Grade 11 / Second degree: functions, variations, inequalities 81 exercises (100% corrected)

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x−∞∞Variationsdef−b=2a−∞−∞−4ax−∞∞Variationsdef−b=2a∞∞−4aPoura<0Poura>0 ChingQuizz : 8 exercises available for Quizz assessment : 1. Reminders E.8375 Complete the sign tables below : 1 x −∞ + ∞ 1 − x 2 x + 1 (1 − x )(2 x +1) 2 x −∞ + ∞ x − 3 − 2 x + 4 ( x − 3)( − 2 x +4) 2. Variation table E.3080 Proposal: Table showing the variation of a quadratic func-tion as a function of the coefficient a of the quadratic term : Draw up a table showing the variations of the quadratic poly-nomial functions below : a f ( x ) = 3 · x 2 − 3 · x + 2 b g ( x ) = − x 2 − 2 · x + 3 E.2976 For each of the functions, draw up the table of variations and give the characteristics of their extrema: a f : x ↦−→ 2 x 2 + 8 x + 1 b g : x ↦−→ − x 2 + 2 x + 1 E.8380 For each of the functions, draw up the table of variations and give the characteristics of their extrema: 1 f ( x ) = − 3 x 2 + 9 x − 2 2 g ( x ) = 3 x 2 + 2 x + 2 E.8522 For each of the functions, draw up the table of variations and give the characteristics of their extrema: 1 f : x ↦−→ 1 6 · x 2 + 1 4 x + 1 2 g : x ↦−→ − x 2 + 2 3 x − 1 E.4502 1 Consider the function f defined by: f ( x ) = a · x 2 + 3 x + 2 où a ∈ R Knowing that its representative curve passes through the point with coordinates A ( − 2 ; − 12) , determine the com-plete expression of the function f . 2 Let g be the function whose image of a real number x is defined by: g ( x ) = 3 x 2 + b · x + 1 où b ∈ R Knowing that the vertex of the representative parabola of the function g has abscissa 1 , determine the complete expression of the function g . 3. Table of variations and roots E.2977 Let h be the function defined by the relation: h ( x ) = 4 x 2 + 2 x + 1 1 Draw up the table of variations of the function h . 2 Justify that the function h never cancels at R . E.8520 Let f be the function defined by the relation: f ( x ) = 2 x 2 + 3 x + 1 1 Draw up the table of variations of the function f . 2 Justify that the function f cancels in two values. https://chingmath.fr chapExoCorrec/8375 sacados/8375 chapExoCorrec/3080 sacados/3080 x−∞∞Variationsdef−b=2a−∞−∞−4ax−∞∞Variationsdef−b=2a∞∞−4aPoura<0Poura>0 chapExoCorrec/2976 sacados/2976 chapExoCorrec/8380 sacados/8380 chapExoCorrec/8522 sacados/8522 chapExoCorrec/4502 sacados/4502 chapExoCorrec/2977 sacados/2977 chapExoCorrec/8520 sacados/8520
IJOCf IJOCg IJOCh IJOCj IJOCk IJOC 3yxABCD ABMD1D2 2myx5m E.8521 Let g be the function defined by the relation: g ( x ) = − 4 x 2 + 4 x − 1 1 Draw up the table of variations of the function g . 2 Justify that the function g cancels at a single value to be specified. E.4469 Associate its graphical representa-tion with each of the polynomials below : A = x 2 +4 x +5 ; B = − x 2 +2 x +2 ; C =4 x 2 − 4 x +1 D = − x 2 − 1 ; E = x 2 − 2 x − 1 ; F = − x 2 +2 x − 1 4. Problems and extremums E.8379 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 playing area is surrounded on three sides by a 3 m wide alley as shown in the sketch below. The set is fenced on three sides [ AB ] , [ BC ] and [ CD ] . We are interested in the length L of the fence : L = AB + BC + CD . Let x and y be the dimensions in meters of the playing area (the value of x and y are necessarily positive) . We have 100 meters of fence that we want to use entirely: 1 a Express, under these conditions, the value of y as a function of x . b Justify that the value of x must be less than 44. 