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(1)×÷x[×{~{mæÛ F |Ý}ÿ×÷[×{N²yy¹z{î|»|\²
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"#" wr»sx²{{ÿ(i)×÷~oßþ~¿º~cgòüñóø(ii)% > 0 ~x}ÿ×÷{Ý % ýU~Û²÷{{¸ÿ(iii)×÷~ÝßÛ×{·y» % {ttv~Û×ý{_²{{ÿ(iv)×÷~ÞîÛ×{·y» & {ttv~Û×ý{_²{{ÿ(v)$ = 2 " "'( xz»sx²{{ÿ(vi)ßÿÛ×~o_|_ut|\{s~Û×|[o×{z»sx²{{ÿ~ÿ"!

s~[o×~| T ²n±¸ÿ(2)×÷x[×{~{mæÛ|Ý{zt|\²x»ÿßÿÛ×~o_|_ut|\ÿ×÷~Û×[o×{z»ÿ}~|_(1)(vi)~txÿyvyvz»{ÿ_(1)~¯Ûx~Õt²ÿ½{ÿvuxx²{x¸ÿ÷ÿÿ ÿ ÿ ïO~_t{x¸ÿßzYu²Ûÿwv}wv¸tÿ(1)ÿòû~vó_ÿ_Ûu Rÿ²_m_xwvÿïO~þßözÞ{ÿ{¹ÿ~µ´¿û{¹z»ñ·²x»ÿÞ{ÿ oþ² T1 {ßqz|¹ÿwÛ| P1 {¹ P2 {YWy»{o©Þ{ÿ oþ| T1 {¹ T2 {ÿwÛ| P2 {¹ P3 {YWy»íñ©Þ{ÿ oþ² T2 {ßqz|¹ÿwÛ| P3 {¹ P4 {YWy»{ow~
Þ{ÿ oþ| T2 {¹ T1 {ÿwÛ| P4 {¹ P1 {YWy»íñw~k~_t{x¸ÿ

(i)

Þ{ÿ{ztvÿÕø¸ýû¾ü~YWxñ·|zyßÏ² R, T1, P1, P2 ²sv}¼~¼è{ÿ(ii)vó_²_NÛwYWu{»|\~ÿñ Cv xÿwÛNÛwYWu{»|\~ÿñ Cp ~{ Cp = Cv + R ~·Ï|r»ÿ³ = Cp / Cv xwv, Þ{ÿxÿw P 1-³ T ³|NÛxz»sx²{{ÿ(iii)jþ²¸óøýôüÿ~þ²oþxwvÿs~ñ·~µ´¿û~ë²÷{{¸ÿ~ÿÞ{ÿ{¹ÿ|÷o~y~øV{ÿßy»{{{ÿ(iv)s~ñ·~ñû÷²n±¸ÿ(v)µ´¿û~TÞ{{ztvñ·|zyßÏÿïO~d{¹g~vqÿy~ñÛ·u~YW{ÿßy»{ÿÞ{ÿ, ÿ, ÿ, ÿ{ttvÿ}¼~¼x¸ÿd Õø¸ýû¾üe ¸ó¿ûôüf øûðûûô~ëu¸ýû¾ü
g ¾ö¹~ëu¸ýû¾ü

(vi)_ (q) w}t ³ ÿ[ß[V[w 5/3 {z»sx²ÿ_~V[Û×~»ù¸º{{ÿÿ ÿ ÿkúü¸{ÿ(2)N{r»~Ôwó¸²_ (1) ~¸vzvó_~ñ·xwvò÷ûWy»ÿó¸~oß{¹_t{U{svjW»Ûöí÷ÿ÷ÿÿ{u¼ÿt~÷sÿA<Dÿÿ_ (1) ~µ´¿û~vqÿtz¼{~Þ{{ÿßy»zZ~¶sºxw×}²èyÿñ·|ÌvßÏ W |ó¸Õø~Û×²ß×w,÷sÿA<Dÿxöîy»ýU{¸ v ²²t
~|Û×²z»xy»ÿO|~o² TDÿN|~o² TUÿ÷|o² TS xy».u¹{,ûó¸{[Ofrº{ìuu¼»ñ Q {ttv

