امتحانات الشهادة الثانوية العامة فرع: العلوم العامة

وزارة التربية والتعليم العالي المديرية العامة للتربية دائرة االمتحانات امتحانات الشهادة الثانوية العامة فرع: العلوم العامة االسم: الرقم: مسابقة في مادة الرياضيات المدة أربع ساعات عدد المسائل: ست مالحظة: - يسمح باستعمال آلة حاسبة غير قابلة للبرمجة او اختزان المعلومات او رسم البيانات. - يستطيع المرش ح اإلجابة بالترتيب الذي يناسبه ( دون االلتزام بترتيب المسائل الواردة في المسابقة(. دورة العام االستثنائي ة الثالثاء آب I- ( poits) I th tabl blow, oly o of th proposd aswrs to ach qustio is corrct. Writ dow th umbr of ach qustio ad giv, with justificatio, th aswr corrspodig to it. Qustios Aswrs a b c d Th particular solutio f() of th diffrtial quatio y '' + y = such that f () = ad f ' (π) = is si si cos si cos For all >, t dt lim = If f is a odd fuctio,cotiuous ovr IR ad such that f ()d, th f ()d Lt M with affi z (z ) b a variabl poit i th compl pla rfrrd to a dirct orthoormal systm. If z z is ral, th M movs o: th circl with ctr O ad radius cpt th poit with affi th -ais cpt th poit with affi th li with quatio y = th y-ais (;)

II- ( poits) I th spac rfrrd to a dirct orthoormal systm O; i, j,k, cosidr th poits E ( ; ; ), F( ; ; ), ad th li (d) with paramtric quatios = t, y = t +, z = t whr t is a ral paramtr. Dot by (P) th pla dtrmid by th poit F ad th li (d). ) Vrify that E is o (d). ) Show that y + = is a quatio of (P). ) Cosidr i th pla (P) th circl (C) with ctr F ad radius FE. a- Dtrmi th coordiats of H, th orthogoal projctio of F o (d). b- Dtrmi th coordiats of L, th scod poit of itrsctio of (d) ad (C). c- Writ a systm of paramtric quatios of th bisctor of th agl EFL. ) Lt (Q) b th pla cotaiig (d) ad prpdicular to (P). Dot by ( ) th prpdicular bisctor of th sgmt [EL] i th pla (Q). Writ a systm of paramtric quatios of ( ). III- ( poits) A ur cotais whit balls ad black balls. A gam cosists of two coscutiv drawigs as follows: A ball is slctd radomly i th first drawig. If th ball slctd is whit, it is put back i th ur; othrwis, it is kpt outsid th ur. Two balls ar slctd simultaously ad radomly i th scod drawig. Cosidr th followig vts: W: «Th ball slctd i th first drawig is whit» E: «Th balls slctd i th scod drawig ar whit» F: «Th balls slctd i th scod drawig ar black» G: «Th balls slctd i th scod drawig ar of diffrt colors». ) Calculat P(E/W) ad P(E / W). Dduc that P(E) = 6 9. ) Calculat P(F) ad P(G). ) Kowig that th balls slctd i th scod drawig hav th sam color, calculat th probability that th ball slctd i th first drawig is black. ) I this part, w mark - poits for ach black ball slctd, ad + poits for ach whit ball slctd. Dot by S th sum of poits markd for th two balls slctd i th scod drawig. Calculat th probability that S is positiv.

IV- ( poits) Th pla is rfrrd to a dirct orthoormal systm (O ; i, j). Dot by (C) th circl with ctr I(; ) ad radius, ad by (d) th li with quatio y =. Lt L( ; ) b a variabl poit o (C). Dot by N th orthogoal projctio of L o (d) ad by M th midpoit of sgmt [LN]. ) Writ a quatio of (C). ) a- Dtrmi th coordiats of M i trms of α ad β. b- As L movs o (C), prov that M movs o th llips (E) with quatio c- Draw (E). ) Lt (P) b th parabola with vrt V (; ) ad focus F;. a- Show that y is a quatio of (P). b- Draw (P) i th sam systm as (E). ) a- Calculat d. y. b- Dduc th ara of th rgio that is abov th -ais ad boudd by (E) ad (P). ) Lt G ( ; ) b a poit o (P) ad ( ) th tagt at G to (P). Dot by H th poit of (P) whr th tagt to (P) is prpdicular to ( ). Prov that G, H ad F ar colliar. V- ( poits) Cosidr a dirct quilatral triagl ODA with sid qual to. Lt R b th rotatio with ctr O ad agl. Dot by B = R (A), D'= R (D). Lt C b th poit so that D = R(C). (C is th pr-imag of D) ) a- Mak a figur. b- Show that O is th midpoit of [CD'] ad that BC=. ) a- Justify that (AC) is prpdicular to (BD) ad that AC = BD. b- Show that (AD) is paralll to (BC). ) Dot by E th poit of itrsctio of lis (AC) ad (BD). Lt h b th dilatio with ctr E that trasforms A oto C. a- Dtrmi h (D). b- Calculat th ratio of h. ) Lt L b th midpoit of [AD] ad F = h (L). Show that O, E, F ad L ar colliar. ) Lt R b th rotatio with ctr E ad agl. Cosidr S = h R. a- Dtrmi th atur of S ad so its lmts. b- Prov that S (A) = B.

