FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Transcript of FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
http://slidepdf.com/reader/full/fea-question-with-class-notes-and-formula-book 1/374
LK9:;: NHDHUK KFKLKDU @D@FXZHZ
^H ZKL LKJC@DHJ@F KDMHDKKYHDM
YKM[F@UHBD 9 3
P[KZUHBD I@DA, JF@ZZ DBUKZ
& NBYL[F@ IBBAWYKW@YKE IX
@ZCBA A[L@Y. Y L.K (J@E)
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 0
Y.L.A JBFFKMK BN KDMHDKKYHDM @DE
UKJCDBFBMX
YZL D@M@Y, W[E[^BX@F->7097>
@ZZHMDLKDU P[KZUHBDZ
LK9:;: ‚ NHDHUK KFKLKDU @D@FXZHZ
HHH I.K. ^H Zkl. LK
EKJ‘ : ‚ L@X‘ 5
[DHU ‚ H ‚ NHDHUK KFKLKDU NBYL[F@UHBD BN IB[DE@YX ^@F[K
WYBIFKLZ
W@YU ‚
@0.0) ]c`t hs tck nhdhtk kfklkdt lktcbe2
0.9) Cbw ebks tck nhdhtk kfklkdt lktcbe wbra2
0.:) ]c`t `rk tck l`hd stkps hdvbfvke hd NK@. Q@[, @prhf / L`y ‚ 9700\
0.5) ]rhtk tck stkps hdvbfvke hd ekvkfbphdm nhdhtk kfklkdt lbekf.
0.;) ]c`t `rk tck i`shj `pprb`jcks tb hlprbvk ` nhdhtk kfklkdt lbekf2
Q@[, Dbv / Ekj ‚
9707\
0.>) ]rhtk `dy twb `ev`dt`mks bn NKL @d`fyshs. Q@[, Dbv / Ekj ‚ 9709\
0.=) ]c`t `rk tck lktcbes mkdkr`ffy `ssbjh`tke whtc nhdhtk kfklkdt `d`fyshs2
0.3) Fhst `dy nbur `ev`dt`mks bn nhdhtk kfklkdt lktcbe. Q@[, @prhf / L`y ‚ 9773\
0.8) ]c`t `rk tck `ppfhj`thbds bn NK@2 Q@[, @prhf / L`y ‚ 9700\
0.07) Eknhdk nhdhtk ehnnkrkdjk lktcbe.
0.00) ]c`t hs tck fhlht`thbd bn ushdm ` nhdhtk ehnnkrkdjk lktcbe2 Q@[, @prhf / L`y ‚ 9707\
0.09) Eknhdk nhdhtk vbfulk lktcbe.
0.0:) Ehnnkrkdth`tk nhdhtk kfklkdt lktcbe nrbl nhdhtk ehnnkrkdjk lktcbe.
0.05) Ehnnkrkdth`tk nhdhtk kfklkdt lktcbe nrbl nhdhtk vbfulk lktcbe.
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 9
0.0;) ]c`t eb ybu lk`d iy ehsjrkthz`thbd hd nhdhtk kfklkdt lktcbe2
0.0>) ]c`t hs ehsjrkthz`thbd2 Q@[, Dbv / Ekj ‚ 9707\
0.0=) Fhst tck typks bn dbeks. Q@[, L`y / Oudk ‚ 9709\
0.03) Eknhdk ekmrkk bn nrkkebl.
0.08) ]c`t hs lk`dt iy ekmrkks bn nrkkebl2 Q@[, Dbv / Ekj ‚
9709\
0.97) Zt`tk tck `ev`dt`mk bn nhdhtk kfklkdt lktcbe bvkr btckr dulkrhj`f `d`fyshs
lktcbes.
0.90) Zt`tk tck nhkfes tb wchjc NK@ sbfvhdm prbjkeurk hs `ppfhj`ifk.
0.99) ]c`t hs ` strujtur`f ̀ de dbd-strujtur`f prbifkl2
0.9:) Ehsthdmuhsc iktwkkd 0E i`r kfklkdt `de 0E ik`l kfklkdt.
Q@[, Dbv / Ekj ‚ 9778, L`y / Oudk ‚ 9700\
0.95) ]rhtk tck kquhfhirhul kqu`thbd nbr `d kfklkdt`f vbfulk hd :E hdjfuehdm tck ibey
nbrjk.
0.9;) Cbw tb wrhtk tck kquhfhirhul kqu`thbd nbr ` nhdhtk kfklkdt2 Q@[, Dbv / Ekj ‚ 9709\
0.9>) Jf`sshny ibude`ry jbdehthbds. Q@[, Dbv / Ekj ‚ 9700\
0.9=) ]c`t `rk tck typks bn ibude`ry jbdehthbds2
0.93) ]c`t eb ybu lk`d iy ibude`ry jbdehthbd `de ibude`ry v`fuk prbifkl2
0.98) ]rhtk tck ehnnkrkdjk iktwkkd hdhth`f v`fuk prbifkl `de ibude`ry v`fuk prbifkl.
0.:7) ]c`t `rk tck ehnnkrkdt typks bn ibude`ry jbdehthbds2 Mhvk kx`lpfks.
Q@[, L`y / Oudk ‚ 9709\
0.:0) Fhst tck v`rhbus lktcbes bn sbfvhdm ibude`ry v`fuk prbifkls.
Q@[, @prhf / L`y ‚ 9707\
0.:9) ]rhtk ebwd tck ibude`ry jbdehthbds bn ` j`dthfkvkr ik`l @I bn sp`d F nhxke `t @
`de nrkk `t I suiokjtke tb ` udhnbrlfy ehstrhiutke fb`e bn W tcrbumcbut tck sp`d.
Q@[, L`y / Oudk ‚ 9778, 9700\
0.::) Irhknfy kxpf`hd nbrjk lktcbe ̀ de sthnndkss lktcbe.
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) :
0.:5) ]c`t hs `spkjt r`thb2
0.:;) ]rhtk ` scbrt dbtk bd strkss ‚ str`hd rkf`thbd.
0.:>) ]rhtk ebwd tck strkss str`hd rkf`thbdschp nbr ̀ tcrkk ehlkdshbd`f strkss nhkfe.
Q@[, @prhf / L`y ‚ 9700\
0.:=) Zt`tk tck knnkjt bn Wbhssbd‘s r`thb.
0.:3) Eknhdk tbt`f pbtkdth`f kdkrmy bn `d kf`sthj ibey.
0.:8) ]rhtk tck pbtkdth`f kdkrmy nbr ik`l bn sp`d F shlpfy suppbrtke `t kdes, suiokjtke
tb ` jbdjkdtr`tke fb`e W `t lhe sp`d. @ssulk KH jbdst`dt.
Q@[, @prhf / L`y, Dbv / Ekj ‚ 9773\
0.57) Zt`tk tck prhdjhpfk bn lhdhlul pbtkdth`f kdkrmy.
Q@[, Dbv / Ekj ‚ 977=, 970:, @prhf / L`y ‚ 9778\
0.50) Cbw whff ybu bit`hd tbt`f pbtkdth`f kdkrmy bn ` strujtur`f systkl2
Q@[, @prhf / L`y ‚ 9700, L`y / Oudk ‚ 9709\
0.59) ]rhtk ebwd tck pbtkdth`f kdkrmy nudjthbd nbr ` tcrkk ehlkdshbd`f eknbrl`ifk ibey
hd tkrls bn str`hd `de ehspf`jklkdts. Q@[, L`y / Oudk ‚ 9778\
0.5:) ]c`t scbufe ik jbdshekrke eurhdm phkjkwhsk trh`f nudjthbds2
Q@[, @prhf / L`y ‚ 9700\
0.55) ]c`t eb ybu udekrst`de iy tck tkrl ‖phkjkwhsk jbdthdubus nudjthbd“2
Q@[, Dbv / Ekj ‚ 970:\
0.5;) D`lk tck wkhmctke rksheu`f lktcbes. Q@[, Dbv / Ekj ‚ 9700\
0.5>) ]c`t hs tck usk bn Yhtz lktcbe2 Q@[, Dbv / Ekj ‚ 9700\
0.5=) Lkdthbd tck i`shj stkps bn Y`yfkhmc-Yhtz lktcbe. Q@[, @prhf / L`y ‚
9700\
0.53) Chmcfhmct tck kquhv`fkdjk `de tck ehnnkrkdjk iktwkkd Y`yfkhmc Yhtz lktcbe `de tck
nhdhtk kfklkdt lktcbe. Q@[, Dbv / Ekj ‚ 9709\
0.58) Ehsthdmuhsc iktwkkd Y`yfkhmc Yhtz lktcbe `de nhdhtk kfklkdt lktcbe.
Q@[, Dbv / Ekj ‚ 970:\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 5
0.;7) Ehsthdmuhsc iktwkkd Y`yfkhmc Yhtz lktcbe `de nhdhtk kfklkdt lktcbe whtc rkm`re
tb jcbbshdm ehspf`jklkdt nudjthbd. Q@[, Dbv / Ekj ‚ 9707\
0.;0) ]cy `rk pbfydblh`f typks bn hdtkrpbf`thbd nudjthbds prknkrrke bvkr trhmbdblktrhj
nudjthbds2 Q@[, @prhf / L`y ‚ 9778, L`y / Oudk ‚ 970:\
0.;9) ]c`t hs lk`dt iy wk`a nbrluf`thbd2 Q@[, L`y / Oudk ‚
970:\
0.;:) Eknhdk tck prhdjhpfk bn vhrtu`f wbra.
0.;5) Ehnnkrkdth`tk ̂ bd Lhsks strkss ̀ de prhdjhpfk strkss.
0.;;) ]c`t eb ybu lk`d iy jbdsthtuthvk f`w2 Q@[, Dbv / Ekj ‚ 977=, @prhf / L`y ‚ 9778\
0.;>) ]c`t `rk c `de p vkrshbds bn nhdhtk kfklkdt lktcbe2
0.;=) ]c`t hs tck ehnnkrkdjk iktwkkd st`thj `de eyd`lhj `d`fyshs2
0.;3) Lkdthbd twb shtu`thbds wckrk M`fkrahd‘s lktcbe hs prknkr`ifk tb Y`yfkhmc ‚ Yhtz
lktcbe. Q@[, Dbv / Ekj ‚ 970:\
0.;8) ]c`t hs M`fkrahd lktcbe bn ̀ pprbxhl`thbd2 Q@[, Dbv / Ekj ‚ 9778\
0.>7) ]c`t hs ` wkhmctke rksuhe`f lktcbe2 Q@[, Dbv / Ekj ‚ 9707\
0.>0) Ehsthdmuhsc iktwkkd pbtkdth`f kdkrmy ̀ de pbtkdth`f kdkrmy nudjthbd`f.
0.>9) D`lk `dy nbur NK@ sbntw`rk
W@YU ‚ I
0.>:) Kxpf`hd tck stkp iy stkp prbjkeurk bn NK@. Q@[, Dbv / Ekj ‚ 9707\
0.>5) Kxpf`hd tck mkdkr`f prbjkeurk bn nhdhtk kfklkdt `d`fyshs. Q@[, Dbv / Ekj ‚ 9700\
0.>;) Irhknfy kxpf`hd tck st`mks hdvbfvke hd NK@.
0.>>) Kxpf`hd tck stkp iy stkp prbjkeurk bn NKL. Q@[, Dbv / Ekj ‚ 9700\
0.>=) Fhst but tck mkdkr`f prbjkeurk nbr NK@ prbifkls. Q@[, L`y / Oudk ‚ 9709\
0.>3) Jblp`rk NKL whtc btckr lktcbes bn `d`fyshs. Q@[, Dbv / Ekj ‚ 9707\
0.>8) Eknhdk ehsjrkthz`thbd. Kxpf`hd lksc rknhdklkdt. Q@[, Dbv / Ekj ‚ 9707\
0.=7) Kxpf`hd tck v`rhbus `spkjts pkrt`hdhdm tb ehsjrkthz`thbd, prbjkss hd nhdhtk kfklkdt
lbekfhdm ̀ d`fyshs. Q@[, Dbv / Ekj ‚ 970:\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) ;
0.=0) Kxpf`hd tck prbjkss bn ehsjrkthz`thbd bn ` strujturk hd nhdhtk kfklkdt lktcbe hd
ekt`hf, whtc suht`ifk hffustr`thbds nbr k`jc `spkjt ikhdm & ehsjusske.
Q@[, Dbv / Ekj ‚ 9709\
0.=9) Ehsjuss prbjkeurk ushdm tck jbllkrjh`f p`ja`mk (W.J. Wrbmr`ls) `v`hf`ifk tbe`y
nbr sbfvhdm prbifkls bn NKL. U`ak ` strujtur`f prbifkl tb kxpf`hd tck s`lk. Q@[, Dbv / Ekj ‚ 9700\
0.=:) Zt`tk tck hlpbrt`djk bn fbj`thdm dbeks hd nhdhtk kfklkdt lbekf.
Q@[, Dbv / Ekj ‚ 9700\
0.=5) ]rhtk ` irhkn dbtk bd tck nbffbwhdm.
(`) hsbtrbphj l`tkrh`f
(i) brtcbtrbphj l`tkrh`f
(j) `dhsbtrbphj l`tkrh`f
0.=;) ]c`t `rk hdhth`f `de nhd`f ibude`ry v`fuk prbifkls2 Kxpf`hd.
Q@[, Dbv / Ekj ‚ 9707\
0.=>) Kxpf`hd tck Wbtkdth`f Kdkrmy @pprb`jc Q@[, Dbv / Ekj ‚ 9707\
0.==) Kxpf`hd tck prhdjhpfk bn lhdhlhz`thbd bn pbtkdth`f kdkrmy. Q@[, Dbv / Ekj ‚ 9700\
0.=3) Kxpf`hd tck nbur wkhmctke rksheu`f lktcbes. Q@[, Dbv / Ekj ‚ 9700\
0.=8) Kxpf`hd Yhtz lktcbe whtc `d kx`lpfk. Q@[, @prhf / L`y ‚ 9700\
0.37) Kxpf`hd Y`yfkhmc Yhtz ̀ de M`fkrahd nbrluf`thbd whtc kx`lpfk.
Q@[, L`y / Oudk ‚ 9709\
0.30) ]rhtk scbrt dbtks bd M`fkrahd lktcbe2 Q@[, @prhf / L`y ‚ 9778\
0.39) Ehsjuss strkssks ̀ de kquhfhirhul bn ̀ tcrkk ehlkdshbd`f ibey.
Q@[, L`y / Oudk ‚ 9709\
0.3:) Ekrhvk tck kfklkdt fkvkf kqu`thbd nbr bdk ehlkdshbd`f i`r kfklkdt i`ske bd tck
st`thbd- bn ` nudjthbd`f. Q@[, L`y / Oudk ‚ 9709\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) >
0.35) Ekrhvk tck jc`r`jtkrhsthj kqu`thbds nbr tck bdk ehlkdshbd`f i`r kfklkdt iy ushdm
phkjk-whsk eknhdke hdtkrpbf`thbds `de wk`a nbrl bn tck wkhmctke rksheu`f lktcbe2
Q@[, L`y / Oudk ‚ 9709\
0.3;) Ekvkfbp tck wk`a nbrl `de ektkrlhdk tck ehspf`jklkdt nhkfe nbr ` j`dthfkvkr ik`l
suiokjtke tb ` udhnbrlfy ehstrhiutke fb`e `de ` pbhdt fb`e `jthdm `t tck nrkk kde. Q@[, Dbv / Ekj ‚ 970:\
0.3>) Kxpf`hd M`ussh`d kfhlhd`thbd lktcbe bn sbfvhdm kqu`thbds.
Q@[, @prhf / L`y ‚ 9700\
0.3=) ]rhtk irhknfy ̀ ibut M`ussh`d kfhlhd`thbd2 Q@[, @prhf / L`y ‚ 9778\
0.33) Uck nbffbwhdm ehnnkrkdth`f kqu`thbd hs ̀ v`hf`ifk nbr ` pcyshj`f pckdblkdbd.
( )
Ibude`ry jbdehthbds ̀ rk, x < 0 u < 9
x < 9
Nhde tck v`fuk bn tck p`r`lktkr `, iy tck nbffbwhdm lktcbes.
(h) Jbffbj`thbd (hh) Zui ‚ Ebl`hd (hhh) Fk`st Zqu`rk
(hv) M`fkrahd
0.38) Uck nbffbwhdm ehnnkrkdth`f kqu`thbd hs ̀ v`hf`ifk nbr ` pcyshj`f pckdblkdbd.
Urh`f nudjthbd hs
Ibude`ry jbdehthbds ̀ rk, y (7) < 7 y (07) < 7
Nhde tck v`fuk bn tck p`r`lktkr `, iy tck nbffbwhdm lktcbes.
(h) Jbffbj`thbd (hh) Zui ‚ Ebl`hd (hhh) Fk`st Zqu`rk
(hv) M`fkrahd
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) =
0.87) Ehsjuss tck nbffbwhdm lktcbes tb sbfvk tck mhvkd ehnnkrkdth`f kqu`thbd ?
whtc tck ibude`ry jbdehthbd y(7) < 7 `de y( C ) < 7
(h) ^`rh`dt lktcbe (hh) Jbffbj`thbd lktcbe. Q@[, @prhf / L`y ‚
9707\
0.80) @ j`dthfkvkr ik`l bn fkdmtc F hs fb`eke whtc ` pbhdt fb`e `t tck nrkk kde. Nhde tck
l`xhlul eknfkjthbd `de l`xhlul ikdehdm lblkdt ushdm Y`yfkhmc-Yhtz lktcbe
ushdm tck nudjthbd Mhvkd? KH hs jbdst`dt.
Q@[, @prhf / L`y ‚ 9773\
0.89)
@ shlpfy suppbrtke ik`l j`rrhks udhnbrlfy ehstrhiutke fb`e bvkr tck kdthrk sp`d.J`fjuf`tk tck ikdehdm lblkdt `de eknfkjthbd. @ssulk KH hs jbdst`dt `de jblp`rk
tck rksufts whtc btckr sbfuthbd. Q@[, Dbv / Ekj ‚ 9709\
0.8:) Ektkrlhdk tck kxprksshbd nbr eknfkjthbd `de ikdehdm lblkdt hd ` shlpfy suppbrtke
ik`l suiokjtke tb udhnbrlfy ehstrhiutke fb`e bvkr kdthrk sp`d. Nhde tck eknfkjthbd
`de lblkdt `t lhesp`d `de jblp`rk whtc kx`jt sbfuthbd ushdm Y`yfkhmc-Yhtz
lktcbe. [sk Q@[, Dbv / Ekj ‚
9773\
0.85) Jblputk tck v`fuk bn jkdtr`f eknfkjthbd hd tck nhmurk ikfbw iy `ssulhdm
Uck ik`l hs udhnbrl tcrbumcbut `de j`rrhks ` jkdtr`f pbhdt fb`e W.
Q@[, Dbv / Ekj ‚ 977=, @prhf / L`y ‚ 9778\
0.8;) Hn ` ehspf`jklkdt nhkfe hs eksjrhike iy
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 3
Ektkrlhdk tck ehrkjt str`hds hd x `de y ehrkjthbds `s wkff tck sck`r str`hd `t tck pbhdt
x < 0, y <7. Q@[, @prhf / L`y ‚ 9700\
0.8>) Hd ` sbfhe ibey, tck shx jblpbdkdts bn tck strkss `t ` pbhdt `rk mhvkd iy x< 57
LW`, y < 97 LW`, z < :7 LW`, yz < -:7 LW`, xz < 0; LW` `de xy < 07 LW`.
Ektkrlhdk tck dbrl`f strkss `t tck pbhdt, bd ` pf`dk nbr wchjc tck dbrl`f hs (d x , d y ,
d z ) < ( ¼, ¼,9
0 )
0.8=) Hd ` pf`dk str`hd prbifkl, wk c`vk
x < 97,777 psh y < - 07,777 psh K < :7 x 07 > psh, < 7.:.
Ektkrlhdk tck v`fuk bn tck strkss z.
0.83) Nbr tck sprhdm systkl scbwd hd nhmurk, j`fjuf`tk tck mfbi`f sthnndkss l`trhx,
ehspf`jklkdts bn dbeks 9 `de :, tck rk`jthbd nbrjks `t dbek 0 `de 5. @fsb j`fjuf`tk
tck nbrjks hd tck sprhdm 9. @ssulk, a 0 < a : < 077 D/l, a 9 < 977 D/l, u0 < u5< 7 `de
W <;77 D . Q@[, @prhf / L`y ‚ 9707\
0.88) [sk tck Y`yfkhmc ‚ Yhtz lktcbe tb nhde tck ehspf`jklkdt bn tck lhepbhdt bn tck rbe
scbwd hd nhmurk. Q@[, @prhf / L`y ‚ 9700\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 8
0.077) Jbdshekr tck ehnnkrkdth`f kqu`thbd
suiokjt
tb ibude`ry jbdehthbds Uck nudjthbd`f jbrrkspbdehdm tb tchs
prbifkl, tb ik kxtrklhzke hs mhvkd iy ∡
0.070) Nhde tck sbfuthbd bn tck prbifkl ushdm Y`yfkhmc-Yhtz lktcbe iy jbdshekrhdm ` twb-
tkrl sbfuthbd ̀ s Q@[, Dbv / Ekj ‚ 9778\
0.079)
@ i`r bn udhnbrl jrbss skjthbd hs jf`lpke `t bdk kde `de fknt nrkk `t tck btckr kde. Hths suiokjtke tb ` udhnbrl fb`e `xh`f fb`e W `s scbwd hd nhmurk. J`fjuf`tk tck
ehspf`jklkdt `de strkss hd tck i`r ushdm tcrkk tkrls pbfydblh`f nbffbwhdm Yhtz
lktcbe. Jblp`rk tck rksufts whtc kx`jt sbfuthbds. Q@[, L`y / Oudk ‚ 9700\
0.07:) @ shlpfy Zuppbrtke ik`l suiokjtke tb udhnbrlfy ehstrhiutke fb`e bvkr kdthrk sp`d
`de ht hs suiokjtke tb ` pbhdt fb`e `t tck jkdtrk bn tck sp`d. J`fjuf`tk tck eknfkjthbd
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 07
ushdm Y`yfkhmc-Yhtz lktcbe ̀ de jblp`rk whtc kx`jt sbfuthbds.
