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

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1 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 اإلستثنائية للعام 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 الوارد في الوسابق ة I- points) In the table below, only one of the proposed answers to each question is correct Write down the number of each question and give, with justification, the answer corresponding to it N Questions Answers a b c Let f be the function defined over by f ) and let g be the function defined over IR by g) - The domain of definition of g f is: IR IR IR IR ; p q is equivalent to : p q) p) q) q) p) A, M and N are three distinct points of respective affies i, zand z If z iz i, then, triangle AMN is : equilateral semiequilateral right isosceles With distinct points situated on a circle, we can determine: 7 triangles triangles 5 triangles 5 The function f defined over ] ;] by f ) has an inverse function g defined by : g g g If z sin icos, then arg z 5 7

2 II- points) The space is referred to a direct orthonormal system O ; i, j, k ) Consider the point A- ; ; ), the plane P) of equation y + z = and the straight line D) defined by the system = m ; y = m ; z = m m is a real parameter) ) a- Verify that A does not belong to P) and calculate the distance from A to P) b- Prove that D) passes through A and is parallel to P) ) a- Determine a system of parametric equations of the straight line d) passing through A and perpendicular to P) b- Determine the coordinates of B, the point of intersection of d) and P) c- Determine a system of parametric equations of the straight line ) passing through B and parallel to D) and prove that ) lies in P) ) Let ) be a straight line, other than ), passing through B and lying in P) a- Prove that ) and D) are skew not coplanar) b- Prove that AB) is perpendicular to ) and to D) III- points) In an oriented plane, consider the rectangle ABCD such that: AB;AD mod, AB = and AD = Let H be the orthogonal projection of A on BD) and h be the dilation, of center H, that transforms D to B ) a- Determine the image of the straight line AD) by h b- Deduce the image E of point A by h Plot E c- Construct the point F image of B by h and the point G image of C by h, then determine the image of rectangle ABCD by h ) Let S be the direct similitude that transforms A onto B and D onto A a- Determine an angle of S b- Determine the image of the straight line AH) by S and the image of the straight line BD) by S c- Deduce that H is the center of S ) Show that SB E and deduce that S S A ) Show that SS h ha)

3 IV- points) An urn contains three white balls and two black balls A player draws randomly and successively three balls from this urn, respecting the following rule: In each draw: if the drawn ball is black, he replaces it back in the urn ; if it is white, he doesn t replace it back in the urn ) a- Calculate the probability of drawing, in the following order: one black ball, one black ball then one white ball b- Show that the probability of obtaining one white ball only, among the three drawn balls, is 8 equal to 5 ) Among the three drawn balls, the player marks three points for each white ball drawn and two points for each black ball drawn Designate by X the random variable equal to the sum of points marked for the three drawn balls a- Show that the possible values of X are:, 7, 8 and 9 b- Determine the probability distribution of X and calculate its epected value ) The player now draws randomly and successively n balls from the urn n > ) respecting the same rule a- Calculate, in terms of n, the probability of the event: the player draws n black balls b- Calculate, in terms of n, the probability P n of the event: the player obtains at least one white ball c- What is the minimum number of balls to be drawn by the player so that P n 99? V- points) In a plane, given two parallel straight lines d) and ) at a distance from each other equal to 5 cm and a point A situated between d) and ) at a distance of cm from ) M is a variable point in the plane and H is its orthogonal projection on ) ) Show that if MA + MH = 5 cm, then M moves on a parabola S ) of focus A In what follows, the plane is referred to a direct orthonormal system O ; i, j ) such that A ; ) ) a- Prove that y is an equation of the parabola S ) b- Draw S ) ) Let E be a point on S ) of ordinate a such that a Show that ay a is an equation of the tangent d ) to S ) at E ) Let G be a point on S ) of ordinate b such that EOG 9 a- Prove that ab = b- The tangent d ) to S ) at G cuts d ) at a point L Prove that, as E and G vary on S ) such that EOG 9, the point L moves on a straight line to be determined d) A M ) H

4 VI- 7 points) A- Consider the function f defined on IR by f) e e Designate by C) its representative curve in an orthonormal system O; i, j ) f ) ) a- Determine lim f ), lim f ) and lim b- Solve the equation f) = ) Calculate f and set up the table of variations of f ) Show that O is a point of inflection of C) ) Write an equation of the tangent T at O to C) 5) Let h be the function defined on IR by h f a- Show that h) for every real number b- Deduce, according to the values of, the relative positions of C) and T ) Draw T and C) 7) Calculate the area of the region bounded by C, the ais of abscissas and the two lines of equations = and = ln 8) a- Show that f has, on ln ; b- Show that the equation f f, an inverse function f has a unique solution α and verify that < α < B- Let g be the function given by lnf g Designate by ) its representative curve in an orthonormal system ) Justify that the domain of definition of g is ; ln; ) Determine lim g) ) Show that the line d of equation Deduce an asymptote D) of ) y is asymptote to ) at ) Determine the coordinates of the points of intersection of ) with d) and D) 5) Set up the table of variations of g ) Draw )

