatT(h|)=3di?5th.
T(b;)=-d|+6d-,-.T(b;)=Jdq.FindthematrixforTrelativetoBandD.
LetA=PDP"andcomputeA?.whereI?=[
Let?D={dbd?andB={bpbg}bebasesforvectorqiiicesVandll".respectively.Let
T:V?>WbealineartransfonnationwiththepropertythatT(d|)=3b;?13b1,T(d;)=
-2b|+5b,.FindthematrixforTrelativetoBand?D.
LetB={b|.bg.b;)beabassforavectorspaceVandletT:V?>R7bealinear
transformationwiththepropertythat
_2:1?3r;+1,
T(.r|b|+13b;+.'r,b;)-_2rl+51.?
FindthematrixforTrelativetoBandthestandardbasisforR2.
LetT:P;?>P;bethetmnsformationthatmapsapolynomialp(t)intothepolynomial
(f+3)P(')-
a.Findtheiniageofp(t)=3-2t+l?.
26.
27.
28.
29.
30.
b.ShowthatTisalineartransformation.
c.FindthematrixforTrelativetothebases{LL17}and{LLFJJ}.
AssumethemappingT:1?;?>ll?;de?nedbyT(?Q+?|l+l|;lz)=3ao+(5no?2zii)l+(~lrii+01)!?
islinear.FindthematrixrepresentationofTrdativetothebasisB=(l.Li?).
P(-1)
9(3)
PU)J
9(0)
a.ShowthatTisalineartransformation.
b.FindthematrixforTrelativetothebases(l,t.t?.t?)forP;andthestandardbasisfor
R4.
De?iieT2P;?>R?byT(p)=l
LetAbea2x2matrixwitheigenvalues-3and-landcorrespondingeigenvcctorsvi=
l_:1andv;=li
Letx(t)bethepositionofaparticleattimet.Solvetheinitialvalue
problemx'=Ax.x(0)=[g
Constructthegeneralsolutionofx?=Ax.A=i
_?
1g1
ComputetheorthogonalprojectionoflQ1ontothelinethroughl_;1andtheorigin.
2.13.1PracticalsQuestionBank
r~
ll.
SolidGeometry
Unit-l
.FindthecquationofthespherethroughthefourpointsH.?l.2).(0.?2.3).(1.?5.?1).(2.0.l).
.Findtheequationofthespherethroughthe[ourpoints(0.0.0).(?a.b.c).(a.?b.c).(a.b.?r).
.Findthecentreandtheradiusofthecircle1+2y+2=15.124-yz+22-2y?4:=ll.
.Showthatthefollowingpointsaremncyclie:
(a)(s.o.2).(2.-c.o).(n-rus).(4~_9.6).
(u)(-s.s.2).(-s.2.2)(-'r.o.o).(-4.a.s).
.Findthecentresofthetwoqihenswhichtouchtheplane41+3y+=47ntthepoints(8,5.4)
andwhichto\|chthespherer?+y?+z?=1.
.Showthatthespheres
1?+y?+=?=25
1?+y?+=?-24:-40y-1s=+22s=o
touchexternallyand?ndthepointofthecontact.
.Findtheequationofthespherethatpassesthroughthetwopoints(0,3.0)_(?2,?l,?1l)and
cutsorthogonallythetwospheres
.r?+y?+:'+.r-3:-2=0.2(:?+y'+:?)+:+3y+4=0.
.Findthelimitingpointsoftheco-axalsystemofspheres
;?+y?+=?-2o?+aoy-4o=+29+,\(21-sy+4=)=0.
.Findthetquationtothetwospheresoftheco-axalsystems
:'+y?+;'-s+,\(2:+y+a=-s)=0.
whichtouchtheplane
3r+-ly=l5.
.Showthattheradicalplanesofthesphereofaco-axalsystemandofanygivenspherepass
throughaline.
Unit-H
Findtheequationoftheconewhosevertmcisthepoint(l.I.0)andwhomguidingcurveis
l4.
l6.
l8.
20.
.Thesectionofaconewhosevertu:isPandguidingcurvetheellipse13/02+yz/b:=L:=0
bytheplane.r=0isarectangularhyperbola.ShowthatthelocusofPis
r_2+y2+:7=l
a:b:'
.Findtheenvelopingconeofthesphere
.r?+y?+z?-2:+4:=
withitsvatmcat(l.l.1).
Findtheequationofthequadricconewhosevertexisattheoriginandwhichpassesthrough
thecurvegivenbytheequations
ax?+6512+0.22=1.1:+my+nz=
.Findtheequationoftheconewithvertexattheoriginanddirectioncosinesofitsgenerators
satisfyingtherelation
312?4m:+5n:=0.
Findtheequationofthecylinderwhosegeneratorsareparallelto
andwhoseguidingcurveistheellipse
12+2y2=l.:=3.
.Findtheequationoftherightcircularcylinderofmdius2whoseaxisistheline
(I-l)(=-3)
i=-2=.
2(u)2
Theaxisofarightcircularcylinderofradius2is
1'?l_y_z-3'
2'a
'1'
showthatitsequationis
|0r?+5112+13:?-l2ry?(iyz--lz.r-8.'r+30y-"u:+ss=0.
.Findtheequationofthecircularcylinderwhoseguidingcircleis
12+yz+z?9=0.r?y+z=
Obtaintheequationoftheriglicircularcylinderdescribedonthecirclethroughthethree
points(l.0.0).(0.1.0).(0.0.l)asguidingcircle.
2.12.1PracticalsQuestionBank
LinearAlgebra
Unit-I
I
'2?l
l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=
l4.
.Letv;=
.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li
v7=
iii
Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that
m
v=
{lS]=1
3
0.51
z
o}
a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch
l?2
?--2andv;=l7
Determineif{vbv?isabasicforR?.ls{vbv?a
3?9
basislorlll2'.?
.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof
p(l)=l+~lf+7|?relativetoB.
Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.
l_?
v;=[__3I1spmtlllzbutdonotfomiabasis.Find
.l...twodifferentwaystoexpressiasalinearcombinationofv?.v;.v;.
.Findthe(liineiisioiioftliesubspaceofallwctorsin1R3whose?rstandthirdentriesareequal.
...21?2?3
.FindthediineiisonofthesubspaceHofIRspannedby_5,w.l5.
.LetHbeanll-(llIllCllSlOllIllsubspaceo!?anll-tlllll0ll?0ll?lvectorqmceV.ShowthatH=V.
.ExplainwhythespacePol?allpolynomialsismiin?nitediiiietisioiialspace.
Unit-II
.Ifa-lx7iiiatrixAhasrank3.?iuldiiii.\'ulA.dimRowA.iiiidniiikAT.
.lla7x5niatrixAhasrank2.?nddiiii.\'iilA.diinRowA.niidmiikAT.
.lftheiiullspaeeofan8x5inatrixAisZl-diiueiiiional.whatisthedimensionoftherowspace
ofA?
IfAisa3x7matrix.whatisthesniallmtpossiblediniensioiiof.\'iilA?
20.
2L
22.
23.
24.
25.
.Letu=l?
FindvinR?suchthatl
l?3~l
=T
2438]uv.
.lfAisa7X5matrix.whatisthelargestpossiblerankofA?lfAisa5x7matrix.whatis
thelargestpossiblerankofA?Explain_vouranswers.
.Withoutcalculations.listrankAanddiin.\'ulA
26?6636
A=?2?36-30?6
-l9?l293l2'
i-23633?(ii
.UseapropertyofdcteriiiinantstoshowthatAandAThavethesamecharacteristicpolynomial.
.Findthecharacteristicequationof
if;?Z?0
Azlo054]
000i
Findthecharacteristicpolynomialandtherealeigenvaluesofl
iii-FH-
$3111i
._.-QIOQNI
b?
Unit-lll
5712
2ain=i23l
LetB={bhbmba}andD={dhdg}bebasesforvectorspacesVandW.respectively.