Q ? v (TS - TD)ûó¸Õø~Û×|[Ofrº{yvÛ×¸ýû¾ü · {ttv

· ?v3

xtv·Ï|rº,W = · ²îyÛ~ök{r»xy»ÿTS = 301 K, TD = 300 K, TU = 251 K ~x}ÿv = 35 m/s wrs.k~_t{x¸ÿ

÷ÿÿ

(i)

÷ÿ~÷s D ÿO|~zZ|ÿNÛ~oþ TD ²ßqz|¹ÿó¸~oß{U{vÞ{²èyÿs¼_ (1) ~Þ{ÿ{¹ÿ~vqÿy¼{ÿßy»{x¸ÿ

(ii)

W = · wr»sx²{}w, v, TSÿTD z¸s TU ~~·Ï²{{.

(iii)

TD x TU }~~~wÿTS | 304 K wr»|\~ v ²n±¸.

(iv)

TS x TD }~~~wÿTU | 236 K wr»|\~ v ²n±¸.

ÿ Ā ÿßw~ügm_~ÿkSözm¼²x»ÿsswÿm¼~ÿ{ÿ×÷ÿ{ÿr, », zĀ²}t»ÿ
m¼~þ~ÿ×ßÿrĀýUrV �㗆�㗅 ÿ_Ûÿ»ĀýUrV �㗆�㗰 ÿz ýUrV �㗆�㗆 |ÿ}¼~¼ÿ�㗆�㗅 = 0ÿ�㗆�㗰 = �㗆ÿ �㗆�㗆 = 0 xè{»x}ÿïO~_t{x¸ÿsswÿ�㗆 ÿr ~ÿr»tÿr xf t ~~·uwr»ÿßþ² �㗰ÿüg÷² �㗰 ÿwÛ² �㗅 xy»ÿnþù¿øû~ z ýUrV² �㗱 xy»ÿïOÿ�㗱 ²nþx|vĀÿ_Ûqy»ÿßwr¼k~}_²}t¸ÿ}_r, » ýU~úó¸û¹øü¿¹~_ÿ_Û²qĀÿ�㗱�㗆�㗅"�㗆�㗅 �㗆�㗰 "�㗆�㗅"�㗆�㗅 �㗆�㗰21 �㗱�㗅 �㗰�㗆�㗅 2 �㗱�㗆�㗰+ �㗆�㗅++ �㗆�㗆
2=2+ m&�㗆�㗅 2 2 2 2}�㗱�㗆"�㗅�㗅 "�㗰"�㗆�㗅�㗰 �㗱�㗅 �㗰�㗅�㗅 �㗱�㗰"�㗆�㗰 �㗆�㗰 "�㗆�㗰"�㗆�㗰 �㗆�㗅 �㗆�㗰
1 �㗱�㗅 �㗰�㗆�㗰 2 �㗱�㗆�㗅�㗱�㗆�㗰+ �㗆�㗅++ �㗆�㗆+=2+ m&�㗆�㗰 2 2 + 2}�㗱�㗆"�㗅�㗅 "�㗰"�㗆�㗅�㗰�㗅 �㗱�㗰 �㗰�㗅�㗅 �㗱�㗰
&c(A)

(B)