2 Determine dimensions so that all 100 meters of fencing can be used and the play area maximized. E.8104 Consider a segment [ AB ] of length 1 m , a point M belonging to the segment [ AB ] and the two disks D 1 and D 2 of diameters [ AM ] and [ MB ] respectively. Determine the location(s) of the point M such that the sum of the areas of the disks D 1 and D 2 is minimal. E.4867 In his field, a farmer has a rectan-gular chicken coop with dimensions 5 m and 2 m . He wishes to build an enclosure as shown in the figure below with 17 m of fencing : The numbers x and y represent the dimensions of this field. The hen house is represented by the hatched area, the fence is shown in blanks and the outdoor area dedicated to the hens is represented by the white area. The area of the outdoor part is A . 1 Establish the following relationship between x and y : x + y = 12 2 Demonstrate that the area of the outer space has the expression : A ( x )= − x 2 +12 · x − 10 3 Draw up the table of variations of the function A on R . 4 Determine the values of x and y so that the area of the outdoor space reserved for hens is maximum. https://chingmath.fr chapExoCorrec/8521 sacados/8521 chapExoCorrec/4469 sacados/4469 IJOCf IJOCg IJOCh IJOCj IJOCk IJOC chapExoCorrec/8379 sacados/8379 3yxABCD chapExoCorrec/8104 sacados/8104 ABMD1D2 chapExoCorrec/4867 sacados/4867 2myx5m
xcmycm3cm ABMC1C2C3 -3-2-123456I-12-8-448JOCf E.8381 A carpenter has a stick of wood 100 centimetres long and 3 centimetres wide. He wants to use the full length of this stick to make a wooden frame like the one shown in the drawing below : Note A ( x ) the interior area of the frame as a function of x . 1 Draw up the table of variations of the function A . 2 Deduce the dimensions of the frame so that the interior area is maximum. E.8430 The figure below is composed of the segment [ AB ] measuring 6 cm and a point M apartment at the segment [ AB ] . The semicircle C 1 (resp. C 2 , C 3 ) admits the segment [ AB ] (resp. [ MB ] , [ AM ] ) as its diameter. Let x be the length of segment [ AM ] . Determine for quelle (s) valeur (s) of x the area of the hatched domain is maximum. 5. Canonical form and factorization E.1829 Consider the second-degree polyno-mial: P = x 2 − 6 x − 16 1 Determine the canonical form of the polynomial P . 2 Deduce the factorization : P = x − 8 x + 2 3 Deduce that the table of signs : x −∞ − 2 − 8 + ∞ x 2 − 6 x − 16 + 0 − 0 + E.8373 Consider the function f defined on R by the relation: f ( x ) = 2 · x 2 − 16 · x + 30 1 Establish that the function f admits as vertex form f ( x ) = 2 · x − 4 2 − 1 2 Deduce factorize the expression of the function f in the form of two factors of degree 1 . Hint: the function f admits a factorisaiton of the form : f ( x )=2 · a · x + b c · x + d where a; b; c; d ∈ R 3 Draw up the sign table for the function f . E.2249 Consider the function f defined on R by the relation: f ( x ) = 6 x 2 − 9 x − 6 1 Show that the expression of f ( x ) can be written : f ( x ) = 6 x − 3 4 2 − 25 16 2 Noting that 25 16 = 5 4 2 , factor the expression for the func-tion f in the form of two factors of degree 1 . 3 Draw up the sign table for the function f . E.6397 Consider the polynomial of the sec-ond degree : P = − 2 x 2 − 13 x − 15 1 Determine the canonical form of the polynomial P . 2 Deduce the factorization : P = − 2 x − 3 x + 5 3 Deduce that the sign table of the polynomial P E.8374 Consider the function f defined for any x ∈ R by: f ( x ) = 2 · x 2 − 6 · x − 8 1 Below is given the curve C f representative of the function f in a reference frame O ; I ; J orthogonal : Conjecture the sign table of the function f . 2 a Determine the vertex form then the factorized form of the function f . b Establish the sign table of the function f . 6. Introduction: roots, factoring and signs https://chingmath.fr chapExoCorrec/8381 sacados/8381 xcmycm3cm chapExoCorrec/8430 sacados/8430 ABMC1C2C3 chapExoCorrec/1829 sacados/1829 chapExoCorrec/8373 sacados/8373 chapExoCorrec/2249 sacados/2249 chapExoCorrec/6397 sacados/6397 chapExoCorrec/8374 sacados/8374 -3-2-123456I-12-8-448JOCf