LetT:V->ll"benlineartransformationwiththepropertythatT(h|)=3di?5th.
T(b;)=-d|+6d-,-.T(b;)=Jdq.FindthematrixforTrelativetoBandD.
LetA=PDP"andcomputeA?.whereI?=[
Let?D={dbd?andB={bpbg}bebasesforvectorqiiicesVandll".respectively.Let
T:V?>WbealineartransfonnationwiththepropertythatT(d|)=3b;?13b1,T(d;)=
-2b|+5b,.FindthematrixforTrelativetoBand?D.
LetB={b|.bg.b;)beabassforavectorspaceVandletT:V?>R7bealinear
transformationwiththepropertythat
_2:1?3r;+1,
T(.r|b|+13b;+.'r,b;)-_2rl+51.?
FindthematrixforTrelativetoBandthestandardbasisforR2.
LetT:P;?>P;bethetmnsformationthatmapsapolynomialp(t)intothepolynomial
(f+3)P(')-
a.Findtheiniageofp(t)=3-2t+l?.
26.
27.
28.
29.
30.
b.ShowthatTisalineartransformation.
c.FindthematrixforTrelativetothebases{LL17}and{LLFJJ}.
AssumethemappingT:1?;?>ll?;de?nedbyT(?Q+?|l+l|;lz)=3ao+(5no?2zii)l+(~lrii+01)!?
islinear.FindthematrixrepresentationofTrdativetothebasisB=(l.Li?).
P(-1)
9(3)
PU)J
9(0)
a.ShowthatTisalineartransformation.
b.FindthematrixforTrelativetothebases(l,t.t?.t?)forP;andthestandardbasisfor
R4.
De?iieT2P;?>R?byT(p)=l
LetAbea2x2matrixwitheigenvalues-3and-landcorrespondingeigenvcctorsvi=
l_:1andv;=li
Letx(t)bethepositionofaparticleattimet.Solvetheinitialvalue
problemx'=Ax.x(0)=[g
Constructthegeneralsolutionofx?=Ax.A=i
_?
1g1
ComputetheorthogonalprojectionoflQ1ontothelinethroughl_;1andtheorigin.
2.13.1PracticalsQuestionBank
r~
ll.
SolidGeometry
Unit-l
.FindthecquationofthespherethroughthefourpointsH.?l.2).(0.?2.3).(1.?5.?1).(2.0.l).
.Findtheequationofthespherethroughthe[ourpoints(0.0.0).(?a.b.c).(a.?b.c).(a.b.?r).
.Findthecentreandtheradiusofthecircle1+2y+2=15.124-yz+22-2y?4:=ll.
.Showthatthefollowingpointsaremncyclie:
(a)(s.o.2).(2.-c.o).(n-rus).(4~_9.6).
(u)(-s.s.2).(-s.2.2)(-'r.o.o).(-4.a.s).
.Findthecentresofthetwoqihenswhichtouchtheplane41+3y+=47ntthepoints(8,5.4)
andwhichto\|chthespherer?+y?+z?=1.
.Showthatthespheres
1?+y?+=?=25
1?+y?+=?-24:-40y-1s=+22s=o
touchexternallyand?ndthepointofthecontact.
.Findtheequationofthespherethatpassesthroughthetwopoints(0,3.0)_(?2,?l,?1l)and
cutsorthogonallythetwospheres
.r?+y?+:'+.r-3:-2=0.2(:?+y'+:?)+:+3y+4=0.
.Findthelimitingpointsoftheco-axalsystemofspheres
;?+y?+=?-2o?+aoy-4o=+29+,\(21-sy+4=)=0.
.Findthetquationtothetwospheresoftheco-axalsystems
:'+y?+;'-s+,\(2:+y+a=-s)=0.
whichtouchtheplane
3r+-ly=l5.
.Showthattheradicalplanesofthesphereofaco-axalsystemandofanygivenspherepass
throughaline.
Unit-H
Findtheequationoftheconewhosevertmcisthepoint(l.I.0)andwhomguidingcurveis
l4.
l6.
l8.
20.
.Thesectionofaconewhosevertu:isPandguidingcurvetheellipse13/02+yz/b:=L:=0
bytheplane.r=0isarectangularhyperbola.ShowthatthelocusofPis
r_2+y2+:7=l
a:b:'
.Findtheenvelopingconeofthesphere
.r?+y?+z?-2:+4:=
withitsvatmcat(l.l.1).
Findtheequationofthequadricconewhosevertexisattheoriginandwhichpassesthrough
thecurvegivenbytheequations
ax?+6512+0.22=1.1:+my+nz=
.Findtheequationoftheconewithvertexattheoriginanddirectioncosinesofitsgenerators
satisfyingtherelation
312?4m:+5n:=0.
Findtheequationofthecylinderwhosegeneratorsareparallelto
andwhoseguidingcurveistheellipse
12+2y2=l.:=3.
.Findtheequationoftherightcircularcylinderofmdius2whoseaxisistheline
(I-l)(=-3)
i=-2=.
2(u)2
Theaxisofarightcircularcylinderofradius2is
1'?l_y_z-3'
2'a
'1'
showthatitsequationis
|0r?+5112+13:?-l2ry?(iyz--lz.r-8.'r+30y-"u:+ss=0.
.Findtheequationofthecircularcylinderwhoseguidingcircleis
12+yz+z?9=0.r?y+z=
Obtaintheequationoftheriglicircularcylinderdescribedonthecirclethroughthethree
points(l.0.0).(0.1.0).(0.0.l)asguidingcircle.
21.
22.
23.
24.
26.
27.
Unit-lll
Findthepointsofintermctionoftheline
l<:+5)?
s?i?1'
withthecoiiieoid
12a?-l7y2+n?=1.
Findtheequationstothetangentplanesto
r?-a?-=?+21=0.
whichpassthroughtheline.
71-Gy+9=3.:=3.
Obtainthetangentplanestotheellipsoid
1
u
Iu||
i-
+
ada
?EH.
+
?u
whichareparalleltothepliuie
lr+my+n:=0.
Showthattheplane3r+l2y?G:?l7=0touchestheconicoid3.1-2?6y?+9:2+l7=U.
and?ndthepointofcontact.
.Findtheequationstotlietangentpliuiestothesurface
i1?may?+1=?+l3=o.
paralleltothepliuie
~lr+20y?2l:=0.
Findtheirpointsofcontactalso.
Findthelocusoftheperpendicularsfromtheorigintothetangentplanestothesnr?ice
whichcutoflfromitsaxesinterceptsthesumofwhosereciprocalsisequaltoaconstantl/k.
llthesectionoftheenvelopingconeoftheellipsoid
1.2y::2l
n2+b2+c7_I
whosevertexisPbytheplanez=0isarectangularhyperbola.showthatthelocusofPis
2.12.1PracticalsQuestionBank
LinearAlgebra
Unit-I
I
'2?l
l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=
l4.
.Letv;=
.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li
v7=
iii
Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that
m
v=
{lS]=1
3
0.51
z
o}
a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch
l?2
?--2andv;=l7
Determineif{vbv?isabasicforR?.ls{vbv?a
3?9
basislorlll2'.?
.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof
p(l)=l+~lf+7|?relativetoB.
Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.
l_?
v;=[__3I1spmtlllzbutdonotfomiabasis.Find
.l...twodifferentwaystoexpressiasalinearcombinationofv?.v;.v;.
.Findthe(liineiisioiioftliesubspaceofallwctorsin1R3whose?rstandthirdentriesareequal.
...21?2?3
.FindthediineiisonofthesubspaceHofIRspannedby_5,w.l5.
.LetHbeanll-(llIllCllSlOllIllsubspaceo!?anll-tlllll0ll?0ll?lvectorqmceV.ShowthatH=V.
.ExplainwhythespacePol?allpolynomialsismiin?nitediiiietisioiialspace.