�㗱 2 1 �㗱
1 �㗱 2�㗱 2+++�㗱�㗅 2 �㗅 �㗱�㗅 �㗅 2 �㗱�㗰 2 �㗱�㗆 2mù¿øû �㗂 ~Þîÿ�㗁�㗅 , �㗁�㗰 , �㗁�㗆 : r, », z ýU~[Où¿øûĀÿ1 �㗱�㗆�㗆 �㗱�㗅�㗆�㗰�㗱�㗆�㗅 �㗱�㗆�㗆
1 �㗱�㗅�㗆�㗰 �㗱�㗆�㗅
' × �㗂 = m2} �㗁�㗅 + m2} �㗁�㗰 + m2} �㗁�㗅 �㗱�㗰�㗱�㗆�㗱�㗆�㗱�㗅�㗅 �㗱�㗅�㗱�㗰 �㗆

(1)
(2)
(3)

(4)

(C)

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= 0 xz»sx²{{ÿ�㗱�㗰nþ �㗱 {ttvÿk~d_|rºûtsx²{{ÿ�㗱�㗱 �㗰 �㗱 2 �㗱 1 �㗱�㗱= (+)
�㗱�㗆 �㗰 �㗱�㗅 2 �㗅 �㗱�㗅d1
m¼Û~xy»ÿ �㗱 = �㗃�㗅 �㗅 ~x}~m¼|ÿnzwnÿ �㗆 ? �㗅 Āx[_özÞîÿ�㗆 ? �㗅 Ā~}\¿{xz»sx²{{ÿwÿ�㗃 ßu~Ûuÿ�㗃 b 0 Āÿ
n uuwr»ÿÿ Ā ÿkúü¸{ÿ(5)m¼ßÛ~xy»ÿf{ tÿt > 0Ā{ztv�㗆 =�㗮
�㗰�㗅 2[1 2 exp (2)]2�㗰�㗅4�㗰�㗆ewr»x}ÿk~_t{x¸ÿwÿ k~Ûuÿ�㗮 > 0Āwrºÿlim �㗆 = 0 wr»ÿ�㗅³0(i)4�㗰�㗆�㗰

0 < �㗅 j :~m¼ÿßö{ÿ[_özÞîxz{»{ÿ}¼x²nzwnxz{»{ÿyâ²ð{wvx¸ÿßwr¼�㗅 �㗆 j 1 + �㗆 (|�㗆| j 1) ²}t¸ÿ(ii)r»f{ t 1 {zq»mV_~ë²ÿjþ² rÿ~þ² �㗆 xwv÷{{¸ÿzzÿ÷wÿf{ t {ztv»sx²}tv¸tÿ(iii)�㗅�㗆5�㗰�㗆
= 0 xz» �㗅 = �㗅0 | �㗅0 j : �㗰 xz�㗅�㗅_(ii)~f{ t 1 ~ök{¹f|}Þwÿf{ t 2 xzsxy»ÿ�㗆 |y~¸v{YWy»{²®ny»xx²{ÿf{ t 2 {zq»mV_~ë²_(ii)~÷{ï}òx¸ÿ}~ÿÿ~~Þ²Yx»zywvÿÿt~m
V_~Õt|nÿxz»¸v{ý{²y»sxÿÿ ÿ ÿ÷ 1 ~¸v{ÿzoÿÿ÷õ0ÿù÷ý0ĀwÿÌ³ó÷óµÿßÿûß C Āx÷÷ R ²^ÿwçÿy»ÿÿ{ÿ×÷ÿ{ÿ (r, ö, z) ²x»ÿer , eö , ez }¼~
¼ r, ö, z ýU~[Où¿øûwr»ÿ³ó÷óµ~×} PAÿPBÿ[ß aĀ~oßÿz þN~ù (0, 0, d ) z¸sßùwþÛu¼vt»ÿ³ó÷óµ~}z¸s^ÿ~÷÷ 0 wr
»xwÿ~ÿ³ó÷óµÿ÷÷ÿ^ÿ~´óÀ¿¿ó¹z¸s÷÷x^ÿ~ßÿûßSV_uÿqw}»xy»ÿ^ÿ~o{¯q¹´óÁ² OFF {wvÿ} PA { +Q0ÿ} PB { 2Q0ÿQ0 > 0Ā~ÿ÷²Nxÿf{ t = 0 {¹´óÁ² ON {wÿïO~_t{x¸ÿ¹´óÁzßU3^ÿaßU1PA (0, 0, d )