Q@[, L`y / Oudk ‚ 970:\
0.075) @ shlpfy suppbrtke ik`l (sp`d F `de nfkxur`f rhmhehty KH) j`rrhks twb kqu`f
jbdjkdtr`tke fb`es `t k`jc bn tck qu`rtkr sp`d pbhdts. [shdm Y`fkhmc ‚ Yhtz lktcbe
ektkrlhdk tck eknfkjthbds udekr tck twb fb`es `de tck twb kde sfbpks. Q@[, @prhf / L`y ‚ 9778\
0.07;) @d`fyzk ` shlpfy suppbrtke ik`l suiokjtke tb ` udhnbrlfy ehstrhiutke fb`e
tcrbumcbut ushdm Y`yfkhmc Yhtz lktcbe. @ebpt bdk p`r`lktkr trhmbdblktrhj
nudjthbd. Kv`fu`tk tck l`xhlul eknfkjthbd `de ikdehdm lblkdt `de jblp`rk whtc
kx`jt sbfuthbd. Q@[, Dbv / Ekj ‚ 9707\
0.07>)
Zbfvk nbr tck ehspf`jklkdt nhkfe nbr ` shlpfy suppbrtke ik`l, suiokjtke tb `udhnbrlfy ehstrhiutke fb`e ushdm Y`yfkhmc ‚ Yhtz lktcbe. Q@[, Dbv / Ekj ‚ 970:\
0.07=) [sk tck Y`yfkhmc ‚ Yhtz lktcbe tb nhde tck ehspf`jklkdt nhkfe u(x) bn tck rbe `s
scbwd ikfbw. Kfklkdt 0 hs l`ek bn `fulhdul `de kfklkdt 9 hs l`ek bn stkkf. Uck
prbpkrthks `rk
K`f < =7 MW` @0 < 877 ll9 F0 < 977 ll
Kst < 977 MW` @9 < 0977 ll9 F9 < :77 ll
Fb`e < W < 07,777 D . @ssulk ` phkjkwhsk fhdk`r ehspf`jklkdt.
Nhkfe u < `0 + `9 x nbr 7 x 977 ll, `de u < `: + `5 x nbr 977 x ;77 ll.
0.073) @ nhxke ik`l fkdmtc bn 9F l j`rrhks ` udhnbrlfy ehstrhiutke fb`e bn ` w(hd D / l)
wchjc rud bvkr ` fkdmtc bn –F‘ l nrbl tck nhxke kde, `s scbwd hd Nhmurk. J`fjuf`tk
tck rbt`thbd `t pbhdt I ushdm NK@. Q@[, Dbv / Ekj ‚ 9700\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 00
0.078) @d`fyzk tck ik`l scbwd hd nhmurk ushdm nhdhtk kfklkdt tkjcdhquk. Ektkrlhdk tck
rbt`thbds `t tck suppbrts. Mhvk K < 977MW` `de H < 5 * 07>ll5
Q@[, Dbv / Ekj ‚ 970:\
0.007) @ rbe nhxke `t hts kdes hs suiokjtke tb ` v`ryhdm ibey nbrjk `s scbwd hd Nhmurk. [sk
tck Y`yfkhmc-Yhtz lktcbe whtc `d `ssulke ehspf`jklkdt nhkfe
tb nhde tck ehspf`jklkdt u(x) `de strkss ϊ(x). Wfbt tck v`rh`thbd bn tck
strkss hd tck rbe. Q@[, Dbv / Ekj ‚ 9709\
0.000) @ udhnbrl rbe suiokjtke tb ` udhnbrl `xh`f fb`e hs hffustr`tke hd Nhmurk. Uck
eknbrl`thbd bn tck i`r hs mbvkrdke iy tck ehnnkrkdth`f kqu`thbd mhvkd ikfbw.
Ektkrlhdk tck ehspf`jklkdt ushdm wkhmctke rksheu`f lktcbe.
Q@[, @prhf / L`y ‚ 9700\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 09
0.009) @ stkkf rbe hs `tt`jcke tb rhmhe w`ffs `t k`jc kde `de hs suiokjtke tb ` ehstrhiutke
fb`e U( x) `s scbwd ikfbw.
`) ]rhtk tck kxprksshbd nbr pbtkdth`f kdkrmy.
i) Ektkrlhdk tck ehspf`jklkdt u(x) ushdm tck Y`yfkhmc ‚ Yhtz lktcbe.
@ssulk ̀ ehspf`jklkdt nhkfe u(x) < `7 + `0 x + `9 x9.
0.00:) Ekrhvk tck strkss ‚ str`hd rkf`thbd `de str`hd ‚ ehspf`jklkdt rkf`thbd nbr `d kfklkdt
hd sp`jk.
0.005) Ekrhvk tck kqu`thbd bn kquhfhirhul hd j`sk bn ` tcrkk ehlkdshbd`f strkss systkl.
Q@[, Dbv / Ekj ‚ 9773\
0.00;) ]c`t hs jbdsthtuthvk rkf`thbdschp2 Kxprkss tck jbdsthtuthvk rkf`thbds nbr ` fhdk`r
kf`sthj hsbtrbphj l`tkrh`f hdjfuehdm hdhth`f strkss ̀ de str`hd. Q@[, Dbv / Ekj ‚ 9778\
0.00>) Mhvk ` ekt`hfke dbtk bd tck nbffbwhdm?
(`) Y`yfkhmc Yhtz lktcbe (i) M`fkrahd lktcbe
(j) Fk`st squ`rk lktcbe `de (e) Jbffbj`thbd lktcbe
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 0:
0.00=) Nhde tck `pprbxhl`tk eknfkjthbd bn ` shlpfy suppbrtke ik`l udekr ` udhnbrlfy
ehstrhiutke fb`e –W– tcrbumcbut hts sp`d. [shdm M`fkrahd `de Fk`st squ`rk rksheu`f
lktcbe. Q@[, L`y / Oudk ‚ 9700\
0.003) Zbfvk tck ehnnkrkdth`f kqu`thbd nbr ` pcyshj`f prbifkl kxprksske `s whtc ibude`ry jbdehthbds `s y (7) < 7 `de y (07) < 7 ushdm
(h) Wbhdt jbffbj`thbd lktcbe
(hh) Zui ebl`hd jbffbj`thbd lktcbe
(hhh) Fk`st squ`rks lktcbe ̀ de
(hv) M`fkrahd lktcbe. Q@[, L`y / Oudk ‚ 970:\
0.008) Zbfvk tck ehnnkrkdth`f kqu`thbd nbr ` pcyshj`f prbifkl kxprksske `s whtc ibude`ry jbdehthbds `s y (7) < 7 `de y (07) < 7 ushdm tck
tr`hf nudjthbd Nhde tck v`fuk bn tck p`r`lktkrs `0 iy tck
nbffbwhdm lktcbes.
(h) Wbhdt jbffbj`thbd lktcbe
(hh) Zui ebl`hd jbffbj`thbd lktcbe
(hhh) Fk`st squ`rks lktcbe ̀ de
(hv) M`fkrahd lktcbe. Q@[, Dbv / Ekj ‚ 9700\
0.097) Zbfvk tck nbffbwhdm kqu`thbd ushdm ` twb ‚ p`r`lktkr trh`f sbfuthbd iy tck
(`) Jbffbj`thbd lktcbe
(i) M`fkrahd lktcbe.
Uckd, jblp`rk tck twb sbfuthbds whtc tck kx`jt sbfuthbd
y (7) < 0
0.090) Ektkrlhdk tck M`fkrahd ̀ pprbxhl`thbd sbfuthbd bn tck ehnnkrkdth`f kqu`thbd
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 05
0.099) Zbfvk tck nbffbwhdm ehnnkrkdth`f kqu`thbd ushdm M`fkrahd‘s lktcbe.
Q@[, @prhf / L`y ‚ 9700\
0.09:) @ pcyshj`f pckdblkdbd hs mbvkrdke iy tck ehnnkrkdth`f kqu`thbd
Uck ibude`ry jbdehthbds `rk mhvkd iy
. Iy t`ahdm twb-tkrl trh`f sbfuthbd `s
whtc,
nhde tck sbfuthbd bn
tck prbifkl ushdm tck M`fkrahd lktcbe. Q@[, Dbv / Ekj ‚ 9778\
0.095) Ektkrlhdk tck twb p`r`lktkr sbfuthbd bn tck nbffbwhdm ushdm M`fkrahd lktcbe. Q@[, Dbv / Ekj ‚ 9709\
0.09;) Mhvk ` bdk ‚ p`r`lktkr M`fkrahd sbfuthbd bn tck nbffbwhdm kqu`thbd, nbr tck twb
ebl`hd‘s scbwd ikfbw. .09
9
9
9
y
u
x
u
0.09>) Eksjrhik tck M`ussh`d kfhlhd`thbd lktcbe bn sbfvhdm kqu`thbds.
Q@[, @prhf / L`y ‚ 9700\
0.09=) Kxpf`hd tck M`ussh`d kfhlhd`thbd lktcbe nbr tck sbfvhdm bn shluft`dkbus fhdk`r
`fmkir`hj kqu`thbds whtc ̀ d kx`lpfk. Q@[, @prhf / L`y ‚ 9773\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 0;
0.093) Zbfvk tck nbffbwhdm systkl bn kqu`thbds ushdm M`uss kfhlhd`thbd lktcbe.
Q@[, Dbv / Ekj ‚ 9707\
x0 ‚ x9 + x: < 0
-:x0 + 9x9 ‚ :x: < ->
9x0 ‚ ;x9 + 5x: < ;
0.098) Zbfvk tck nbffbwhdm systkl bn kqu`thbds iy M`uss Kfhlhd`thbd lktcbe.
9x0 ‚ 9x9 ‚ x5 < 0
9x9 + x: + 9x5 < 9
x0 ‚ 9x9 + :x: ‚ 9x5 < : Q@[, L`y / Oudk ‚ 9709\
x9 + 9x
: + 9x
5 < 5
0.0:7) Zbfvk tck nbffbwhdm kqu`thbds iy M`uss kfhlhd`thbd lktcbe.
93r 0 + >r 9 < 0
>r 0 + 95r 9 + >r : < 7
>r 9 + 93r : + 3r 5 < -0
3r : + 0>r 5 < 07 Q@[, Dbv / Ekj ‚ 9707, 9709\
0.0:0) [sk tck M`ussh`d kfhlhd`thbd lktcbe tb sbfvk tck nbffbwhdm shluft`dkbus
kqu`thbds?
5x0 + 9x9 ‚ 9x: ‚ 3x5 < 5
x0 + 9x9 + x: < 9
7.;x0 ‚ x9 + 5x: + 5x5 < 07
‚ 5x0 ‚ 9x9 ‚ x5 < 7 Q@[, @prhf / L`y ‚
9778\
0.0:9) Zbfvk tck nbffbwhdm systkl bn kqu`thbds ushdm M`uss kfhlhd`thbd lktcbe.
x0 + :x9 + 9x: < 0:
‚ 9x0 + x9 ‚ x: < ‚ :
- ;x0 + x9 + :x: < > Q@[, Dbv / Ekj ‚ 9778\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 0>
[DHU ‚ HH ‚ BDK EHLKDZHBD@F NHDHUK KFKLKDU @D@FXZHZ
W@YU ‚ @
9.0) ]rhtk ` dbtk bd dbek dulikrhdm sjcklk.
9.9) ]c`t eb ybu lk`d iy dbek `de kfklkdt2
9.:) ]c`t `rk tck typks bn prbifkls trk`tke `s bdk ehlkdshbd`f prbifkl2
Q@[, L`y / Oudk ‚ 970:\
9.5) Chmcfhmct `t fk`st twb rufks tb muhek tck pf`jklkdt bn tck dbeks wckd bit`hdhdm
`pprbxhl`tk sbfuthbd tb ̀ ehnnkrkdth`f kqu`thbd. Q@[, @prhf / L`y ‚ 9707\
9.;) Eknhdk sc`pk nudjthbd. Q@[, Dbv / Ekj ‚ 977=, @prhf / L`y ‚ 9778\
9.>) ]c`t hs ` sc`pk nudjthbd2 Q@[, Dbv / Ekj ‚ 9778\
9.=) Ehnnkrkdth`tk sc`pk nudjthbd nrbl ehspf`jklkdt lbekf.
9.3) Er`w tck sc`pk nudjthbd bn ` twb dbeke fhdk kfklkdt. Q@[, @prhf / L`y ‚ 9778\
9.8) Er`w tck sc`pk nudjthbd bn ` twb dbeke fhdk kfklkdt whtc bdk ekmrkk bn nrkkebl `t
k`jc dbek. Q@[, Dbv / Ekj ‚ 9707\
9.07) Er`w tck sc`pk nudjthbd nbr bdk ehlkdshbd`f fhdk kfklkdt whtc tcrkk dbeks.
Q@[, @prhf / L`y ‚ 9778\
9.00) Zt`tk tck prbpkrthks bn sthnndkss l`trhx. Q@[, Dbv / Ekj ‚ 9778, 9707, 9700\
9.09) Fhst but tck sthnndkss l`trhx prbpkrthks. Q@[, L`y / Oudk ‚ 9709\
9.0:) Zt`tk tck jc`r`jtkrhsthjs bn sc`pk nudjthbd. Q@[, L`y / Oudk ‚ 9700\
9.05) Fhst tck jc`r`jtkrhsthjs bn sc`pk nudjthbds. Q@[, @prhf / L`y ‚ 9707\
9.0;) ]ckd ebks tck sthnndkss l`trhx bn ` strujturk ikjblk shdmuf`r2
Q@[, Dbv / Ekj ‚
9709\
9.0>) Zt`tk tck shmdhnhj`djk bn sc`pk nudjthbd.
9.0=) ]rhtk tck kfklkdt sthnndkss l`trhx nbr ` twb dbeke fhdk`r kfklkdt suiokjtke tb `xh`f
fb`ehdm.
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 0=
9.03) ]rhtk tck sthnndkss l`trhx nbr tck shlpfk ik`l kfklkdt mhvkd ikfbw.
Q@[, Dbv / Ekj ‚ 9773\
9.08) ]c`t `rk tck prbpkrthks bn mfbi`f sthnndkss l`trhx2 Q@[, @prhf / L`y ‚ 9700\
9.97) ]rhtk tck prbpkrthks bn Mfbi`f Zthnndkss L`trhx bn ` bdk ehlkdshbd`f kfklkdt.
Q@[, L`y / Oudk ‚ 9709\
9.90) Ehnnkrkdth`tk mfbi`f sthnndkss l`trhx nrbl kfklkdt`f sthnndkss l`trhx.
9.99) ]c`t eb ybu lk`d iy i`deke l`trhx2
9.9:)
Cbw whff ybu nhde tck whetc bn ` i`de2
9.95) Cbw eb ybu j`fjuf`tk tck shzk bn tck mfbi`f sthnndkss l`trhx2
9.9;) Fhst tck prbpkrthks bn tck mfbi`f sthnndkss l`trhx. Q@[, @prhf / L`y ‚ 9707\
9.9>) ]c`t hs tck knnkjt bn dbek dulikrhdm bd `ssklifke sthnndkss l`trhx2
Q@[, Dbv / Ekj ‚ 970:\
9.9=) Mhvk ` irhkn dbtk bd tck nbffbwhdm
(`) kfhlhd`thbd ̀ pprb`jc (i) pkd`fty ̀ pprb`jc.
9.93) D`lk tck n`jtbrs wchjc `nnkjt tck dulikr kfklkdt hd tck mhvkd ebl`hd.
9.98) Zt`tk tck rkquhrklkdts tb ik nufnhffke iy tck `pprbxhl`tk sbfuthbd nbr hts
jbdvkrmkdjk tbw`res tck ̀ jtu`f sbfuthbd.
9.:7) ]c`t eb ybu lk`d iy jbdthduhty wk`akdhdm2
9.:0) Jblp`rk tck fhdk`r pbfydblh`f `pprbxhl`thbd `de qu`er`thj pbfydblh`f
`pprbxhl`thbd.
9.:9) ]cy pbfydblh`fs `rk mkdkr`ffy uske `s sc`pk nudjthbd2 Q@[, Dbv / Ekj ‚ 9700\
9.::) ]cy `rk pbfydblh`f tkrls prknkrrke nbr sc`pk nudjthbds hd nhdhtk kfklkdt lktcbe2
Q@[, @prhf / L`y ‚ 9700\
9.:5) ]c`t eb ybu lk`d iy krrbr hd NK@ sbfuthbd2
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NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 03
9.:;) ]c`t `rk tck typks bn fb`e `jthdm bd tck strujturk2
9.:>) Eknhdk tr`jthbd nbrjk (U).
9.:=) Zt`tk tck `ssulpthbds `rk l`ek wchfk nhdehdm tck nbrjks hd ` truss.
Q@[, Dbv / Ekj ‚ 9700\
9.:3) Cbw `rk tckrl`f fb`es hdput hd nhdhtk kfklkdt `d`fyshs2
Q@[, Dbv / Ekj ‚ 977=, @prhf / L`y ‚ 9778\
9.:8) ]c`t hs `d hdtkrpbf`thbd nudjthbd2 Q@[, L`y / Oudk ‚ 9709\
9.57) ]cy `rk pbfydblh`f typks bn hdtkrpbf`thbd nudjthbds prknkrrke bvkr trhmbdblktrhj
nudjthbds2 Q@[, Dbv / Ekj ‚ 977=, @prhf / L`y ‚ 9778\
9.50) ]c`t hs `d kquhv`fkdt dbe`f nbrjk2 Q@[, @prhf / L`y ‚ 9773\
9.59) ]c`t `rk j`ffke chmckr brekr kfklkdts2
Q@[, @prhf / L`y ‚ 9773, Dbv / Ekj ‚ 9707, 9700\
9.5:) ]c`t hs chmckr brekr kfklkdt2 Q@[, Dbv / Ekj ‚ 9700\
9.55) ]c`t eb ybu lk`d iy chmckr brekr kfklkdts2 Q@[, Dbv / Ekj ‚ 9773\
9.5;) ]cy chmckr brekr kfklkdts `rk rkquhrke nbr NK `d`fyshs2 Q@[, Dbv / Ekj ‚ 9709\
9.5>) ]c`t `rk chmckr brekr kfklkdts `de wcy `rk tcky prknkrrke2
Q@[, @prhf / L`y ‚ 9700\
9.5=) ]c`t `rk tck jc`r`jtkrhsthjs bn sc`pk nudjthbds2
9.53) Wfbt tck v`rh`thbds bn sc`pk nudjthbd nbr 0 ‚ E ik`l kfklkdt.
Q@[, Dbv / Ekj ‚ 9707\
9.58) Hffustr`tk kfklkdt jbddkjthvhty hdnbrl`thbd jbdshekrhdm ik`l kfklkdts.
Q@[, Dbv / Ekj ‚
970:\
9.;7) ]ckd eb wk rksbrt tb 0 E qu`er`thj sp`r kfklkdts2 Q@[, @prhf / L`y ‚ 9700\
9.;0) Bit`hd `dy bdk sc`pk nudjthbd nbr ` qu`er`thj juihj sp`r kfklkdt.
Q@[, Dbv / Ekj ‚ 970:\
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NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 08
9.;9) Lkdthbd twb `ev`dt`mks bn qu`er`thj sp`r kfklkdt bvkr fhdk`r sp`r kfklkdt.
Q@[, Dbv / Ekj ‚ 970:\
9.;:) Mhvk ` irhkn dbtk bd tck sburjks bn krrbr hd NK@.
9.;5) Zt`tk tck shmdhnhj`djk bn pbst prbjksshdm tck sbfuthbd hd NK@.
9.;;) ]c`t eb ybu adbw `ibut r`eh`ffy syllktrhj prbifkl2
9.;>) ]rhtk tck ibude`ry jbdehthbd nbr ` j`dthfkvkr ik`l suiokjtke tb pbhdt fb`e `t hts
nrkk kde.
9.;=) Nbr ` bdk ehlkdshbd`f nhd prbifkl, wc`t `rk `ff tck ibude`ry jbdehthbds tc`t j`d ik
spkjhnhke `t tck nrkk kde2
9.;3) Ektkrlhdk tck fb`e vkjtbr nbr tck ik`l kfklkdt scbwd hd Nhmurk
Q@[, Dbv / Ekj ‚ 9709\
9.;8) ]rhtk tck kfklkdt sthnndkss l`trhx bn ` truss kfklkdt. Q@[, L`y / Oudk ‚ 9709\
9.>7) Zaktjc ̀ typhj`f truss kfklkdt scbwhdm fbj`f mfbi`f tr`dsnbrl`thbd.
Q@[, @prhf / L`y ‚ 9700\
9.>0) Ehnnkrkdth`tk mfbi`f ̀ de fbj`f jbbrehd`tks. Q@[, L`y / Oudk ‚ 970:\
9.>9) Zt`tk tck ehnnkrkdjks iktwkkd ` i`r kfklkdt `de ` truss kfklkdt.
W@YU ‚ I
9.>:) ]c`t `rk tck ehnnkrkdt typks bn kfklkdts2 Kxpf`hd tck shmdhnhj`djk bn k`jc. Q@[, Dbv / Ekj ‚ 9707\
9.>5) Ekrhvk `de saktjc tck qu`er`thj sc`pk nudjthbd nbr tck i`r kfklkdt.
Q@[, L`y / Oudk ‚ 9700\
9.>;) Ekrhvk tck sc`pk nudjthbd bn ` qu`er`thj 0 ‚ E kfklkdt. Q@[, Dbv / Ekj ‚ 9700\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 97
9.>>) Ekrhvk tck sc`pk nudjthbds nbr bdk ehlkdshbd`f fhdk`r kfklkdt ushdm ehrkjt lktcbe.
Q@[, L`y / Oudk ‚ 970:\
9.>=) Ektkrlhdk tck sc`pk nudjthbd `de kfklkdt l`trhjks nbr qu`er`thj i`r kfklkdt.
Q@[, L`y / Oudk ‚ 9709\
9.>3) Ekrhvk tck sthnndkss l`trhx `de nhdhtk kfklkdt kqu`thbd nbr bdk ehlkdshbd`f i`r.
Q@[, Dbv / Ekj ‚ 9700\
9.>8) Ekrhvk tck sthnndkss l`trhx `de ibey nbrjk vkjtbr nbr ` qu`er`thj sp`r kfklkdt.
Q@[, Dbv / Ekj ‚ 970:\
9.=7) Bit`hd `d kxprksshbd nbr tck sc`pk nudjthbd bn ` fhdk`r i`r kfklkdt.