Unit-II
.Ifa-lx7iiiatrixAhasrank3.?iuldiiii.\'ulA.dimRowA.iiiidniiikAT.
.lla7x5niatrixAhasrank2.?nddiiii.\'iilA.diinRowA.niidmiikAT.
.lftheiiullspaeeofan8x5inatrixAisZl-diiueiiiional.whatisthedimensionoftherowspace
ofA?
IfAisa3x7matrix.whatisthesniallmtpossiblediniensioiiof.\'iilA?
20.
2L
22.
23.
24.
25.
.Letu=l?
FindvinR?suchthatl
l?3~l
=T
2438]uv.
.lfAisa7X5matrix.whatisthelargestpossiblerankofA?lfAisa5x7matrix.whatis
thelargestpossiblerankofA?Explain_vouranswers.
.Withoutcalculations.listrankAanddiin.\'ulA
26?6636
A=?2?36-30?6
-l9?l293l2'
i-23633?(ii
.UseapropertyofdcteriiiinantstoshowthatAandAThavethesamecharacteristicpolynomial.
.Findthecharacteristicequationof
if;?Z?0
Azlo054]
000i
Findthecharacteristicpolynomialandtherealeigenvaluesofl
iii-FH-
$3111i
._.-QIOQNI
b?
Unit-lll
5712
2ain=i23l
LetB={bhbmba}andD={dhdg}bebasesforvectorspacesVandW.respectively.
LetT:V->ll"benlineartransformationwiththepropertythatT(h|)=3di?5th.
T(b;)=-d|+6d-,-.T(b;)=Jdq.FindthematrixforTrelativetoBandD.
LetA=PDP"andcomputeA?.whereI?=[
Let?D={dbd?andB={bpbg}bebasesforvectorqiiicesVandll".respectively.Let
T:V?>WbealineartransfonnationwiththepropertythatT(d|)=3b;?13b1,T(d;)=
-2b|+5b,.FindthematrixforTrelativetoBand?D.
LetB={b|.bg.b;)beabassforavectorspaceVandletT:V?>R7bealinear
transformationwiththepropertythat
_2:1?3r;+1,
T(.r|b|+13b;+.'r,b;)-_2rl+51.?
FindthematrixforTrelativetoBandthestandardbasisforR2.
LetT:P;?>P;bethetmnsformationthatmapsapolynomialp(t)intothepolynomial
(f+3)P(')-
a.Findtheiniageofp(t)=3-2t+l?.
26.
27.
28.
29.
30.
b.ShowthatTisalineartransformation.
c.FindthematrixforTrelativetothebases{LL17}and{LLFJJ}.
AssumethemappingT:1?;?>ll?;de?nedbyT(?Q+?|l+l|;lz)=3ao+(5no?2zii)l+(~lrii+01)!?
islinear.FindthematrixrepresentationofTrdativetothebasisB=(l.Li?).
P(-1)
9(3)
PU)J
9(0)
a.ShowthatTisalineartransformation.
b.FindthematrixforTrelativetothebases(l,t.t?.t?)forP;andthestandardbasisfor
R4.
De?iieT2P;?>R?byT(p)=l
LetAbea2x2matrixwitheigenvalues-3and-landcorrespondingeigenvcctorsvi=
l_:1andv;=li
Letx(t)bethepositionofaparticleattimet.Solvetheinitialvalue
problemx'=Ax.x(0)=[g
Constructthegeneralsolutionofx?=Ax.A=i
_?
1g1
ComputetheorthogonalprojectionoflQ1ontothelinethroughl_;1andtheorigin.
2.13.1PracticalsQuestionBank
r~
ll.
SolidGeometry
Unit-l
.FindthecquationofthespherethroughthefourpointsH.?l.2).(0.?2.3).(1.?5.?1).(2.0.l).
.Findtheequationofthespherethroughthe[ourpoints(0.0.0).(?a.b.c).(a.?b.c).(a.b.?r).
.Findthecentreandtheradiusofthecircle1+2y+2=15.124-yz+22-2y?4:=ll.
.Showthatthefollowingpointsaremncyclie:
(a)(s.o.2).(2.-c.o).(n-rus).(4~_9.6).
(u)(-s.s.2).(-s.2.2)(-'r.o.o).(-4.a.s).
.Findthecentresofthetwoqihenswhichtouchtheplane41+3y+=47ntthepoints(8,5.4)
andwhichto\|chthespherer?+y?+z?=1.
.Showthatthespheres
1?+y?+=?=25
1?+y?+=?-24:-40y-1s=+22s=o
touchexternallyand?ndthepointofthecontact.
.Findtheequationofthespherethatpassesthroughthetwopoints(0,3.0)_(?2,?l,?1l)and
cutsorthogonallythetwospheres
.r?+y?+:'+.r-3:-2=0.2(:?+y'+:?)+:+3y+4=0.
.Findthelimitingpointsoftheco-axalsystemofspheres
;?+y?+=?-2o?+aoy-4o=+29+,\(21-sy+4=)=0.
.Findthetquationtothetwospheresoftheco-axalsystems
:'+y?+;'-s+,\(2:+y+a=-s)=0.
whichtouchtheplane
3r+-ly=l5.
.Showthattheradicalplanesofthesphereofaco-axalsystemandofanygivenspherepass
throughaline.
Unit-H
Findtheequationoftheconewhosevertmcisthepoint(l.I.0)andwhomguidingcurveis
l4.
l6.
l8.
20.
.Thesectionofaconewhosevertu:isPandguidingcurvetheellipse13/02+yz/b:=L:=0
bytheplane.r=0isarectangularhyperbola.ShowthatthelocusofPis
r_2+y2+:7=l
a:b:'
.Findtheenvelopingconeofthesphere
.r?+y?+z?-2:+4:=
withitsvatmcat(l.l.1).
Findtheequationofthequadricconewhosevertexisattheoriginandwhichpassesthrough
thecurvegivenbytheequations
ax?+6512+0.22=1.1:+my+nz=
.Findtheequationoftheconewithvertexattheoriginanddirectioncosinesofitsgenerators
satisfyingtherelation
312?4m:+5n:=0.
Findtheequationofthecylinderwhosegeneratorsareparallelto
andwhoseguidingcurveistheellipse
12+2y2=l.:=3.
.Findtheequationoftherightcircularcylinderofmdius2whoseaxisistheline
(I-l)(=-3)
i=-2=.
2(u)2
Theaxisofarightcircularcylinderofradius2is
1'?l_y_z-3'
2'a
'1'
showthatitsequationis
|0r?+5112+13:?-l2ry?(iyz--lz.r-8.'r+30y-"u:+ss=0.
.Findtheequationofthecircularcylinderwhoseguidingcircleis
12+yz+z?9=0.r?y+z=
Obtaintheequationoftheriglicircularcylinderdescribedonthecirclethroughthethree
points(l.0.0).(0.1.0).(0.0.l)asguidingcircle.
21.
22.
23.
24.
26.
27.
Unit-lll
Findthepointsofintermctionoftheline
l<:+5)?
s?i?1'
withthecoiiieoid
12a?-l7y2+n?=1.
Findtheequationstothetangentplanesto
r?-a?-=?+21=0.
whichpassthroughtheline.
71-Gy+9=3.:=3.
Obtainthetangentplanestotheellipsoid
1
u
Iu||
i-
+
ada
?EH.
+
?u
whichareparalleltothepliuie
lr+my+n:=0.
Showthattheplane3r+l2y?G:?l7=0touchestheconicoid3.1-2?6y?+9:2+l7=U.
and?ndthepointofcontact.
.Findtheequationstotlietangentpliuiestothesurface
i1?may?+1=?+l3=o.
paralleltothepliuie
~lr+20y?2l:=0.
Findtheirpointsofcontactalso.