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ö÷SC
(r = a, 0 < z < d )RCPB(0, 0, 0)ùX (a, ö, z)÷÷³ó÷óµ^ÿ÷ÿ(1)f{ t = 0 {³ó÷óµ|Äxvt»ßÿ¸ýû¾ü²n¸ÿ(2)f{ tÿt > 0Āw~} PA ~ÿ÷ Q(t) ²n¸ÿ

(3)÷÷ R wí¿u¼»¸õüûñV2ÿV ÷÷{{{»ÿwĀ²f{ t = 0 {¹>~ÿRöwVwÿ}¼|_(1)wn³ó÷óµ~ßÿ¸ýû¾üxNõy»sx²{{ÿ~ÿ}~{vöow²}{x¸ÿ(4)PAÿPB w~¼³ó÷óµ~_|÷ÿ r = aÿ0 < z < d Ā²ö÷ SC xy»ÿPAÿPB ~ÿ||Nùxzyxÿö÷ SC N~ù X (a, ö, z) {zq»f{ tÿt > 0ĀQ (t )
e xz»ÿ~ÿ| E = 2
E |¼¶¹~ýW²îysx²ú{¸ÿë{ÿõ 0ð a 2 z¼¶¹~ýW²ù}wßxö÷²÷{y»sxÿÿ ÿ ÿkúü¸{ÿ(5)ö÷ SC N~ù X (a, ö, z) w~f{ tÿt > 0Ā~ù| B = 2

ý0 dQ ( t )eö xz»ÿ2ð a dt

B |²óúüûûþ¿¹¶·û~ýW²îysx²ú{¸ÿë{ÿ²óúüûûþ¿¹¶·û~ýW²ù}wö÷xöÿ²÷{y»sxÿ(6)_(2)ÿ(4)ÿ(5)~}ÿ²x{ÿö÷ SC {¹_{U{vý´óö³óÀù¿øûEôB

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222B öEô Bd ö õ0 E
÷÷ dV = ò+÷ ndS + ò ( E ÷ j) dVS ýUdt òU ÷ø 22ý0 ÷ø0deÿý´óö³óÀ~ÛvĀfwÿU ßÿS ß U ²öö÷ÿn ö÷ S ~ýÿù¿øûÿE 
ÿ|ÿB ù|ÿj ÿmßþwr»ÿℎÿ ÿ ÿ k~_t{x¸ÿℎ ÷ùó¿Ûu (6.6 × 10−34 m2 kg s−1 , ℏ = 2!
ÿÿ𝜀0 

zo~ÿ÷ÿ𝑚 ÿ[~îßÿ𝑒 }ÿ÷wr»ÿ(1)|}ß[~oß{¹~Ýö² 𝑟 xy»xÿÿßök{r»ÿ[~î×·u 𝜓(𝑟)

 𝑟 ~~·uxwvd_~¸v{èu¼»ÿ𝐶 k~Ûuÿÿk~_t{x¸ÿßwr¼pO~}_²}t¸ÿ𝜓(𝑟) = 𝐶 e2 𝑎 ,𝑎 = 𝑚𝑒02𝑟(}_)

4𝜋𝜀 ℏ2∞

∫0∞

∫0(i)

𝑟𝑒−𝑟 𝑑𝑟 = 1

𝑟2 𝑒−𝑟 𝑑𝑟 = 2

|Wgþ²}tv 𝐶 ²n±¸ÿ𝑝̂𝑒Û×¸ýû¾ü 𝑇 ̂ = 2𝑚
xýöó·óû¸ýû¾ü 𝑉 ̂ = − 4𝜋𝜀~ßtÿ𝑟2(ii)d20z¹s{ÿ}¼¹~Üwr»y¸ýû¾ü²n±¸ÿssw 𝑝̂ Û×ß𝜕 2 𝜕ÿ[wrºÿ𝑝̂2 = −ℏ2 (𝑟12 𝜕𝑟[𝑟 𝜕𝑟]) ²}tv¸t.