Q@[, @prhf / L`y ‚ 9700\
9.=0) Ekrhvk sc`pk nudjthbds `de sthnndkss l`trhx nbr ` 9E rkjt`dmuf`r kfklkdt.
Q@[, Dbv / Ekj ‚ 9709\
9.=9) Jbdshekr tck rbe (` rbibt `rl) `s scbwd ikfbw, wchjc hs rbt`thdm `t jbdst`dt
`dmuf`r vkfbjhty < :7 r`e/skj. Ektkrlhdk tck `xh`f strkss ehstrhiuthbd hd tck rbe,
ushdm twb qu`er`thj kfklkdts. Jbdshekr bdfy tck jkdtrhnum`f nbrjk. Hmdbrk ikdehdm
bn tck rbe.
9.=:) @ fhda bn 9l, phd ‚ obhdtke `t bdk kde, hs rbt`thdm `t `dmuf`r vkfbjhty ; r`e / skj. tck
jrbss ‚ skjthbd`f `rk` bn fhda hs 9 * 07 -: l9. Ektkrlhdk tck dbe`f ehspf`jklkdts
ushdm twb fhdk`r sp`r kfklkdts. U`ak K < 977MW` `de π < =3;7 am/l:.
Q@[, Dbv / Ekj ‚ 970:\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 90
9.=5) @ stkkf rbe bn fkdmtc 0l hs suiokjtke tb `d `xh`f fb`e bn ; aD `s scbwd hd nhmurk.
@rk` bn jrbss skjthbd bn tck rbe hs 9;7 ll9. [shdm 0 ‚ E kfklkdt kqu`thbd sbfvk nbr
tck eknfkjthbd bn tck i`r, K < 9*07; D/ll9. [sk nbur kfklkdts.
Q@[, Dbv / Ekj ‚ 9707\
9.=;) @ jbfuld bn fkdmtc ;77ll hs fb`eke `xh`ffy `s scbwd hd nhmurk. @d`fyzk tck
jbfuld `de kv`fu`tk tck strkss `de str`hd `t s`fhkdt pbhdts. Uck Xbudm‘s lbeufus
j`d ik t`akd `s K. U`ak @0 < >9.;ll9 `de @9 < 09;ll9
Q@[, @prhf / L`y ‚ 9778\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 99
9.=>) Jbdshekr ` i`r `s scbwd hd nhmurk. Xbudm‘s Lbeufus K < 9*07; D/ll9. @0 < 9 jl9,
@9 < 0 jl9 `de nbrjk bn 077 D. Ektkrlhdk tck dbe`f ehspf`jklkdt.
Q@[, Dbv / Ekj ‚ 9707\
9.==) Jbdshekr tck i`r scbwd hd Nhmurk @xh`f nbrjk W < :7 aD hs `ppfhke `s scbwd.
Ektkrlhdk tck dbe`f ehspf`jklkdt, strkssks hd k`jc kfklkdt `de rk`jthbd nbrjks
Q@[, L`y / Oudk ‚ 9709\
9.=3) Jbdshekr tck i`r `s scbwd hd nhmurk. @xh`f nbrjk W0 < 97 aD `de W9 < 0; aD hs
`ppfhke `s scbwd hd nhmurk. Ektkrlhdk tck dbe`f ehspf`jklkdts, strkssks hd k`jc
kfklkdt ̀ de rk`jthbd nbrjks. Q@[, @prhf / L`y ‚ 9700\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 9:
9.=8) Nhde tck dbe`f ehspf`jklkdt `de kfklkdt`f strkssks nbr tck i`r scbwd hd Nhmurk.
Q@[, @prhf / L`y ‚
9700\
9.37) @d `xh`f fb`e W < :77 x 07: D hs `ppfhke `t 977J tb tck rbe `s scbwd ikfbw. Uck
tklpkr`turk hs tckd r`hske tb >77J
`) @ssklifk tck sthnndkss (A) `de fb`e (N) l`trhjks.
i) Ektkrlhdk tck dbe`f ehspf`jklkdts `de kfklkdt strkssks.
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 95
9.30) Uck stkppke i`r scbwd hd nhm hs suiokjtke tb `d hdjrk`sk hd tklpkr`turk, U<37b J .
Ektkrlhdk tck ehspf`jklkdts, kfklkdt strkssks ̀ de suppbrt rk`jthbds.
Q@[, Dbv / Ekj ‚ 9778\
9.39) @xh`f fb`e bn ;77D hs `ppfhke tb ` stkppke sc`nt, `t tck hdtkrn`jk bn twb i`rs. Uck
kdes `rk nhxke. Bit`hd tck dbe`f ehspf`jklkdts `de strkssks wckd tck kfklkdt hs
suiokjtke tb `ff hd tklpkr`turk bn 077˟J. U`ak K0 < =7*07: D/ll9, K9 < 977*07:
D/ll9, @0 < 877ll9, @9 < 0977ll9, ΰ0 < 9:*07-> / ˟J, ΰ9 < 00.=*07-> / ˟J, F0 <
977ll, F9 < :77ll. Q@[, Dbv / Ekj ‚ 9700\
9.3:) Jbdshekr ` i`r `s scbwd ikfbw c`vhdm ` jrbss skjthbd`f `rk` @k < 0.9 hd9 `de
Xbudm‘s lbeufus K < :7 x 07> psh Hn q0 < 7.79 hd `de q9 < 7.79; hd, ektkrlhdk tck
nbffbwhdm?
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 9;
`) Uck ehspf`jklkdt ̀ t tck pbhdt W i) Uck str`hd `de strkss
j) Uck kfklkdt sthnndkss l`trhx `de e) Uck str`hd kdkrmy hd tck kfklkdt.
@ nhdhtk kfklkdt sbfuthbd ushdm bdk ‚ ehlkdshbd`f, twb ‚ dbeke kfklkdts c`s ikkd
bit`hdke nbr ` rbe `s scbwd ikfbw.
Ehspf`jklkdt `rk `s nbffbwsU
llP 7.0\-7,7.>,Q-7.9, , K < 0 D/ll9, `rk` bn k`jc
kfklkdt < 0 ll9 , F0-9 < ;7 ll, F9-: < 37 ll, F:-5 < 077 ll.
h) @jjbrehdm tb tck nhdhtk kfklkdt tckbry, pfbt tck ehspf`jklkdt u(x) vkrsus x.
hh) @jjbrehdm tb tck nhdhtk kfklkdt tckbry, pfbt tck str`hd ( x) vkrsus x.
hhh) Ektkrlhdk tck I l`trhx nbr kfklkdt 9-:.
hv) Ektkrlhdk tck str h̀d kdkrmy hd tck kfklkdt 0-9 ushdm .9
0aqq[ U
9.35) Jbdshekr tck i`r, fb`eke `s scbwd ikfbw. Ektkrlhdk tck dbe`f ehspf`jklkdts,
kfklkdt strkssks `de suppbrt rk`jthbds. Zbfvk tchs prbifkl iy `ebpthdm kfhlhd`thbd
lktcbe nbr c`defhdm ibude`ry jbdehthbds. (v`fuk bn K < 977 x 078 D/l9).
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 9>
9.3;) Hd tck nhmurk scbwd ikfbw fb`e W < >7aD hs `ppfhke. Ektkrlhdk tck ehspf`jklkdt
nhkfe, strkss `de suppbrt rk`jthbds hd tck ibey. U`ak K < 97 aD/ll9
Q@[, L`y / Oudk ‚ 9700\
9.3>) Jbdshekr tck i`r `s scbwd ikfbw. Ektkrlhdk tck dbe`f ehspf`jklkdts, kfklkdt
strkssks `de suppbrt rk`jthbds. (K < 977 x 078 D/l9)
9.3=) @d `xh`f fb`e W < :3; AD hs `ppfhke tb tck jblpbshtk ifbja `s scbwd ikfbw.
Ektkrlhdk tck strkss hd k`jc l`tkrh`f.
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 9=
9.33) Nbr ` vkrthj`f rbe `s scbwd ikfbw, nhde tck eknfkjthbd `t @ `de tck strkss
ehstrhiuthbd. K < 077 LW` `de wkhmct pkr udht vbfulk < 7.7> D/jl:. Jbllkdt bd
tck strkss ehstrhiuthbd.
9.38) Jbdshekr ` twb-i`r suppbrtke iy ` sprhdm scbwd hd nhmurk. Ibtc i`rs c`vk K < 907
MW` `de @<;.7 x07-5 l9. I`r bdk c`s ` fkdmtc bn ;l `de i`r twb c`s ` fkdmtc bn 07
l. Uck sprhdm sthnndkss hs a< 9 aD/l. Ektkrlhdk tck cbrhzbdt`f `de vkrthj`f
ehspf`jklkdts `t tck obhdt 0 `de strkssks hd k`jc i`r. Q@[, Dbv / Ekj ‚
9778\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 93
9.87) K`jc bn tck tcrkk i`rs bn tck phd ‚ obhdtke nr`lk scbwd hd nhmurk c`s ` jrbss
skjthbd`f `rk` bn 0777ll9 whtc K < 977MW`. Zbfvk nbr ehspf`jklkdts.
Q@[, Dbv / Ekj ‚
970:\
9.80) Nhde tck eknfkjthbd `t tck nrkk kde udekr hts bwd wkhmct, ushdm ehvhshbds bn
`) 0 kfklkdt i) 9 kfklkdts j) 5 kfklkdts e) 3 kfklkdts `de k) 0> kfklkdts
Uckd pfbt tck dulikr bn kfklkdts vkrsus eknfkjthbd.
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 98
9.89) Nbr tck ehsjrkthz`thbd bn ik`l kfklkdts `s scbwd ikfbw, dulikr tck dbeks sb `s tb
lhdhlhzk tck i`dewhetc bn tck `ssklifke sthnndkss l`trhx (A)
9.8:) Uck kfklkdts bn ` rbw br jbfuld bn tck sthnndkss l`trhx bn ` i`r kfklkdt sul up tb
zkrb, iut dbt sb nbr ` ik`l kfklkdt. Kxpf`hd wcy tchs hs sb.
9.85) Nbr tck ik`l prbifkl scbwd ikfbw, ektkrlhdk tck thp eknfkjthbd `de tck sfbpk `t
tck rbffkr suppbrt.
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) :7
9.8;) Nbr tck ik`l `de fb`ehdm `s scbwd hd nhmurk. Ektkrlhdk tck sfbpks `t tck twb kdes
bn tck ehstrhiutke fb`e `de tck vkrthj`f eknfkjthbd `t tck lhe-pbhdt bn tck ehstrhiutke
fb`e. U`ak K < 977MW` `de H < 5*07> ll5 Q@[, L`y / Oudk ‚ 9700\
9.8>) Nhde tck eknfkjthbd `de sfbpk nbr tck nbffbwhdm ik`l skjthbd `t wchjc pbhdt fb`e hs
`ppfhke.
9.8=) Zbfvk tck nbffbwhdm ik`l `s scbwd ikfbw, jf`lpke `t bdk kde `de sprhdm suppbrt
`t btckr kde. @ fhdk`rfy v`ryhdm tr`dsvkrsk fb`e bn l`xhlul l`mdhtuek bn 077
D/jl `ppfhke bvkr tck sp`d bn 5 jl tb 07 jl. U`ak KH < 9 x 07= D/jl9,9
07
KH
A .
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) :0
9.83) Bit`hd tck eknfkjthbd `t tck lhepbhdt bn tck ik`l scbwd ikfbw `de ektkrlhdk tck
rk`jthbd.
9.88) Uck shlpfy suppbrtke ik`l scbwd hd nhmurk hs suiokjtke tb ` udhnbrl tr`dsvkrsk
fb`e, `s scbwd. [shdm twb kqu`f-fkdmtc kfklkdts `de wbra-kquhv`fkdt dbe`f fb`es
bit`hd ` nhdhtk kfklkdt sbfuthbd nbr tck eknfkjthbd `t lhe-sp`d `de jblp`rk ht tb tck
sbfuthbd mhvkd iy kfklkdt`ry ik`l tckbry. Q@[, @prhf / L`y - 9707\
9.077) Ektkrlhdk tck ehspf`jklkdts `de sfbpks `t tck dbeks nbr tck ik`l scbwd hd nhmurk.
U`ak a<977aD / l, K<=7MW` `de H<9x07-5l5. Q@[, Dbv / Ekj ‚ 9709\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) :9
9.070) Ektkrlhdk tck dbe`f ehspf`jklkdts `de sfbpks nbr tck ik`l scbwd hd Nhmurk. Nhde
tck lblkdt `t tck lhepbhdt bn kfklkdt 0. Q@[, Dbv / Ekj ‚
9709\
9.079) Ektkrlhdk tck ehspf`jklkdt bn dbek 0 `de tck strkss hd kfklkdt :, nbr tck tcrkk-i`r
truss `s scbwd ikfbw. U`ak @ < 9;7 ll9, K < 977 MW` nbr `ff kfklkdts.
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) ::
9.07:) Ektkrlhdk tck nbrjk hd tck lklikrs bn tck truss `s scbwd hd nhmurk.
U`ak K < 977 MW` Q@[, L`y / Oudk ‚ 9709\
9.075)
Ektkrlhdk tck dbe`f ehspf`jklkdts `de tck kfklkdt strkssks nbr tck twb ehlkdshbd`ffb`eke pf`tk `s scbwd hd nhmurk. @ssulk pf`dk strkss jbdehthbd. Ibey nbrjk l`y ik
dkmfkjtke hd jblp`rhsbd tb tck kxtkrd`f nbrjks. U`ak K < 907MW`, » < 7.9;,
Uchjadkss t < 07ll. Q@[, L`y / Oudk ‚ 9700\
9.07;) Uck fb`ehdm `de btckr p`r`lktkrs nbr ` twb i`r truss kfklkdt hs scbwd hd nhmurk
Ektkrlhdk Q@[, L`y / Oudk ‚ 970:\
(h) Uck kfklkdt sthnndkss l`trhx nbr k`jc kfklkdt
(hh) Mfbi`f sthnndkss l`trhx
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) :5
(hhh) Dbe`f ehspf`jklkdts
(hv) Yk`jthbd nbrjks
(v) Uck strkssks hdeujke hd tck kfklkdts. @ssulk K < 977 MW`.
9.07>) J`fjuf`tk dbe`f ehspf`jklkdt `de kfklkdt`f strkssks nbr tck truss scbwd hd Nhmurk.
K< =7Mp`.jrbss-skjthbd`f ̀ rk` @ < 9jl9 nbr `ff truss lklikrs.
Q@[, @prhf / L`y ‚ 9700\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) :;
9.07=) Nhde tck cbrhzbdt`f `de vkrthj`f ehspf`jklkdts bn dbek 0 nbr tck truss scbwd ikfbw.
U`ak @ < :77 ll9, K < 9 x07; D/ll9 nbr k`jc kfklkdt.
9.073) K`jc bn tck nhvk i`rs bn tck phd obhdtke truss scbwd hd nhmurk ikfbw c`s ` jrbss
skjthbd`f `rk` 97 sq. jl. `de K < 977 MW`.
(h) Nbrl tck kqu`thbd N < A[ wckrk A hs tck `ssklifke sthnndkss l`trhx bn tck
strujturk.
(hh)
Nhde tck nbrjks hd `ff tck nhvk lklikrs. Q@[, @prhf / L`y ‚
9773\
9.078) @d`fyzk tck truss scbwd hd nhmurk `de kv`fu`tk tck strkss rksuft`dts hd lklikr (9).
@ssulk `rk` bn jrbss skjthbd bn `ff tck lklikrs hd s`lk. K < 9 * 07; D/ll9
Q@[, Dbv / Ekj ‚ 9707\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) :>
9.007) Ektkrlhdk tck obhdt ehspf`jklkdts, tck obhdt rk`jthbds, kfklkdt nbrjks `de kfklkdt
strkssks bn tck mhvkd truss kfklkdts. Q@[, @prhf / L`y - 9707\
Kfklkdts @
jl9
K
D /l9
F
l
Mfbi`f
Dbek
jbddkjthbd
Ekmrkk
0 :9.9 >.8k 07 9.;5 9 tb : 87
9 :3.= 97.=k07 9.;5 9 tb 0 7
: 9;.3 97.=k07 :.;8 0 tb : 0:;
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) :=
9.000) Ektkrlhdk tck nbrjk hd tck lklikrs bn tck truss scbwd hd nhmurk.
Q@[, @prhf / L`y ‚ 9700\
9.009) Nhde tck dbe`f ehspf`jklkdt ekvkfbpke hd tck pf`dkr truss scbwd hd Nhmurk wckd `
vkrthj`ffy ebwdw`re fb`e bn 0777 D hs `ppfhke `t dbek 5. Uck rkquhrke e`t` `rk
mhvkd hd tck U`ifk. Q@[, L`y / Oudk ‚ 9709\
Kfklkdt Db.
–k‘
Jrbss ‚ Zkjthbd`f `rk` @
jl9 Fkdmtc f (k) jl
Xbudm‘s Lbeufus
K(k) D/ll9
0 9 ∟9 ;7 9 * 07>
9 9 9 ;7 9 * 07>
: 0 ∟9.; 077 9 * 07>
5 0 ∟9 077 9 * 07>
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) :3
9.00:) Ekrhvk tck sc`pk nudjthbd nbr ` 9 dbeke ik`l kfklkdt `de ` : dbeke i`r kfklkdt.
Q@[, Dbv / Ekj ‚ 9773\
9.005) ]cy hs chmckr brekr kfklkdts dkkeke2 Ektkrlhdk tck sc`pk nudjthbds bn `d khmct
dbeke rkjt`dmuf`r kfklkdt. Q@[, Dbv / Ekj ‚ 977=, @prhf / L`y ‚ 9778\
9.00;) Ekrhvk tck sc`pk nudjthbds nbr ` 9E ik`l kfklkdt.
Q@[, Dbv / Ekj ‚ 977=, @prhf / L`y ‚ 9773, 9778\
9.00>) Ekrhvk tck sc`pk nudjthbds nbr ` 9E truss kfklkdt.
Q@[, Dbv / Ekj ‚ 977=, @prhf / L`y ‚ 9773, 9778\
9.00=) Ekrhvk tck hdtkrpbf`thbd nudjthbd nbr tck bdk ehlkdshbd`f fhdk`r kfklkdt whtc `
fkdmtc ‖ F“ `de twb dbeks, bdk `t k`jc kde, ekshmd`tke `s ‖h“ `de “ o“. @ssulk tck
brhmhd bn tck jbbrehd`tk systkl hs tb tck fknt bn dbek ‖h“.
Q@[, @prhf / L`y - 9707\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) :8
Nhmurk scbws tck bdk-ehlkdshbd`f fhdk`r kfklkdt
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 57
[DHU ‚ HHH ‚ U]B EHLKDZHBD@F NHDHUK KFKLKDU @D@FXZHZ
W@YU ‚ @
:.0) D`lk nkw 9-E kfklkdts `fbdm whtc ` dk`t saktjc.
:.9) Zt`tk tck ehnnkrkdjks iktwkkd 9E kfklkdt `de 0E kfklkdt.
:.:) Eknhdk F`mr`dmk‘s hdtkrpbf`thbd.
:.5) ]c`t hs mkblktrhj Hsbtrbpy2 Q@[, L`y / Oudk ‚ 970:\
:.;) ]rhtk tck F`mr`dmk`d sc`pk nudjthbds nbr ` 0E, 9 dbeke kfklkdts.
Q@[, Dbv / Ekj ‚ 9773\
:.>) ]rhtk tck rkf`thbd tb bit`hd tck shzk bn tck sthnndkss l`trhx nbr ` fhdk`r qu`erhf`tkr`f
kfklkdt c`vhdm [x `de [y `s ebn .
:.=) ]cy hs tck : dbeke trh`dmuf`r kfklkdt j`ffke `s ` JZU kfklkdt2
Q@[, Dbv / Ekj ‚ 9707\
:.3) ]rhtk ebwd tck hdtkrpbf`thbd nudjthbd bn ` nhkfe v`rh`ifk nbr tcrkk-dbek trh`dmuf`r
kfklkdt. Q@[, @prhf / L`y ‚ 9707\
:.8) ]c`t hs ` JZU kfklkdt2 Q@[, @prhf / L`y ‚ 9700\
:.07) Er`w tck sc`pk nudjthbds bn ` JZU kfklkdt. Q@[, Dbv / Ekj ‚ 9707\
:.00) Kxpf`hd tck hlpbrt`dt prbpkrthks bn JZU kfklkdts. Q@[, Dbv / Ekj ‚ 9773\
:.09) ]rhtk ` dbtk bd JZU kfklkdt. Q@[, L`y / Oudk ‚ 9700\
:.0:) ]rhtk irhknfy `ibut FZU `de PZU kfklkdts.
:.05) ]c`t `rk JZU `de FZU kfklkdts2 Q@[, Dbv / Ekj ‚ 9778\
:.0;) Eknhdk FZU kfklkdt. Q@[, Dbv / Ekj ‚ 9709\
:.0>) ]rhtk tck ehspf`jklkdt nudjthbd kqu`thbd nbr JZU kfklkdt.
:.0=) ]rhtk tck str`hd ‚ ehspf`jklkdt l`trhx nbr JZU kfklkdt.
:.03) Ehnnkrkdth`tk JZU ̀ de FZU kfklkdts. Q@[, Dbv / Ekj ‚ 977=, @prhf / L`y ‚ 9778\
:.08) Mhvk tck O`jbih`d l`trhx nbr ` JZU kfklkdt `de st`tk hts shmdhnhj`djk.
Q@[, Dbv / Ekj ‚ 970:\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 50
:.97) Kv`fu`tk tck nbffbwhdm `rk` hdtkmr`fs nbr tck tcrkk dbek trh`dmuf`r kfklkdt∡ Q@[, L`y / Oudk ‚ 9709\
:.90) @ trh`dmuf`r kfklkdt hs scbwd hd Nhmurk `de tck dbe`f jbbrehd`tks `rk kxprksske hd
ll. Jblputk tck str`hd ehspf`jklkdt l`trhx. Q@[, Dbv / Ekj ‚ 9709\
:.99) ]c`t eb ybu lk`d iy tck tkrls ? j7,j0 `de jd jbdthduhty2
Q@[, @prhf / L`y ‚ 9707\
:.9:) Ehsthdmuhsc iktwkkd J7, J0 `de J9 jbdthduhty kfklkdts.