Findthelocusoftheperpendicularsfromtheorigintothetangentplanestothesnr?ice
whichcutoflfromitsaxesinterceptsthesumofwhosereciprocalsisequaltoaconstantl/k.
llthesectionoftheenvelopingconeoftheellipsoid
1.2y::2l
n2+b2+c7_I
whosevertexisPbytheplanez=0isarectangularhyperbola.showthatthelocusofPis
28.Findthelocusofpointsfromwhichthreemutuallyperpendiculartangentlinescanhedravni
totheconieoidor?+by?+cz?=l.
29.P(l.3.2)isapointontheconicoid.
1?-2y?+s=?+s=o.
Findthelocusoi?themid-pointsofchordsdrawnparalleltoOP.
2.12.1PracticalsQuestionBank
LinearAlgebra
Unit-I
I
'2?l
l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=
l4.
.Letv;=
.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li
v7=
iii
Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that
m
v=
{lS]=1
3
0.51
z
o}
a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch
l?2
?--2andv;=l7
Determineif{vbv?isabasicforR?.ls{vbv?a
3?9
basislorlll2'.?
.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof
p(l)=l+~lf+7|?relativetoB.
Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.
l_?
v;=[__3I1spmtlllzbutdonotfomiabasis.Find
.l...twodifferentwaystoexpressiasalinearcombinationofv?.v;.v;.
.Findthe(liineiisioiioftliesubspaceofallwctorsin1R3whose?rstandthirdentriesareequal.
...21?2?3
.FindthediineiisonofthesubspaceHofIRspannedby_5,w.l5.
.LetHbeanll-(llIllCllSlOllIllsubspaceo!?anll-tlllll0ll?0ll?lvectorqmceV.ShowthatH=V.
.ExplainwhythespacePol?allpolynomialsismiin?nitediiiietisioiialspace.
Unit-II
.Ifa-lx7iiiatrixAhasrank3.?iuldiiii.\'ulA.dimRowA.iiiidniiikAT.
.lla7x5niatrixAhasrank2.?nddiiii.\'iilA.diinRowA.niidmiikAT.
.lftheiiullspaeeofan8x5inatrixAisZl-diiueiiiional.whatisthedimensionoftherowspace
ofA?
IfAisa3x7matrix.whatisthesniallmtpossiblediniensioiiof.\'iilA?
20.
2L
22.
23.
24.
25.
.Letu=l?
FindvinR?suchthatl
l?3~l
=T
2438]uv.
.lfAisa7X5matrix.whatisthelargestpossiblerankofA?lfAisa5x7matrix.whatis
thelargestpossiblerankofA?Explain_vouranswers.
.Withoutcalculations.listrankAanddiin.\'ulA
26?6636
A=?2?36-30?6
-l9?l293l2'
i-23633?(ii
.UseapropertyofdcteriiiinantstoshowthatAandAThavethesamecharacteristicpolynomial.
.Findthecharacteristicequationof
if;?Z?0
Azlo054]
000i
Findthecharacteristicpolynomialandtherealeigenvaluesofl
iii-FH-
$3111i
._.-QIOQNI
b?
Unit-lll
5712
2ain=i23l
LetB={bhbmba}andD={dhdg}bebasesforvectorspacesVandW.respectively.
LetT:V->ll"benlineartransformationwiththepropertythatT(h|)=3di?5th.
T(b;)=-d|+6d-,-.T(b;)=Jdq.FindthematrixforTrelativetoBandD.
LetA=PDP"andcomputeA?.whereI?=[
Let?D={dbd?andB={bpbg}bebasesforvectorqiiicesVandll".respectively.Let
T:V?>WbealineartransfonnationwiththepropertythatT(d|)=3b;?13b1,T(d;)=
-2b|+5b,.FindthematrixforTrelativetoBand?D.
LetB={b|.bg.b;)beabassforavectorspaceVandletT:V?>R7bealinear
transformationwiththepropertythat
_2:1?3r;+1,
T(.r|b|+13b;+.'r,b;)-_2rl+51.?
FindthematrixforTrelativetoBandthestandardbasisforR2.
LetT:P;?>P;bethetmnsformationthatmapsapolynomialp(t)intothepolynomial
(f+3)P(')-
a.Findtheiniageofp(t)=3-2t+l?.
26.
27.
28.
29.
30.
b.ShowthatTisalineartransformation.
c.FindthematrixforTrelativetothebases{LL17}and{LLFJJ}.
AssumethemappingT:1?;?>ll?;de?nedbyT(?Q+?|l+l|;lz)=3ao+(5no?2zii)l+(~lrii+01)!?
islinear.FindthematrixrepresentationofTrdativetothebasisB=(l.Li?).
P(-1)
9(3)
PU)J
9(0)
a.ShowthatTisalineartransformation.
b.FindthematrixforTrelativetothebases(l,t.t?.t?)forP;andthestandardbasisfor
R4.
De?iieT2P;?>R?byT(p)=l
LetAbea2x2matrixwitheigenvalues-3and-landcorrespondingeigenvcctorsvi=
l_:1andv;=li
Letx(t)bethepositionofaparticleattimet.Solvetheinitialvalue
problemx'=Ax.x(0)=[g
Constructthegeneralsolutionofx?=Ax.A=i
_?
1g1
ComputetheorthogonalprojectionoflQ1ontothelinethroughl_;1andtheorigin.
2.13.1PracticalsQuestionBank
r~
ll.
SolidGeometry
Unit-l
.FindthecquationofthespherethroughthefourpointsH.?l.2).(0.?2.3).(1.?5.?1).(2.0.l).
.Findtheequationofthespherethroughthe[ourpoints(0.0.0).(?a.b.c).(a.?b.c).(a.b.?r).
.Findthecentreandtheradiusofthecircle1+2y+2=15.124-yz+22-2y?4:=ll.
.Showthatthefollowingpointsaremncyclie:
(a)(s.o.2).(2.-c.o).(n-rus).(4~_9.6).
(u)(-s.s.2).(-s.2.2)(-'r.o.o).(-4.a.s).
.Findthecentresofthetwoqihenswhichtouchtheplane41+3y+=47ntthepoints(8,5.4)
andwhichto\|chthespherer?+y?+z?=1.
.Showthatthespheres
1?+y?+=?=25
1?+y?+=?-24:-40y-1s=+22s=o
touchexternallyand?ndthepointofthecontact.
.Findtheequationofthespherethatpassesthroughthetwopoints(0,3.0)_(?2,?l,?1l)and
cutsorthogonallythetwospheres
.r?+y?+:'+.r-3:-2=0.2(:?+y'+:?)+:+3y+4=0.
.Findthelimitingpointsoftheco-axalsystemofspheres
;?+y?+=?-2o?+aoy-4o=+29+,\(21-sy+4=)=0.
.Findthetquationtothetwospheresoftheco-axalsystems
:'+y?+;'-s+,\(2:+y+a=-s)=0.
whichtouchtheplane
3r+-ly=l5.
.Showthattheradicalplanesofthesphereofaco-axalsystemandofanygivenspherepass
throughaline.
Unit-H
Findtheequationoftheconewhosevertmcisthepoint(l.I.0)andwhomguidingcurveis
l4.
l6.
l8.
20.
.Thesectionofaconewhosevertu:isPandguidingcurvetheellipse13/02+yz/b:=L:=0
bytheplane.r=0isarectangularhyperbola.ShowthatthelocusofPis
r_2+y2+:7=l
a:b:'
.Findtheenvelopingconeofthesphere
.r?+y?+z?-2:+4:=
withitsvatmcat(l.l.1).
Findtheequationofthequadricconewhosevertexisattheoriginandwhichpassesthrough
thecurvegivenbytheequations
ax?+6512+0.22=1.1:+my+nz=
.Findtheequationoftheconewithvertexattheoriginanddirectioncosinesofitsgenerators
satisfyingtherelation
312?4m:+5n:=0.
Findtheequationofthecylinderwhosegeneratorsareparallelto
andwhoseguidingcurveistheellipse
12+2y2=l.:=3.