<0Aucunefactorisation0a·xb2·a2>0a·x−¸x−˛où¸et˛sontles deux racines du polynômes E.10223 Consider the polynomial: P = 3 · x 2 + 3 · x − 18 1 Determine the roots of the polynomial P . Let x 1 and x 2 be the two roots of the polynomial P . 2 a Expand the expression x − x 1 x − x 2 . b Deduce a factorization of the polynomial P . c Draw up the sign table for the expression P at R . E.10224 Consider the polynomial: P = − 9 · x 2 + 6 · x + 15 1 Determine the roots of the polynomial P . Let x 1 and x 2 be the two roots of the polynomial P . 2 a Expand the expression x − x 1 x − x 2 . b Deduce a factorization of the polynomial P . c Draw up the sign table for the expression P at R . E.10323 Consider the polynomial: P = − 2 · x 2 − 8 · x + 16 1 Determine the roots of the polynomial P . Hint: these roots will be given in the form ˇ a + b · √ c ı where a;b;c ∈ Z Let x 1 and x 2 be the two roots of the polynomial P . 2 a Expand the expression x − x 1 x − x 2 . b Deduce a factorization of the polynomial P . c Draw up the sign table for the expression P at R . 7. Factorizations E.2527 Proposition: The factorization of a second-degree poly-nomial a · x 2 + b · x + c depends on the value of its discriminant Δ : Factorize, if possible, the following expressions : a x 2 − 3 x + 2 b − 2 x 2 − 2 x + 4 c − x 2 + 2 x − 1 d 4 x 2 + x + 3 E.9518 Give the factorized form of the fol-lowing expressions : a 3 · x 2 − 3 · x − 6 b 2 · x 2 + 12 · x + 18 E.9521 Factor, if possible, the second-degree polynomials below : a x 2 + 2 x + 1 b 3 x 2 − 4 x + 2 c − 3 x 2 + 4 x − 1 Indication : present results as : a · x + b c · x + d or a · x + b 2 with a;b;c;d ∈ Z E.9520 If possible, factor the following ex-pressions : a 8 x 2 − 24 x + 18 b 3 x 2 + x + 1 c − 4 x 2 + x + 3 Indication : present results as : a · x + b c · x + d or a · x + b 2 with a;b;c;d ∈ Z E.8427 Factor the following expressions : a 6 · x 2 − 7 · x − 3 b 4 · x 2 +12 · x +9 Indication : present results as : a · x + b c · x + d or a · x + b 2 with a;b;c;d ∈ Z E.10594 If possible, factor each of the poly-nomials below : a 3 x 2 − 12 x + 12 b − 5 x 2 + 2 x − 1 c 6 x 2 + x − 15 E.8290 Factor the following expressions : a 4 x 2 + 4 x − 5 b x 2 − 2 x − 4 Hint: Simplify the factored expression of these polynomi-als as much as possible, paying particular attention to the expression of their roots. E.10579 Factor the following expressions : a 2 x 2 − 6 x + 2 b − x 2 − 4 x + 8 Hint: Simplify the factored expression of these polynomi-als as much as possible, paying particular attention to the expression of their roots. E.10595 Factor the expression : A = 12 x 2 + 12 x − 3 Hint: we factor the expression into the form : A = 2 x + ¸ 6 x + ˛ where ¸ and ˛ are two real numbers. https://chingmath.fr chapExoCorrec/10223 sacados/10223 chapExoCorrec/10224 sacados/10224 chapExoCorrec/10323 sacados/10323 chapExoCorrec/2527 sacados/2527 <0Aucunefactorisation0a·xb2·a2>0a·x−¸x−˛où¸et˛sontles deux racines du polynômes chapExoCorrec/9518 sacados/9518 chapExoCorrec/9521 sacados/9521 chapExoCorrec/9520 sacados/9520 chapExoCorrec/8427 sacados/8427 chapExoCorrec/10594 sacados/10594 chapExoCorrec/8290 sacados/8290 chapExoCorrec/10579 sacados/10579 chapExoCorrec/10595 sacados/10595