5 وزارة التربية والتعلين العالي الوديرية العاهة للتربية دائرة االهتحانات هشروع هعيار التصحيح اهتحانات الشهادة الثانىية العاهة الفرع : علىم عاهة مسابقة في مادة الرياضيات الدورة اإلستثنائية للعام 9 QI Answer M IR {} and f), so the domain of g f is IR ; c) 5 p q is p q then p qis equivalent to p q a) 5 z AN zi iz i i iz i i z z AM i z i z i So AM AN andam; AN,the triangle AMN is right isosceles at A c) With distinct points situated on a circle, we can determine C triangles b) 5 5 y gives y then y, hence z sin i cos i, argz g c) 5 b) 5 QII Answer M a = 9, A does not belong to P ) ; d A ; P)) 5 b A is the point of D ) corresponding to m ; D ) P) OR n ; ; ) P), u ; ; ) // D) and n u 5 a n ; ; ) is a direction vector of d ) ; d ) : t ; y t ; z t 5 d ) P) B ; ; ) 5 b c a b u ; ; ) is a direction vector of ) ; ) : ; y ; z D ) is parallel to P ) and ) passes through the point B of P ) and is parallel to D ) ; hence, ) lies in P ) 5 ) is not parallel to D ) since ) is parallel to D ) ) and ) ) and D ) do not intersect since D ) is parallel to P ) and ) is a straight line in P ) Hence ) and D ) are not coplanar AB) is perpendicular to P ) at B ;then AB) is perpendicular to ) and to ) at B But ) is parallel to D ) ; hence AB) is perpendicular to D ) at A and to ) at B 5 QIII Answer M d) hd B, so the image of ADis the line passing in B and parallel to AD, so A a 5 h AD BC D) ) B ) P

6 E AH and A AD, then BC thus E AH BC E, b 5 c F BH B AB, so F d passing through E and parallel to F BH d AB, so G is the intersection of HC) and the parallel through B to DC)The image of rectangle ABCD by h is the rectangle EFGB AD ; BA 5 SA B, so the image of AH is line passing through B and perpendicular to AH, so it is BD SD A, so the image of BD is line passing through A and perpendicular to 5 BD, so it is AH H AH BD, so S H BD AH, thus SH H so H is the center of S 5 a b c B BD, so B AH thus B E S SA SSA SB E ha S and SAB) = BC) S intersection of the two lines AH) and BC) SoS is a similitude of center H and angle, thus SoS is a negative homothecy S S A h A, then SoS = h Since 5 QIV Answer M a P r BBW)= 5 5 = 5 5 b P r BBW)+P r BWB)+P r BBW)= a In three draws the possible outcomes are black, or white and black, or W and B, or W; Thus the possible values of X are: ;7;8 and 9 5 b 8 P r X=)= P r white balls)=p r BBB)= 5 5 P r X=7)=P r white ball)= 8 5 ; 5 P r X=8)=P r W)= P r WWB)+P r WBW)+P r BWW)= P r X=9)=P r white balls)= 5 EX)=

7 a P r n black balls in the n draws) = n 5 5 b c P r at least a white ball)=- 5 = P n n n n P n99 99 n ln n ln) Hence, the minimum number of balls is 5 QV Answer M MH) is perpendicular to d ) at K and MA MH MK MH 5 then MA MK d M, d )) Then M moves on a parabola S) of focus A and directri d) a A ; ) and d ) : then S) : y, since the origin is the verte and p = b 5 yy =; y y ; Equation of d ): a y a ; then d ) : a ay a a a b E ; a ) and G ; b ) OE OG gives ab b a b d ) : by b So d ) d ) L ; ) When E and G vary on S), y L describes IRand L moves on the line of equation 5 VI Answer M f ) e lim f ), lim f ) lim e e lim lim Aa e Ab f) =, e or e ; = or = ln 5 f e e ln f + A f - A e e f for, f for f, thus O is a point of inflection of C ) f, and f for,

8 A y f ) and f A5a A5b then y 5, for every h f e ; then h h f ), h is strictly increasing and h then for, h so above T and for, h so C is below T C and T intersect at point O C is A The line of equation y = is asymptote at y y is an asymptotic direction at 5 ln A7 S f d e e ln ln ) units of area A8a f is continuous and strictly increasing on ln ;, so it has an inverse function A8b The curves of the functions f and The line of equation f 5 f intersect on the line with equation y f, y cuts C at one point only of abscissa Let, then ; B f for or ln,so the domain of definition of g is ln ; B lim g ) ln thus the line of equation y =ln is a horizontal asymptote of ) B lim g) lim lne e ln e lim ln e e B g ) ln gives e e, so e then ln g ) gives e then J ln ;ln ln g ) + +, ln ;ln I ; B5 g) ln + 5 B

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دورة الؼام اهتحا ات الشهادة الثا ىية الؼاهة وزارة التربية والتؼلين الؼالي اإلنث يي 07 حسيراى 7102 فرع: الؼلىم الؼاهة

70 الؼادي ة دورة الؼام اهتحا ات الشهادة الثا ىية الؼاهة وزارة التربية والتؼلين الؼالي اإلنث يي 07 حسيراى 70 فرع: الؼلىم الؼاهة الوديرية الؼاهة للتربية دائرة االهتحا ات الرسوية االسن: هسابقة في هادة الرياضيات