(2)

OúÛg·Ï²|}ß[{ù}wvÿÿ[~ÎÓ[ß²x»ÿÿ[ÎÓ[ß 𝑟 ~wÿù~ß{x¿±¹¼vt»xy»ÿ~ÿÿ[~Û×ß~}u² 𝑝 xy¼pÿOúÛg·Ï 𝑟 ; 𝑝 j / wßö{èÿu¼»xy»ÿk~_t{x¸ÿ
(i)ÿ[|²ty¸ýû¾ü 𝐸 ² 𝑚, 𝑒, 𝜀0 , ℏ, 𝑟 ²}tvè{ÿ(ii)

ÿ[|ÿßök{r»x}ÿ𝑟 = 𝑚𝑒02 xz»sx²{{ÿ(iii)OúÛg·Ï{ÿutvÿ|}ß[~ÿ[|ÿ[{½q¿~ztvu² 80

4𝜋𝜀 ℏ2_{þw®n{¸ÿ(3)úøú¶ðß[|óÿök 2𝑃3/2 {¹ÿßök 2𝑆1/2 {÷ûy»x}ÿoßî÷

| 5.9 × 10−7 m ~ÿ²\y»ÿßwxs½ÿs~óÿök~Wÿ} ∆𝑡 1.0 × 10−8 sÿ~ÿ~o×u~_ÿëw_ÿ 1.6 × 107 s−1 xnu¼ÿ(i)ÿ{ëw_|r»sxÿóÿ¸ýû¾ü 𝐸 {_|r»sx²owy(ii)∆𝑡 x ∆𝐸 OúÛg·Ï²îysx²{{ÿ»ÿs~_ ∆𝐸 ~t²oûu_ 2 iwn±¸ÿÿ ÿ ÿÿkSöîöÿÿ{ÿ O 2 �㗆�㗆�㗆 {zq»g_~ÿ_Y{·y»k~_t{x¸ÿ
(1)

g_|_Û²ØqvYy»xÿ}~Õø{_Û|zwßÏ{þ_y»rz¸ýû¾üÿ�㗄 xy»Ā|Äx¹¼»ÿ�㗄 {·y»k~_t{x¸ÿsswÿ_Û{¸ºg_Õ{xßÛxrz² �㗰�㗅�㗅 x �㗅�㗅�㗅 ÿw, �㗅, �㗅, �㗅, �㗅 = �㗆, �㗆, �㗆Āxwÿ}¼¹~{ÿ~·Ï|rºûÿÿÜ}²}tv �㗰�㗅�㗅 = �㗃�㗅�㗅�㗅�㗅 �㗅�㗅�㗅 ÿwÿ�㗃�㗅�㗅�㗅�㗅 ÛuĀxþq»²~xy»ÿ(i) ÷ÿ{{y¸v{ÿT¿~÷u|�㗅�㗆ÿ�㗅�㗆ÿ�㗅�㗆 ~ÿ_öý_{ÿwvÿ{³íßÛ �㗰�㗆�㗆 z¸s �㗰�㗆�㗆 ~|xÿ�㗅�㗅�㗆�㗆 z¸s �㗅�㗅�㗆�㗆 qrz|ßòw|\{ttvx»ÿs~x}~[O_rº~rz¸ýû¾ü~ßV²n±¸ÿ(ii) yyv~ßÛrV|x»|\~[O_rº~rz¸ýû¾ü~ßV²ÿrzx}~ßVz¸s �㗃�㗅�㗅�㗅�㗅 {¸svÿÿÜ}²}tv{{ÿ(iii) ëwökÿ�㗅�㗅�㗅 = 0Ā{¹ÿr»rz~ökÿ�㗅�㗅�㗅 b 0Ā{YWwx}ÿs~