<00>0¸et˛sontlesdeuxracinesa>0a<0x−∞∞x−∞∞−x−∞∞−b/2a0x−∞∞−b/2a0−−x−∞∞αβ00−x−∞∞αβ00−− E.8376 1 Factorize the expression : − 2 · x 2 − 3 · x +5 . 2 For each statement, only one answer is correct. Check the corresponding box. Hint: we will use the result from question 1 a The factored form of − x 2 − 3 2 · x + 5 2 is : x + 5 1 − x x + 5 2 1 − x x + 5 1 − 1 2 · x x + 5 2 1 2 − 1 2 x b The factored form of − 2 · x 2 − 3 · x +5+ 1 − x is : 2 · x + 5 1 − x 2 · x + 5 · x 2 · x + 6 1 − x 2 · x + 6 2 − x c The factored form of − 2 · x +1 2 − 3 · x +1 +5 is : 2 · x + 6 1 − x 2 · x + 6 2 − x − 2 · x + 7 · x 2 · x + 7 2 − x 8. Factorizations (degree 3) E.9527 Consider the function f defined on R by the relation: f ( x ) = x 3 − 7 · x − 6 1 Verify that the number 3 is a zero of the function f . Hint: the polynomial x 3 − 7 · x − 6 has the number 3 as a root. It therefore admits a factorization of the form : x 3 − 7 · x − 6= x − 3 a · x 2 + b · x + c where a;b;c ∈ R 2 a For a , b , c real numbers, verify the following iden-tity: x − 3 a · x 2 + b · x + c = a · x 3 + b − 3 a · x 2 + c − 3 b · x − 3 c b Determine the values of a and b that satisfy : a =1 ; b − 3 a = 0 ; c − 3 b = − 7 ; − 3 c = − 6 c Deduce the factorized form of the function f into fac-tors of degree 1 . E.9522 Consider the function f defined on R whose expression is : f ( x )=2 · x 3 − 7 · x 2 +4 · x +3 1 Determine the real numbers a , b , c realizing the equality: f ( x ) = 2 · x − 3 a · x 2 + b · x + c 2 Deduce the factorized form of the function f as a product of factors of degree 1 . 9. Sign table E.2277 The sign chart for a quadratic polynomial depends on the sign of the coefficient a of the quadratic term and the sign of the discriminant Δ . The six possibilities are shown below : Draw the sign chart for the following quadratic polynomials: a x 2 + 3 x + 4 b 4 x 2 + 3 x − 10 c 4 x 2 − 16 x + 16 E.5712 Draw up the sign table for the fol-lowing expressions : a 3 x 2 +4 x − 4 b − 4 x 2 +2 x +6 E.10597 At R , draw up the sign table for each of the following functions defined at R : a f ( x ) = 5 x 2 − 4 x + 1 b g ( x ) = − 3 x 2 + 7 x + 20 E.10580 Establish the sign table of the fol-lowing expressions on R : a 2 x 2 +11 x +5 b − 3 x 2 + 4 x + 4 E.6505 Draw up the sign table for each of the expressions below : a 2 · x 2 +9 · x +10 b 12 x 2 − 31 x +20 c − 5 x 2 − 3 x − 1 E.8377 Establish the sign table for the fol-lowing quadratic polynomials: a − x 2 + 2 x + 6 b 2 x 2 − 8 x − 8 Note: The roots of the polynomials will be indicated in the form : a + b c where a;b ∈ R and c ∈ R ∗ + . 10. Sign table and inequation https://chingmath.fr chapExoCorrec/8376 sacados/8376 chapExoCorrec/9527 sacados/9527 chapExoCorrec/9522 sacados/9522 chapExoCorrec/2277 sacados/2277 <00>0¸et˛sontlesdeuxracinesa>0a<0x−∞∞x−∞∞−x−∞∞−b/2a0x−∞∞−b/2a0−−x−∞∞αβ00−x−∞∞αβ00−− chapExoCorrec/5712 sacados/5712 chapExoCorrec/10597 sacados/10597 chapExoCorrec/10580 sacados/10580 chapExoCorrec/6505 sacados/6505 chapExoCorrec/8377 sacados/8377
-5-4-3-2-123I-1234JOCf E.4460 Solve the following inequalities: a x 2 − x − 2 < 0 b − 9 x 2 +12 x − 4 0 E.8378 Solve the following inequalities: a − 4 x 2 +2 x +2 0 b 3 · x 2 + x +1 < 0 E.8428 Solve the following inequalities: a − 2 · x 2 + x + 3 < 0 b 3 · x 2 − 6 · x − 3 0 E.10581 Solve the following inequalities: a x 2 − 3 x +2 > 0 b 5 x 2 +4 x − 1 < 0 E.9519 Solve inequalities: a 6 · x 2 + x − 1 0 b − x 2 + x − 3 > 0 E.9436 Solve the following inequalities: a − 10 x 2 − 13 x + 3 0 b (3 x + 1)( x 2 + x + 1) < 0 E.9437 Solve the inequation : 2 x 2 − 8 x +2 0 11. Sign table and inequation (degree 3) E.9523 Consider the function f defined on R by the relation: f ( x ) = x 3 − 4 · x 2 − 4 · x + 16 1 Verify that the number 2 is a zero of the function f . Hint: the polynomial x 3 − 4 · x 2 − 4 · x +16 has the number 2 as a root. It therefore admits a factorization of the form : x 3 − 4 · x 2 − 4 · x +16= x − 2 a · x 2 + b · x + c where a;b;c ∈ R 2 a For a , b , c real numbers, verify the following iden-tity: x − 2 a · x 