Grade 12 / Other annals, MCQs, assertions ... 5 exercises (including 4 corrected)

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4mCamion7mCCABD„ 1. Trigonometry E.3163 A rabbit wants to cross a road that is 4 meters wide. A truck, occupying the entire road, is approaching at a speed of 60 km = h . The rabbit decides at the last moment to cross, when the truck is only 7 meters away. It takes off at lightning speed and we assume that it crosses in a straight line at its maximum speed, i.e. 30 km = h ! The front of the truck is represented by the segment [ CC ] in the diagram below. The rabbit starts from point A and heads towards d . This direction is marked by angle „ = BAD with : 0 „< ı 2 (in radians) 1 Determine the distances AD and CD as a function of „ and the times t 1 and t 2 taken by the rabbit and the truck to travel the distances AD and CD , respectively. 2 We set : f ( „ )= 7 2 +2 · tan „ − 4 cos „ . Show that the rabbit will have crossed the road before the truck passes if and only if f ( „ ) > 0 . 3 Conclude. Reminder: The function x ↦−→ tan x is differentiable on 0 ; ı 2 and has the function x ↦−→ 1 cos x 2 as its derivative. 2. Claims E.3169 Ten statements, divided into three themes and numbered from 1 a to 3 d are suggested below. Candidates should mark their copy with the word TRUE or FALSE opposite the statement number. Each correct answer earns 0.4 point. Each incorrect answer deducts 0.1 point. No answer is disregarded. Any negative total is reduced to 0. 1 For any real x , e x denotes the image of x by the expo-nential function. Affirmation a Pour all reals a and b : e a b =e a b Affirmation b Pour all reals a and b : e a − b = e a e b Assertion c La line of equation y = x +1 is the tan-gent to the representative curve of the exponential function at its point of ab-scissa 1. 2 Let f be a numerical function defined on an open interval I and let a be an element of I . Assertion a Si f is derivable in a , then f is continu-ous in a Assertion b Si f is continuous in a , then f derivable in a Assertion c Si f is derivable in a , then the function h ↦−→ f ( a + h ) − f ( a ) h admits a finite limit in 0 . 3 Consider two sequences u n and v n defined on N : Affirmation a Si lim u n =+ ∞ and if lim v n = −∞ then lim( u n + v n )=0 Assertion b Si ( u n ) converges to a non-zero real and if lim v n =+ ∞ then the sequence u n × v n does not converge. Assertion c Si ( u n ) converges to a non-zero real, if ( v n ) is positive and if lim v n =0 , then the sequence u n v n does not converge. Assertion d Si ( u n ) and ( v n ) converge then the se-quence u n v n converges. https://chingmath.fr chapExoCorrec/3163 sacados/3163 4mCamion7mCCABD„ chapExoCorrec/3169 sacados/3169
ABCDEFGHIJMN ABCDEFGH E.6352 For each of the questions below, only one of the four answers is correct. In each case, indicate the correct answer without justification. 1 In a spatial reference frame, consider the three points : A 1 ; 2 ; 3 ; B − 1 ; 5 ; 4 ; C − 1 ; 0 ; 4 . The line parallel to the line ( AB ) passing through the point C has the parametric representation : a x = − 2 t − 1 y = 3 t z = t + 4 ;t ∈ R b x = − 1 y = 7 t z = 7 t + 4 ;t ∈ R c x = − 1 − 2 t y = 5 + 3 t z = 4 + t ;t ∈ R d x = 2 t y = − 3 t z = − t ;t ∈ R 2 Space is provided with an O ; −→ i ; −→ j ; −→ k orthonormal reference frame. The straight line ( D ) is defined by the parametric repre-sentation : x = 5 − 2 t y = 1 + 3 t z = 4 , t ∈ R We note D the straight line that passes through the point A of coordinates 3 ; 1 ; 1 and has as director vector: −→ u = 2 · −→ i − −→ j + 2 · −→ k . The straight lines D and D are: a parallèles b confondues c non coplanaire d sécantes 3 The figure below represents a ABCDEFGH cube. The points I and J are the respective middles of the edges [ GH ] and [ FG ] . Points M and N are the respective cen-ters of faces ABFE and BCGF . The straight lines ( IJ ) and ( MN ) are: a perpendiculaires b orthogonales c sécantes, not perpendicular d parallèles 4 The set of complex numbers z such that z = z +1 z − 1 is a real is : a the set of real numbers whose imaginary part is equal to the real part ; b the set of pure imaginary numbers ; c the set of real numbers deprived of the number 1 ; d the number i 3. Unclassified financial years E.6050 For each of the following four propositions, indicate whether it is true or false and justify the answer chosen. One point is awarded for each correct answer correctly justi-fied. An unjustified answer is not taken into account. Ab-sence of an answer is not penalized. 1 Proposition 1: In the plane provided with an orthonor-mal reference frame, the set of points M whose affix z verifies the equality | z − i | = | z +1 | is a straight line. 2 Proposition 2: The complex number 1+i · 3 4 is a real number. 3 Let ABCDEFGH be a cube. Proposition 3: The straight lines ( EC ) and ( BG ) are orthogonal. 4 Space is provided with an orthonormal reference frame O ; −→ i ; −→ j ; −→ k . Let the plane P have standard form : x + y + 3 z + 4 = 0 . Note S the point with coordinates 1 ; − 2 ; − 2 . Proposition 4: The straight line ( d ) that passes through S and is perpendicular to the plane P has the parametric representation : x = 2 + t y = − 1 + t z = 1 + 3 t où t ∈ R . E.8147 Antilles-Guyane September 2018 5 points https://chingmath.fr chapExoCorrec/6352 sacados/6352 ABCDEFGHIJMN chapExoCorrec/6050 sacados/6050 ABCDEFGH sacados/8147 Antilles-Guyane Septembre 2018 5 points