:.95) ]c`t `rk tck ehnnkrkdt prbifkls mbvkrdke iy 9E sj`f`r nhkfe v`rh`ifks2
:.9;) [sk v`rhbus dulikr bn trh`dmuf`r kfklkdts tb lksc tck mhvkd ebl`hd hd tck brekr bn
hdjrk`shdm sbfuthbd rknhdklkdt.
:.9>) Eknhdk W`sj`f trh`dmfk.
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 59
:.9=) ]rhtk tck shmdhnhj`djk bn W`sj`f trh`dmfk hd ekvkfbphdm trh`dmuf`r kfklkdts.
:.93) Ehsthdmuhsc bdk nrbl tck btckr bn tck nbffbwhdm
`) Fhdk`r ̀ de qu`er`thj trh`dmuf`r kfklkdts. i) Fhdk`r `de qu`er`thj F`mr`dmk
kfklkdts.
:.98) ]c`t eb ybu lk`d iy `rk` jb-brehd`tk lktcbe2
:.:7) Zt`tk tck `ev`dt`mk bn skrkdehphty kfklkdt.
:.:0) ]c`t eb ybu lk`d iy wr`pphdm2
:.:9) ]rhtk tck dbek dulikrhdm `de kfklkdt jbddkjthvhty t`ifk nbr tck mhvkd ebl`hd
ushdm suht`ifk ehsjrkthz`thbd.
:.::) Wfbt tck v`rh`thbd bn sc`pk nudjthbd whtc rkspkjt dbek bn ` : dbeke trh`dmuf`r
kfklkdt.
:.:5) ]rhtk ebwd tck dbe`f ehspf`jklkdt kqu`thbds nbr ` twb ehlkdshbd`f trh`dmuf`r
kf`sthjhty kfklkdt. Q@[, @prhf / L`y ‚ 9707\
:.:;) Eknhdk ` pf`dk strkss jbdehthbd. Q@[, Dbv / Ekj ‚
9700\
:.:>) Zt`tk tck jbdehthbd nbr pf`dk strkss prbifkl.
:.:=) Mhvk bdk kx`lpfk k`jc nbr pf`dk strkss `de pf`dk str`hd prbifkls.
Q@[, Dbv / Ekj ‚ 9773\
:.:3) Ehsthdmuhsc iktwkkd pf`dk strkss ̀ de pf`dk str`hd prbifkls. Q@[, Dbv / Ekj ‚ 9778\
:.:8) Ehsthdmuhsc pf`dk strkss ̀ de pf`dk str`hd jbdehthbds. Q@[, Dbv / Ekj ‚ 9707\
:.57) Eknhdk pf`dk str`hd whtc suht`ifk kx`lpfk. Q@[, Dbv / Ekj ‚ 9709\
:.50) Eknhdk pf`dk str`hd `d`fyshs. Q@[, Dbv / Ekj ‚ 9700\
:.59) Eknhdk ` pf`dk strkss prbifkl whtc ` suht`ifk kx`lpfk. Q@[, L`y / Oudk ‚ 970:\
:.5:) Kxpf`hd pf`dk strkss prbifkl whtc `d kx`lpfk. Q@[, @prhf / L`y ‚ 9700\
:.55) Kxpf`hd pf`dk strkss jbdehthbds whtc kx`lpfk. Q@[, L`y / Oudk ‚ 9700\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 5:
:.5;) ]rhtk ebwd tck str`hd ehspf`jklkdt rkf`thbd. Q@[, @prhf / L`y ‚ 9700\
:.5>) Zt`tk wcktckr pf`dk strkss br pf`dk str`hd kfklkdts j`d ik uske tb lbekf tck
nbffbwhdm strujturks. Ousthny ybur ̀ dswkr. Q@[, Dbv / Ekj ‚ 9709\
(`) @ w`ff suiokjtke tb whde fb`e
(i) @ wrkdjc suiokjtke tb ` nbrjk hd tck pf`dk bn tck wrkdjc.
:.5=) ]rhtk tck `ssulpthbds uske tb eknhdk tck mhvkd prbifkl `s pf`dk strkss prbifkl.
:.53) ]rhtk tck `ssulpthbds uske tb eknhdk tck mhvkd prbifkl `s pf`dk str`hd prbifkl.
:.58) [shdm mkdkr`f strkss - str`hd rkf`thbd, bit`hd pf`dk strkss kqu`thbd.
:.;7) Ikmhddhdm whtc mkdkr`f kf`sthj strkss-str`hd rkf`thbd, ekrhvk tck pf`dk str`hd
jbdehthbd.
:.;0) ]c`t `rk tck ehnnkrkdjks iktwkkd 9 Ehlkdshbd`f sj`f`r v`rh`ifk `de vkjtbr v`rh`ifk
kfklkdts2 Q@[, Dbv / Ekj ‚ 9778\
:.;9) ]c`t `rk tck w`ys iy wchjc ` tcrkk ehlkdshbd`f prbifkl j`d ik rkeujke tb ` twb
ehlkdshbd`f prbifkl2
:.;:) Cbw tb rkeujk ` :E prbifkl hdtb ` 9E prbifkl2 Q@[, Dbv / Ekj ‚ 9709\
:.;5)
Mhvk tck sthnndkss l`trhx kqu`thbd nbr ̀ d ̀ xhsyllktrhj trh`dmuf`r kfklkdt.
:.;;) ]c`t hs `xhsyllktrhj kfklkdt2
:.;>) Mhvk kx`lpfks bn `xhsyllktrhj prbifkls. Q@[, L`y / Oudk ‚ 9709\
:.;=) ]c`t hs `d ̀ xhsyllktrhj prbifkl2 Q@[, @prhf / L`y ‚ 9700\
:.;3) ]rhtk scbrt dbtks bd `xhsyllktrhj prbifkls.
Q@[, Dbv / Ekj ‚ 977=, @prhf / L`y ‚ 9778\
:.;8) ]c`t hs lk`dt iy `xh-sylktrhj nhkfe prbifkl2 Mhvkd `d kx`lpfk.
Q@[, Dbv / Ekj ‚ 9778\
:.>7) ]ckd ̀ rk `xhsyllktrhj kfklkdts prknkrrke2 Q@[, Dbv / Ekj ‚ 970:\
:.>0) Zt`tk tck shtu`thbds wckrk tck ̀ xhsyllktrhj nbrluf`thbd j`d ik ̀ ppfhke.
Q@[, @prhf / L`y ‚ 9700\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 55
:.>9) Mhvk nbur `ppfhj`thbds wckrk `xhsyllktrhj kfklkdts j`d ik uske.
Q@[, @prhf / L`y ‚ 9700\
:.>:) Zt`tk tck `ppfhj`thbds bn ̀ xhsyllktrhj kfklkdts. Q@[, Dbv / Ekj ‚ 9707\
:.>5) ]rhtk ebwd tck jbdsthtuthvk rkf`thbdschp nbr ̀ xhsyllktrhj prbifkl.
Q@[, @prhf / L`y ‚
9778\
:.>;) ]rhtk ebwd tck jbdsthtuthvk rkf`thbdschp nbr tck pf`dk strkss prbifkl.
Q@[, Dbv / Ekj ‚ 9707\
:.>>) ]c`t eb ybu lk`d iy jbdsthtuthvk f`w `de mhvk tck jbdsthtuthvk f`w nbr `xh-
syllktrhj prbifkls2 Q@[, @prhf / L`y, Dbv / Ekj ‚ 9773\
:.>=) Zpkjhny tck ibey nbrjk tkrl `de tck ibey nbrjk vkjtbr nbr `xhsyllktrhj trh`dmuf`r
kfklkdt. Q@[, Dbv / Ekj ‚
970:\
:.>3) Mhvk bdk kx`lpfk k`jc nbr pf`dk strkss `de pf`dk str`hd prbifkls.
Q@[, @prhf / L`y - 9773\
:.>8) Kxpf`hd pf`dk str`hd prbifkl whtc `d kx`lpfk. Q@[, L`y / Oudk ‚ 9709\
:.=7) Mhvk ` irhkn dbtk bd st`thj jbdekds`thbd.
:.=0) Wrbvk tc`t 9 7 nbr pf`dk str`hd jbdehthbd.
:.=9) Ehnnkrkdth`tk ̀ xh ‚ syllktrhj `de jyjfhj ‚ syllktrhj strujturks.
:.=:) Ehnnkrkdth`tk ̀ xh-syllktrhj fb`e ̀ de `syllktrhj fb`e whtc kx`lpfks.
:.=5) Zt`tk tck jbdehthbd nbr ̀ xh-syllktrhj prbifkl.
:.=;) Fhst tck rkquhrke jbdehthbds nbr ` prbifkl ̀ ssulke tb ik `xhsyllktrhj.
Q@[, @prhf / L`y ‚ 9707\
:.=>) ]c`t `rk tck nbur i`shj skts bn kf`sthjhty kqu`thbds2 Q@[, L`y / Oudk ‚
9709\
:.==) Mhvk kx`lpfks nbr tck nbffbwhdm j`sks.
`) pf`dk strkss prbifkl i) pf`dk str`hd prbifkl j) `xh-syllktrhj prbifkl
:.=3) Eknhdk tck nbffbwhdm tkrls whtc suht`ifk kx`lpfks Q@[, @prhf / L`y ‚ 9707\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 5;
h) Wf`dk strkss, pf`dk str`hd hh) Dbek, kfklkdt `de sc`pk nudjthbds
hhh) @xhsyllktrhj `d`fyshs hv) Hsb ‚ p`r`lktrhj kfklkdt
:.=8) Eknhdk tck tkrl hdhth`f str`hd.
:.37) Zt`tk tck knnkjt bn Wbhssbd‘s r`thb hd pf`dk str`hd prbifkl.
:.30) Cbw whff tck strkss nhkfe v`ry fhdk`rfy2
:.39) Jblp`rk tck jc`dmks hd tck E l`trhx kvbfvke but bn pf`dk str`hd, pf`dk strkss `de
`xh-syllktrhj prbifkl.
:.3:) ]c`t eb ybu lk`d iy Hsbp`r`lktrhj nbrluf`thbds2
Q@[, Dbv / Ekj ‚ 977=, @prhf / L`y ‚ 9778\
:.35) Kxprkss tck sc`pk nudjthbds bn nbur dbek qu`erhf`tkr`f kfklkdt.
Q@[, L`y / Oudk ‚ 9709\
:.3;) ]c`t eb ybu udekrst`de iy ` d`tur`f jb ‚ brehd`tk systkl2
Q@[, @prhf / L`y ‚ 9700\
:.3>) ]c`t eb ybu lk`d iy d`tur`f jb-brehd`tk systkl2 Q@[, L`y / Oudk ‚ 9700\
:.3=) ]c`t `rk tck `ev`dt`mks bn d`tur`f jb-brehd`tks2
Q@[, Dbv / Ekj ‚ 977=, @prhf / L`y ‚ 9778\
:.33) ]c`t `rk tck `ev`dt`mks bn d`tur`f jbbrehd`tks bvkr mfbi`f jb-brehd`tks2
Q@[, Dbv / Ekj ‚ 9773\
:.38) Mhvk ` irhkn dbtk bd d`tur`f jb-brehd`tk systkl.
:.87) ]rhtk tck d`tur`f jb-brehd`tks nbr tck pbhdt ‖W“ bn tck trh`dmuf`r kfklkdt. Uck pbhdt
–W‘ hs tck J.M. bn tck trh`dmfk. Q@[, Dbv / Ekj ‚ 9773\
:.80) Zcbw tck tr`dsnbrl`thbd nbr l`pphdm x-jbbrehd`tk systkl bdtb ` d`tur`f jbbrehd`tk
systkl nbr ` fhdk`r sp`r kfklkdt `de nbr ` qu`er`thj sp`r kfklkdt.
Q@[, Dbv / Ekj ‚ 9709\
:.89) Eknhdk ` fbj`f jb ‚ brehd`tk systkl. Q@[, Dbv / Ekj ‚ 9700\
:.8:) ]c`t hs `rk` jb ‚ brehd`tks2 Q@[, Dbv / Ekj ‚ 9700\
:.85) ]c`t eb ybu udekrst`de iy `rk` jb ‚ brehd`tks2 Q@[, @prhf / L`y ‚ 9700\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 5>
:.8;) Zt`tk tck i`shj f`ws bd wchjc Hsbp`r`lktrhj jbdjkpt hs ekvkfbpke.
Q@[, @prhf / L`y ‚ 9773\
:.8>) Ehnnkrkdth`tk? fbj`f `xhs `de mfbi`f `xhs. Q@[, @prhf / L`y ‚ 9773\
:.8=) Eknhdk supkr p`r`lktrhj kfklkdt. Q@[, @prhf / L`y ‚ 9778\
:.83) Kxpf`hd supkr p`r`lktrhj kfklkdt. Q@[, Dbv / Ekj ‚
9707\
:.88) Eknhdk Hsbp`r`lktrhj kfklkdts2 Q@[, Dbv / Ekj ‚ 9773\
:.077) Eknhdk Hsbp`r`lktrhj kfklkdts whtc suht`ifk kx`lpfks Q@[, @prhf / L`y ‚ 9707\
:.070) Eknhdk Hsbp`r`lktrhj kfklkdt nbrluf`thbds. Q@[, Dbv / Ekj ‚ 9709\
:.079) ]c`t eb ybu lk`d iy Hsbp`r`lktrhj nbrluf`thbd2 Q@[, @prhf / L`y ‚ 9700\
:.07:) ]c`t hs tck purpbsk bn Hsbp`r`lktrhj kfklkdts2
:.075) ]c`t hs tck s`fhkdt nk`turk bn `d Hsbp`r`lktrhj kfklkdt2 Mhvk `d kx`lpfk.
Q@[, Dbv / Ekj ‚ 970:\
:.07;) ]c`t ̀ rk tck `ppfhj`thbds bn Hsbp`r`lktrhj kfklkdts2 Q@[, @prhf / L`y ‚ 9700\
:.07>) Ehnnkrkdth`tk x ‚ y sp`jk `de - sp`jk.
:.07=) ]rhtk tck `ev`dt`mks bn jb-brehd`tk tr`dsnbrl`thbd nrbl J`rtksh`d jb-brehd`tks tb
d`tur`f jb-brehd`tks.
:.073) Eknhdk O`jbih`d. Q@[, Dbv / Ekj ‚ 970:\
:.078) ]c`t hs ` O`jbih`d2 Q@[, Dbv / Ekj ‚ 9707\
:.007) ]c`t hs tck dkke bn O`jbih`d2 Q@[, @prhf / L`y ‚ 9700\
:.000) ]rhtk ebwd tck O`jbih`d l`trhx. Q@[, Dbv / Ekj ‚ 9707\
:.009) ]rhtk ̀ ibut O`jbih`d tr`dsnbrl`thbd uske hd jb-brehd`tk tr`dsnbrl`thbd.
:.00:) ]c`t hs tck shmdhnhj`djk bn O`jbih`d bn tr`dsnbrl t̀hbd2 Q@[, L`y / Oudk ‚ 9709\
:.005) Ehnnkrkdth`tk iktwkkd sui-p`r`lktrhj, hsb- p`r`lktrhj `de supkr ‚ p`r`lktrhj
kfklkdts.
:.00;) Mhvk twb kx`lpfks nbr sui p`r`lktrhj kfklkdts. Q@[, Dbv / Ekj ‚ 970:\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 5=
:.00>) Ykprkskdt tck v`rh`thbd bn sc`pk nudjthbd whtc rkspkjt tb dbeks nbr qu`er`thj
kfklkdts hd tkrls bn d`tur`f jb-brehd`tks.
:.00=) Jblp`rk fhdk`r lbekf, qu`er`thj lbekf `de juihj lbekf hd tkrls bn d`tur`f jb-
brehd`tk systkl.
:.003) ]rhtk ` irhkn dbtk bd jbdthduhty `de jblp`thihfhty.
:.008) ]rhtk ebwd tck kfklkdt nbrjk vkjtbr kqu`thbd nbr ` nbur dbeke qu`erhf`tkr`f
kfklkdt.
:.097) ]rhtk ebwd tck O`jbih`d l`trhx nbr ` nbur dbeke qu`erhf`tkr`f kfklkdt
:.090) ]rhtk tck sc`pk nudjthbd nbr tck qu`erhf`tkr`f kfklkdt hd , sp`jk.
:.099) ]cy hs nbur dbeke qu`erhf`tkr`f kfklkdt hs prknkrrke nbr `xh-syllktrhj prbifkl
tc`d tcrkk dbeke trh`dmuf`r kfklkdt2
:.09:) Zaktjc ` nbur dbek qu`erhf`tkr`f kfklkdt `fbdm whtc dbe`f ekmrkks bn nrkkebl.
Q@[, @prhf / L`y ‚ 9700\
:.095) ]rhtk ebwd tck sthnndkss l`trhx nbr nbur dbeke qu`erhf`tkr`f kfklkdts.
Q@[, L`y / Oudk ‚ 9700\
:.09;) Ehsthdmuhsc iktwkkd ksskdth`f ibude`ry jbdehthbds `de d`tur`f ibude`ry
jbdehthbds. Q@[, Dbv / Ekj ‚ 9778\
:.09>) ]rhtk tck `ev`dt`mks bn chmckr brekr kfklkdts hd d`tur`f jb ‚ brehd`tk systkl.
:.09=) ]c`t `rk tck typks bn dbd-fhdk`rhty2
Q@[, Dbv / Ekj ‚ 977=, @prhf / L`y ‚ 9778, L`y / Oudk ‚ 9709\
:.093) Zt`tk tck `ev`dt`mk bn M`ussh`d hdtkmr`thbd.
:.098) Zt`tk tck nbur-pbhdt M`ussh`d qu`er`turk rufk.
:.0:7) Irhknfy kxpf`hd M`ussh`d qu`er`turk. Q@[, @prhf / L`y ‚ 9700\
:.0:0) ]c`t `rk tck `ev`dt`mks bn M`ussh`d qu`er`turk2 Q@[, Dbv / Ekj ‚ 9709\
:.0:9) ]c`t `rk tck wkhmcts `de s`lpfhdm pbhdts bn twb pbhdt nbrluf` bn M`uss
qu`er`turk nbrluf`2 Q@[, L`y / Oudk ‚ 9709\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 53
:.0::) ]cy dulkrhj`f hdtkmr`thbd hs rkquhrke nbr kv`fu`thbd bn sthnndkss l`trhx bn `d
Hsbp`r`lktrhj kfklkdt2 Q@[, Dbv / Ekj ‚ 9700\
:.0:5) ]ckd eb wk rksbrt tb dulkrhj`f hdtkmr`thbd hd 9E kfklkdts2
Q@[, Dbv / Ekj ‚ 970:\
:.0:;) ]rhtk tck M`uss pbhdts `de wkhmcts nbr twb pbhdt nbrluf` bn dulkrhj`f hdtkmr`thbd.
Q@[, @prhf / L`y ‚ 9700\
:.0:>) ]rhtk ebwd tck M`uss hdtkmr`thbd nbrluf` nbr trh`dmuf`r ebl`hds.
Q@[, @prhf / L`y ‚ 9778\
:.0:=) Kv`fu`tk tck hdtkmr`f ∡ ushdm M`ussh`d qu`er`turk lktcbe.
Q@[, Dbv / Ekj ‚ 9709\
:.0:3) D`lk tck jbllbdfy uske hdtkmr̀ thbd lktcbe hd d`tur`f ‚ jb-brehd`tk systkl.
:.0:8) ]rhtk tck rkf`thbd iktwkkd wkhmcts ̀ de M`uss pbhdts hd M`uss-Fkmkderk qu`er`turk.
W@YU ‚ I
:.057) Ektkrlhdk tck sc`pk nudjthbds nbr ` jbdst`dt str`hd trh`dmuf`r (JZU) kfklkdt hd
tkrls bn d`tur`f jbbrehd`tk systkl. Q@[, Dbv / Ekj ‚ 9773\
:.050) ]c`t `rk sc`pk nudjthbds2 Ekrhvk tck sc`pk nudjthbd nbr tck tcrkk dbeke trh`dmuf`r
kfklkdts. Q@[, Dbv / Ekj ‚ 9700\
:.059) Ekrhvk tck kfklkdt str`hd ehspf`jklkdt l`trhx `de kfklkdt sthnndkss l`trhx bn `
JZU kfklkdt. Q@[, @prhf / L`y ‚ 9700\
:.05:) Kxpf`hd tck tkrls pf`dk strkss `de pf`dk str`hd prbifkls. Mhvk tck jbdsthtuthvk
f`ws nbr tcksk j`sks. Q@[, Dbv / Ekj ‚ 977=, @prhf / L`y ‚ 9778\
:.055) Ekrhvk tck kqu`thbds bn kquhfhirhul hd tck j`sk bn ` tcrkk ehlkdshbd`f systkl.
Q@[, Dbv / Ekj ‚ 977=, 9773, @prhf / L`y ‚ 9778\
:.05;) Ekrhvk tck kxprksshbd nbr jbdsthtuthvk strkss-str`hd rkf`thbdschp `de `fsb rkeujk ht tb
tck prbifkl bn pf`dk strkss `de pf`dk str`hd. Q@[, Dbv / Ekj - 9773\
:.05>) Ekrhvk tck jbdst`dt-str`hd trh`dmuf`r kfklkdt‘s sthnndkss l`trhx `de kqu`thbds.
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 58
Q@[, @prhf / L`y - 9773\
:.05=) Ekrhvk tck fhdk`r ‚ str`hd trh`dmuf`r kfklkdt‘s sthnndkss l`trhx `de kqu`thbds.
Q@[, @prhf / L`y ‚ 9773\
:.053) Ekrhvk tck sthnndkss l`trhx `de kqu`thbds nbr ` FZU kfklkdt.
Q@[, Dbv / Ekj ‚
9709\
:.058) Ekrhvk tck kfklkdt str`hd ehspf`jklkdt l`trhx `de kfklkdt sthnndkss l`trhx bn `
trh`dmuf`r kfklkdt. Q@[, L`y / Oudk ‚ 9709\
:.0;7) @ twb dbeke fhdk kfklkdt whtc bdk tr`dsf`thbd`f ekmrkk bn nrkkebl hs suiokjtke tb `
udhnbrlfy v`ryhdm fb`e bn hdtkdshty W0 `t dbek 0 `de W9 `t dbek 9. Kv`fu`tk tck
dbe`f fb`e vkjtbr ushdm dulkrhj`f hdtkmr`thbd. Q@[, Dbv / Ekj ‚ 9709\
:.0;0) J`fjuf`tk tck kfklkdt strkssks 90 `de,,, xy y x `de tck prhdjhpfk `dmfk p nbr
tck kfklkdt scbwd ikfbw.