.Findtheequationoftherightcircularcylinderofmdius2whoseaxisistheline
(I-l)(=-3)
i=-2=.
2(u)2
Theaxisofarightcircularcylinderofradius2is
1'?l_y_z-3'
2'a
'1'
showthatitsequationis
|0r?+5112+13:?-l2ry?(iyz--lz.r-8.'r+30y-"u:+ss=0.
.Findtheequationofthecircularcylinderwhoseguidingcircleis
12+yz+z?9=0.r?y+z=
Obtaintheequationoftheriglicircularcylinderdescribedonthecirclethroughthethree
points(l.0.0).(0.1.0).(0.0.l)asguidingcircle.
21.
22.
23.
24.
26.
27.
Unit-lll
Findthepointsofintermctionoftheline
l<:+5)?
s?i?1'
withthecoiiieoid
12a?-l7y2+n?=1.
Findtheequationstothetangentplanesto
r?-a?-=?+21=0.
whichpassthroughtheline.
71-Gy+9=3.:=3.
Obtainthetangentplanestotheellipsoid
1
u
Iu||
i-
+
ada
?EH.
+
?u
whichareparalleltothepliuie
lr+my+n:=0.
Showthattheplane3r+l2y?G:?l7=0touchestheconicoid3.1-2?6y?+9:2+l7=U.
and?ndthepointofcontact.
.Findtheequationstotlietangentpliuiestothesurface
i1?may?+1=?+l3=o.
paralleltothepliuie
~lr+20y?2l:=0.
Findtheirpointsofcontactalso.
Findthelocusoftheperpendicularsfromtheorigintothetangentplanestothesnr?ice
whichcutoflfromitsaxesinterceptsthesumofwhosereciprocalsisequaltoaconstantl/k.
llthesectionoftheenvelopingconeoftheellipsoid
1.2y::2l
n2+b2+c7_I
whosevertexisPbytheplanez=0isarectangularhyperbola.showthatthelocusofPis
28.Findthelocusofpointsfromwhichthreemutuallyperpendiculartangentlinescanhedravni
totheconieoidor?+by?+cz?=l.
29.P(l.3.2)isapointontheconicoid.
1?-2y?+s=?+s=o.
Findthelocusoi?themid-pointsofchordsdrawnparalleltoOP.
2.14.1PracticalsQuestionBank
IntegralCalculus
Unit-I
l.LetR=[-3.3]x[-2.2].Withoutexplicitlyevaluatinganyiteratedintegrals.detenninethe
valueof
f/(1%.mm
n
2.Integratethefunctionf(x.y)=31yovertheregionboundedbyy=3213midy=
l0.
l3.
.f(:.y.:)=2:?y+z:ll"istheregionhoundedbythecylinder:=
.Integratethefunctionf(x.y)=1+yovertheregionhoundedbyz+y=2andy?-2y-.'r=0.
.Evaluatefforyrlzl.whereDistheregionboundedby.r=y?andy=1".
.Evaluateffo?zd/i.whereDisthetriangularregionwithvertices(0.0).(l.0)and(1.1).
.Evaluateffbilyrl/l.whereDistheregionboundedbyry?=1.y=r.r=0andy=3.
.EvaluateffD(.-r?2y)d?l.whereDistheregionboundedb_vy=:2+2andy=212?2.
.EvaluateffD(r"?+y')d/l.whereDistheregioninthe?rstquadrantboundedbyy=4r.y=3:
and.'ry=3.
.Considertheintegral
22:
/(2:+l)rlyd.r
oa
a)Evaluatethisintegral.
b)Sketchtheregionofintegratiou.
c)Writeanequivalentiteratedintegralwiththeorderofintegrationreverse.Evaluatethis
newintegralandcheek_vouransweragreeswithpart(a).
Findthevolumeoftheregionunderthegraphof
!(1-u)=2?|1|?|y|
andabovethery-plmre
Unit-II
IntegratethefollowingovertheindicatrxlregionW.
y?.thery-plane.the
planes1=0.:=l.l!?/&.y=2.
.j(z.y.:)=y:Wistheregionboundedbytheplane1+y+z=2.thecylinder12+:2=l
andy=0.
[(r.y.:)=8ryzzWistheregionhoundedbythecylindery=r2.theplaney+:=9and
thezy-plane.
2.12.1PracticalsQuestionBank
LinearAlgebra
Unit-I
I
'2?l
l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=
l4.
.Letv;=
.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li
v7=
iii
Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that
m
v=
{lS]=1
3
0.51
z
o}
a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch
l?2
?--2andv;=l7
Determineif{vbv?isabasicforR?.ls{vbv?a
3?9
basislorlll2'.?
.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof
p(l)=l+~lf+7|?relativetoB.
Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.
l_?
v;=[__3I1spmtlllzbutdonotfomiabasis.Find
.l...twodifferentwaystoexpressiasalinearcombinationofv?.v;.v;.
.Findthe(liineiisioiioftliesubspaceofallwctorsin1R3whose?rstandthirdentriesareequal.
...21?2?3
.FindthediineiisonofthesubspaceHofIRspannedby_5,w.l5.
.LetHbeanll-(llIllCllSlOllIllsubspaceo!?anll-tlllll0ll?0ll?lvectorqmceV.ShowthatH=V.
.ExplainwhythespacePol?allpolynomialsismiin?nitediiiietisioiialspace.
Unit-II
.Ifa-lx7iiiatrixAhasrank3.?iuldiiii.\'ulA.dimRowA.iiiidniiikAT.
.lla7x5niatrixAhasrank2.?nddiiii.\'iilA.diinRowA.niidmiikAT.
.lftheiiullspaeeofan8x5inatrixAisZl-diiueiiiional.whatisthedimensionoftherowspace
ofA?
IfAisa3x7matrix.whatisthesniallmtpossiblediniensioiiof.\'iilA?
20.
2L
22.
23.
24.
25.
.Letu=l?
FindvinR?suchthatl
l?3~l
=T
2438]uv.
.lfAisa7X5matrix.whatisthelargestpossiblerankofA?lfAisa5x7matrix.whatis
thelargestpossiblerankofA?Explain_vouranswers.
.Withoutcalculations.listrankAanddiin.\'ulA
26?6636
A=?2?36-30?6
-l9?l293l2'
i-23633?(ii
.UseapropertyofdcteriiiinantstoshowthatAandAThavethesamecharacteristicpolynomial.
.Findthecharacteristicequationof
if;?Z?0
Azlo054]
000i
Findthecharacteristicpolynomialandtherealeigenvaluesofl
iii-FH-
$3111i
._.-QIOQNI
b?
Unit-lll
5712
2ain=i23l
LetB={bhbmba}andD={dhdg}bebasesforvectorspacesVandW.respectively.
LetT:V->ll"benlineartransformationwiththepropertythatT(h|)=3di?5th.
T(b;)=-d|+6d-,-.T(b;)=Jdq.FindthematrixforTrelativetoBandD.
LetA=PDP"andcomputeA?.whereI?=[
Let?D={dbd?andB={bpbg}bebasesforvectorqiiicesVandll".respectively.Let
T:V?>WbealineartransfonnationwiththepropertythatT(d|)=3b;?13b1,T(d;)=
-2b|+5b,.FindthematrixforTrelativetoBand?D.
LetB={b|.bg.b;)beabassforavectorspaceVandletT:V?>R7bealinear
transformationwiththepropertythat
_2:1?3r;+1,
T(.r|b|+13b;+.'r,b;)-_2rl+51.?
FindthematrixforTrelativetoBandthestandardbasisforR2.
LetT:P;?>P;bethetmnsformationthatmapsapolynomialp(t)intothepolynomial
(f+3)P(')-
a.Findtheiniageofp(t)=3-2t+l?.
26.
27.
28.
29.
30.
b.ShowthatTisalineartransformation.
c.FindthematrixforTrelativetothebases{LL17}and{LLFJJ}.