2 + b · x + c = a · x 3 + b − 2 a · x 2 + c − 2 b · x − 2 c b Determine the values of a and b that satisfy : a =1 ; b − 2 a = − 4 ; c − 2 b = − 4 ; − 2 c = 16 Propose a factored form of the function f . c Establish the sign table for the function f . E.1643 Consider the polynomial P whose expression is : P = 2 · x 3 + 7 · x 2 − 7 · x − 12 1 Establish the following factorization where b is a real number to be determined : P = x + 1 · 2 · x 2 + b · x − 12 2 Deduce the sign table of the polynomial P . E.1158 Consider the third-degree polyno-mial: P = 3 x 3 + 5 x 2 − 5 x + 1 We know that the polynomial P admits a factorization of the form : P = 3 x − 1 a · x 2 + b · x + c 1 Determine the values of a , b , c verifying this factoriza-tion. 2 Deduce the set of roots of the polynomial P . 3 Draw up the sign table for P . E.2965 1 a Establish that the polynomial P ( x )=2 x 2 − x +1 is strictly positive on R . b Deduce the sign of the polynomial: Q ( x ) = (2 x 2 − x + 1) 2 + 3 · (2 x 2 − x + 1) + 1 2 a Give the reduced developed form of the polynomial Q . b Justify that the equation below admits no solution : 4 x 4 − 4 x 3 + 11 x 2 − 5 x + 5 = 0 E.4615 1 Establish the following equality: ( − 2 · x 2 + 4 · x − 2)( a · x 2 + b · x + c ) = − 2 ax 4 + (4 a − 2 b ) x 3 + ( − 2 c +4 b − 2 a ) x 2 + (4 c − 2 b ) x − 2 c 2 Give, without justification, the values of a , b , c achieving the following equality: ( − 2 x 2 + 4 x − 2)( a · x 2 + b · x + c ) = − 18 · x 4 + 18 · x 3 + 14 · x 2 − 10 · x − 4 3 Draw up the sign table of the function f defined on R by: f ( x )= − 18 · x 4 +18 · x 3 +14 · x 2 − 10 · x − 4 12. Relative positions of curves E.5742 Consider the function f defined on R by the relation: f ( x ) = − x 2 − 2 x + 3 Below is given the curve C f representative of the function f in a O ; I ; J orthonormal reference frame : https://chingmath.fr chapExoCorrec/4460 sacados/4460 chapExoCorrec/8378 sacados/8378 chapExoCorrec/8428 sacados/8428 chapExoCorrec/10581 sacados/10581 chapExoCorrec/9519 sacados/9519 chapExoCorrec/9436 sacados/9436 chapExoCorrec/9437 sacados/9437 chapExoCorrec/9523 sacados/9523 chapExoCorrec/1643 sacados/1643 chapExoCorrec/1158 sacados/1158 chapExoCorrec/2965 sacados/2965 chapExoCorrec/4615 sacados/4615 chapExoCorrec/5742 sacados/5742 -5-4-3-2-123I-1234JOCf
-1234567I-2246JOCfCg -3-2-12345I-1234JO -4-3-2-101234-4-224CfCg -2-1234I-12JOCfCg Consider the linear function g defined by the relation: g ( x ) = − x + 1 1 Draw in the reference frame below the straight line ( d ) representative of the function g . 2 a Establish the sign table for the expression : f ( x ) − g ( x ) . b Deduce the relative position of the curves C g and C f on R . E.6506 Consider the two functions f and g defined on R by: f ( x ) = x 2 − 6 · x + 7 ; g ( x ) = 2 · x 2 − 2 · x + 2 In the plane provided with an orthogonal reference frame O ; I ; J , we give the curves C f and C g representative of the functions f and g respectively. Determine the relative position of the curves C f and C g . E.5054 Consider the function f whose im-age of any real number x is defined by the relation: f ( x )= − 2 x 2 +4 x +2 In the plane provided with a reference frame O ; I ; J , con-sider the curve C representative of the function f and the straight line (Δ) first bisector of the plane admitting as equa-tion y = x . Algebraically, investigate the relative position of the curve C f and the straight line (Δ) . E.7105 In the plane provided with a ( O ; I ; J , consider the curves C f and C g representative of the functions f and g defined by: f ( x ) = − 2 · x 2 − 5 x + 1 ; g ( x ) = 6 · x 2 + x − 4 Below is the graphical representation of these two curves : Determine the relative position of these two curves. E.7084 In the plane provided with a refer-ence frame O ; I ; J , consider the curves C f and C g repre-sentative of the functions f and g defined by: f ( x ) = − x 2 + 4 · x − 3 ; g ( x ) = 7 2 · x 2 − 5 x + 1 Determine the relative position of these two curves. 