:.0;9) Uck dbe`f jb-brehd`tks bn tck trh`dmuf`r kfklkdt hs s̀ scbwd ikfbw. @t tck hdtkrhbr
pbhdt W, tck x- jb-brehd`tk hs :.: `de D0 < 7.:. Ektkrlhdk D9, D: `de tck y ‚ jb-
brehd`tk `t pbhdt W.
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) ;7
:.0;:) Uck ( x,y) jb-brehd`tks bn dbeks h, o `de a bn ` trh`dmuf`r kfklkdt `rk mhvkd iy (7,7),
(:,7) `de (0.;,5) ll rkspkjthvkfy. Kv`fu`tk tck sc`pk nudjthbds D 0, D 9 `de D : `t `d
hdtkrhbr pbhdt W (9, 9.;) ll nbr tck kfklkdt. Nbr tck s`lk trh`dmuf`r kfklkdt, bit`hd
tck str`hd-ehspf`jklkdt rkf`thbd l`trhx I. Q@[, Dbv / Ekj ‚ 9778\
:.0;5)
Nbr tck trh`dmuf`r kfklkdt scbwd ikfbw, bit`hd tck str`hd ‚ ehspf`jklkdt rkf`thbdl`trhx I `de ektkrlhdk tck str`hds x ,y `de xy.
:.0;;) Ekrhvk tck kxprksshbd nbr dbe`f vkjtbr hd ` JZU kfklkdt suiokjtke tb prkssurks W x0 ,
W y0 bd shek 0, W x9, W y9, bd shek 9 `de W x:, W y: bd shek : `s scbwd hd nhmurk.
Q@[, Dbv / Ekj ‚
970:\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) ;0
:.0;>) Jbdshekr tck trh`dmuf`r kfklkdt scbw hd Nhmurk. Uck kfklkdt hs kxtr`jtke nrbl `
tchd pf`tk bn tchjadkss 7.; jl. Uck l`tkrh`f hs cbt rbffke fbw j`ribd stkkf. Uck
Dbe`f jb-brehd`tks `rk xh <7, yh < 7, x o <7, y o < -0 , xa <7, ya < -0 jl,. Ektkrlhdk tck
kfklkdt`f sthnndkss l`trhx. @ssulhdm pf`dk strkss `d`fyshs. U`ak » < 7.: `de K <
9.0*07= D/jl9 Q@[, L`y / Oudk ‚ 9709\
:.0;=) Ekrhvk tck hdtkrpbf`thbd nudjthbd 05 nbr tck qu`er`thj trh`dmuf`r kfklkdt `s scbwd
ikfbw.
:.0;3) Ekrhvk tck hdtkrpbf`thbd nudjthbd bn ` jbrdkr dbek hd ` juihj skrkdehphty kfklkdt.
:.0;8) Nhde tck kxprksshbd nbr dbe`f vkjtbr hd ` JZU kfklkdt scbwd hd nhmurk suiokjtke tb
prkssurks W x0 bd shek 0. Q@[, Dbv / Ekj ‚ 9773\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) ;9
:.0>7) Nbr tck JZU kfklkdt mhvkd ikfbw, `ssklifk st`hd ehspf`jklkdt l`trhx. U`ak t < 97
ll `de K < 9*07; D/ll9 Q@[, Dbv / Ekj - 9773\
:.0>0)
J`fjuf`tk tck v`fuk bn prkssurk `t tck pbhdt @ wchjc hs hdshek tck : dbeke trh`dmuf`rkfklkdts `s scbwd hd nhmurk. Uck dbe`f v`fuks `rk ώ0 < 57 LW`, ώ9 < :5 LW` `de ώ:
< 5> LW`, Wbhdt @ hs fbj`tke `t (9, 0.;) @ssulk prkssurk hs fhdk`rfy v`ryhdm hd tck
kfklkdt. @fsb ektkrlhdk tck fbj`thbd bn 59 LW` jbdtbur fhdk.
Q@[, L`y / Oudk ‚ 970:\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) ;:
:.0>9) Bit`hd tck mfbi`f sthnndkss l`trhx nbr tck pf`tk scbwd hd nhmurk. U`ahdm twb
trh`dmuf`r kfklkdts. @ssulk pf`dk strkss jbdehthbd. Q@[, L`y / Oudk ‚ 9709\
:.0>:) Nbr tck jbdst`dt str`hd trh`dmuf`r kfklkdt scbwd hd nhmurk ikfbw, `ssklifk tck
str`hd ‚ ehspf`jklkdt l`trhx. U`ak t < 97 ll `de K < 9 x 07; D/ll9.
Q@[, Dbv / Ekj ‚ 977=, 970:, @prhf / L`y ‚ 9778\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) ;5
:.0>5) @ JZU kfklkdt c`s dbe f̀ jbbrehd`tks 0 (7,7), 9 (;,7), : (7,5). Uck kfklkdt hs
suiokjtke tb ` ibey nbrjk n < x: D/l:. Ektkrlhdk tck dbe`f nbrjk vkjtbr. U`ak tck
kfklkdt tchjadkss ̀ s 7.:l. Q@[, Dbv / Ekj ‚ 970:\
:.0>;) Nbr tck pf`dk str`hd kfklkdt scbwd hd tck nhmurk, tck dbe`f ehspf`jklkdts `rk mhvkd
`s ? u0< 7.77; ll, u9 < 7.779 ll, u:<7.7ll, u5 < 7.7 ll, u; < 7.775 ll, u>
< 7.7 ll. Ektkrlhdk tck kfklkdt strkssks. U`ak K < 977 Mp` `de < 7.:. [sk udht
tchjadkss nbr pf`dk str`hd. Q@[, @prhf / L`y - 9707\
:.0>>) Nbr tck twb-ehlkdshbd`f fb`eke pf`tk `s scbwd hd Nhmurk. Ektkrlhdk tck dbe`f
ehspf`jklkdts `de kfklkdt strkss ushdm pf`dk str`hd jbdehthbd jbdshekrhdm ibey
nbrjk. U`ak Xbudm‘s lbeufus `s 977 MW`, Wbhssbd‘s r`thbd `s 7.: `de ekdshty `s
=377 am/l:. Q@[, @prhf / L`y ‚ 9700\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) ;;
:.0>=) Ekrhvk kfklkdt nbrjk vkjtbr wckd fhdk`rfy v`ryhdm prkssurk `jts bd tck shek obhdhdm
dbeks oa bn ` trh`dmuf`r kfklkdt scbwd hd Nhmurk `de ibey nbrjk bn 9;D/ll9
`jts
ebwdw`res. Uchjadkss < ;ll. Q@[, @prhf / L`y ‚ 9700\
:.0>3)
Nbr tck pf`dk strkss kfklkdt wcbsk jbbrehd`tks `rk mhvkd iy (077,077), (577, 077)`de (977, 5777, tck dbe`f ehspf`jklkdts `rk u0 < 9.7ll, v0 <f.7ll, u9 <f.7ll, v9
<0.;ll, u: < 9.;ll, v: < 7.;ll. Ektkrlhdk tck kfklkdt strkssks. @ssulk K <
977 MD/l9, » < 7.: `de t < 07 ll. @ff jbbrehd`tks `rk hd ll.
Q@[, L`y / Oudk ‚ 970:\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) ;>
:.0>8) @ tchd pf`tk ` suiokjtke tb surn`jk nr`jthbd `s scbwd hd Nhmurk. J`fjuf`tk tck mfbi`f
sthnndkss l`trhx. U`ifk t < 9; ll, K < 9 *07; D/ll9 `de κ < 7.:7. @ssulk pf`dk
strkss jbdehthbd. Q@[, Dbv / Ekj ‚ 9700\
:.0=7) Ektkrlhdk tck eknfkjthbd bn ` tchd pf`tk suiokjtke tb kxtkdshbd`f fb`e `s scbwd.
:.0=0) J`fjuf`tk dbe`f ehspf`jklkdt `de kfklkdt`f strkssks nbr tck truss scbwd hd Nhmurk.
K < =7 MW` jrbss-skjthbd`f `rk` @ < 9 jl9 nbr `ff truss lklikrs.
Q@[, Dbv / Ekj ‚ 9709\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) ;=
:.0=9) @ tchd kf`sthj pf`tk suiokjtke tb udhnbrlfy ehstrhiutke kemk fb`e `s scbwd ikfbw.
Nhde tck sthnndkss `de nbrjk l`trhx bn tck kfklkdt.
:.0=:) Nbr tck jbdnhmur`thbd `s scbwd hd nhmurk ektkrlhdk tck eknfkjthbd `t tck pbhdt fb`e
`ppfhj`thbds. [sk bdk lbekf lktcbe. @ssulk pf`dk strkss jbdehthbd.
Q@[, @prhf / L`y ‚ 9700\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) ;3
:.0=5) Ekrhvk tck kxprksshbd nbr tck kfklkdt sthnndkss l`trhx nbr `d `xhsyllktrhj sckff
kfklkdt. Q@[, Dbv / Ekj ‚ 977=, @prhf / L`y ‚ 9778\
:.0=;) Eksjrhik tck stkp iy stkp prbjkeurk bn sbfvhdm `xhsyllktrhj prbifkl iy nhdhtk
kfklkdt nbrluf`thbd. Q@[, L`y / Oudk ‚
9709\
:.0=>) Ekrhvk `d kxprksshbd nbr tck sthnndkss l`trhx bn `d `xhsyllktrhj kfklkdt.
Q@[, @prhf / L`y ‚ 9700\
:.0==) Nbr `d `xhsyllktrhj trh`dmuf`r kfklkdt. Bit`hd tck QI\ l`trhx `de jbdsthtuthvk
l`trhx Q@[, Dbv / Ekj ‚ 9707\
:.0=3) Ekrhvk tck strkss-str`hd rkf`thbdschp l`trhx (E) nbr tck `xhsyllktrhj trh`dmuf`r
kfklkdt. Q@[, Dbv / Ekj ‚ 9709\
:.0=8) Kxpf`hd tck lbekfhdm bn jyfhdekrs suiokjtke tb hdtkrd`f `de kxtkrd`f prkssurk ushdm
`xhsyllktrhj. Q@[, Dbv / Ekj ‚ 9700\
:.037) Nbr ` tchja jyfhdekr suiokjtke tb hdtkrd`f `de kxtkrd`f prkssurk, hdehj`tk tck stkps bn
nhdehdm tck r`eh`f strkss. Q@[, Dbv / Ekj ‚ 9707\
:.030) Ekrhvk tck l`tkrh`f prbpkrty l`trhx nbr ̀ xhsyllktrhj kf`sthjhty.
Q@[, Dbv / Ekj ‚ 9700\
:.039) Kxpf`hd M`fkrahd‘s lktcbe bn nbrluf`thbd nbr ektkrlhdhdm tck sthn ndkss l`trhx nbr
`d ̀ xhsyllktrhj trh`dmuf`r kfklkdt. Q@[, Dbv / Ekj ‚ 970:\
:.03:) Uck ( x, y) jb- brehd`tks bn dbeks h, o `de a bn `d `xhsyllktrhj trh`dmuf`r kfklkdt
`rk mhvkd iy (:, 5), (>, ;), `de (;, 3) jl rkspkjthvkfy. Uck kfklkdt ehspf`jklkdt (hd
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) ;8
jl) vkjtbr hs mhvkd `s q < Q7.779, 7.770, 7.770, 7.775, -7.77:, 7.77=\U. Ektkrlhdk
tck kfklkdt str`hds. Q@[, Dbv / Ekj ‚ 9778\
:.035) @ fbdm jyfhdekr bn hdshek eh`lktkr 37 ll `de butshek eh`lktkr 097 ll sdumfy nhts
hd ` cbfk bvkr hts nuff fkdmtc. Uck jyfhdekr hs tckd suiokjtke tb `d hdtkrd`f prkssurk
bn 9 LW`. [shdm twb kfklkdts bd tck 07 ll fkdmtc scbwd, nhde tck ehspf`jklkdt`t tck hddkr r`ehus.
:.03;) Ektkrlhdk tck sthnndkss l`trhx nbr tck `xhsyllktrhj kfklkdt scbwd hd nhm, U`ak K `s
9.0* 07> D/ll9 `de Wbhssbd's r`thb `s 7.:. Q@[, Dbv / Ekj ‚ 9709\
:.03>) Ektkrlhdk tck kfklkdt strkssks nbr tck `xhsyllktrhj kfklkdt `s scbwd ikfbw. U`ak
K < 9.0 x 07; D/ll9 `de < 7.9;.
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NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) >7
[sk tck dbe`f ehspf`jklkdts `s
u0 < 7.7; ll w0 < 7.7: ll
u9 < 7.79 ll w9 < 7.79 ll
u: < 7 ll w: < 7 ll
:.03=) Jblputk tck str`hd ehspf`jklkdt l`trhx nbr tck nbffbwhdm `xhsyllktrhj kfklkdt.
@fsb j`fjuf`tk tck kfklkdt strkss vkjtbrs. Hn
Qq\ < Q :.535 7 :.:90 7 7 7\U * 07-: jl
QE\ < Q@[, @prhf / L`y ‚ 9700\
:.033) @d bpkd kdeke stkkf jyfhdekr c`s ` fkdmtc bn 977ll `de tck hddkr `de butkr
eh`lktkrs `s >3ll `de 077ll rkspkjthvkfy. Uck jyfhdekr hs suiokjtke tb `d
hdtkrd`f prkssurk bn 9LW`. Ektkrlhdk tck eknbrlke sc`pk `de ehstrhiuthbd bn
prhdjhpfk strkssks. U`ak K < 977MW` `de Wbhssbd‘s r`thb < 7.:
Q@[, @prhf / L`y ‚ 9700\
:.038) Ekrhvk tck Hsbp`r`lktrhj rkprkskdt`thbd nbr ̀ trh`dmuf`r kfklkdt.
Q@[, Dbv / Ekj ‚ 9707\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) >0
:.087) Ekrhvk kfklkdt sthnndkss l`trhx nbr ̀ fhdk`r Hsbp`r`lktrhj qu`erhf`tkr f̀ kfklkdt
Q@[, Dbv / Ekj ‚ 977=, @prhf / L`y ‚ 9773\
:.080) Ekrhvk sthnndkss l`trhx nbr ̀ fhdk`r Hsbp`r`lktrhj kfklkdt. Q@[, Dbv / Ekj ‚ 9709\
:.089) Kxpf`hd tck tkrls Hsbp`r`lktrhj sui p`r`lktrhj ̀ de supkr p`r`lktrhj kfklkdts.
Q@[, Dbv / Ekj ‚
970:\
:.08:) Ehsthdmuhsc iktwkkd sui p`r`lktrhj ̀ de supkr p`r`lktrhj kfklkdts.
Q@[, Dbv / Ekj ‚ 9778, 9707\
:.085) Kst`ifhsc tck sc`pk nudjthbds bn `d khmct dbek qu`erhf`tkr`f kfklkdt `de rkprkskdt
tckl mr`pchj`ffy. Q@[, @prhf / L`y ‚ 9700\
:.08;) Kst`ifhsc `dy twb sc`pk nudjthbds jbrrkspbdehdm tb bdk jbrdkr dbek `de bdk lhe ‚
dbek nbr `d khmct dbek qu`erhf`tkr`f kfklkdt. Q@[, Dbv / Ekj ‚
970:\
:.08>) Ekrhvk tck sc`pk nudjthbd nbr `d khmct dbeke irhja kfklkdt.
Q@[, @prhf / L`y ‚ 9778\
:.08=) Ekrhvk tck sc`pk nudjthbds bn ̀ dhdk dbek qu`erhf`tkr`f Hsbp`r`lktrhj kfklkdt.
Q@[, @prhf / L`y ‚ 9700, L`y / Oudk ‚ 9709\
:.083) Ekrhvk kfklkdt sthnndkss l`trhx nbr fhdk`r Hsbp`r`lktrhj qu`erhf`tkr`f kfklkdt.
Q@[, @prhf / L`y ‚ 9778, L`y / Oudk ‚ 9700\
:.088) Eksjrhik tck kfklkdt str`hd ehspf`jklkdt l`trhx bn ` nbur dbek qu`erhf`tkr`f
kfklkdt. Q@[, L`y / Oudk ‚ 9709\
:.977) Ekrhvk tck sc`pk nudjthbd nbr `d khmct ‚ dbeke qu`erhf`tkr`f kfklkdt hd , sp`jk.
:.970) Kst`ifhsc tck ibey nbrjk `de tr`jthbd nbrjk (udhnbrlfy ehstrhiutke) vkjtbr nbr `
fbwkr brekr qu`erhf`tkr`f kfklkdt. Q@[, Dbv / Ekj ‚ 970:\
:.979) Nbr tck Hsbp`r`lktrhj qu`erhf`tkr`f kfklkdt `s scbwd ikfbw, tck J`rtksh`d jb-
brehd`tks bn tck pbhdt W `rk (>,5). Uck fb`es 07 AD `de 09 AD `rk `jthdm hd S `de
X ehrkjthbds bd tc`t pbhdt W. Kv`fu`tk tck dbe`f kquhv`fkdt nbrjks.
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NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) >9
:.97:) Jbdshekr tck qu`erhf`tkr`f kfklkdt `s scbwd ikfbw ushdm tck fhdk`r hdtkrpbf`thbd
nudjthbds bn ` rkjt`dmuf`r kfklkdt, tr`dsnbrl tck kfklkdt tb tck fbj`f jb-brehd`tk
systkl ̀ de saktjc tck tr`dsnbrlke kfklkdt.
:.975) @ nbur dbeke rkjt`dmuf`r kfklkdt hs scbwd hd nhmurk. Ektkrlhdk tck nbffbwhdm? (h)
O`jbih`d L`trhx (hh) Ztr`hd Ehspf`jklkdt L`trhx (hhh) Kfklkdt Ztrkss. U`ak K <
97*07; D/ll9, ε < 7.;.
u < Q 7 7 7.77: 7.775 7.77> 7.775 7 7\U λ < 7, ν < 7.
@ssulk pf`dk strkss jbdehthbd. Q@[, L`y / Oudk ‚ 9709\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) >:
:.97;) @ nbur dbe`f qu`erhf`tkr`f pf`dk strkss Hsbp`r`lktrhj kfklkdt hs eknhdke iy dbeks 0
(7,7), 9 (57,7), : (57, 0;) `de 5 (7,0;). Ektkrlhdk tck O`jbih`d l`trhx
jbrrkspbdehdm tb tck M`uss pbhdt (7.;==:;, 7.;==:;) nbr tck `ibvk kfklkdt. Q@[, Dbv / Ekj ‚ 970:\
:.97>) Hd ` nbur-dbeke rkjt`dmuf`r kfklkdt, tck dbe`f ehspf`jklkdts hd ll `rk mhvkd iy
u0 < 7 u9 < 7.09= u: < 7.7>:; u5 < 7
v0 < 7 v9 < 7.7>:; v: < -7.7>:; v5 < 7
Nbr i < ;7 ll, c < 9; ll, K < 9*07; D / ll9 `de Wbhssbd's r`thb < 7.:,
ektkrlhdk tck kfklkdt str`hds `de strkssks `t tck jkdtrbhe bn tck kfklkdt `de `t tck
jbrdkr dbeks. Q@[, Dbv / Ekj ‚ 9709\
:.97=) Nhde tck O`jbih`d l`trhx nbr tck dhdk-dbek rkjt`dmuf`r kfklkdt `s scbwd ikfbw.
]c`t hs tck ektkrlhd`dt bn tck O`jbih`d l`trhx2
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NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) >5
:.973) Ektkrlhdk tck O`jbih`d nbr tck ( x, y) ‚ ( , ) tr`dsnbrl`thbd nbr tck kfklkdt scbwd
ikfbw. @fsb nhde tck ̀ rk` bn trh`dmfk ushdm ektkrlhd`dt lktcbe.
:.978) Jblputk tck kfklkdt `de nbrjk l`trhx nbr tck nbur dbeke rkjt`dmuf`r kfklkdts `s
scbwd ikfbw.
:.907) Uck J`rtksh`d (mfbi`f) jbbrehd`tks bn tck jbrdkr dbeks bn ` qu`erhf`tkr`f kfklkdt `rk
mhvkd iy (7,-0), (-9, :), (9, 5) `de (;, :). Nhde tck jbbrehd`tk tr`dsnbrl`thbd iktwkkd
tck mfbi`f `de fbj`f (d`tur`f) jbbrehd`tks. [shdm tchs, ektkrlhdk tck J`rtksh`d
jbbrehd`tks bn tck pbhdt eknhdke iy (r,s) < (7.;, 7.;) hd tck mfbi`f jbbrehd`tk systkl.
Q@[, Dbv / Ekj ‚ 9778\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) >;
:.900) Jbdshekr ` rkjt̀ dmuf`r kfklkdt `s scbwd ikfbw. @ssulk pf`dk strkss jbdehthbd,
U q psh K \7,7,77:9.7,77>.7,77:.7,779.7,7,7Q,:.7,07x:7
> hdjcks Kv`fu`tk
tck O`jbih`d tr`dsnbrl`thbd (O), I l`trhx, `de `t < 7 `de < 7.
:.909) Uck J`rtksh`d (mfbi`f) jbbrehd`tks bn tck jbrdkr dbeks bn `d Hsbp`r`lktrhj
qu`erhf`tkr`f kfklkdt `rk mhvkd iy (0,7), (9,7), (9.;,0.;) `de(0.;,0). Nhde htsO`jbih`d l`trhx. Q@[, Dbv / Ekj ‚ 9778\
:.90:) @ rkjt`dmuf`r kfklkdt c`s hts dbeks `t tck nbffbwhdm pbhdts hd J`rtksh`d jbbrehd`tk
systkl (7, 7), (;, 7), (;, ;), `de (7, ;). Bit`hd tck kxprksshbds nbr tck sc`pk
nudjthbds bn tck jbrrkspbdehdm Hsbp`r`lktrhj kfklkdt. [shdm tckl bit`hd tck
kfklkdts hn O`jbih`d l`trhx bn tr`dsnbrl`thbd. Q@[, Dbv / Ekj ‚ 9700\
:.905) Hn tck jbbrehd`tks bn tck qu`erhf`tkr`f `rk (0, 9), (07, 9), (3, >) `de (9, 07). Bit`hdtck O`jbih`d `de ckdjk, nhde tck `rk` bn tck kfklkdt. Q@[, Dbv / Ekj ‚ 9700\
:.90;) Ektkrlhdk tck O`jbih`d l`trhx nbr tck nbffbwhdm qu`erhf`tkr`f kfklkdt `t x < 5.:;
ll `de y < : ll. Q@[, @prhf / L`y ‚ 9700\
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NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) >>
:.90>) Jbdshekr tck qu`erhf`tkr`f kfklkdt ̀ s scbwd hd nhmurk. Kv`fu`tk `de
`t
(δ, ν), (7, 7) `de ( ) ushdm Hsbp`r`lktrhj nbrluf`thbd.
