Grade 11 - STMG / Second-degree polynomial 32 exercises (including 31 corrected)

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ABCDMNx23EFGHPQRSOx22x3 -4-3-2-123I-3-2-12JOCfCg ..................xVariationde... ..................xVariationde... 1. Algebra reminders E.7182 Consider the two rectangles ABCD and EFGH shown below : 1 The rectangle ABCD has been cut into two rectangles : AMND and BCNM . The dimensions are given : AM =2 ; MN = x ; AD =3 Justify that the area A of the rectangle ABCD can be expressed in two ways : A = 3 · x + 6 ; A = 3 · x + 2 2 The rectangle EFGH has been cut into four rectangles : EPOR , PFSO , SGQO and QHRO . The following measurements are given : EP = x ; PF =2 ; ER =3 ; RH =2 x Justify that the area A of the rectangle EFGH can be expressed in two ways : A = 2 x 2 + 7 x + 6 ; A = 2 x + 3 x + 2 E.7129 Expand the following expressions : a 2 x + 1 x + 1 b 3 · 1 + x 2 · x + 3 c 2 x + 1 2 d 5 x + 2 x + 2 + 2 1 + x E.7176 Expand and reduce the following expres-sions : a 3 · 2 · x + 3 + 2 · 4 x b 2 · x 1 2 3 · x E.7130 For each of the equations below, check whether the number 2 is a solution : a 2 x + 5 = 5 x 1 b 2 x + 3 + 1 = 15 c 2 1 x 2 4 = 4 4 x d x 2 4 x + 1 = 3 E.7131 Solve the equations below : a 2 x + 3 = 6 b 5 x + 1 = 2 x + 7 c 3 x 4 = 7 x + 4 d x + 1 2 = 9 E.7177 Solve the following equations : a 3 · x + 2 = 7 x b 3 · ( x 2) 2 · (1 x ) = 0 E.7132 Consider two functions f and g repre-sented below in a O ; I ; J respectively by the curves C f and C g : Below are two tables of variations. Complete the dotted lines in each of these tables of variations. E.11428 1 a Consider the linear function f defined by: f ( x ) = 3 x 4 Complete the sign table for the function f : x −∞ + f ( x ) b Consider the linear function g defined by: g ( x ) = 2 x + 1 Complete the sign table for the function g : x −∞ + g ( x ) 2. Writing a second-degree polynomial E.7158 Write each of the polynomials below in the form : a · x 2 + b · x + c a 3 · x 2 + 5 2 · x b 5 x + 3 · x 2 c 2 · x + 1 x 2 + 3 · x d 3 · x 2 1 + x + 3 e 2 · x 2 + x + 3 · 3 x f ( x + 1)(2 x ) https://chingmath.fr chapExoCorrec/7182 sacados/7182 ABCDMNx23EFGHPQRSOx22x3 chapExoCorrec/7129 sacados/7129 chapExoCorrec/7176 sacados/7176 chapExoCorrec/7130 sacados/7130 chapExoCorrec/7131 sacados/7131 chapExoCorrec/7177 sacados/7177 chapExoCorrec/7132 sacados/7132 -4-3-2-123I-3-2-12JOCfCg ..................xVariationde... ..................xVariationde... chapExoCorrec/11428 sacados/11428 chapExoCorrec/7158 sacados/7158
E.7159 1 Expand the following expressions : a ( x 3)( x 1) b x 2 2 1 2 Expand the following expressions : a 2 · x + 2 x + 4 b 2 · x + 3 2 2 3 Expand the following expressions : a ( x 5)( x 1) b 4 x 3 2 3. Canonical form E.7160 Definition: For any second-degree polynomial a · x 2 + b · x + c , there exist two real numbers ¸ and ˛ such that : a · x 2 + b · x + c = a · x + ¸ 2 + ˛ This expression is called the canonical form of this poly-nomial. 