اهتحانات الشهادة الثانىية العاهة الفرع : علىم عاهة هسابقت في هادة الزياضياث االسن: الودة أربع ساعاث

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1 وزارة التربية والتعلين العالي الوديرية العاهة للتربية دائرة االهتحانات اهتحانات الشهادة الثانىية العاهة الفرع : علىم عاهة الدورة العادية للعام هسابقت في هادة الزياضياث االسن: الودة أربع ساعاث عدد الوسائل:سث الرقن: ارشاداث عاهت :- يسوح باستعوال آلت حاسبت غيز قابلت للبزهجت او اختزاى الوعلوهاث او رسن البياناث - يستطيع الوزشح اإلجابت بالتزتيب الذي يناسبه دوى االل تزام بتزتيب الوسائل الواردة في الوسابقت I- ( poits) I the table below, oly oe of the proposed aswers to each questio is correct Write dow the umber of each questio ad give, with justificatio, the aswer correspodig to it swers N Questios a b c a 5 a x si x dx a a i e arg i The roots of the equatio z + z = + i are: + i ad i + i ad + i + i ad i If u z z i, the iu iz iz iz iz iz iz x 5 lim x e x 7π If arcsi si 5, the = II- ( poits) Cosider the cube BCDEFGH represeted H G i the adjacet figure The space is referred to a direct orthoormal system ( ; B, D,E ) Desigate by I the midpoit of EF ad by K the ceter of the E K D I F C square DHE ) a- Calculate the area of triagle IG b- Calculate the volume of the tetrahedro BIG B c- Deduce that the distace from poit B to the plae (IG) is ) a- Write a equatio of the plae (FH ) b- The lie (CE) cuts the plae (FH) at a poit L Calculate the coordiates of L c- Prove that L belogs to the lie (FK) What does the poit L represet for the triagle FH?

2 III-( poits) Cosider two urs U ad U U cotais four red balls ad three gree balls U cotais two red balls ad oe gree ball - We draw at radom a ball from U ad we put it i U, the we draw at radom a ball from U Desigate by X the radom variable that is equal to the umber of red balls remaiig i the ur U after the two precedig draws ) Prove that the probability P(X = ) is equal to 9 ) Fid the three values of X ad determie the probability distributio of X B- I this part, each red ball carries the umber ad each gree ball carries the umber We choose at radom a ur the we draw simultaeously ad at radom two balls from the chose ur Cosider the followig evets: E: «The chose ur is U» F: «The sum of the umbers carried by the two draw balls is equal to» ) a- Calculate the probabilities P(F/E) ad P(F/ E ) b- Deduce that P(F) = ) Desigate by G the evet «The sum of the umbers carried by the two draw balls is equal to» Calculate P(G) IV- (poits) I the plae referred to a orthoormal system (O ; i, j ), cosider the lie (d) with equatio x = ad the parabola (P) with focus O ad directrix (d) ) a-show that a equatio of (P) is y = 8x + Determie the vertex S of (P) b- Draw (P) c- Let D be the regio bouded by (P) ad the axis of ordiates Calculate the area of D d- Calculate the volume of the solid geerated by the rotatio of D about the axis of abscissas ) Let ( ; 8) be a poit o (P) a- Write a equatio of the taget (T ) at to (P) b-the lie (O) itersects (P) agai at a poit B Calculate the coordiates of B ad write a equatio of the taget (T B ) at B to (P)

3 c-verify that (T ) ad (T B ) are perpedicular ad that they itersect o the directrix of (P) ) Let M(x o ; y o ) be a poit o (P), distict from S N is the orthogoal projectio of M o the taget through S to (P) The perpedicular through N to the lie (MS) itersects the axis of abscissas at I Show that the abscissa of I is idepedet of x o ad y o V- ( poits) F cm D E cm C cm 5cm G I the figure above, BCD ad EFG are two direct rectagles so that ( B, D ) = (mod ) B S is the direct plae similitude that trasforms B oto E ad C oto F; ; T is the traslatio with vector EF f is the similitude defied by T o S ) a- Determie the ratio k ad a agle of S b- Determie the image of D by S c- Prove that is the ceter of S ) a- Fid f(b) ad f() b- Specify the ratio ad a agle of the similitude f c- Costruct the ceter W of f ) The complex plae is referred to a direct orthoormal system ( ; B, E ) a- Write the complex form of f b-deduce the affix of poit W ) Let F be the image of F by S, ad for ay ozero atural iteger, let F + be the image of F by S Determie the values of so that, F ad F are colliear