:.90=) Kst`ifhsc tck str`hd ‚ ehspf`jklkdt l`trhx nbr tck fhdk`r qu`erhf`tkr`f kfklkdt `s
scbwd hd nhmurk ikfbw `t M`uss pbhdt r < 7.;==:; `de s < -;==:;.
Q@[, Dbv / Ekj ‚ 977=, @prhf / L`y ‚ 9778\
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NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) >=
:.903) ]rhtk scbrt dbtks bd Q@[, Dbv / Ekj ‚ 9773\
(h) [dhqukdkss bn l`pphdm bn Hsbp`r`lktrhj kfklkdts.
(hh) O`jbih`d l`trhx.
(hhh) M`ussh`d Pu`er`turk hdtkmr`thbd tkjcdhquk.
:.908) Ekrhvk tck M`uss pbhdts `de wkhmcts hd j`sk bn bdk pbhdt nbrluf` `de twb pbhdt
nbrluf` bn M`uss dulkrhj`f hdtkmr`thbd. Q@[, @prhf / L`y ‚ 9700\
:.997) Ekrhvk tck wkhmcts `de M`uss pbhdts bn twb pbhdt nbrluf` bn M`uss qu`er`turk
rufk. Q@[, L`y / Oudk ‚ 9709\
:.990) Hdtkmr`tk iktwkkd 3 `de
09. [sk M`ussh`d qu`er`turk rufk. Q@[, @prhf / L`y ‚ 9773\
:.999) Kv`fu`tk tck hdtkmr`f < ∡ `de jblp`rk whtc kx`jt rksufts.
Q@[, Dbv / Ekj ‚ 9778\
:.99:) Dulkrhj`ffy kv`fu`tk tck nbffbwhdm hdtkmr`f `de jblp`rk whtc kx`jt bdk.
∡ ∡ Q@[, @prhf / L`y ‚
9700\
:.995) [shdm d`tur`f jbbrehd`tks ekrhvk tck sc`pk nudjthbd nbr ` fhdk`r qu`erhf`tkr`f
kfklkdt. Q@[, Dbv / Ekj ‚ 9773\
:.99;) [sk M`uss qu`er`turk rufk (d<9) tb dulkrhj`ffy hdtkmr`tk Q@[, Dbv / Ekj ‚ 9773\
∡ ∡
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:.99>) [sk M`ussh`d qu`er`turk rufk (d < 9) tb dulkrhj`ffy hdtkmr`tk∡ ∡ Q@[, Dbv / Ekj ‚ 9709\
:.99=) Kv`fu`tk
∡
ushdm M`uss qu`er`turk nbrluf`.
Q@[, L`y / Oudk ‚
9709\
:.993) Kv`fu`tk tck hdtkmr`f ∡ ushdm bdk pbhdt `de twb pbhdt
M`uss qu`er`turk nbrluf`. Q@[, @prhf / L`y ‚ 9700\
:.998) Kv`fu`tk tck hdtkmr`f ∡ ushdm tcrkk pbhdt M`uss qu`er`turk `de
jblp`rk whtc kx`jt sbfuthbd. Q@[, Dbv / Ekj ‚ 9700\
:.9:7) Kv`fu`tk ∡ ∡ ushdm M`uss dulkrhj`f hdtkmr`thbd.
Q@[, @prhf / L`y ‚ 9700\
:.9:0) [sk M`ussh`d qu`er`turk tb bit`hd `d kx`jt v`fuk bn tck hdtkmr`f.
Q@[, @prhf / L`y ‚ 9707\
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NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) >8
[DHU ‚ H^ ‚ EXD@LHJ @D@FXZHZ [ZHDM NHDHUK KFKLKDU LKUCBE
W@YU ‚ @
5.0) ]c`t hs eyd`lhj ̀ d`fyshs2 Mhvk kx`lpfks Q@[, Dbv / Ekj ‚ 9707\
5.9) ]c`t hs lk`d iy eyd`lhj `d`fyshs2 Q@[, Dbv / Ekj ‚ 9700\
5.:) Fhst tck typks bn eyd`lhj `d`fyshs prbifkls. Q@[, L`y / Oudk ‚ 9709\
5.5) Eknhdk dbrl`f lbeks. Q@[, L`y / Oudk ‚ 970:\
5.;) ]c`t hs tck prhdjhpfk bn lbek supkrpbshthbd tkjcdhquk2 Q@[, Dbv / Ekj ‚ 970:\
5.>) Zaktjc twb :E kfklkdts kxchihthdm fhdk`r str`hd ikc`vhbr. Q@[, @prhf / L`y ‚ 9700\
5.=) ]c`t hs tck hdnfukdjk bn kfklkdt ehstbrthbd bd tck `d`fyshs rksufts2
Q@[, @prhf / L`y ‚ 9700\
5.3) Ektkrlhdk tck kfklkdt l`ss l`trhx nbr bdk-ehlkdshbd`f, eyd`lhj strujtur`f `d`fyshs
prbifkls. @ssulk tck twb-dbek, fhdk`r kfklkdt. Q@[, Dbv / Ekj ‚ 9700\
5.8) ]c`t `rk jbdshstkdt `de fulpke l`ss tkjcdhquks2 Q@[, Dbv / Ekj ‚ 970:\
5.07) Zpkjhny tck jbdshstkdt l`ss l`trhx nbr ` ik`l kfklkdt. Q@[, Dbv / Ekj ‚ 970:\
5.00) Jbllkdt bd tck `jjur`jy bn tck v`fuks bn d`tur`f nrkqukdjhks bit`hdke iy ushdm
fulpke l`ss l`trhjks ̀ de jbdshstkdt l`ss l`trhjks. Q@[, Dbv / Ekj ‚ 9709\
5.09) Kxpf`hd jbdshstkdt fb`e vkjtbr. Q@[, Dbv / Ekj ‚ 9707\
5.0:) ]c`t eb ybu lk`d iy Fulpke l`ss l`trhx2 Q@[, L`y / Oudk ‚
9700\
5.05) ]rhtk ebwd tck fulpke l`ss l`trhx nbr tck truss kfklkdt. Q@[, @prhf / L`y ‚ 9778\
5.0;) ]c`t `rk tck typks bn Khmkd v`fuk prbifkls2 Q@[, L`y / Oudk ‚ 9709\
5.0>) ]c`t hs lk`dt iy lbek supkrpbshthbd tkjcdhquk2 Q@[, L`y / Oudk ‚ 970:\
W@YU ‚ I
5.0=) Ekrhvk tck kqu`thbd bn' lbthbd i`ske bd wk`a nbrl nbr tr`dsvkrsk vhir`thbd bn `
ik`l. Q@[, L`y / Oudk ‚ 9709\
5.03) Ekrhvk tck mbvkrdhdm kqu`thbd nbr fbdmhtuehd`f vhir`thbd.
5.08) Ekrhvk tck wk`a nbrluf`thbd nbr fbdmhtuehd`f vhir`thbd.
5.97) Ekrhvk tck wk`a nbrluf`thbd nbr tr`dsvkrsk vhir`thbd.
5.90) Ekrhvk tck kfklkdt kqu`thbd nbr fbdmhtuehd`f nrkk vhir`thbd.
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) =7
5.99) Ekrhvk tck kfklkdt kqu`thbd nbr tr`dsvkrsk nrkk vhir`thbd.
5.9:) Ekrhvk tck jbdshstkdt l`ss l`trhx nbr i`r kfklkdt
5.95) Ekrhvk tck jbdshstkdt l`ss l`trhx nbr truss kfklkdt
5.9;) Ekrhvk tck jbdshstkdt l`ss l`trhx nbr JZU kfklkdt
5.9>) Kxpf`hd tck ehrkjt hdtkmr`thbd lktcbe ushdm jkdtr`f ehnnkrkdjk sjcklk nbr prkehjthdm
tck tr`dshkdt eyd`lhj rkspbdsk bn ` strujturk. Q@[, Dbv / Ekj ‚ 970:\
5.9=) Ekrhvk tck jbdshstkdt l`ss l`trhx nbr ` truss kfklkdt hd hts fbj`f jbbrehd`tk systkl.
Q@[, Dbv / Ekj ‚ 9709\
5.93) Ekrhvk tck nhdhtk kqu`thbds nbr tck thlk ‚ ekpkdekdt strkss ̀ d`fyshs bn bdk ehlkdshbd`f
i`r. Q@[, L`y / Oudk ‚ 9700\
5.98) Nhde tck d`tur`f nrkqukdjhks bn tr`dsvkrsk vhir`thbds bn tck j`dthfkvkr ik`l scbwd hd
Nhmurk ushdm bdk ik`l kfklkdt. Q@[, Dbv / Ekj ‚
9709\
5.:7) Jbdshekr ` udhnbrl jrbss ‚ skjthbd i`r `s scbwd hd nhmurk bn fkdmtc ‖F“ l`ek up bn `
l`tkrh`f wcbsk Xbudm‘s lbeufus `de ekdshty `rk mhvkd iy K `de π. Ksthl`tk tck
d`tur`f nrkqukdjhks bn `xh`f vhir`thbd bn tck i`r ushdm ibtc fulpke `de jbdshstkdt
l`ss l`trhx. Q@[, L`y / Oudk ‚ 977;\
5.:0) Ektkrlhdk tck d`tur`f nrkqukdjhks `de lbek sc`pks bn tr`dsvkrsk vhir`thbd nbr ` ik`l
nhxke `t ibtc kdes. Uck ik`l l`y ik lbekffke iy twb kfklkdts, k`jc bn fkdmtc F `de
jrbss-skjthbd`f ̀ rk` @. Jbdshekr fulpke l`ss l`trhx ̀ pprb`jc.
Q@[, @prhf / L`y ‚ 9700\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) =0
5.:9) [shdm twb kqu`f-fkdmtc nhdhtk kfklkdts, ektkrlhdk tck d`tur`f nrkqukdjhks bn tck sbfhe
jhrjuf`r sc`nt nhxke `t bdk kde scbwd hd nhmurk. Q@[, Dbv / Ekj ‚ 9700\
5.::) Bit`hd tck d`tur`f nrkqukdjhks bn vhir`thbd nbr ` stkppke stkkf i`r bn `rk` >9;ll9 nbr
tck fkdmtc bn 9;7ll `de :09.;ll9 nbr tck fkdmtc bn 09;ll. Uck kfklkdt hs nhxke `t
f`rmkr kde Q@[, Dbv / Ekj ‚ 977=\
5.:5) Ektkrlhdk tck Khmkd v`fuks `de nrkqukdjhks nbr tck stkppke i`r scbwd hd tck nhmurk.
U`ak K < 97 * 0707 D/l9 `de skfn-wkhmct < 3;77 am.l: Q@[, L`y / Oudk ‚ 977;\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) =9
5.:;) Nhde tck d`tur`f nrkqukdjy bn fbdmhtuehd`f vhir`thbd bn tck udjbdstr`hdke stkppke i`r
`s scbwd hd nhmurk. Q@[, Dbv / Ekj ‚ 977>\
5.:>) Jblputk l`tkrh`f nrkqukdjhks bn nrkk tr`dsvkrsk vhir`thbd bn ` stkppke ik`l scbwd hd
nhmurk. Q@[, L`y / Oudk ‚
977:\
5.:=) Ektkrlhdk tck d`tur`f nrkqukdjhks bn tr`dsvkrsk vhir`thbd nbr ` ik`l nhxke `t ibtc
kdes. Uck ik`l l`y ik lbekffke iy twb kfklkdts k`jc bn fkdmtc F `de jrbss ‚
skjthbd`f `rk` @. Uck usk bn syllktry ibude`ry jbdehthbd hs bpthbd`f.
Q@[, L`y / Oudk ‚ 9773\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) =:
5.:3) Ektkrlhdk tck d`tur`f nrkqukdjhks `de lbek sc`pks bn ` systkl wcbsk sthnndkss `de
l`ss l`trhjks `rk mhvkd ikfbw Q@[, L`y / Oudk ‚ 9773\ 5.:8) @ vkrthj`f pf`tk bn tchjadkss 57 ll hs t`pkrke whtc whetcs bn 7.0;l `de 7.7=;l `t
tbp `de ibttbl kdes rkspkjthvkfy. Uck pf`tk hs nhxke `t tck tbp kde. Uck fkdmtc bn tck
pf`tk hs 7.3l. U`ak Xbudm's lbeufus `s 977 MW` `de ekdshty `s =377 am/l:. Lbekf
tck pf`tk whtc twb sp`r kfklkdts. Ektkrlhdk tck d`tur`f nrkqukdjhks bn fbdmhtuehd`f
vhir`thbd `de tck lbek sc`pks. Q@[, Dbv / Ekj ‚ 970:\
5.57) Nhde tck rkspbdsk bn tck systkl mhvkd ikfbw ushdm lbe`f supkrpbshthbd lktcbe.
Q@[, @prhf / L`y ‚ 9700\
5.50) Ektkrlhdk tck d`tur`f nrkqukdjhks nbr tck truss scbwd hd nhmurk ushdm nhdhtk kfklkdt
lktcbe. Q@[, L`y / Oudk ‚ 977=\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) =5
5.59) Nhde tck d`tur`f nrkqukdjhks bn vhir`thbd bn twb kfklkdt shlpfy suppbrtke ik`l iy
t`ahdm `ev`dt`mk bn tck syllktry `ibut tck lhe-pbhdt Q@[, L`y / Oudk ‚ 977=\
5.5:) Nbrluf`tk tck l`ss l`trhx nbr twb-ehlkdshbd`f rkjt`dmuf`r kfklkdt ekphjtke hd
nhmurk. Uck kfklkdt c`s udhnbrl tchjadkss ; ll `de ekdshty π < =.3: * 07->am/ll:.
Q@[, Dbv / Ekj ‚ 9700\
5.55) Jbdshekr tck ude`lpke 9 ekmrkk bn nrkkebl systkl `s scbwd hd nhmurk. Nhde tck
rkspbdsk bn tck systkl wckd tck nhrst l`ss `fbdk hs mhvkd `d hdhth`f ehspf`jklkdt bn
udhty `de rk`fhske nrbl rkst.
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) =;
Uck l`tckl`thj`f rkprkskdt`thbd bn tck systkl nbr nrkk, C`rlbdhj vhir`thbd hs mhvkd
iy Q@[, L`y / Oudk ‚ 9709\
5.5;) J`fjuf`tk tck jbdshstkdt `de fulpke fb`e vkjtbr nbr tck kfklkdt scbwd hd nhmurk.
Q@[, Dbv / Ekj ‚ 9707\
5.5>) Jbdshekr ` udhnbrl jrbss skjthbd i`r, `s scbwd hd nhmurk bn fkdmtc F l`ek up bn
l`tkrh`f wcbsk ybudm's lbeufus `de ekdshty hs mhvkd iy K `de π . Ksthl`tk tck
d`tur`f nrkqukdjhks bn `xh`f vhir`thbd bn tck i`r ushdm ibtc jbdshstkdt `de fulpke
l`ss l`trhjks. Q@[, L`y / Oudk ‚ 970:\
5.5=) Ektkrlhdk tck Khmkd v`fuks `de Khmkd vkjtbrs nbr tck stkppke i`r `s scbwd hd nhmurk.
Q@[, L`y / Oudk ‚
9700\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) =>
5.53) Nhde tck Khmkd v`fuks `de Khmkd vkjtbrs bn tck l`trhx.
Q
\
5.58) Nhde tck Khmkd v`fuks `de Khmkd vkjtbrs bn tck l`trhx.
Q \
5.;7) [sk htkr`thvk prbjkeurks tb ektkrlhdk tck nhrst `de tchre Khmkd v`fuks nbr tck strujturk
scbwd hd nhmurk. Ckdjk ektkrlhdk tck skjbde Khmkd v`fuk `de tck d`tur`f nrkqukdjhks
bn iuhfehdm. Nhd`ffy, kst`ifhsc tck Khmkd vkjtbrs `de jckja tck rkst iy `ppfyhdm tck
brtcbmbd`fhty prbpkrthks bn Khmkd vkjtbrs. Q@[, L`y / Oudk ‚ 970:\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) ==
[DHU ‚ ^ ‚ @WWFHJ@UHBDZ HD CK@U UY@DZNKY & NF[HE
LKJC@DHJZ
W@YU ‚ @
;.0) ]rhtk ebwd tck bdk ehlkdshbd`f ck`t jbdeujthbd kqu`thbd. Q@[, @prhf / L`y ‚ 9700\
;.9) Ehsthdmuhsc iktwkkd cblbmkdbus ̀ de dbd ‚ cblbmkdbus ibude`ry jbdehthbds. Q@[, Dbv / Ekj ‚ 970:\
;.:) ]rhtk ebwd tck kxprksshbd bn sc`pk nudjthbd `de tklpkr`turk nudjthbd nbr bdk
ehlkdshbd`f ck`t jbdeujthbd. Q@[, L`y / Oudk ‚ 9700\
;.5) ]rhtk ebwd tck mbvkrdhdm ehnnkrkdth`f kqu`thbd nbr tck stk`ey st`tk bdk ehlkdshbd`f
jbdeujthbd ck`t tr`dsnkr. Q@[, Dbv / Ekj ‚ 9707, 9709\
;.;) ]rhtk ebwd tck mbvkrdhdm ehnnkrkdth`f kqu`thbd nbr ` twb ehlkdshbd`f stk`ey-st`tk
ck`t tr`dsnkr prbifkl. Q@[, Dbv / Ekj ‚ 9778\
;.>) ]rhtk ebwd tck sthnndkss l`trhx kqu`thbd nbr bdk ehlkdshbd`f ck`t jbdeujthbd
kfklkdt. Q@[, Dbv / Ekj ‚ 9700\
;.=) Zaktjc ` twb ehlkdshbd`f ehnnkrkdth`f jbdtrbf kfklkdt nbr ck`t tr`dsnkr `de bit`hd tck
ck`t ehnnushbd kqu`thbd. Q@[, Dbv / Ekj ‚ 9709\
;.3) Eknhdk kfklkdt j`p`jht`djk l`trhx nbr udstk`ey st`tk ck`t tr`dsnkr prbifkls.
Q@[, L`y / Oudk ‚ 970:\
;.8) D`lk ` nkw ibude`ry jbdehthbds hdvbfvke hd `dy ck`t tr`dsnkr `d`fyshs.
Q@[, @prhf / L`y ‚ 9707\
;.07) Lkdthbd twb d`tur`f ibude`ry jbdehthbds `s `ppfhke tb tckrl`f prbifkls.
Q@[, @prhf / L`y ‚ 9700\
;.00) Jbdshekr ` w`ff bn ` t`da jbdt`hdhdm ` cbt fhquhe `t ` tklpkr`turk U7 whtc `d `hr
strk`l bn tklpkr`turk Ux p`sske bd tck butshek, l`hdt`hdhdm ` w`ff tklpkr`turk bn UF
`t tck ibude`ry. Zpkjhny tck ibude`ry jbdehthbds. Q@[, @prhf / L`y ‚ 9778\
;.09) Eknhdk st`thj jbdekds`thbd. Q@[, Dbv / Ekj ‚ 9707\
;.0:) Mhvk tck mbvkrdhdm kqu`thbd bn tbrshbd prbifkl. Q@[, L`y / Oudk ‚ 9709\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) =3
;.05) ]rhtk tck stkp iy stkp prbjkeurk bn sbfvhdm ` tbrshbd prbifkl iy nhdhtk kfklkdt
lktcbe. Q@[, @prhf / L`y ‚ 9700\
;.0;) Butfhdk tck stkp iy stkp prbjkeurk bn c`defhdm tbrshbd prbifkl ushdm tck nhdhtk
kfklkdt lktcbe. Q@[, L`y / Oudk ‚ 9709\
;.0>) Eknhdk strk`lfhdk. Q@[, L`y / Oudk ‚
9709\
;.0=) Eknhdk tck strk`l nudjthbd nbr ̀ bdk-ehlkdshbd`f hdjblprksshifk nfbw.
Q@[, @prhf / L`y ‚ 9700\
;.03) Fhst tck `ppfhj`thbds bn tck pbtkdth`f nfbw. Q@[, Dbv / Ekj ‚ 9700\
;.08) Fhst tck lktcbe bn eksjrhihdm tck lbthbd bn nfuhe. Q@[, L`y / Oudk ‚ 9709\
;.97) Zt`tk tck rkf`thbd iktwkkd tck vkfbjhty bn nfuhe nfbw `de tck cyer`ufhj mr`ehkdt
`jjbrehdm tb E`rjy's f`w, kxpf`hdhdm tck tkrls hdvbfvke. Q@[, Dbv / Ekj ‚ 9709\
;.90) Eknhdk tck strk`l nudjthbd nbr ` bdk ehlkdshbd`f hdjblprksshifk nfbw.
Q@[, Dbv / Ekj ‚ 970:\
;.99) Eknhdk tck strk`l nudjthbd nbr ` twb ehlkdshbd`f hdjblprksshifk nfbw.
Q@[, L`y / Oudk ‚ 970:\
W`rt ‚ I
;.9:) ]rhtk tck l`tckl`thj`f nbrluf`thbd nbr ` stk`ey st`tk ck`t tr`dsnkr jbdeujthbd
prbifkl `de ekrhvk tck sthnndkss `de nbrjk l`trhjks nbr tck s`lk.
Q@[, Dbv / Ekj ‚ 9773\
;.95) Jbdshekr ` pf`dk w`ff whtc udhnbrlfy ehstrhiutke ck`t sburjk. Bit`hd tck nhdhtk
kfklkdt nbrluf`thbd nbr tck `ibvk j`sk i`ske bd tck st`thbd`rhty bn ` nudjthbd`f.
Q@[, Dbv / Ekj ‚ 970:\
;.9;) Ekrhvk ` nhdhtk kfklkdt kqu`thbd nbr bdk ehlkdshbd`f ck`t jbdeujthbd whtc nrkk kde
jbdvkjthbd. Q@[, L`y / Oudk ‚ 970:\
;.9>) Uck tklpkr`turk `t tck nbur jbrdkrs bn ` nbur ‚ dbeke rkjt`dmfk `rk U0, U9 U: `de U5.