1 a Show that x 3 2 4 is the canonical form of the polynomial x 2 6 · x +5 b Show that x +1 2 4 is the canonical form of the poly-nomial x 2 +2 · x 3 c Show that x 3 2 25 is the canonical form of the poly-nomial x 2 6 · x 16 Definition: Let a · x 2 + b · x + c be a second-degree polyno-mial. The roots of this polynomial are the numbers x whose evaluation by the polynomial is 0 : a · x 2 + b · x + c = 0 Method : To determine the roots of a polynomial, we use its canonical form. Let’s take the expression in the ques-tion as an example: a : x 2 6 · x + 5 = 0 x 3 2 4 = 0 x 3 2 = 4 x 3 2 = 2 2 We use the following property: ˇ Two numbers whose squares are equal are either equal or opposite ı. From this, we deduce the two equations : x 3 = 2 x = 2 + 3 x = 5 x 3 = 2 x = 2 + 3 x = 1 Thus, the polynomial x 2 6 · x +5 has two roots, 1 and 5 . 2 Using the same method, b Show that the polynomial x 2 +2 · x 3 has roots 3 and 1 . c Show that the polynomial x 2 6 · x 16 has roots 2 and 8 . E.7178 Copy and complete the dotted lines of each equality to obtain the canonical form of each of the poly-nomials : a x 2 + 4 · x 1 = x + 2 2 + : : : b 2 · x 2 8 · x + 10 = 2 · x : : : 2 + 2 E.7179 We wish to solve the equation : x 2 + 6 x 7=0 1 Establish equality: x 2 + 6 x 7 = x + 3 2 16 2 Deduce the two solutions of the equation x 2 +6 · x 7=0 4. Discriminant E.7706 Definition: le The discriminant of a second-degree polynomial a · x 2 + b · x + c is a number that can be calcu-lated using the coefficients of the polynomial: Δ = b 2 4 × a × c Complete the table below for each of the second-degree poly-nomials : a b c Δ= b 2 4 · a · c x 2 + x +1 0 ; 5 · x 2 +2 · x +1 x 2 +4 2 · x 2 3 · x 1 https://chingmath.fr chapExoCorrec/7159 sacados/7159 chapExoCorrec/7160 sacados/7160 chapExoCorrec/7178 sacados/7178 chapExoCorrec/7179 sacados/7179 chapExoCorrec/7706 sacados/7706
<0Aucune solution01solutionb2·a>02solutionsb2·a;b2·a ABCDMNP32x2x E.7161 Complete the table below for each of the second-degree polynomials: a b c Δ= b 2 4 · a · c 2 · x 2 +5 · x +1 x 2 +7 · x +3 x 2 5 · x +4 2 · x 2 4 · x 1 x 2 x 1 x 2 +7 E.7162 Determine the discriminant of the second-degree polynomials below : a x 2 + 2 x + 4 b 2 x 2 + 4 x + 1 c x 2 2 x + 1 d 2 x 2 + 2 x + 1 e x 2 x 1 f 3 x 2 + x 2 E.7180 Determine the discriminant of the poly-nomials below : a 2 · x 2 3 · x + 3 b x 2 + 5 · x 4 5. Second-degree equation E.7163 Definition: The roots of a polynomial are the values that cancel out this polynomial. Proposition : for a second-degree polynomial a · x 2 + b · x + c , the number of existing roots depends on the discriminant : Solve the following equations : a x 2 + 2 x 35 = 0 b 2 x 2 + 8 x + 6 = 0 c 5 x 2 3 x + 2 = 0 d 9 x 2 24 x + 16 = 0 e 2 x 2 + 3 x 5 = 0 f 3 x 2 + 12 x + 9 = 0 E.7164 Solve the following equations : a 3 x 2 + 9 x + 6 = 0 b 3 x 2 4 x + 2 = 0 c x 2 + 2 x + 3 = 0 d 2 x 2 4 x + 2 = 0 e x 2 + 12 x + 27 = 0 f 2 x 2 + 12 x + 10 E.7181 Solve the following equations : a 2 x 2 + 2 x + 4 = 0 b 2 · x 2 4 · x + 2 = 0 E.7170 Let x be an indeterminate measure. Con-sider the rectangle ABCD shown below where : AM =3 ; MB =2 x ; AN =2 ; ND = x 1 In the particular case where x =2 , establish that the area A of the rectangle ABCD has the measure 28 . 