13. Relative positions of curves (degree 3) E.9524 Consider the two functions f and g defined on R \{ 5 } by the relations : f ( x ) = 5 4 x 2 + 1 ; g ( x ) = − x + 2 Study the relative position of the curves C f and C g represen-tative of the functions f and g respectively. 14. Problems and inequalities https://chingmath.fr chapExoCorrec/6506 sacados/6506 -1234567I-2246JOCfCg chapExoCorrec/5054 sacados/5054 -3-2-12345I-1234JO chapExoCorrec/7105 sacados/7105 -4-3-2-101234-4-224CfCg chapExoCorrec/7084 sacados/7084 -2-1234I-12JOCfCg chapExoCorrec/9524 sacados/9524
ABCDEFGHI2xx4cm4cm ABCDMNPRQ ABCDMNPQ 16cmxx10cm E.8383 Consider the figure below composed : square AEFG , of two rectangles ABCD and CIFH . The points B , D , I , H belong to the sides of the square AEFG . Consider the shaded area shown opposite and note its area A : (measurements are in centimeters) Determine the set of values of x realizing the inequation : A 37 4 Any trace of research or initiative will be taken into account in the assessment. E.6443 Consider the configuration below where : where : the quadrilaterals ABCD and CRNQ are rectangles and AMNP is a square the points M , Q , R , P belong respectively to the seg-ments [ AB ] , [ BC ] , [ CD ] , [ DA ] AB = 10 cm and AD = 5 cm Note x the length of segment [ AM ] and note A the area of the unshaded part of this figure : 1 Show that the area A is expressed as a function of x by: A = − 2 · x 2 + 15 · x 2 a Solve the equation A = 27 b Determine the positions of the point M so that the unshaded surface has an area greater than or equal to 27 cm 2 3 Similarly, we’ll determine the positions of the point M so that the unshaded surface has an area greater than or equal to 18 cm 2 E.8429 Consider the figure opposite, where ABCD is a square whose sides measure 5 cm . Consider a point M on the segment [ AB ] and place the point P on the segment [ AD ] and the points N and Q so that AMNP is a square and BCQM is a rectangle. Let x be the measure of segment [ AM ] . Determine the set of values of x so that the area of the square AMNP is strictly greater than the area of the rectangle BCQM . E.8382 A rectangular box without a lid is to be made in the pattern below. The lengths are expressed in cm . 1 a When the box is built, the number x will represent which dimension? Length, width or height? b What values can the variable x take in this problem? c Give the expression for the volume V as a function of the value of x . 2 In this question, we investigate for what values of ˇ x ı, this box has a volume equal to 144 cm 3 : a Determine the value of the reals of a and b verifying the following factorization : 4 x 3 − 52 x 2 + 160 x − 144 = ( a · x + b )(2 x − 4) 2 b Deduce the values of x for which V ( x ) has the value 144. https://chingmath.fr chapExoCorrec/8383 sacados/8383 ABCDEFGHI2xx4cm4cm chapExoCorrec/6443 sacados/6443 ABCDMNPRQ chapExoCorrec/8429 sacados/8429 ABCDMNPQ chapExoCorrec/8382 sacados/8382 16cmxx10cm