4 VI- (7 poits) Cosider the fuctio f defied over ] ; 5[ by f(x) = l(5 x) Desigate by (C) the represetative curve of f i a orthoormal system (O ; i, j ) f (x) ) a- Calculate limf (x), lim f (x) ad lim Iterpret, graphically, the results thus obtaied x5 x x x b- Set up the table of variatios of f over ] ; 5[ ) a- Determie a equatio of the taget ( T ) to ( C) at the poit with abscissa b- Draw ( T ) ad ( C) c- The curve (C) itersects the lie with equatio y = x at a poit with abscissa Verify that < < ) f has a iverse fuctio f Desigate by (C') the represetative curve of i the same system of ( C) a- Prove that the taget ( T ) to ( C) is also taget to ( C' ) b- Draw (C') f ) Let h be the fuctio defied o ] ; 5[ by h(x) = (5 x) l(5 x) a- Verify that h '(x) f (x) ad deduce a atiderivative of the fuctio f b- Desigate by the area of the regio bouded by (C), the axis of abscissas ad the two lies with equatios x ad x = Prove that ( ) 5) Let I be the iterval[ ; ] a- Prove that f (I) is icluded i I b- Prove, for all x i I, that f '(x) c- Deduce that, for all x i I, f (x) x ) Cosider the sequece U defied by: U ad, for all, U f (U ) a- Prove by mathematical iductio that, for all, U belogs to I b- Show that, for all, U U c- Prove, for all, that U ad deduce that the sequece U is coverget

5 وزارة التربية والتعلين العالي هشروع هعيار التصحيح اهتحانات الشهادة الثانىية العاهة الفرع : علىم عاهة الدورة العادية للعام هسابقة في هادة الرياضيات Q Solutio G aa, is zero c The itegral of a odd fuctio o i e b i i x + iy + x + y = + i, the y = ad x + y + x= ; x + x = b Thus, x = or x= Therefore, z = + i or z = +i or by verificatio arg arg arg i u z z i z z i ; iu iz iz b x t t t x t t t e a 7 arcsi si sice ; ad 7 si si 5 5 c 5 lim x e lim t e lim e Q Solutio G I( ;;), G (;;) ; IG ( ; ; ) ad I (-/ ; ; -) a IG I i j k rea of (IG ) / / B ( ; ; ) ; B(IG I ) b The volume of the tetrahedro BIG is V B(IG I ) Let d be the distace of B to plae (IG ) c V rea of (IG ) d d Thus, d a F H i j k (FH ) : x y z b c OR M F H (C E ) : x t; y t;z t (CE ) (FH ) : t + t + t = So, t ad L( ; ; ) FL ; ; ad FK ; ;, the FL = FK Thus, L belogs to [FK ], the media i triagle FH Therefore, L is the ceter of gravity of triagle FH Q Solutio G X = occurs whe we draw oe red ball from U the oe red ball from U or oe gree ball 9 5 from U the oe gree ball of U Thus: P(X ) 7 7 The values of X are, ad X = occurs whe oe red ball remais i the ur U That is drawig a gree ball from U 5 the a red ball from U Thus, P(X ) 7 X = occurs whe we draw a red ball from U ad a gree ball from U

B a 9 P(X ) Or : P(X ) 7 7 7 To get a sum, we should draw a red ball ad a gree ball P(F / E) ; P(F / E) C 7 C B b B 7 P(F) P F E P F E P(E) P(F / E) P(E) P(F / E) 7 G occurs whe we draw two gree