2 We place ourselves in the general case where x represents an indeterminate number: a Show that the area A of the rectangle ABCD has the value : A = 2 · x 2 + 7 · x + 6 b Determine the value(s) of x so that the area A of the rectangle ABCD has the value 36 . https://chingmath.fr chapExoCorrec/7161 sacados/7161 chapExoCorrec/7162 sacados/7162 chapExoCorrec/7180 sacados/7180 chapExoCorrec/7163 sacados/7163 <0Aucune solution01solutionb2·a>02solutionsb2·a;b2·a chapExoCorrec/7164 sacados/7164 chapExoCorrec/7181 sacados/7181 chapExoCorrec/7170 sacados/7170 ABCDMNP32x2x
ABCDxx64 IJOCfCg <00>0¸et˛sontlesdeuxracinesa>0a<0x−∞x−∞x−∞b/2a0x−∞b/2a0x−∞αβ00x−∞αβ00 E.7171 Let x be an indeterminate measure and consider the rectangle ABCD , of dimensions 6 and 4 , shown below : where a rectangle, area shown striped, obtained by reducing the dimensions of ABCD by x . Note A the area of the rectangle shown striped. 1 Demonstrate that the area A is expressed as : A = x 2 10 · x +24 2 Determine the value of x so that A has the value 8 . 6. Reminders on sign tables E.7173 consider the two functions f and g de-fined by: f ( x ) = x + 1 ; g ( x ) = 0.5 · x 1 In the reference frame O ; I ; J given below, are represented the curves C f and C g representative of the functions f and g . Without justification, complete the sign tables for the func-tions f and g : x −∞ + f ( x ) x −∞ + g ( x ) E.7172 Complete the table of signs for each expression E : 1 x −∞ 3 1 2 + 2 x + 1 0 3 + x 0 E =(2 x +1)(3+ x ) 0 0 2 x −∞ 3 4 2 + x 2 4 x 3 E =( x 2)(4 x 3) 3 x −∞ + 2 + x 2 x E = 2+ x 2 x 7. Sign table E.7174 Proposal: The sign table for a quadratic polynomial de-pends on the sign of the coefficient of the quadratic term and the sign of the discriminant. The six possibilities are shown below : https://chingmath.fr chapExoCorrec/7171 sacados/7171 ABCDxx64 chapExoCorrec/7173 sacados/7173 IJOCfCg chapExoCorrec/7172 sacados/7172 chapExoCorrec/7174 sacados/7174 <00>0¸et˛sontlesdeuxracinesa>0a<0x−∞x−∞x−∞b/2a0x−∞b/2a0x−∞αβ00x−∞αβ00
2m12xx5m 31;5·xxABCD Draw the sign chart for each of the following expressions : a x 2 + x 6 b 2 · x 2 3 · x + 1 c 3 x 2 + 3 x 6 d x 2 + x + 2 e 2 x 2 + 12 x 18 f 3 x 2 5 x 2 E.7175 Draw up the sign table for the following expressions : a 2 · x 2 + 3 · x 5 b 6 · x 2 + x 1 c 2 · x 2 + 4 x + 6 d 3 · x 2 3 · x 1 8. Equations E.7198 Solve the following inequations : a x 2 3 x + 2 > 0 b x 2 x 2 < 0 c 9 x 2 + 12 x 4 0 d 5 x 2 + 4 x 1 < 0 e 4 x 2 + 2 x + 2 0 f x 2 + x 3 > 0 E.7199 Solve the following inequations : a x 2 + 5 x + 4 < 0 b x 2 x + 6 < 0 c 4 x 2 + 4 x + 1 > 0 d 2 x 2 + 5 x + 3 > 0 e 4 x 2 3 x + 2 0 f 12 x 2 + 12 x + 3 0 9. Problems E.7200 In his field, a farmer has a rectangu-lar chicken coop with dimensions 5 m and 2 m . He wishes to build an enclosure (shown dotted) as shown in the figure below with 17 m of fencing : The henhouse is represented by the hatched area, the fence is shown in dotted lines and the outdoor area dedicated to the hens is represented by the white area. The area of the outdoor part is A . 