g_{Äx¹¼»[O_rº~rz¸ýû¾ü²ÿrzrVxßÛrV²}tvþ}O{ÿ

(iv) d_{{yßÛxrz~·Ï_²}tvÿ �㗄 = 0 {z»gþ²{{ÿ�㗰�㗅�㗅 =�㗃�㗰(�㗯 �㗅 + �㗅�㗅�㗅 )1 + �㗰 1 2 2�㗰 �㗅�㗅 �㗅�㗅dsswÿ�㗃 ôóÀ÷ÿ �㗰 ý²½óÿÿ�㗯�㗅�㗅 ¿ýýó»ü~÷û¿²{yÿ(v) rz¸ýû¾ü~gî²}tvÿ �㗃 |×ºv»ÿö²{{ÿ÷1�㗉 = (�㗆�㗉 , �㗆�㗉 , �㗆�㗉 ) YO²{yÿÿ ÿ ÿkúü¸{ÿ(2)_Û²qw|\ÿ{ýg_~Û×ý{_e_~ºþq»ÿk~_t{x¸ÿ�㗰�㗱 2 �㗉= (�㗰+�㗰)�㗯(�㗯 ; �㗉) + �㗰�㗯 2 �㗉�㗱�㗆 2esswÿ�㗉 YOù¿øûÿ�㗰 Úî~ßþÿ�㗰 , �㗰 ùñ~Ûu²{yÿ(i) e_ÿo¿~~gz¸sÞî²×»sxwf_xg_w{u¼»î×ý{_~{Yw}»ÿssw �㗰 x �㗴 }¼~¼¹»ùüxù¿øûwr»ÿf_z¸sg_²^qÿ�㗱 2 �㗰 �㗰+2�㗰 2=�㗯 �㗰�㗱�㗆 2�㗰�㗱 2 �㗴 �㗰 2= �㗯 �㗴�㗱�㗆 2�㗰

fg(ii) f_xg_w{u¼»î×~{ÿ²ÿ�㗰 z¸s �㗴 | �㗉 xy~¸vz·Ï{r»{{wþwvÿ}¼~¼ÿy¸ÿÿ ÿ ÿ(1)(2)

k~_t{x¸ÿ

k~ÿVý{_²ëqÿzzÿ_ (ii) ßënë²n±»sxÿ

(i)dy y 2 1=dxxy(ii)ù dy1
ü dx = 6 y1 2 3 y2 2 7 y3
ü
ü dy2
= 2 y1 + 2 y2 + y3
údx
ü
ü dy3
ü dx = 5 y1 2 3 y2 2 6 y3û

÷ 1 ~¸v{÷ß 2aÿÿß 2b ~}Ö{Õçy»÷ý~÷~gt²ÿù°ùó¸õ~oÛ_uý²}tvn±¸ÿwÿa, b k~Ûuxy»ÿ

÷1>

(3)

I c + e2 ax dx xy»x}ÿ I 2 =22>Ãwr»sx²{{ÿwÿa k~Ûux
ay»ÿ

(4)

r»øÕ²tÛ±v{¹ey»~w~f t |guV_ p (t ) = » exp ( 2» t ) {ßvxy»ÿwÿ » > 0, t g 0 xy»ÿ(i)øÕ²tÛ±v{¹f{ T ~w{e|ÿs¹ztú÷²n±¸ÿ

(ii)

øÕ²tÛ±v{¹f{ T ~we|ÿs¹z{sx~gþ~Owÿ}~ßÿf{ T+ &T ~w~{e|ÿs»ú÷²n±¸ÿ

(iii)

øÕ²tÛ±v{¹ey»~w~Wf²n±¸ÿ