IJOCfCgx8A1A2AB DA23456I234JO(d DA-4-3-2-12345678I-6-5-4-3-2-123456JO(d E.2956 In the plane provided with a refer-ence frame ( O ; I ; J ) orthonormal, consider the representation of the two functions f and g whose image of x is defined by: f ( x ) = 8 x + 1 ; g ( x ) = − 6 x + 1 + 6 The number x belongs to the interval [0 ; 8] . Consider the points A and B of abscissa x belonging respectively to the representative curves C f and C g . Parallel to the axes, we construct two rectangles shown above ; we note A 1 and A 2 each of their areas. 1 Determine the expression for the areas A 1 and A 2 as a function of the value of x . 2 Determine for which values of x , we have : A 2 A 1 15. Problems, inequalities and square roots E.8105 In the plane provided with a refer-ence frame O ; I ; J , consider the disk D of center A (3 ; 1) and radius 2 and the line ( d ) passing through the points B (0 ; 3.5) and C (1.5 ; 3) Determine the set of abscissas of the points on the line ( d ) included in the disk D . Hint : we will be interested in the set of points M on the line ( d ) such that AM 2 4 E.8110 In the plane provided with a ref-erence frame O ; I ; J , consider the disk D with center A 3 2 ; 0 and radius 5 and the straight line ( d ) passing through the points B ( − 2 ; 5) and C (1 ; − 1) Determine the set of abscissas of the points on the line ( d ) included in the disk D . Hint : we will be interested in the set of points M on the line ( d ) such that AM 2 25 https://chingmath.fr chapExoCorrec/2956 sacados/2956 IJOCfCgx8A1A2AB chapExoCorrec/8105 sacados/8105 DA23456I234JO(d chapExoCorrec/8110 sacados/8110 DA-4-3-2-12345678I-6-5-4-3-2-123456JO(d
-4-3-2-1234I-2-12JOCfCg E.5821 Consider the function f whose im-age of a number x is defined by: f ( x ) = 2 x − − x 2 + 6 x − 8 1 a Solve the inequation : − x 2 +6 x − 8 0 . b Demonstrate that the following equation admits no so-lution : x = − x 2 + 6 x − 8 c Give the defining set D f of the function f 2 Show that the function f is positive on its defining set. 16. Rational fractions and simplifications E.7154 Consider the function f defined by: f ( x )= x 2 − 6 · x − 7 1 Determine the factorized form of the function f . 2 Consider the function g defined on R \{ 3 ; 7 } by: g ( x ) = x 2 − 6 · x − 7 x − 3 x − 7 . a Simplify the expression of the function g . b Draw up the sign table for the function g . E.7102 Simplify the following rational frac-tion : x 2 − x − 2 2 x 2 − 3 x − 2 E.2528 Simplify the expression of the ratio- nal fractions below : a 3 x − 1 3 x 2 + 2 x − 1 b 6 x 2 − 5 x + 1 1 − 4 x 2 E.2746 Consider the function f whose image of x is defined by the rational fraction below : f ( x ) = 8 x 2 + 6 x − 5 14 x 2 − 13 x + 3 Give the definition set of the function f , then determine, if it exists, the simplified form of f ( x ) . E.7103 Simplify the rational expression be-low : 3 x 2 − 6 x − 6 x 2 − 3 + 2 x + 3 + 1 17. Table of variations and sign table E.2276 Consider the functions f and g de-fined on R defined by the relations : f ( x ) = x 2 + x + 1 ; g ( x ) = − 2 x 2 − 3 x + 5 1 Draw up the table of variations for each of these func-tions. 2 Draw up the table of signs for each of these functions. E.7083 Determine the sign table of the fol-lowing expressions at R : a 2 x 2 − 3 x − 2 b (2 x + 1)(3 x 2 − 2 x − 1) 18. Share E.2973 In the plane provided with a refer-ence frame O ; I ; J , consider the curves C f and C g repre-sentative of the functions f and g defined by: f ( x ) = x 2 + 3 2 · x − 1 ; g ( x ) = − 1 2 · x 2 + x + 1 The questions below will be answered algebraically: 1 Determine the zeros of the functions f and g . (i.e. the antecedents of 0 by each of these two functions) 2 Determine, algebraically, the relative position of the curves C f and C g . E.2279 Consider the parabola P with equa-tion y = x 2 − x − 10 and the straight line D with equation y =2 x − 1 . 1 Determine the coordinates of the intersection points of D and P . 2 Give the values of x for which the point P having abscissa x lies above the point of D having the same abscissa. https://chingmath.fr chapExoCorrec/5821 sacados/5821 chapExoCorrec/7154 sacados/7154 chapExoCorrec/7102 sacados/7102 chapExoCorrec/2528 sacados/2528 chapExoCorrec/2746 sacados/2746 chapExoCorrec/7103 sacados/7103 chapExoCorrec/2276 sacados/2276 chapExoCorrec/7083 sacados/7083 chapExoCorrec/2973 sacados/2973 -4-3-2-1234I-2-12JOCfCg chapExoCorrec/2279 sacados/2279