1 Justify that the area A of the outdoor space has the ex-pression : A ( x )= x 2 +12 · x 10 2 For what values of x , the outer space has an area of 25 m 2 . E.7201 We want to construct a rectangular play area along the side of a building. Furthermore, we want the dimensions of this rectangle to be greater than or equal to 10 m . This play area is surrounded on three sides by a 3 m -wide walkway, as shown in the sketch below. 1 Express the total area of the plot (including driveways) . 2 For what value of x , the area A measures 84 m 2 . 10. Share E.11622 Sur un axe gradué en mètres, on organise une course en-tre une tortue et un escargot. La tortue part du point d’abscisse x = 0 . Elle se déplace vers la droite à une vitesse de 2 mètres par minute. L’escargot part du point d’abscisse x = 12 . Il se déplace vers la droite à une vitesse de 50 centimètres par minute. Les deux concurrents partent en même temps. À quel endroit la tortue rattrapera-t-elle l’escargot ? (Toute trace de recherche, même infructueuse, sera prise en compte). https://chingmath.fr chapExoCorrec/7175 sacados/7175 chapExoCorrec/7198 sacados/7198 chapExoCorrec/7199 sacados/7199 chapExoCorrec/7200 sacados/7200 2m12xx5m chapExoCorrec/7201 sacados/7201 31;5·xxABCD chapExoCorrec/11622 sacados/11622
E.11623 1 Le volume d’un glacier diminue de 3 % chaque année. Si V ( n ) désigne le volume du glacier pour l’année n , on a: a V ( n + 1) = V ( n ) 0 ; 03 b V ( n + 1) = 0 ; 03 × V ( n ) c V ( n + 1) = 0 ; 97 × V ( n ) d V ( n + 1) = V ( n ) 0 ; 97 2 Dans un repère du plan on a représen une droite. Le coefficient directeur de cette droite est égal à: a 3 b 1 c 2 d 3 3 Dix stylos coûtent en tout 13 euros. Le prix de trois stylos est égal à: a 3,60 euros b 6,90 euros c 3,90 euros d 6,50 euros 4 Une athlète parcourt 1 km en 5 minutes. Quelle est sa vitesse moyenne ? a 8 km/h b 10 km/h c 12 km/h d 14 km/h 5 Sur 60 personnes présentes à une exposition, on distingue trois groupes : groupe A: 30 personnes groupe B : 12 personnes groupe C : les autres. Quelle représentation décrit la situation ? 6 Donner un ordre de grandeur de 101 × 99 : a 100 b 1 000 c 10 000 d 100 000 7 Un prix augmente de 20% puis diminue de 20%. Après ces deux évolutions, on peut affirmer que : a Le prix est égal à sa valeur de départ. b Le prix est strictement supérieur à sa valeur de départ. c Le prix est strictement inférieur à sa valeur de départ. d On ne peut pas savoir: cela dépend de la valeur de départ. 8 Par combien faut-il multiplier une quantité positive pour que celle-ci diminue de 2,3% ? a 1,23 b 0,977 c 0,77 d 1,023 9 Dans un lycée, 50 élèves étudient le Grec, ce qui représente 4% du nombre d’élèves inscrits dans ce lycée. Le nombre d’élèves inscrits dans ce lycée est égal à: a 2 b 200 c 125 d 1250 10 On considère la fonction f définie sur R par : f ( x ) = 2 x 2 + 3 x + 1 . L’image de 1 par la fonction f est égale à: a 0 b 2 c 2 d 4 https://chingmath.fr sacados/11623
11 On considère la fonction f définie sur R par : f ( x ) = 2 x 2 5 x + 3 . Un antécédent de 0 par la fonction f est : a 1 b 1 c 0 d 2 12 On considère : A = 1 2 1 2 × 4 3 a A = 0 b A = 1 6 c A = 2 3 d A = 1 https://chingmath.fr