AssumethemappingT:1?;?>ll?;de?nedbyT(?Q+?|l+l|;lz)=3ao+(5no?2zii)l+(~lrii+01)!?
islinear.FindthematrixrepresentationofTrdativetothebasisB=(l.Li?).
P(-1)
9(3)
PU)J
9(0)
a.ShowthatTisalineartransformation.
b.FindthematrixforTrelativetothebases(l,t.t?.t?)forP;andthestandardbasisfor
R4.
De?iieT2P;?>R?byT(p)=l
LetAbea2x2matrixwitheigenvalues-3and-landcorrespondingeigenvcctorsvi=
l_:1andv;=li
Letx(t)bethepositionofaparticleattimet.Solvetheinitialvalue
problemx'=Ax.x(0)=[g
Constructthegeneralsolutionofx?=Ax.A=i
_?
1g1
ComputetheorthogonalprojectionoflQ1ontothelinethroughl_;1andtheorigin.
2.13.1PracticalsQuestionBank
r~
ll.
SolidGeometry
Unit-l
.FindthecquationofthespherethroughthefourpointsH.?l.2).(0.?2.3).(1.?5.?1).(2.0.l).
.Findtheequationofthespherethroughthe[ourpoints(0.0.0).(?a.b.c).(a.?b.c).(a.b.?r).
.Findthecentreandtheradiusofthecircle1+2y+2=15.124-yz+22-2y?4:=ll.
.Showthatthefollowingpointsaremncyclie:
(a)(s.o.2).(2.-c.o).(n-rus).(4~_9.6).
(u)(-s.s.2).(-s.2.2)(-'r.o.o).(-4.a.s).
.Findthecentresofthetwoqihenswhichtouchtheplane41+3y+=47ntthepoints(8,5.4)
andwhichto\|chthespherer?+y?+z?=1.
.Showthatthespheres
1?+y?+=?=25
1?+y?+=?-24:-40y-1s=+22s=o
touchexternallyand?ndthepointofthecontact.
.Findtheequationofthespherethatpassesthroughthetwopoints(0,3.0)_(?2,?l,?1l)and
cutsorthogonallythetwospheres
.r?+y?+:'+.r-3:-2=0.2(:?+y'+:?)+:+3y+4=0.
.Findthelimitingpointsoftheco-axalsystemofspheres
;?+y?+=?-2o?+aoy-4o=+29+,\(21-sy+4=)=0.
.Findthetquationtothetwospheresoftheco-axalsystems
:'+y?+;'-s+,\(2:+y+a=-s)=0.
whichtouchtheplane
3r+-ly=l5.
.Showthattheradicalplanesofthesphereofaco-axalsystemandofanygivenspherepass
throughaline.
Unit-H
Findtheequationoftheconewhosevertmcisthepoint(l.I.0)andwhomguidingcurveis
l4.
l6.
l8.
20.
.Thesectionofaconewhosevertu:isPandguidingcurvetheellipse13/02+yz/b:=L:=0
bytheplane.r=0isarectangularhyperbola.ShowthatthelocusofPis
r_2+y2+:7=l
a:b:'
.Findtheenvelopingconeofthesphere
.r?+y?+z?-2:+4:=
withitsvatmcat(l.l.1).
Findtheequationofthequadricconewhosevertexisattheoriginandwhichpassesthrough
thecurvegivenbytheequations
ax?+6512+0.22=1.1:+my+nz=
.Findtheequationoftheconewithvertexattheoriginanddirectioncosinesofitsgenerators
satisfyingtherelation
312?4m:+5n:=0.
Findtheequationofthecylinderwhosegeneratorsareparallelto
andwhoseguidingcurveistheellipse
12+2y2=l.:=3.
.Findtheequationoftherightcircularcylinderofmdius2whoseaxisistheline
(I-l)(=-3)
i=-2=.
2(u)2
Theaxisofarightcircularcylinderofradius2is
1'?l_y_z-3'
2'a
'1'
showthatitsequationis
|0r?+5112+13:?-l2ry?(iyz--lz.r-8.'r+30y-"u:+ss=0.
.Findtheequationofthecircularcylinderwhoseguidingcircleis
12+yz+z?9=0.r?y+z=
Obtaintheequationoftheriglicircularcylinderdescribedonthecirclethroughthethree
points(l.0.0).(0.1.0).(0.0.l)asguidingcircle.
21.
22.
23.
24.
26.
27.
Unit-lll
Findthepointsofintermctionoftheline
l<:+5)?
s?i?1'
withthecoiiieoid
12a?-l7y2+n?=1.
Findtheequationstothetangentplanesto
r?-a?-=?+21=0.
whichpassthroughtheline.
71-Gy+9=3.:=3.
Obtainthetangentplanestotheellipsoid
1
u
Iu||
i-
+
ada
?EH.
+
?u
whichareparalleltothepliuie
lr+my+n:=0.
Showthattheplane3r+l2y?G:?l7=0touchestheconicoid3.1-2?6y?+9:2+l7=U.
and?ndthepointofcontact.
.Findtheequationstotlietangentpliuiestothesurface
i1?may?+1=?+l3=o.
paralleltothepliuie
~lr+20y?2l:=0.
Findtheirpointsofcontactalso.
Findthelocusoftheperpendicularsfromtheorigintothetangentplanestothesnr?ice
whichcutoflfromitsaxesinterceptsthesumofwhosereciprocalsisequaltoaconstantl/k.
llthesectionoftheenvelopingconeoftheellipsoid
1.2y::2l
n2+b2+c7_I
whosevertexisPbytheplanez=0isarectangularhyperbola.showthatthelocusofPis
28.Findthelocusofpointsfromwhichthreemutuallyperpendiculartangentlinescanhedravni
totheconieoidor?+by?+cz?=l.
29.P(l.3.2)isapointontheconicoid.
1?-2y?+s=?+s=o.
Findthelocusoi?themid-pointsofchordsdrawnparalleltoOP.
2.14.1PracticalsQuestionBank
IntegralCalculus
Unit-I
l.LetR=[-3.3]x[-2.2].Withoutexplicitlyevaluatinganyiteratedintegrals.detenninethe
valueof
f/(1%.mm
n
2.Integratethefunctionf(x.y)=31yovertheregionboundedbyy=3213midy=
l0.
l3.
.f(:.y.:)=2:?y+z:ll"istheregionhoundedbythecylinder:=
.Integratethefunctionf(x.y)=1+yovertheregionhoundedbyz+y=2andy?-2y-.'r=0.
.Evaluatefforyrlzl.whereDistheregionboundedby.r=y?andy=1".
.Evaluateffo?zd/i.whereDisthetriangularregionwithvertices(0.0).(l.0)and(1.1).
.Evaluateffbilyrl/l.whereDistheregionboundedbyry?=1.y=r.r=0andy=3.
.EvaluateffD(.-r?2y)d?l.whereDistheregionboundedb_vy=:2+2andy=212?2.
.EvaluateffD(r"?+y')d/l.whereDistheregioninthe?rstquadrantboundedbyy=4r.y=3:
and.'ry=3.
.Considertheintegral
22:
/(2:+l)rlyd.r
oa
a)Evaluatethisintegral.
b)Sketchtheregionofintegratiou.
c)Writeanequivalentiteratedintegralwiththeorderofintegrationreverse.Evaluatethis
newintegralandcheek_vouransweragreeswithpart(a).
Findthevolumeoftheregionunderthegraphof
!(1-u)=2?|1|?|y|
andabovethery-plmre
Unit-II
IntegratethefollowingovertheindicatrxlregionW.
y?.thery-plane.the
planes1=0.:=l.l!?/&.y=2.
.j(z.y.:)=y:Wistheregionboundedbytheplane1+y+z=2.thecylinder12+:2=l
andy=0.
