M2-BAC 2011 VARIANTA 25.docx

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    Subiectul I

    1. Să se arate că numărul {tex}(1-i)^24{/tex} este real.Rezolvare: om scrie numărul su! "ormă tri#onometrică. $ie{tex}z%1-i{/tex}.&tunci

    {tex}r%'z'%srt{a^2*!^2}%srt{1^2*(-1)^2}%srt{2}{/tex}+iar ima#inea lui z este un ,unct in caranul ,rin urmare{tex}0^%2,i-arctan le"t'"rac{!}{a}ri#t'%2,i-arctan 1%2,i-"rac{,i}{4}%"rac{3,i}{4}{/tex}.$orma tri#onometrică a numărului com,lex z este:{tex}z%r(cos 0^*isin 0^)%srt{2}le"t(cos "rac{3,i}{4}*isin "rac{3,i}{4}ri#t){/tex}iar{tex}z^24%r^24(cos 240^*isin 240^)%{/tex}{tex}%(srt{2})^24le"tcos le"t(24cot "rac{3,i}{4}ri#t)*isin le"t(24cot "rac{3,i}{4}ri#t)ri#t5%2^{12}(cos 42,i*isin 42,i)%2^{12}cos (6*2cot21,i)*isin (6*2cot21,i)5%2^{12}in R{/tex}.7rin urmare numărul at este e#al cu {tex}2^{12}{/tex} 8i este un număr real.

     

    2. Să se rezolve 9n mulimea numerelor reale ecuaia {tex}"rac{;x-1}{x*1}*"rac{x*1}{2x-1}%;{/tex}.Rezolvare: {/tex}{tex}?x^2-;x-2x*1*x^2*2x*1%;(2x^2*2x-x-1)=e"tri#tarro>{/tex}{tex}3x^2-;x*2%?x^2*;x-;=e"tri#tarro>{/tex}{tex}3x^2-;x*2-?x^2-;x*;%6=e"tri#tarro>{/tex}{tex}x^2-?x*@%6{/tex}.&vem{tex}

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    Hxistă B numere naturale e ouă ci"re cu ci"rele ientice: 11+22+;;+44+@@+??+33+II 8i BB. &vem {tex}(BB-B%)B6{/tex}e cazuri ,osi!ile (cGte numere naturale e ouă ci"re sunt) 8i (B6-B%I1) e cazuri "avora!ile (cGte numere naturale eouă ci"re i"erite sunt).7rin urmare{tex}7(&)%"rac{I1}{B6}%"rac{B}{16}{/tex}.

     

    @. Să se calculeze lun#imea meianei in & a triun#iului &JK une &(-2+-1) J(2+6) 8i K(6+?).Rezolvare: $ie D miElocul se#mentului JK. &tunci{tex}xC{D}%"rac{xC{J}*xC{K}}{2}%"rac{2*6}{2}%1{/tex}+{tex}FC{D}%"rac{FC{J}*FC{K}}{2}%"rac{6*?}{2}%;{/tex}.&tunci{tex}&D%srt{(FC{D}-FC{&})^2*(xC{D}-xC{&})^2}%srt{(;*1)^2*(1*2)^2}%srt{1?*B}%srt{2@}%@{/tex}.{u}%moverri#tarro>{i}*;overri#tarro>{E}{/tex} 8i{tex}overri#tarro>{v}%(m-2)overri#tarro>{i}-overri#tarro>{E}{/tex}. Să se etermine mL6 ast"el 9ncGt vectorii{tex}overri#tarro>{u}{/tex} 8i {tex}overri#tarro>{v}{/tex} să 0e ,er,eniculari.Rezolvare: ectorii {tex}overri#tarro>{u}{/tex} 8i {tex}overri#tarro>{v}{/tex} sunt ,er,eniculari acă 8i numai

    acă ,rousul lor scalar este nul aică{tex}overri#tarro>{u}cot overri#tarro>{v}%6=e"tri#tarro>{/tex}{tex}uC{1}vC{1}*uC{2}vC{2}%6=e"tri#tarro>{/tex}{tex}m(m-2)*;cot (-1)%6=e"tri#tarro>{/tex}{tex}m(m-2)-;%6=e"tri#tarro>{/tex}{tex}m^2-2m-;%6{/tex}.&vem{tex}{v}{/tex} sunt,er,eniculari.