-5-4-3-2-12I-2246JOCf(d x9cm4cmABCDEF 19. Unclassified financial years E.4614 Consider the function f defined on R by the relation: f ( x ) = − x 2 − 7 2 x + 2 In the plane provided with a reference frame O ; I ; J or-thonormal, is given the curve C f representative of the func-tion f and the straight line ( d ) of equation : y = − 3 4 · x + 5 4 1 a Determine the coordinates of the points of intersec-tion of the curve C f with the x-axis. b Determine the coordinates of the intersection point of the curve C f with the y-axis. 2 Determine the relative position of the curves C f and ( d ) on R . E.11675 Consider the triangle ABC right-angled A such that : AB = 9 cm ; AC = 4 cm Consider a point D belonging to segment [ AB ] and note x the length of segment [ BD ] . From the point D we construct a rectangle DEFA such that : E ∈ [ BC ] ; F ∈ [ AC ] Note A the area of the rectangle DEFA . 1 a Determine the expression for length FA as a func-tion of x . b Justify, briefly, that the real number x belongs to the interval 0 ; 9 . 2 Establish that the area A is expressed as a function of x by: A ( x ) = 4 x − 4 9 · x 2 3 a Draw up the table of variations of the function A on the interval 0 ; 9 . b What is the maximum area reached by the A area? 4 We wish to know the values of x for which the area A has greater than or equal to 5 cm 2 : a Establish the following factorization : − 4 x 2 + 36 x − 45 = (2 x − 15)(3 − 2 x ) b Solve the inequation : A ( x ) 5 . https://chingmath.fr chapExoCorrec/4614 sacados/4614 -5-4-3-2-12I-2246JOCf(d chapExoCorrec/11675 sacados/11675 x9cm4cmABCDEF
-4-3-2-1I-12JOCf -2-1234I-2-12JOAMCf 4cm6cmyxABCDEFG E.7329 Consider the function f defined on R by the relation: f ( x ) = 3 · x 2 + 4 · x + 1 Below is given the representative curve C f of the function f in a O ; I ; J orthonormal coordinate system : 1 Consider the straight line ( d ) passing through the points A ( − 3 ; 2) and B ( − 2 ; 1) . a Draw the straight line ( d ) in the reference frame below. b Determine the slope-intercept formof the line ( d ) . 2 Determine the relative position of the curve C f and the straight line ( d ) . E.2838 Consider the function f defined by the relation: f ( x ) = − x + 1 The graphical representation is given below : Consider the point A with coordinates (3 ; 1) and M a point on the curve C f . Determine the position of the point M on C f so that the length AM is minimal. E.10596 Let ABCD be a rectangle of dimension 6 cm and 4 cm . Consider the points E and G , located outside the rectangle ABCD , belonging respectively to the half-lines [ AB ) and [ AD ) and the point F such that the quadrilateral AGFE is a rectangle. Note x and y the following two distances : x = DG ; y = BE The points E and G are required to form a rectangle AEFG with a perimeter of 28 cm . 1 a Show that the length y is expressed as a function of x by: y =4 − x b Deduce the possible values of x . Note A the area of the hatched part (that of the polygon BEFGDC ) . 2 Establish that the area of the hatched part written as a function of x is obtained by the equality: A ( x ) = − x 2 + 2 x + 24 3 Draw up the table of variations of the function A on 0 ; 4 . (the value of the local extremum will be indicated) . 4 Give the value of the maximum area of the hatched part? For what values of x is it reached? https://chingmath.fr chapExoCorrec/7329 sacados/7329 -4-3-2-1I-12JOCf chapExoCorrec/2838 sacados/2838 -2-1234I-2-12JOAMCf chapExoCorrec/10596 sacados/10596 4cm6cmyxABCDEFG