6 B a 9 P(X ) Or : P(X ) To get a sum, we should draw a red ball ad a gree ball P(F / E) ; P(F / E) C 7 C B b B 7 P(F) P F E P F E P(E) P(F / E) P(E) P(F / E) 7 G occurs whe we draw two gree balls which is oly possible from the ur U sice the ur C U cotais oly oe gree ball, thus P(G) C Q Solutio G a MO d(m (d)) ; MO d (M (d)); x y (x ) ; y 8x y 8x ; (y ) 8(x ) The vertex is S ( ; ) 7 b c The product of the slopes of the tagets c d a b 8x dx 8x u V y dx 8x dx u yy' = 8 ; y' ; y' The equatio of y (T ) is y = x+5 (O) : y x The abscissas of the poits of itersectio of (O) ad (P) verifies the equatio : x 8x, x 9x 8 ; 9 x' = ad x" = = x B B( ; ) The equatio of (T B) is y + = y' B (x + ) ; y = x 5 (T ) ad (T B) is equal to thus (T ) ad (T B) are perpedicular Moreover, x+5 = x 5 ; x = ad y =, thus (T ) ad (T B) itersect o the directrix (d) Let I( a ; ) We have N( ; y ) ; MS( x ; y ) ; MI(a ; y ) MS MI = ; ( x )(a ) y ; ( x )(a ) 8(x ) ; (x )( a) a = ( x ) Therefore, the abscissa of I is idepedet of x ad y ;

7 Q5 Solutio G S = sim(k ; ) ; B S E ; C S F a EF = k BC ; k = = ; = BC, EF = BC, D + D, EF = (), 5 b Triagle EFG is similar to triagle BCD ad i the same sese Thus, S(D)= G c S(BCD) is the direct rectagle EFG, S() =, the is the ceter of S a f(b) = T(S(B)) = T(E) = F ; f() = T(S()) = T() = G b f = similitude of ratio ad agle c WB, WF = ad W, WG = ; W is the poit of itersectio of the two circles of diameters [BF] ad [G] other tha G a f : M(z) M'(z') ; z' = iz + b ; zg = = iz + b ; b = The complex form of f is z' = iz b z W = izw ; z W iz W = 9; z W = i i F,F = F,F + F,F + + F, F = ( ), F ad F are colliear for ( ) = k ; the = k+ where k is a atural iteger( is odd) Q Solutio G f (x) lim f ( x), limf (x) ad lim x x5 x x a 5 The straight lie of equatio x = 5 is a asymptote to (C) ad the curve (C) has a horizotal asymptotic directio at f '(x) with x 5 over ] ; 5[ x - x 5 f (x) b + f(x) - a (C) cuts x ' x at the poit ( ; ) ad y ' y at the poit B( ; 5) (T) is taget at to (C) ; (T ) :y x

8 y (T) 5 ' (D) b (C) B E O 5 x 5 c f() = l > ad f() = l < the (C ') a (C ') is symmetric to (C ) with respect to the straight lie (D ) of equatio y = x (C ) cuts x ' x at ( ; ) ad admits (T ) as a taget at By symmetry with respect to (D ), (C ') cuts y ' y at the poit '( ; ) ad admits the symmetric of (T ) with respect to (D ), as a taget at ' But (T ) (D ), thus (T ) is the symmetric of itself with respect to (D ) s a result, (T ) is the taget at ' to (C ') Check the figure part b b (C ) ad (C ') are symmetric with respect to the straight lie of equatio y = x a h' ( x) (5 x) ; so h '( x) f ( x), therefore, F(x) = h(x) x b ( ) f(x)dx x h(x) ( 5 ) ( 5 ) But, (5 ) ; thus ( ) 5 u 5a f is cotiuous ad strictly decreasig ; the f(i)= [f(),f()]=[l,l5] I 5b f '( x) with x 5 ; the f '( x) x 5 5 x but x, so 5 x 5 ad Cosequetly, f '(x) 5 5 x 5c Usig the mea value iequality, we ca write : f (x) f ( ) x with f ( ) therefore, f (x) x a U ; the U I If U I, the f ( U ) f ( I) Hece, U I b U I, the f ( U ) U Cosequetly, U U U ad ; the, U ad U If U, the U U Or by usig multiplicatio ad c 5 simplificatio lim x ; the lim U ad lim U x x

اهتحانات الشهادة الثانىية العاهة الفرع : علىم عاهة هسابقت في هادة الزياضياث الودة أربع ساعاث

9 وزارة التربية والتعلين العالي الوديرية العاهة للتربية دائرة االهتحانات عدد الوسائل : سث اهتحانات الشهادة الثانىية العاهة الفرع : علىم عاهة هسابقت في هادة الزياضياث الودة أربع ساعاث االسن: الرقن: الدورة