[(r.y.:)=8ryzzWistheregionhoundedbythecylindery=r2.theplaney+:=9and
thezy-plane.
l7.
l9.
20.
2
r~
22.
23.
24.
25.
.f(.'r.y.:)=z:Wistheregioninthe?rstoctantboundvxlbythecylindery:+z?=9andthe
planesy=1.:=0andz=0.
.f(r.y.z)=l?s?:Wisthetetrahedronwithvertiees(0.0.0).(1.0.0).(0.2.0)and(0.0.3).
.j(r.y.:)=3r;II"istheregioninthe?rstoctantboundedby:.=r2+5/2.:=0.y=0and
z=-l.
j(x.y.z)=r+y:Wistheregionboundedbythecylinderr2+3:2=9andtheplaney=0.
r+y=3.
.[(1.y.z)=z:II?istheregionboundedbyz=0.I?:+4y?=-lturdz=1+2.
Unit-III
IntegratethefollowingovertheindicatedregionW.
](.-r.y.:)=-l.r+yzWistheregionboundedby1'=y?.y=:..r=yaadz=0.
j(z.y.z)=.r:Wistheregioninthe?rstoctantboundedbyz=r2+2f.z=G
?
:2
?
y?.
r=0arrdy=0.
LetT(u.r')=(3ll.?v)
.WriteT(u.u)asAIt;lforasuitablematrixA.
DeztribetheimageD=T(D?).whereD?istheunitsquare[0.1]x[0.1].
f/?li?
1?2y
D
whereDistheregioninR2mrelosetlbytheIirresy=
y=0and
I+
y=I.
Deterrrrinethevalueof
Evaluate
(2.r+y?3)?
tld.
(2y?:r+6)?Iy
whereDisthesquarewithwrtices(0.0).(2.l).(3.-l)iurd(l.-2).(Hint:FirstdtetchDand
?ndtheequationsofitssides)
Evaluate
f]cw?rz+yzM/l.
0
whereDistheshadedregioninthefollowing?gure-l.
2.12.1PracticalsQuestionBank
LinearAlgebra
Unit-I
I
'2?l
l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=
l4.
.Letv;=
.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li
v7=
iii
Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that
m
v=
{lS]=1
3
0.51
z
o}
a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch
l?2
?--2andv;=l7
Determineif{vbv?isabasicforR?.ls{vbv?a
3?9
basislorlll2'.?
.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof
p(l)=l+~lf+7|?relativetoB.
Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.
l_?
v;=[__3I1spmtlllzbutdonotfomiabasis.Find
.l...twodifferentwaystoexpressiasalinearcombinationofv?.v;.v;.
.Findthe(liineiisioiioftliesubspaceofallwctorsin1R3whose?rstandthirdentriesareequal.
...21?2?3
.FindthediineiisonofthesubspaceHofIRspannedby_5,w.l5.
.LetHbeanll-(llIllCllSlOllIllsubspaceo!?anll-tlllll0ll?0ll?lvectorqmceV.ShowthatH=V.
.ExplainwhythespacePol?allpolynomialsismiin?nitediiiietisioiialspace.
Unit-II
.Ifa-lx7iiiatrixAhasrank3.?iuldiiii.\'ulA.dimRowA.iiiidniiikAT.
.lla7x5niatrixAhasrank2.?nddiiii.\'iilA.diinRowA.niidmiikAT.
.lftheiiullspaeeofan8x5inatrixAisZl-diiueiiiional.whatisthedimensionoftherowspace
ofA?
IfAisa3x7matrix.whatisthesniallmtpossiblediniensioiiof.\'iilA?
20.
2L
22.
23.
24.
25.
.Letu=l?
FindvinR?suchthatl
l?3~l
=T
2438]uv.
.lfAisa7X5matrix.whatisthelargestpossiblerankofA?lfAisa5x7matrix.whatis
thelargestpossiblerankofA?Explain_vouranswers.
.Withoutcalculations.listrankAanddiin.\'ulA
26?6636
A=?2?36-30?6
-l9?l293l2'
i-23633?(ii
.UseapropertyofdcteriiiinantstoshowthatAandAThavethesamecharacteristicpolynomial.
.Findthecharacteristicequationof
if;?Z?0
Azlo054]
000i
Findthecharacteristicpolynomialandtherealeigenvaluesofl
iii-FH-
$3111i
._.-QIOQNI
b?
Unit-lll
5712
2ain=i23l
LetB={bhbmba}andD={dhdg}bebasesforvectorspacesVandW.respectively.
LetT:V->ll"benlineartransformationwiththepropertythatT(h|)=3di?5th.
T(b;)=-d|+6d-,-.T(b;)=Jdq.FindthematrixforTrelativetoBandD.
LetA=PDP"andcomputeA?.whereI?=[
Let?D={dbd?andB={bpbg}bebasesforvectorqiiicesVandll".respectively.Let
T:V?>WbealineartransfonnationwiththepropertythatT(d|)=3b;?13b1,T(d;)=
-2b|+5b,.FindthematrixforTrelativetoBand?D.
LetB={b|.bg.b;)beabassforavectorspaceVandletT:V?>R7bealinear
transformationwiththepropertythat
_2:1?3r;+1,
T(.r|b|+13b;+.'r,b;)-_2rl+51.?
FindthematrixforTrelativetoBandthestandardbasisforR2.
LetT:P;?>P;bethetmnsformationthatmapsapolynomialp(t)intothepolynomial
(f+3)P(')-
a.Findtheiniageofp(t)=3-2t+l?.
26.
27.
28.
29.
30.
b.ShowthatTisalineartransformation.
c.FindthematrixforTrelativetothebases{LL17}and{LLFJJ}.
AssumethemappingT:1?;?>ll?;de?nedbyT(?Q+?|l+l|;lz)=3ao+(5no?2zii)l+(~lrii+01)!?
islinear.FindthematrixrepresentationofTrdativetothebasisB=(l.Li?).
P(-1)
9(3)
PU)J
9(0)
a.ShowthatTisalineartransformation.
b.FindthematrixforTrelativetothebases(l,t.t?.t?)forP;andthestandardbasisfor
R4.
De?iieT2P;?>R?byT(p)=l
LetAbea2x2matrixwitheigenvalues-3and-landcorrespondingeigenvcctorsvi=
l_:1andv;=li
Letx(t)bethepositionofaparticleattimet.Solvetheinitialvalue
problemx'=Ax.x(0)=[g
Constructthegeneralsolutionofx?=Ax.A=i
_?
1g1
ComputetheorthogonalprojectionoflQ1ontothelinethroughl_;1andtheorigin.
2.13.1PracticalsQuestionBank
r~
ll.
SolidGeometry
Unit-l
.FindthecquationofthespherethroughthefourpointsH.?l.2).(0.?2.3).(1.?5.?1).(2.0.l).
.Findtheequationofthespherethroughthe[ourpoints(0.0.0).(?a.b.c).(a.?b.c).(a.b.?r).
.Findthecentreandtheradiusofthecircle1+2y+2=15.124-yz+22-2y?4:=ll.
.Showthatthefollowingpointsaremncyclie:
(a)(s.o.2).(2.-c.o).(n-rus).(4~_9.6).
(u)(-s.s.2).(-s.2.2)(-'r.o.o).(-4.a.s).
.Findthecentresofthetwoqihenswhichtouchtheplane41+3y+=47ntthepoints(8,5.4)
andwhichto\|chthespherer?+y?+z?=1.
.Showthatthespheres
1?+y?+=?=25
1?+y?+=?-24:-40y-1s=+22s=o
touchexternallyand?ndthepointofthecontact.
.Findtheequationofthespherethatpassesthroughthetwopoints(0,3.0)_(?2,?l,?1l)and
cutsorthogonallythetwospheres
.r?+y?+:'+.r-3:-2=0.2(:?+y'+:?)+:+3y+4=0.
.Findthelimitingpointsoftheco-axalsystemofspheres
;?+y?+=?-2o?+aoy-4o=+29+,\(21-sy+4=)=0.