     

    Subiectul II 

    1. Se consieră matricea {tex}&in DC{2}(R){/tex} {tex}& %le"t( {!e#in{arraF}{cc}2 N 2 1 N 1 en{arraF} } ri#t){/tex}.a) Să se arate că există {tex}ain R{/tex} ast"el 9ncGt {tex}&^2%acot &{/tex}.!) Să se calculeze {tex}(&-&^t) {̂266B}{/tex}.c) Să se rezolve ecuaia {tex}O^@%&{/tex} {tex}Oin DC{2}(R){/tex}.Rezolvare:a) &vem:{tex}&^2%&cot &%{/tex}

    {tex}%le"t( {!e#in{arraF}{cc}2 N 2 1 N 1 en{arraF} } ri#t)cot le"t( {!e#in{arraF}{cc}2 N 2 1 N 1 en{arraF} } ri#t)%{/tex}{tex}%le"t( {!e#in{arraF}{cc}2cot2*2cot1 N 2cot2*2cot1 1cot2*1cot1 N 1cot2*1cot1 en{arraF} } ri#t)%

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    le"t( {!e#in{arraF}{cc}? N ? ; N ; en{arraF} } ri#t)%{/tex}{tex}%;cot le"t( {!e#in{arraF}{cc}2 N 2 1 N 1 en{arraF} } ri#t)%;cot &{/tex}.

    &8aar ,entru {tex}a%;in R{/tex} avem {tex}&^2%acot &{/tex}.!) &vem:{tex}&^t %le"t( {!e#in{arraF}{cc}2 N 1 2 N 1 en{arraF} } ri#t){/tex}8i{tex}&-&^t %le"t( {!e#in{arraF}{cc}2 N 2 1 N 1 en{arraF} } ri#t)- le"t( {!e#in{arraF}{cc}2 N 1 2 N 1 en{arraF} } ri#t)% le"t( {!e#in{arraF}{cc}

    6 N 1 -1 N 6 en{arraF} } ri#t){/tex}.&,oi{tex}(&-&^t)^2 %le"t( {!e#in{arraF}{cc}6 N 1 -1 N 6 en{arraF} } ri#t)cot le"t( {!e#in{arraF}{cc}6 N 1 -1 N 6 en{arraF} } ri#t)%{/tex}{tex}le"t( {!e#in{arraF}{cc}6cot6*1cot (-1) N 6cot1*1cot6 (-1)cot6*6cot (-1) N (-1)cot1*6cot6 en{arraF} } ri#t)% le"t( {!e#in{arraF}{cc}

    -1 N 6 6 N -1 en{arraF} } ri#t)%{/tex}{tex}%(-1)cot C{2}{/tex}.&tunci{tex}(&-&^t)^{266B}%(&-&^t)̂ {266I}cot (&-&^t)%(&-&^t)̂ 25^{1664}cot (&-&^t)%{/tex}{tex}%(-1)cot C{2}5^{1664}cot (&-&^t)%{/tex}{tex}%C{2}cot (&-&^t)%{/tex}{tex}% le"t( {!e#in{arraF}{cc}6 N 1 -1 N 6 en{arraF} } ri#t){/tex}.c) $ie {tex}Oin DC{2}(R){/tex} {tex}O %le"t( {!e#in{arraF}{cc}x N F z N t en{arraF} } ri#t){/tex} o soluie a ecuaiei{tex}O^@%&{/tex}.xt%Fz%P{/tex}.

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    {tex}t%"rac{P}{x}{/tex} 8i {tex}z%"rac{P}{F}{/tex}.7rin urmare{tex}O %le"t( {!e#in{arraF}{cc}x N F "rac{P}{F} N "rac{P}{x} en{arraF} } ri#t){/tex}.H"ectuGn calculele o!inem

    {tex}O^2%"rac{x^2*P}{x}cot le"t( {!e#in{arraF}{cc}x N F "rac{P}{F} N "rac{P}{x} en{arraF} } ri#t){/tex}aică{tex}O^2%"rac{x^2*P}{x}cot O{/tex}.&nalo#{tex}O^;%O^2cot O%"rac{x^2*P}{x}cot O^2%le"t("rac{x^2*P}{O}ri#t)^2cot O{/tex}8i{tex}O^@%O^;cot O^2%{/tex}{tex}%le"tle"t("rac{x^2*P}{x}ri#t) 2̂cot Ori#t5cot le"tle"t("rac{x^2*P}{x}ri#t)cot Ori#t5%{/tex}{tex}%le"t("rac{x^2*P}{x}ri#t)^;cot O^2%{/tex}{tex}%le"t("rac{x^2*P}{x}ri#t)^4cot O{/tex}.&tunci{tex}O^@%&=e"tri#tarro>{/tex}{tex}le"t("rac{x^2*P}{x}ri#t)^4cot le"t( {!e#in{arraF}{cc}