VI- (7 poits) A- Lt h b th fuctio dfid o IR as h(). Dot by (C) its rprstativ curv i a orthoormal systm. ) a- Dtrmi lim h(). b- Dtrmi lim h() ad show that th li (d) with quatio y = is a asymptot to (C). ) a- Calculat h () ad st up th tabl of variatios of h. b- Draw (C) ad (d). c- Dduc that for all. B- Lt f b th fuctio dfid as f (). Dot by C its rprstativ curv i aothr orthoormal systm. ) Show that f is dfid ovr IR. ) Dtrmi th asymptots to C. ) Vrify that f () ad st up th tabl of variatios of f. ) a- Writ a quatio of (T), th tagt toc at th poit E with abscissa. b- Vrify that ( ) f. c- Study, accordig to th valus of, th rlativ positios of C with rspct to (T). d- Draw C ad (T). C- For all atural umbrs, dfi th squc ) Show that th squc (u ) is icrasig. ) a- For, vrify that f (). b- Is th squc (u ) covrgt? Justify. (u ) as u f d.

Aswr Ky- Math SG Scod Sssio - QI Aswrs N y= Acos+Bsi. f () = A=. f ()= Asi+Bcos. f '(π)= B =. b) t dt (Idtrmiat). L H.R. lim t dt lim lim. f ()d f ()d f ()d f ()d. a) z z z ral th which givs z.z z z z.z z z z z z ad z z. Thrfor, M movs o th -ais cpt poit with affi. b) c) QII Aswrs N For t = ; E is a poit o (d). (t ) (t + ) + = + + = ; F yf thus F (P) th th giv quatio is that of (P). a FHt ;t ; t ; FH V(d) ; (t ) (t ) t ; t th 9 H ; ;. 9 9 9 9 b H is th midpoit of EL thus L ; ;. 9 9 9 c 8 (FH) is th bisctor of EFL. FM mfh m, y m, z m. 9 9 9 A dirctor vctor of th prpdicular bisctor is P (; ;) ad th prpdicular bisctor passs i poit H; th systm of paramtric quatios is: ; y ;z whr λ is a ral paramtr. 9 9 9

QIII Aswrs N PE 7 C W C ; P E P(F) P(F W) P(F W) 7 7 P W 6 C 6 ; P(E) P(W E) P(W E). W C 7 7 9 ; P(G) P(E) P(F) 6 6. 7 C G. P W G P(W).P 7 C 8 W G PG P(G) 6 8 7 P(S ) P(S ) 7 7 QIV Aswrs N y. L ;, N ; M ;. a b sic L is o (C). Thus M movs o (E). c a p SF thus p =.y-ais is th focal ais, thrfor OR: Th dirctri is th li (D) with quatio y y. y. Dist(R,(D)) = RF with R(; y) (P) y y y. b S figur i c. a d. b Ara = ab d. y = f() = thus f '( ) =, f '( H) thus H ;. 6 GH GF thrfor G, H t F ar colliar.

QV Aswrs N a b CÔD DÔD 9 th C, O, D ar colliar ad OC = OD = OD, th O is th midpoit of [CD ]. OB = OC = OD thrfor triagl CBD is right at B. Pythagoras givs : CB CD' BD'. a R(C) = D t R(A) = B (AC) is prpdicular to (BD) ad AC = BD. b R(D) = D, R(A) =B thus AD prpdicular to BD ad (BC) is prpdicular to (BD ), th (AD) ad (BC) ar paralll. a h(a) = C ad (AD) ar paralll to (BC), thrfor h(d) = B. b BC KDA So: K. F midpoit of [BC] ; E, F ad L ar colliar. (OF) ad (OL) ar prpdicular to (BC) O, F ad L ar colliar. a S is th similitud S (E,, ). b S (A) = hr (A) h(r (A)) h(d) Bsic EAD is a right isoscls triagl. QVI Aswrs N a lim h() lim. A b lim h() lim ( ) ad (d) is a asymptot to (C). lim[h() y] lim, a h ().

b h() lim thus(c) has a vrtical asymptotic dirctio. c Th miimum of h() is, so h(), th, thus. h() ; so thus D f lim f () lim, lim f () lim. Doc ls droits d'équatios y = t y = sot du asymptots à C. f (). a y = +. b ( )( ) ( ) f () ( ) ( ). c For ; f() ( ),, C itrcpt (T). For ; f() ( ) For = ; C is abov (T). C is blow (T). B d C. u u f d f d f d f d f d kowig f th a b f d ; So (u ) is icrasig. f () thus f () for all. f ()d d si c th u ; lim u lim ; lim u thrfor u is divrgt.