.Findthetquationtothetwospheresoftheco-axalsystems
:'+y?+;'-s+,\(2:+y+a=-s)=0.
whichtouchtheplane
3r+-ly=l5.
.Showthattheradicalplanesofthesphereofaco-axalsystemandofanygivenspherepass
throughaline.
Unit-H
Findtheequationoftheconewhosevertmcisthepoint(l.I.0)andwhomguidingcurveis
l4.
l6.
l8.
20.
.Thesectionofaconewhosevertu:isPandguidingcurvetheellipse13/02+yz/b:=L:=0
bytheplane.r=0isarectangularhyperbola.ShowthatthelocusofPis
r_2+y2+:7=l
a:b:'
.Findtheenvelopingconeofthesphere
.r?+y?+z?-2:+4:=
withitsvatmcat(l.l.1).
Findtheequationofthequadricconewhosevertexisattheoriginandwhichpassesthrough
thecurvegivenbytheequations
ax?+6512+0.22=1.1:+my+nz=
.Findtheequationoftheconewithvertexattheoriginanddirectioncosinesofitsgenerators
satisfyingtherelation
312?4m:+5n:=0.
Findtheequationofthecylinderwhosegeneratorsareparallelto
andwhoseguidingcurveistheellipse
12+2y2=l.:=3.
.Findtheequationoftherightcircularcylinderofmdius2whoseaxisistheline
(I-l)(=-3)
i=-2=.
2(u)2
Theaxisofarightcircularcylinderofradius2is
1'?l_y_z-3'
2'a
'1'
showthatitsequationis
|0r?+5112+13:?-l2ry?(iyz--lz.r-8.'r+30y-"u:+ss=0.
.Findtheequationofthecircularcylinderwhoseguidingcircleis
12+yz+z?9=0.r?y+z=
Obtaintheequationoftheriglicircularcylinderdescribedonthecirclethroughthethree
points(l.0.0).(0.1.0).(0.0.l)asguidingcircle.
21.
22.
23.
24.
26.
27.
Unit-lll
Findthepointsofintermctionoftheline
l<:+5)?
s?i?1'
withthecoiiieoid
12a?-l7y2+n?=1.
Findtheequationstothetangentplanesto
r?-a?-=?+21=0.
whichpassthroughtheline.
71-Gy+9=3.:=3.
Obtainthetangentplanestotheellipsoid
1
u
Iu||
i-
+
ada
?EH.
+
?u
whichareparalleltothepliuie
lr+my+n:=0.
Showthattheplane3r+l2y?G:?l7=0touchestheconicoid3.1-2?6y?+9:2+l7=U.
and?ndthepointofcontact.
.Findtheequationstotlietangentpliuiestothesurface
i1?may?+1=?+l3=o.
paralleltothepliuie
~lr+20y?2l:=0.
Findtheirpointsofcontactalso.
Findthelocusoftheperpendicularsfromtheorigintothetangentplanestothesnr?ice
whichcutoflfromitsaxesinterceptsthesumofwhosereciprocalsisequaltoaconstantl/k.
llthesectionoftheenvelopingconeoftheellipsoid
1.2y::2l
n2+b2+c7_I
whosevertexisPbytheplanez=0isarectangularhyperbola.showthatthelocusofPis
28.Findthelocusofpointsfromwhichthreemutuallyperpendiculartangentlinescanhedravni
totheconieoidor?+by?+cz?=l.
29.P(l.3.2)isapointontheconicoid.
1?-2y?+s=?+s=o.
Findthelocusoi?themid-pointsofchordsdrawnparalleltoOP.
2.14.1PracticalsQuestionBank
IntegralCalculus
Unit-I
l.LetR=[-3.3]x[-2.2].Withoutexplicitlyevaluatinganyiteratedintegrals.detenninethe
valueof
f/(1%.mm
n
2.Integratethefunctionf(x.y)=31yovertheregionboundedbyy=3213midy=
l0.
l3.
.f(:.y.:)=2:?y+z:ll"istheregionhoundedbythecylinder:=
.Integratethefunctionf(x.y)=1+yovertheregionhoundedbyz+y=2andy?-2y-.'r=0.
.Evaluatefforyrlzl.whereDistheregionboundedby.r=y?andy=1".
.Evaluateffo?zd/i.whereDisthetriangularregionwithvertices(0.0).(l.0)and(1.1).
.Evaluateffbilyrl/l.whereDistheregionboundedbyry?=1.y=r.r=0andy=3.
.EvaluateffD(.-r?2y)d?l.whereDistheregionboundedb_vy=:2+2andy=212?2.
.EvaluateffD(r"?+y')d/l.whereDistheregioninthe?rstquadrantboundedbyy=4r.y=3:
and.'ry=3.
.Considertheintegral
22:
/(2:+l)rlyd.r
oa
a)Evaluatethisintegral.
b)Sketchtheregionofintegratiou.
c)Writeanequivalentiteratedintegralwiththeorderofintegrationreverse.Evaluatethis
newintegralandcheek_vouransweragreeswithpart(a).
Findthevolumeoftheregionunderthegraphof
!(1-u)=2?|1|?|y|
andabovethery-plmre
Unit-II
IntegratethefollowingovertheindicatrxlregionW.
y?.thery-plane.the
planes1=0.:=l.l!?/&.y=2.
.j(z.y.:)=y:Wistheregionboundedbytheplane1+y+z=2.thecylinder12+:2=l
andy=0.
[(r.y.:)=8ryzzWistheregionhoundedbythecylindery=r2.theplaney+:=9and
thezy-plane.
l7.
l9.
20.
2
r~
22.
23.
24.
25.
.f(.'r.y.:)=z:Wistheregioninthe?rstoctantboundvxlbythecylindery:+z?=9andthe
planesy=1.:=0andz=0.
.f(r.y.z)=l?s?:Wisthetetrahedronwithvertiees(0.0.0).(1.0.0).(0.2.0)and(0.0.3).
.j(r.y.:)=3r;II"istheregioninthe?rstoctantboundedby:.=r2+5/2.:=0.y=0and
z=-l.
j(x.y.z)=r+y:Wistheregionboundedbythecylinderr2+3:2=9andtheplaney=0.
r+y=3.
.[(1.y.z)=z:II?istheregionboundedbyz=0.I?:+4y?=-lturdz=1+2.
Unit-III
IntegratethefollowingovertheindicatedregionW.
](.-r.y.:)=-l.r+yzWistheregionboundedby1'=y?.y=:..r=yaadz=0.
j(z.y.z)=.r:Wistheregioninthe?rstoctantboundedbyz=r2+2f.z=G
?
:2
?
y?.
r=0arrdy=0.
LetT(u.r')=(3ll.?v)
.WriteT(u.u)asAIt;lforasuitablematrixA.
DeztribetheimageD=T(D?).whereD?istheunitsquare[0.1]x[0.1].
f/?li?
1?2y
D
whereDistheregioninR2mrelosetlbytheIirresy=
y=0and
I+
y=I.
Deterrrrinethevalueof
Evaluate
(2.r+y?3)?
tld.
(2y?:r+6)?Iy
whereDisthesquarewithwrtices(0.0).(2.l).(3.-l)iurd(l.-2).(Hint:FirstdtetchDand
?ndtheequationsofitssides)
Evaluate
f]cw?rz+yzM/l.
0
whereDistheshadedregioninthefollowing?gure-l.
Arr:of|rcircle
ofradiusl.i,
(ceiilenaat'
"Iilli?l
\
Figurel:
2G.Evaluatel
[Itl?.
0
WhereDisthediskofradiuslwithcerterat(0.l).(BecarefulwhenyoudeatribeD.)
27.Determinethevalueof_
iv
whereWisthesolidregionboundedbytheplanez=12andtheparaboloidz=2r?+2y?
?
6.
This post was last modified on 21 November 2019