    x N F "rac{P}{F} N "rac{P}{x} en{arraF} } ri#t)% le"t( {!e#in{arraF}{cc}2 N 2 1 N 1 en{arraF} } ri#t)=e"tri#tarro>{/tex}{tex}!e#in{cases}le"t("rac{x^2*P}{x}ri#t)^4cot x%2 le"t("rac{x^2*P}{x}ri#t)^4cot F%2 le"t("rac{x^2*P}{x}ri#t)^4cot "rac{P}{F}%1 le"t("rac{x^2*P}{x}ri#t)^4cot "rac{P}{x}%1 en{cases}{/tex}.nlocuin {tex}le"t("rac{x^2*P}{x}ri#t)^4{/tex} in ultima ecuaie 9n ,rima ecuaie a sistemului o!inem:{tex}"rac{x^2}{P}%2=e"tri#tarro>{/tex}{tex}x^2%2P=e"tri#tarro>{/tex}{tex}x%srt{2P}{/tex} 9ntrucGt x nu ,oate 0 ne#ativ u,ă cum rezultă in ,rima ecuaie a sistemului.

    nlocuin valoarea #ăsită ,entru x 9n ,rima ecuaie a sistemului o!inem:{tex}le"t("rac{2P*P}{srt{2P}}ri#t)^4cot srt{2P}%2=e"tri#tarro>{/tex}{tex}"rac{I1P^4}{4P^2}cot srt{2P}%2=e"tri#tarro>{/tex}{tex}srt{2P^@}%"rac{I}{I1}'^2=e"tri#tarro>{/tex}{tex}2P^@%"rac{2^?}{;^I}=e"tri#tarro>{/tex}{tex}P%"rac{2}{;srt@5{23}}{/tex}.7rin urmare{tex}x%srt{"rac{4}{srt@5{;^I}}}%"rac{2}{srt@5{srt{;^I}}}%"rac{2}{srt@5{I1}}{/tex}.

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    $ie {tex}a !in D{/tex}. &tunci:{tex}a#e6 !#e6{/tex}8i{tex}e^a#e1 e^!#e1{/tex}.Rezultă atunci că{tex}e^a*e^!#e2=e"tri#tarro>{/tex}{tex}e^a*e^!-1#e1=e"tri#tarro>{/tex}{tex}ln (e^a*e^!-1)#e6=e"tri#tarro>{/tex}

    {tex}a!in D{/tex}.!) A este asociativă acă 8i numai acă {tex}"orall a ! c in D (a!)c%a(!c)=e"tri#tarro>{/tex}{tex}"orall a ! c in D ln(e^a*e^!-1)5c%aln (e^!*e^c-1)5=e"tri#tarro>{/tex}{tex}"orall a ! c in D ln (e^{ln (e^a*e^!-1)}*e^c-1)%ln (e^a*e^{ln (e^!*e^c-1)}-1)=e"tri#tarro>{/tex}{tex}"orall a ! c in D ln (e^a*e^!-1*e^c-1)%ln (e^a*e^!*e^c-1-1)=e"tri#tarro>{/tex}{tex}"orall a ! c in D ln (e^a*e^!*e^c-2)%ln (e^a*e^!*e^c-2){/tex} (&).c) $ie {tex}min T m#e2{/tex} 8i {tex}ain D{/tex}. &tunci{tex}a^2%aa%ln (e^a*e^a-1)%ln (2e^a-1){/tex}+{tex}a^;%a^2a%{/tex}{tex}%ln (2e^a-1)5a%ln (e^{ln (2e^a-1)}*e^a-1)%ln (2e^a-1*e^a-1)%ln (;e^a-2){/tex}+{tex}a^4%a^;a%{/tex}{tex}%ln (;e^a-2)5a%ln (e^{ln (;e^a-2)}*e^a-1)%ln (4e^a-;){/tex}+...{tex}a^m%ln (me^a-(m-1)){/tex}.&8aar avem{tex}a^m%2a=e"tri#tarro>{/tex}

    {tex}ln (me^a-(m-1))%2a=e"tri#tarro>{/tex}{tex}ln (me^a-m*1)%ln e^{2a}=e"tri#tarro>{/tex}{tex}me^a-m*1%e^{2a}=e"tri#tarro>{/tex}{tex}(e^{a})^2-me^a*m-1%6{/tex}.$ie {tex}e^{a}%t t#e1{/tex}.Ultima ecuaie evine atunci{tex}t^2-mt*m-1%6{/tex}{tex}{/tex}

    {tex}aC{n}in (6+1)Ri#tarro> aC{n*1} in (6+1) n in T^ =e"tri#tarro> {/tex}{tex} aC{n} in (6+1) Ri#tarro> aC{n}(1-srt{aC{n}}) in (6+1) n in T^ {/tex}.ntr-aevăr acă {tex}aC{n} in (6+1) n in T^ {/tex} atunci{tex}6MaC{n}M1 n in T^ {/tex}8i{tex} 6M1-srt{aC{n}}M1 n in T^ {/tex}8i ,rin urmare{tex}6MaC{n}(1-srt{aC{n}})M1 n in T^ {/tex}. 7(n*1) n in T^ {/tex}.