Ektkrlhdk tck jbdshstkdt fb`e vkjtbr nbr ` 9-E `d`fyshs, `hlke tb ektkrlhdk tck
tckrl`f strkssks. Q@[, Dbv / Ekj ‚ 977=, @prhf / L`y ‚ 9778\
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) =8
;.9=) Ekrhvk tck sthnndkss l`trhx `de fb`e vkjtbrs nbr nfuhe lkjc`dhjs hd twb ehlkdshbd`f
nhdhtk kfklkdt. Q@[, L`y / Oudk ‚ 9709\
;.93) Mhvk tck bdk-ehlkdshbd`f nbrluf`thbd nbr bdk-ehlkdshbd`f nfbw `de ekrhvk tck
kfklkdt sthnndkss l`trhx nbr tck nfbw tcrbumc ` pbrbus lkehul.
Q@[, Dbv / Ekj ‚
9709\
;.98) Hd tck nhdhtk kfklkdt `d`fyshs bn ` twb ehlkdshbd`f nfbw ushdm trh`dmuf`r kfklkdts, tck
vkfbjhty jblpbdkdts u `de v `rk `ssulke tb v`ry fhdk`rfy whtchd `d kfklkdt (k) `s
u(x ,y) < `0[h(k)
+ `9[ o(k)
+ `:[a (k)
v(x ,y) < `0^h(k)
+ `9 ̂o(k)
+ `:^a (k)
wckrk ([h(k) , ^h
(k)) ekdbtk tck v`fuks bn (u, v ) `t dbek h. Nhde tck rkf`thbdschp iktwkkd
([h(k) , ^h
(k)............. ^a
(k))wchjc hs tb ik s`thsnhke nbr tck nfbw tb ik hdjblprksshifk.
Q@[, L`y / Oudk ‚ 970:\
;.:7) Ekvkfbp sthnndkss jbknnhjhkdts euk tb tbrshbd nbr ` tcrkk ehlkdshbd`f ik`l kfklkdt.
Q@[, @prhf / L`y ‚ 9778\
;.:0) Kst`ifhsc tck nhdhtk kfklkdt kqu`thbds hdjfuehdm nbrjk l`trhjks nbr tck `d`fyshs bn twb
ehlkdshbd`f stk`ey ‚ st`tk nfuhe nfbws tcrbumc ` pbrbus lkehul ushdm trh`dmuf`rkfklkdt. Q@[, Dbv / Ekj ‚ 970:\
;.:9) Kxpf`hd tck pbtkdth`f nudjthbd nbrluf`thbd bn nhdhtk kfklkdt kqu`thbds nbr hek`f nfbw
prbifkls. Q@[, L`y / Oudk ‚ 970:\
;.::) Nhde tck tklpkr`turk `t ` pbhdt W (0,0.;) hdshek tck trh`dmuf`r kfklkdt scbwd whtc tck
dbe`f tklpkr`turks mhvkd `s U 0 < 577J, U O < :57J , `de U A < 5>7J . @fsb ektkrlhdk tck
fbj`thbd bn tck 59
7
J jbdtbur fhdk nbr tck trh`dmuf`r kfklkdt scbwd hd nhmurk ikfbw. Q@[, @prhf / L`y - 9773\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 37
;.:5) Bit`hd tck nhdhtk kfklkdt kqu`thbds nbr tck nbffbwhdm kfklkdt. Uck tckrl`f
jbdeujthvhty (a) bn tck l`tkrh`f bn tck kfklkdt hs 9 ]/ lA. Uck jbdvkjthvk ck`t
tr`dsnkr jbknnhjhkdt (c) hs : ]/l9A. Uck `lihkdt tklpkr`turk (Un ) hs 9;˟ J. Uck
tchjadkss (t) bn tck l`tkrh`f hs 0ll. @ssulk jbdvkjthbd `fbdm tck kemk –oa‘ `fbdk.
Q@[, @prhf / L`y - 9700\
;.:;) Jblputk tck kfklkdt`f strkss vkjtbrs nbr tck nbffbwhdm kfklkdt, `ssulhdm pf`dk strkss
jbdehthbds. Uck dbe`f ehspf`jklkdts hd –ll‘ Qq\ < Q7 0 0 7 0 0\U. Uck
tklpkr`turk hdjrk`sk hd tck kfklkdt hs ;˟J. U`ak K < 977 MW` `de » < 7.:. Uck
tckrl`f jbknnhjhkdt bn kxp`dshbd hs 00 * 07->
/˟J. Uck tchjadkss bn tck l`tkrh`f hs 0ll. Q@[, @prhf / L`y - 9700\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 30
;.:>) J`fjuf`tk tck kfklkdt sthnndkss l`trhx `de tck tckrl`f nbrjk vkjtbr nbr tck
`xhsyllktrhj trh`dmuf`r kfklkdt `s scbwd ikfbw. Uck kfklkdt kxpkrhkdjks ` 0;7 J
hdjrk`sk hd tklpkr`turk. U`ak < 07 x 07-> / 7J, K < 9 x 07; D/ll9 `de < 7.9;
;.:=) Ektkrlhdk tck tklpkr`turk `de ck`t nfuxks `t ` fbj`thbd (9, 0) hd ` squ`rk pf`tk `s
scbwd hd nhmurk. Er`w tck hsbtckrl`f nbr 09;·J. U0 < 077·J, U9 < 0;7·J, U: <
977·J, U5 < ;7·J Q@[, Dbv / Ekj ‚ 9707\
;.:3) Jbdshekr ` irhja w`ff bn tchjadkss 7.: l, a < 7.= ]/l˟J. Uck hddkr surn`jk hs `t 93˟J
`de tck butkr surn`jk hs kxpbske tb jbfe `hr `t -0;˟J. Uck ck`t tr`dsnkr jbknnhjhkdt
`ssbjh`tke whtc tck butshek surn`jk hs 57 ]/l9˟J. Ektkrlhdk tck stk`ey st`tk
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 39
tklpkr`turk ehstrhiuthbd whtchd tck w`ff `de `fsb tck ck`t nfux tcrbumc tck w`ff. [sk
twb 0E kfklkdts `de bit`hd tck sbfuthbd. Q@[, Dbv / Ekj ‚ 970:\
;.:8) Jbdshekr ` irhja w`ff `s scbwd hd nhmurk bn tchjadkss F < :7jl, A < 7.= ]/l˟J. Uck
hddkr surn`jk hs `t 93˟J `de tck butkr surn`jk hs kxpbske tb jbfe `hr `t -0;˟J. Uck ck`t
tr`dsnkr jbknnhjhkdt `ssbjh`tke whtc tck butshek surn`jk hs c < 57 ]/l
9
˟J. Ektkrlhdktck stk`ey st`tk tklpkr`turk ehstrhiuthbd whtchd tck w`ff `de `fsb tck ck`t nfux tcrbumc
tck w`ff. [sk ` twb kfklkdt lbekf. @ssulk bdk ehlkdshbd`f nfbw.
Q@[, @prhf / L`y ‚ 9700\
;.57) @ jblpbshtk w`ff jbdshsts bn tcrkk l`tkrh`fs `s scbwd hd nhmurk. Uck butkr
tklpkr`turk hs U7 < 97˟J. Jbdvkjthbd ck`t tr`dsnkr t`aks pf`jk bd tck hddkr surn`jk bn
tck w`ff whtc U∔ < 377˟J `de c < 9;]/l9˟J. Ektkrlhdk tck tklpkr`turk ehstrhiuthbd
hd tck w`ff. Q@[, L`y / Oudk ‚ 9700\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 3:
;.50) @ jblpbshtk w`ff hs l`ek bn tcrkk ehnnkrkdt l`tkrh`fs. Uck tckrl`f jbdeujthvhty bn tck
v`rhbus skjthbds `rk a 0 < 9 ]/jl ˟J, a 9 < 0 ]/jl ˟ J, a : < 7.9 < ]/jl ˟J. Uck
tchjadkss bn tck w`ff nbr tck skjthbd hs 0jl, ;jl `de 5jl rkspkjthvkfy. Ektkrlhdk tck
tklpkr`turk v`fuks bn dbe`f pbhdts whtchd tck w`ff. @ssulk tck surn`jk `rk` tb udhty.
Uck fknt kemk bn tck w`ff hs suiokjtke tb ` tklpkr`turk bn :7˟J `de tck rhmct shek bn tckw`ff hs `t 07˟J. Q@[, Dbv / Ekj ‚ 9700\
;.59) Nhmurk scbws ` s`dewhjcke jblpbshtk w`ff. Jbdvkjthbd ck`t fbss bjjurs bd tck fknt
surn`jk `de tck tklpkr`turk bd tck rhmct surn`jk hs jbdst`dt. Jbdshekrhdm ` udht `rk`
`de whtc tck p`r`lktkrs mhvkd, usk tcrkk fhdk`r kfklkdts (bdk nbr k`jc f`ykr) `de
(h) Ektkrlhdk tck tklpkr`turk ehstrhiuthbd tcrbumc tck jblpbshtk w`ff `de
(hh) J`fjuf`tk tck nfux bd tck rhmct surn`jk bn tck w`ff. Q@[, Dbv / Ekj ‚
9709\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 35
;.5:) @ w`ff bn 7.>l tchjadkss c`vhdm tckrl`f jbdeujthvhty bn 0.9 ]/l-A tck w`ff hs tb ikhdsuf`tke whtc ` l`tkrh`f bn tchjadkss 7.7> l c`vhdm `d `vkr`mk tckrl`f jbdeujthvhty
bn 7.: ]/l-A. Uck hddkr surn`jk tklp hs 0777˟J `de butshek bn tck hdsuf`thbd hs
kxpbske tb `tlbspckrhj `hr `t :7˟J whtc ck`t tr`dsnkr jb-knnhjhkdt bn :; D/l9 A.
J`fjuf`tk tck dbe`f tklpkr`turk ushdm NK@. Q@[, Dbv / Ekj ‚ 9700\
;.55) @ fbdm i`r bn rkjt`dmuf`r jrbss skjthbd c`vhdm tckrl`f jbdeujthvhty bn 0.; ]/l ˟J hs
suiokjtke tb tck ibude`ry jbdehthbd `s scbwd ikfbw. Uwb bppbshtk sheks `rkl`hdt`hdke `t udhnbrl tklpkr`turk bn 037 7J. Bdk shek hs hdsuf`tke `de tck rkl`hdhdm
shek hs suiokjtke tb ` jbdvkjthbd prbjkss whtc U < 3;˟J `de c < ;7 ]/l9˟
J.
Ektkrlhdk tck tklpkr`turk ehstrhiuthbd hd tck i`r.
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 3;
;.5;) Uck pf`dk w`ff scbwd ikfbw hs 7.; l tchja. Uck fknt surn`jk bn tck w`ff hs l`hdt`hdke
`t ` jbdst`dt tklpkr`turk bn 9777J `de tck rhmct surn`jk hs hdsuf t̀ke. Uck tckrl`f
jbdeujthvhty A < 9; ]/L bJ `de tckrk hs ` udhnbrl ck`t mkdkr`thbd hdshek tck w`ff
bn P < 577 ]/l:. Ektkrlhdk tck tklpkr`turk ehstrhiuthbd tcrbumc tck w`ff tchjadkss
ushdm fhdk`r kfklkdts.
;.5>) Ektkrlhdk tcrkk pbhdts bd tck ;7b J jbdtbur fhdk nbr tck rkjt`dmuf`r kfklkdt scbwd hd
tck nhmurk. Uck dbe`f v`fuks `rk h< 59b J , o<;5b J , a < ;>b J `de l< 5>b J .
;.5=) Jblputk tck stk`ey st`tk tklpkr`turk ehstrhiuthbd nbr tck pf`tk scbwd hd tck nhmurk
ikfbw. @ jbdst`dt tklpkr`turk bn U7 < 0;77 J hs l`hdt`hdke `fbdm tck kemk y < w `de
`ff btckr kemks c`vk zkrb tklpkr`turk. Uck tckrl`f jbdeujthvhthks `rk A x < A y < 0.
@ssulk w < F < 0 `de tchjadkss t < 0.
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 3>
;.53) @ stkkf rbe bn eh`lktkr e < 9 jl, fkdmtc f <; jl `de tckrl`f jbdeujthvhty A < ;7
]/l˟J hs kxpbske `t bdk kde tb ` jbdst`dt tklpkr`turk bn :97˟J. Uck btckr kde hs hd
`lihkdt `hr bn tklpkr`turk 97˟J whtc ` jbdvkjthbd jb-knnhjhkdt bn c < 077 ]/l9˟J.
Ektkrlhdk tck tklpkr`turk `t tck lhepbhdt bn tck rbe ushdm NK@.
Q@[, Dbv / Ekj ‚ 9700\
;.58) Ektkrlhdk tck tklpkr`turk ehstrhiuthbd hd bdk ehlkdshbd`f rkjt`dmuf`r jrbss-skjthbd
`s scbwd hd Nhmurk. Uck nhd c`s rkjt`dmuf`r jrbss-skjthbd `de hs 3jl fbdm 5jl whek
`de 0jl tchja. @ssulk tc`t jbdvkjthbd ck`t fbss bjjurs nrbl tck kde bn tck nhd. U`ak
c < :] / jl˟J, c < 7.0 ] / jl9˟ J,U ∔ < 97˟J. Q@[, @prhf / L`y ‚ 9700\
;.;7) J`fjuf`tk tck tklpkr`turk ehstrhiuthbd hd st`hdfkss stkkf nhd scbwd hd nhmurk. Uck
rkmhbd j`d ik ehsjrkthzke hdtb nhvk kfklkdts `de shx dbeks. Q@[, @prhf / L`y ‚
9778\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 3=
;.;0) J`fjuf`tk tck kfklkdt sthnndkss l`trhx `de tckrl`f nbrjk vkjtbr nbr tck pf`dk strkss
kfklkdt scbwd hd nhmurk ikfbw. Uck kfklkdt kxpkrhkdjks ` rhsk bn 077J.
Q@[, @prhf / L`y - 9773\
;.;9) J`fjuf`tk tck tklpkr`turk `t tck pbhdt nbr ` tcrkk dbeke trh`dmuf`r kfklkdt `s scbwd
hd nhmurk. Uck dbe`f v`fuks `rk U0 < 57˟J, U9 < :5˟J `de U: < 5>˟J. Wbhdt @ hs
fbj`tke `t (9, 0.;). @ssulk tck tklpkr`turk hs fhdk`rfy v`ryhdm hd tck kfklkdt. @fsb
ektkrlhdk tck fbj`thbd bn 59˟J jbdtbur fhdk. Q@[, L`y / Oudk ‚ 9700\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 33
;.;:) Ektkrlhdk tck kfklkdt sthnndkss l`trhx `de tck tckrl`f fb`e vkjtbr nbr tck pf`dk strkss
kfklkdt scbwd hd nhmurk. Uck kfklkdt kxpkrhkdjks 97bJ hdjrk`sk hd tklpkr`turk. U`ak
K < 0;k> D/jl9, < 7.9;, t < 7.; jl `de ` < >k - >/b J . Q@[, @prhf / L`y - 9707\
;.;5) Uck trh`dmuf`r kfklkdt scbwd hd nhmurk hs suiokjtke tb ` jbdst`dt prkssurk 07 D/ll9
`fbdm tck kemk ho. @ssulk K < 977 Mp`, Wbhssbd‘s r`thb < 7.: `de tchjadkss bn tck
kfklkdt < 9 ll. Uck jbknnhjhkdt bn tckrl`f kxp`dshbd bn tck l`tkrh`f < 9
x07->/ bJ `de U < ;7b J . Ektkrlhdk tck jbdsthtuthvk l`trhx (strkss-str`hd rkf`thbdschp
l`trhx E) `de tck dbe`f nbrjk vkjtbr nbr tck kfklkdt. Q@[, Dbv / Ekj - 9778\
;.;;) Jblputk tck kfklkdt sthnndkss l`trhx `de vkjtbrs nbr tck kfklkdt scbwd hd nhmurk
wckd tck kemk 9 ‚ : `de : ‚ 0 kxpkrhkdjk ck`t fbss. Q@[, L`y / Oudk ‚ 9709\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 38
;.;>) Jblputk tck kfklkdt l`trhjks `de vkjtbrs nbr tck kfklkdt scbwd ikfbw, wckd tck
kemks oa `de ha kxpkrhkdjk jbdvkjthbd ck`t fbss.
;.;=) Jblputk kfklkdt l`trhjks `de vkjtbrs nbr tck kfklkdts scbwd hd nhmurk wckd tck
kemk ao kxpkrhkdjks jbdvkjthbd ck`t fbss. Q@[, Dbv / Ekj ‚ 9778\
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Y.L.A JBFFKMK BN KDMM @DE UKJC / @P / Y9773/ LK9:;: / ^H / LKJC / EKJ 970: ‚ L@X 9705
NHDHUK KFKLKDU @D@FXZHZ P[KZUHBD I@DA iy @ZCBA A[[email protected] (@W / Lkjc) 87
;.;3) Nbr tck slbbtc phpk bn v`rh`ifk jrbss-skjthbd `s scbwd hd Nhmurk. Ektkrlhdk tck
pbtkdth`fs `t tck oudjthbds, tck vkfbjhthks hd k`jc phpk. Uck pbtkdth`fs `t tck fknt kde hs
07 l `de tc`t `t tck rhmct kde hs 9l.Uck pkrlk`ihfhty jbknnhjhkdt hs 0 l/skj.
Q@[, @prhf / L`y ‚ 9700\
;.;8) Nbr tck twb ehlkdshbd`f s`dey sbhf rkmhbd `s scbwd hd nhmurk. Ektkrlhdk tck pbtkdth`f
ehstrhiuthbd. Uck pbtkdth f̀ (nfuhe ck`e) bd tck fknt shek hs 07l `de bd rhmct c`de shek hs
7. Uck uppkr `de fbwkr kemks `rk hlpkrlk`ifk A xx < 9;*07-; l/s `de A yy < 9;*07-;
l/s. @ssulk udht tchjadkss.
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EJ F ~ F @ ~BY?. Y k -
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G < 9 ( B S.
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]KHMCUKE YKZHE[@F LKUCBEZ?
Mkdkr`f Urh`f nudjthbd?
Zuisthtutk tck trh`f nudjthbd hd ehnnkrkdth`f kqu`thbd tb mkt tck rksheuk nudjthbd.
Wbhdt Jbffbj`thbd Lktcbe?
Y( x) Yksheuk nudjthbdCkrk tck pbhdt bn x hs iktwkkd tcksk fhlhts. Hn tck fhlht hs 7 tb 0
Nbr twb udadbwds `dy twb pbhdt iktwkkd tcksk fhlhts. Nbr kx`lpfk
x v`fuk hs x v`fuk hs
Nbr tcrkk udadbwds `dy twb pbhdt iktwkkd tcksk fhlhts. Nbr kx`lpfk
x v`fuk hs
x v`fuk hs
x v`fuk hs
Zui Ebl`hd Lktcbe?
Ckrk tck fhlht scbufe ik t`akd hdtb twb hdtkrv`fs. Hn tck fhlht hs 7 tb 0
Uck fhlht v`rhks iktwkkd 7 ‚ 7.;
∡
Uck fhlht v`rhks iktwkkd 7.; ‚ 0 ∡
Fk`st Zqu`rk Lktcbe?
Ehnnkrkdth`tk tck Y( x) whtc rkspkjt tb `0, `9
M`fkrahd Lktcbe?
[DHU H ‚ NHDHUK KFKLKDU NBYL[F@UHBD BN IB[DE@YX
^@F[K WYBIFKLZ
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]khmcthdm nudjthbd (Nudjthbd `ssbjh`tke whtc udadbwd
trh`f nudjthbd) Y@XFKHMC ‚ YHU_ LKUCBE?
Nbr Ik`l tck Nburhkr skrhks kqu`thbd
∐
Nbr J`dthfkvkr I`r (@xh`f Fb`ehdm)
]ckrk `0 `9 & `: `rk Yhtz p`r`lktkrs
Uck Ubt`f Wbtkdth`f Kdkrmy bn tck Ik`l?
[ Ztr`hd Kdkrmy
C ]bra Ebdk iy Kxtkrd`f Nbrjk
Ztr`hd Kdkrmy nbr Ur`dsvkrsk Fb`ehdm?
Ztr`hd Kdkrmy nbr @xh`f Fb`ehdm?
()
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`de Wbhdt Fb`e `t hts lhe ‚ pbhdt bn fkdmtc ∡
∡
Nbr J`dthfkvkr I`r whtc `xh`f fb`ehdm
Ikdehdm Lblkdt?
K Xbudm‘s lbeufus
H Lblkdt bn Hdkrth`
Kx`jt Zbfuthbd?
ZZI L`x Eknfkjthbd whtc [EF tcrbumcbut hts fkdmtc
ZZI L`x Ikdehdm Lblkdt whtc [EF tcrbumcbut hts fkdmtc
ZZI L`x Eknfkjthbd whtc Wbhdt Fb`e `t hts lhe ‚ pbhdt bn fkdmtc
ZZI L`x Ikdehdm Lblkdt whtc Wbhdt Fb`e `t hts lhe ‚ pbhdt bn fkdmtc
ZZI L`x Eknfkjthbd whtc [EF tcrbumcbut hts fkdmtc `de Wbhdt Fb`e `t hts lhe ‚ pbhdt bn fkdmtc
ZZI L`x Ikdehdm Lblkdt whtc [EF tcrbumcbut hts fkdmtc `de Wbhdt Fb`e `t
hts lhe ‚ pbhdt bn fkdmtc
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MKDKY@F UYHMBDBLKUYHJ N[DJUHBDZ [ZKE HD YHU_ LKUCBE?
ZWYHDMZ?
Uck Ubt`f Wbtkdth`f Kdkrmy bn tck Ik`l?
[ Ztr`hd Kdkrmy
C ]bra Ebdk iy Kxtkrd`f Nbrjk
Ztr`hd Kdkrmy?
a Zthnndkss bn Zprhdm
ε Eknfkjthbd
]bra ebdk iy Kxtkrd`f Nbrjk?
N Nbrjk `jthdm bd tck sprhdm
u Ehspf`jklkdt
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Ehspf`jklkdt Nudjthbd?
Nbr 0E fhdk`r i`r kfklkdt
Nbr 0E qu`er`thj kfklkdt
[DHU HH ‚ BDK EHLKDZHBD@F NHDHUK KFKLKDU
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NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) =
D0 Zc`pk Nudjthbd `t dbek 0
u0 Ehspf`jklkdt `t dbek 0
D9 Zc`pk Nudjthbd `t dbek 9
u9 Ehspf`jklkdt `t dbek 9
D: Zc`pk Nudjthbd `t dbek :
u: Ehspf`jklkdt `t dbek 9
Zthnndkss L`trhx?
Nbr 0E fhdk`r i`r kfklkdt
Nbr 0E qu`er`thj kfklkdt
@ @rk` bn tck kfklkdt - ll9
K Xbudm‘s Lbeufus bn tck kfklkdt - D/ll9
f Fkdmtc bn tck kfklkdt - ll
Mkdkr`f Nbrjk Kqu`thbd?
Nbrjk vkjtbr (Mfbi`f)
Mfbi`f sthnndkss l`trhx Ehspf`jklkdt l`trhx
Yk`jthbd Nbrjk? Yk`jthbd nbrjk
Hn Uck Ibey Hs Zuiokjtke Ub Zkfn ‚ ]khmct?
Nbr 0E fhdk`r i`r kfklkdt
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Nbr 0E qu`er`thj kfklkdt
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@ @rk` bn tck kfklkdt - ll9
f Fkdmtc bn tck kfklkdt - ll
Ztrkss Bd Uck Kfklkdt?
NBY U@WKY WF@UK?
Nbr rkjt`dmuf`r jrbss skjthbd
@rk` < ]hetc * Uchjadkss
@t `dy pbhdt bn x
@rk` `t dbek 0?
@rk` `t dbek 9(@ppfhj`ifk bdfy nbr lhe ‚ pbhdt)?
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 8
( )
@rk` `t dbek :?
@vkr`mk `rk` bn kfklkdt 0? @vkr`mk `rk` bn kfklkdt 9?
Nbr jhrjuf`r jrbss skjthbd
UKLWKY@U[YK KNNKJUZ BD ZUY[JU[Y@F WYBIFKL?
Zthnndkss L`trhx?
@ @rk` bn tck kfklkdt - ll9
K Xbudm‘s Lbeufus bn tck kfklkdt - D/ll9
f Fkdmtc bn tck kfklkdt - ll
Uckrl`f Fb`e?
@ @rk` bn tck kfklkdt - ll9
K Xbudm‘s Lbeufus bn tck kfklkdt - D/ll9
ΰ Jbknnhjhkdt bn tckrl`f kxp`dshbd - /˟J Uklpkr`turk ehnnkrkdjk
Uckrl`f Ztrkss?
Uckrl`f Ztr`hd
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 0
Nbr kfklkdt 0
Yk`jthbd Nbrjk?
Yk`jthbd nbrjk Nbrjk vkjtbr (Mfbi`f) Mfbi`f sthnndkss l`trhx Ehspf`jklkdt l`trhx
ZWYHDMZ?
Zthnndkss L`trhx? a Zthnndkss bn tck sprhdm - D/ll
UY[ZZKZ?
Zthnndkss L`trhx?
@k @rk` bn tck kfklkdt - ll9
Kk Xbudm‘s Lbeufus bn tck kfklkdt - D/ll9
f k Fkdmtc bn tck kfklkdt - ll
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 00
Fkdmtc bn kfklkdt 0 ∟
Ztrkss?
Ztrkss bn kfklkdt 0
u l`trhx v`rhks `jjbrehdm tck kfklkdt
IK@LZ?
Ehspf`jklkdt Nudjthbd?
D0 Zc`pk Nudjthbd `t dbek 0
u0 Ehspf`jklkdt `t dbek 0
D9 Zc`pk Nudjthbd `t dbek 9
u9 Ehspf`jklkdt `t dbek 9
D: Zc`pk Nudjthbd `t dbek :
u: Ehspf`jklkdt `t dbek :
D5 Zc`pk Nudjthbd `t dbek 5
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 09
u5 Ehspf`jklkdt `t dbek 5
Zthnndkss L`trhx?
H Lblkdt bn Hdkrth` - ll5
K Xbudm‘s Lbeufus - D/ll9
F Fkdmtc bn tck ik`l - ll
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 0:
Dbe`f Nbrjks `de Ikdehdm Lblkdts?
Z.DbFB@EHDM
J@ZKN0 l0 N9 l9
0)
9)
:)
5)
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 05
Z.DbFB@EHDM
J@ZKN0 l0 N9 l9
;)
>)
=)
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 0;
JBDZU@DU ZUY@HD UYH@DM[F@Y KFKLKDU?
Zc`pk Nudjthbd?
@rk`?
Zc`pk Nudjthbd?
]ckd `ff tck sc`pk nudjthbd hs mhvkd
[DHU HHH ‚ U]B EHLKDZHBD@F NHDHUK KFKLKDU
@D@FXZHZ
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 0>
]ckd twb br bdk sc`pk nudjthbd hs mhvkd
Ztr`hd Ehspf`jklkdt L`trhx?
@ @rk` bn tck trh`dmuf`r kfklkdt - ll9
Ztrkss Ztr`hd Ykf`thbdschp L`trhx?
Nbr Wf`dk Ztrkss jbdehthbd?
Nbr Wf`dk Ztrkss jbdehthbd?
K Xbudm‘s Lbeufus - D/ll9
κ Wbhssbd‘s r`thb
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 0=
Zthnndkss L`trhx? Ztr`hd Ehspf`jklkdt l`trhx
Ztrkss ‚ Ztr`hd Ykf`thbdschp l`trhx
t Uchjadkss bn tck trh`dmuf`r kfklkdt - ll @rk` bn tck trh`dmuf`r kfklkdt - ll9
Ub Jbdvkrt Ur`jthbd Nbrjk Hdtb Dbe`f Nbrjk?
Dbe`f nbrjk - D Zurn`jk tr`jthbd - D / ll9
t Uchjadkss bn tck kfklkdt - ll
i Irk`etc bn tck kfklkdt - ll
@rk` bn tck trh`dmuf`r kfklkdt - ll9
Ub Jbdvkrt Wrkssurk Hdtb Dbe`f Nbrjk?
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 03
Hn shek o - a hs suiokjtke tb prkssurk hd x ehrkjthbd (dbek 9 & :)
{
}
Hn shek o - a hs suiokjtke tb prkssurk hd y ehrkjthbd (dbek 9 & :)
{
}
Hn shek h - o hs suiokjtke tb prkssurk hd x ehrkjthbd (dbek 0 & 9)
{
}
Hn shek h - o hs suiokjtke tb prkssurk hd y ehrkjthbd (dbek 0 & 9)
{
}
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 08
Hn shek h - a hs suiokjtke tb prkssurk hd x ehrkjthbd (dbek 0 & :)
{
}
Hn shek h - a hs suiokjtke tb prkssurk hd y ehrkjthbd (dbek 0 & :)
{
}
Nbrjk vkjtbr - D
Wrkssurk - D / ll9
t Uchjadkss bn tck kfklkdt - ll
F Fkdmtc bn tck kfklkdt - ll
Kfklkdt Ztrkss?
{
}
Dbrl`f strkss hd x ehrkjthbd - D/ll
9
Dbrl`f strkss hd y ehrkjthbd - D/ll9
Zck`r Ztrkss - D/ll9
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 9
u & v Dbe`f ehspf`jklkdt - ll
L`xhlul Dbrl`f Ztrkss?
Lhdhlul Dbrl`f Ztrkss?
Wrhdjhpfk @dmfk?
Kfklkdt Ztr`hd?
{
}
Dbrl`f str`hd hd x ehrkjthbd - D/ll9
Dbrl`f str`hd hd y ehrkjthbd - D/ll9
Zck`r Ztrkss - D/ll9
u & v Dbe`f ehspf`jklkdt - ll
UKLWKY@U[YK KNNKJUZ?
Hdhth`f Ztr`hd?
Nbr Wf`dk Ztrkss jbdehthbd?
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 90
Nbr Wf`dk Ztr`hd jbdehthbd?
ΰ Jbknnhjhkdt bn tckrl`f kxp`dshbd - /˟J Uklpkr`turk ehnnkrkdjk
κ Wbhssbd‘s r`thb
Kfklkdt Uklpkr`turk Nbrjk?
Ztr`hd Ehspf`jklkdt l`trhx Ztrkss ‚ Ztr`hd Ykf`thbdschp l`trhx Hdhth`f str`hd
t Uchjadkss bn tck trh`dmuf`r kfklkdt - ll
@rk` bn tck trh`dmuf`r kfklkdt - ll
9
@SHZ ZXLLKUYHJ KFKLKDUZ?
Zc`pk Nudjthbd?
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
http://slidepdf.com/reader/full/fea-question-with-class-notes-and-formula-book 355/374
EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 99
@rk`?
Ztr`hd Ehspf`jklkdt L`trhx?
@ @rk` bn tck trh`dmuf`r kfklkdt - ll9
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 9:
Ztrkss Ztr`hd Ykf`thbdschp L`trhx?
K Xbudm‘s Lbeufus - D/ll9
κ Wbhssbd‘s r`thb
Zthnndkss L`trhx?
Ztr`hd Ehspf`jklkdt l`trhx
Ztrkss ‚ Ztr`hd Ykf`thbdschp l`trhx
r Y`ehus bn tck trh`dmuf`r kfklkdt - ll @rk` bn tck trh`dmuf`r kfklkdt - ll9
Kfklkdt Ztrkss?
{}
Y`eh`f strkss - D/ll
9
Jhrjulnkrkdth`f strkss - D/ll9
Fbdmhtuehd`f strkss - D/ll9
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 95
Zck`r Ztrkss - D/ll9
u & w Dbe`f ehspf`jklkdt - ll
UKLWKY@U[YK KNNKJUZ?
Uckrl`f Nbrjk ^kjtbr?
Ztr`hd Ehspf`jklkdt l`trhx
Ztrkss ‚ Ztr`hd Ykf`thbdschp l`trhx
Ztr`hd l`trhx
r Y`ehus bn tck trh`dmuf`r kfklkdt - ll @rk` bn tck trh`dmuf`r kfklkdt - ll9
Ztr`hd?
ΰ Jbknnhjhkdt bn tckrl`f kxp`dshbd - /˟J Uklpkr`turk ehnnkrkdjk
YKJU@DM[F@Y KFKLKDU?
Zc`pk Nudjthbd?
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 9;
HZBW@Y@LKUYHJ KFKLKDU?
NB[Y DBEKE P[@EYHF@UKY@F?
Zc`pk Nudjthbd?
O`jbih`d L`trhx?
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 9>
Ztr`hd Ehspf`jklkdt L`trhx?
| |
Q \
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
http://slidepdf.com/reader/full/fea-question-with-class-notes-and-formula-book 360/374
EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 9=
Ztrkss Ztr`hd Ykf`thbdschp L`trhx?
Nbr Wf`dk Ztrkss jbdehthbd?
Nbr Wf`dk Ztrkss jbdehthbd?
K Xbudm‘s Lbeufus - D/ll
9
κ Wbhssbd‘s r`thb
Zthnndkss L`trhx?
Nbr Hsbp`r`lktrhj Pu`erhf`tkr`f Kfklkdt?
∦
Nbr D`tur`f Jb ‚ Brehd`tks
| |
Ztr`hd Ehspf`jklkdt l`trhx Ztrkss ‚ Ztr`hd Ykf`thbdschp l`trhx
| | Ektkrlhd`dt bn tck O`jbih`d l`trhx
λ,ν D`tur`f Jb ‚ Brehd`tks
t Uchjadkss bn tck kfklkdt - ll @rk` bn tck trh`dmuf`r kfklkdt - ll9
ZHS DBEKE P[@EYHF@UKY@F?
Zc`pk Nudjthbd?
( )
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 93
( )
( )
KHMCU DBEKE P[@EYHF@UKY@F?
Zc`pk Nudjthbd?
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 98
DHDK DBEKE P[@EYHF@UKY@F?
Zc`pk Nudjthbd?
M@[ZZH@D P[@EY@U[YK?
∐
]khmct nudjthbd ^`fuks bn tck nudjthbd `t prk ‚ ektkrlhdke s`lpfhdm pbhdts
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) :
Dulikr bn
pbhdts
d
Fbj`thbd
x h
]khmcts
wh
0
9
:
5
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
http://slidepdf.com/reader/full/fea-question-with-class-notes-and-formula-book 364/374
EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) :0
FBDMHU[EHD@F ^HIY@UHBD BN I@Y?Mkdkr`f Nhdhtk Kfklkdt Kqu`thbd? Zthnndkss l`trhx L`ss l`trhx
Kxtkrd`f Nbrjk
Ehspf`jklkdt l`trhx D`tur`f Nrkqukdjy - r`e / skj
Zthnndkss L`trhx?
@ @rk` bn tck kfklkdt - ll9
K Xbudm‘s Lbeufus bn tck kfklkdt - D/ll9
f Fkdmtc bn tck kfklkdt - ll
Fulpke L`ss L`trhx?
π [dht wkhmct ekdshty bn tck kfklkdt - D/ll:
@ @rk` bn tck kfklkdt - ll9
f Fkdmtc bn tck kfklkdt - ll
Jbdshstkdt L`ss L`trhx?
π [dht wkhmct ekdshty bn tck kfklkdt - D/ll:
@ @rk` bn tck kfklkdt - ll9
f Fkdmtc bn tck kfklkdt - ll
[DHU H^ ‚ EXD@LHJ @D@FXZHZ [ZHDM NHDHUK
KFKLKDU LKUCBE
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) :9
UY@DZ^KYZK ^HIY@UHBD BN IK@LZ?
Mkdkr`f Nhdhtk Kfklkdt Kqu`thbd?
Zthnndkss l`trhx
L`ss l`trhx Kxtkrd`f Nbrjk Ehspf`jklkdt l`trhx D`tur`f Nrkqukdjy - r`e / skj
Zthnndkss L`trhx?
H Lblkdt bn Hdkrth` - ll5
K Xbudm‘s Lbeufus - D/ll9
F Fkdmtc bn tck ik`l - ll
Fulpke L`ss L`trhx?
π [dht wkhmct ekdshty bn tck kfklkdt - D/ll:
@ @rk` bn tck kfklkdt - ll9
f Fkdmtc bn tck kfklkdt - ll
Jbdshstkdt L`ss L`trhx?
π [dht wkhmct ekdshty bn tck kfklkdt - D/ll:
@ @rk` bn tck kfklkdt - ll9
f Fkdmtc bn tck kfklkdt - ll
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) ::
KHMKD ^@F[KZ & KHMKD ^KJUBYZ?
Mkdkr`f Nhdhtk Kfklkdt Kqu`thbd?
Zthnndkss l`trhx
L`ss l`trhx Ehspf`jklkdt l`trhx Khmkd v`fuks
Ektkrlhd`dt Lktcbe?
Zthnndkss l`trhx
L`ss l`trhx Khmkd v`fuks
D`tur`f Nrkqukdjy bn I`r Kfklkdt?
Nrkqukdjy bn i`r - Cz L`ss l`trhx Khmkd v`fuks D`tur`f Nrkqukdjy - r`e / skj
UY@DZHKDU ^HIY@UHBD @D@FXZHZ?
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) :5
CK@U UY@DZNKY?
BDK EHLKDZHBD@F CK@U UY@DZNKY BD ]@FF?
Mfbi`f Zthnndkss L`trhx ]ckd ]`ff Zuiokjtke Ub Jbdeujthbd?
Nbrjk vkjtbr (Mfbi`f)
Mfbi`f sthnndkss l`trhx
Uklpkr`turk l`trhx
@ @rk` bn tck w`ff - l
9
a Uckrl`f jbdeujthvhty bn w`ff - ]/lA
f Fkdmtc bn tck w`ff - l
Mfbi`f Zthnndkss L`trhx ]ckd ]`ff Zuiokjtke Ub Jbdeujthbd &
Jbdvkjthbd?
@ @rk` bn tck w`ff - l9
a Uckrl`f jbdeujthvhty bn w`ff - ]/lA
f Fkdmtc bn tck w`ff - l
c Ck`t tr`dsnkr jbknnhjhkdt - ]/l9A
@tlbspckrhj `hr tklpkr`turk - A
[DHU ^ ‚ @WWFHJ@UHBDZ HD CK@U UY@DZNKY & NF[HE
LKJC@DHJZ
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) :;
BDK EHLKDZHBD@F CK@U UY@DZNKY BD NHD?
Mfbi`f Zthnndkss L`trhx ]ckd Nhd Zuiokjtke Ub Jbdeujthbd &
Jbdvkjthbd?
@ @rk` bn tck nhd - l9
W Wkrhlktkr bn tck nhd - l
a Uckrl`f jbdeujthvhty bn nhd - ]/lA
f Fkdmtc bn tck nhd - l
c Ck`t tr`dsnkr jbknnhjhkdt - ]/l9A @tlbspckrhj `hr tklpkr`turk - A
P Ck`t Mkdkr`thbd - ]
U]B EHLKDZHBD@F CK@U UY@DZNKY?
Zthnndkss L`trhx ]ckd Zuiokjtke Ub Jbdeujthbd?
Ztr`hd Ehspf`jklkdt l`trhx L`tkrh`f Wrbpkrty l`trhx
t Uchjadkss bn tck trh`dmuf`r kfklkdt - ll
@rk` bn tck trh`dmuf`r kfklkdt - ll9
Ztr`hd Ehspf`jklkdt L`trhx?
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) :>
@rk`?
L`tkrh`f Wrbpkrty L`trhx?
a xx & a yy Uckrl`f jbdeujthvhty - ]/lA
Zthnndkss L`trhx ]ckd Zuiokjtke Ub Jbdeujthbd?
a Uckrl`f jbdeujthvhty - ]/lA @rk` bn tck trh`dmuf`r kfklkdt - ll9
Zthnndkss L`trhx ]ckd Zuiokjtke Ub Jbdeujthbd & Jbdvkjthbd?
Hn shek 0-9 hs suiokjtke tb jbdvkjthbd
Hn shek 9-: hs suiokjtke tb jbdvkjthbd
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) :=
Hn shek 0-: hs suiokjtke tb jbdvkjthbd
]ckrk ∟ ∟
∟ Ztr`hd Ehspf`jklkdt l`trhx L`tkrh`f Wrbpkrty l`trhx
t Uchjadkss bn tck trh`dmuf`r kfklkdt - ll @rk` bn tck trh`dmuf`r kfklkdt - ll9
f Fkdmtc bn tck shek suiokjtke tb jbdvkjthbd - l
c Ck`t tr`dsnkr jbknnhjhkdt - ]/l
9
AZthnndkss L`trhx ]ckd Zuiokjtke Ub Jbdvkjthbd?
Hn shek 0-9 hs suiokjtke tb jbdvkjthbd
Hn shek 9-: hs suiokjtke tb jbdvkjthbd
Hn shek 0-: hs suiokjtke tb jbdvkjthbd
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) :3
Nbrjk ^kjtbr Nbr Jbdvkjthbd?
Hn shek 0-9 hs suiokjtke tb jbdvkjthbd
Hn shek 9-: hs suiokjtke tb jbdvkjthbd
Hn shek 0-: hs suiokjtke tb jbdvkjthbd
@tlbspckrhj `hr tklpkr`turk - A
t Uchjadkss bn tck trh`dmuf`r kfklkdt - ll
f Fkdmtc bn tck shek suiokjtke tb jbdvkjthbd - l
c Ck`t tr`dsnkr jbknnhjhkdt - ]/l9A
Nbrjk ^kjtbr Hn Ck`t Zburjk Hs Mhvkd?
P Ck`t Zburjk - ]/l
:
^ ^bfulk bn trh`dmuf`r kfklkdt - l
:
@ @rk` bn trh`dmuf`r kfklkdt - l9
t Uchjadkss bn tck trh`dmuf`r kfklkdt - ll
Nbrjk ^kjtbr Hn Ck`t Nfux Hs Mhvkd?
Hn shek 0-9 hs suiokjtke tb ck`t nfux
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) :8
Hn shek 9- : hs suiokjtke tb ck`t nfux
Hn shek 0-: hs suiokjtke tb ck`t nfux
NF[HE NFB] @D@FXZHZ?
BDK EHLKDZHBD@F NF[HE NFB] @D@FXZHZ?
Zthnndkss L`trhx?
@ @rk` bn tck jrbss skjthbd bn tck phpk - l9
a xx Wkrlk`ihfhty jbknnhjhkdt - l/s
f Fkdmtc - l
Nfuhe ^kfbjhty?
Nbr kfklkdt 0
Ztr`hd Ehspf`jklkdt l`trhx
a xx Wkrlk`ihfhty jbknnhjhkdt - l/s
p pbtkdth`f - l9/s
^bfulktrhj Nfbw Y`tk?
P Nfbw r`tk - l
:
/sv ^kfbjhty - l/s
@ @rk` bn tck jrbss skjthbd bn tck phpk - l9
7/22/2019 FEA QUESTION WITH CLASS NOTES AND FORMULA BOOK
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@
NHDHUK KFKLKDU @D@FXZHZ NBYL[F@ IBBA IX @ZCBA A[L@Y .Y (@W / LKJC) 5
U]B EHLKDZHBD@F NF[HE NFB]?
Zthnndkss L`trhx?
Ztr`hd Ehspf`jklkdt l`trhx L`tkrh`f Wrbpkrty l`trhx
t Uchjadkss bn tck trh`dmuf`r kfklkdt - l @rk` bn tck trh`dmuf`r kfklkdt - l9
Ztr`hd Ehspf`jklkdt L`trhx?
@rk`?
L`tkrh`f Wrbpkrty L`trhx?
a xx & a yy Wkrlk`ihfhty jbknnhjhkdt - l/s
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EKW@YULKDU BN LKJC@DHJ@F KDMHDKKYHDM LK9:;: ‚ NK@ Zthnndkss L`trhx?
a xx & a yy Wkrlk`ihfhty jbknnhjhkdt - l/s @rk` bn tck trh`dmuf`r kfklkdt - l9
Mr`ehkdt L`trhx?
Ztr`hd Ehspf`jklkdt l`trhx
p pbtkdth`f l`trhx - l9/s
^kfbjhty Mr`ehkdt L`trhx? L`tkrh`f Wrbpkrty l`trhx - l/s Mr`ehkdt l`trhx
^bfulktrhj Nfbw Y`tk?
Nfuhe nfbw r`tk - l:/s
t Uchjadkss bn tck trh`dmuf`r kfklkdt - l