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    aică {tex} aC{n} in (6+1) "orall n in T^{/tex}.!) Virul {tex} (aC{n})C{n in T^}{/tex} este strict escrescător acă 8i numai acă{tex} aC{n*1} M aC{n} "orall n in T^ =e"tri#tarro> {/tex}{tex} aC{n}(1-srt{aC{n}}) M aC{n} "orall n in T^ =e"tri#tarro> {/tex}{tex} aC{n}(1-srt{aC{n}}) - aC{n} M 6 "orall n in T^ {/tex}{tex} -aC{n}srt{aC{n}} M 6 "orall n in T^ {/tex} (& acă inem cont e ,ct. a)).c) om emonstra că:{tex} aC{n}^2 M aC{n} - aC{n*1} "orall n in T^{/tex}.

    ntr-aevăr{tex} aC{n}^2 M aC{n} - aC{n*1} "orall n in T^ =e"tri#tarro>{/tex}{tex} aC{n}^2 M aC{n} - aC{n}(1-srt{aC{n}}) "orall n in T^ =e"tri#tarro>{/tex}{tex} aC{n}^2 M aC{n}srt{aC{n}} "orall n in T^ ':aC{n}L6 =e"tri#tarro>{/tex}{tex} aC{n} M srt{aC{n}} "orall n in T^{/tex}8i 9ntrucGt am!ii mem!ri ai ine#alităii sunt ,ozitivi avem{tex} aC{n} M srt{aC{n}} "orall n in T^ '^2 =e"tri#tarro>{/tex}{tex} aC{n}^2 M aC{n} "orall n in T^ ':aC{n}L6 =e"tri#tarro>{/tex}{tex} aC{n} M 1 "orall n in T^{/tex} (& c". ,ct. a)).7rin urmare{tex} aC{n}^2 M aC{n} - aC{n*1} "orall n in T^{/tex}.{/tex}{tex} "rac{4}{;} cot "rac{;}{4(x^2*x*1)}%"rac{1}{x^2*x*1} "orall x in R{/tex} (&).!) &ria cerută este:{tex} &%int C{6}^{1} #(x)x%{/tex}{tex}%intC{6}^{1} (2x*1)"(x)x%{/tex}{tex}intC{6}^{1} "rac{2x*1}{x^2*x*1}x%{/tex}{tex}%ln (x^2*x*1)5 'C{6}^{1}%{/tex}{tex} %ln (1^2*1*1)-ln (6^2*6*1)%{/tex}{tex}%ln ;-6%ln ;{/tex}.

    c) Kalculăm mai 9ntGi{tex}C{n}%intC{-n}^{n} "(x)x n in T^{/tex}.&vem:{tex}C{n}%intC{-n}^{n} "rac{1}{x^2*x*1}x%{/tex}{tex}%intC{-n}^{n} "rac{1}{(x*"rac{1}{2})^2*1-"rac{1}{4}}x%{/tex}{tex}%intC{-n}^{n} "rac{1}{(x*"rac{1}{2})^2*("rac{srt{;}}{2})^2}x%{/tex}{tex}%le"t"rac{2}{srt{;}} arctan "rac{2}{srt{;}}le"t(x*"rac{1}{2}ri#t)ri#t5 ri#t'C{-n}^{n}%{/tex}{tex}%le"t"rac{2srt{;}}{;} arctan "rac{2srt{;}(2x*1)}{;}ri#t'C{-n}^{n}%{/tex}{tex}%"rac{2srt{;}}{;}le"t(arctan "rac{srt{;}(2n*1)}{;}-arctan "rac{srt{;}(-2n*1)}{;}ri#t)%{/tex}{tex}"rac{2srt{;}}{;}le"t(arctan "rac{srt{;}(2n*1)}{;}*arctan "rac{srt{;}(2n-1)}{;}ri#t){/tex}.&tunci

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    {tex}limC{n to in"tF} C{n}%{/tex}{tex}%linC{n to in"tF} "rac{2srt{;}}{;}le"t(arctan "rac{srt{;}(2n*1)}{;}*arctan "rac{srt{;}(2n-1)}{;}ri#t)%{/tex}{tex}%"rac{2srt{;}}{;}le"t("rac{,i}{2}*"rac{,i}{2}ri#t)%{/tex}{tex}%"rac{2,i srt{;}}{;}{/tex}.