Grade 11 / Course questions 10 exercises (100% corrected)

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1. Reference conditions E.5991 Show that the square root function is increasing on 0 ; + ∞ . E.5992 Consider the identity function, the square function, the square root function noted respectively: f : x ↦−→ x ; g : x ↦−→ x 2 ; h : x ↦−→ √ x 1 Compare the functions f and g on the interval 0 ; + ∞ . 2 Compare the functions f and h on the interval 0 ; + ∞ . 2. Suites E.5993 1 Let u n be an arithmetic sequence with first term u 0 and reason r . Show that, for any natural number n , we have : u 0 + u 1 + · · · + u n = n + 1 · u 0 + u n 2 2 Let v n be a geometric sequence with first term v 0 and reason q . Show that, for any natural number n , if q =1 , then we have : v 0 + v 1 + · · · + v n = v 0 · 1 − q n +1 1 − q 3. Scalar product E.5994 Let A and B be two points in the plane and I the midpoint of [ AB ] . For any point M of the plane, we have equality: MA 2 + MB 2 = 2 · MI 2 + AB 2 2 E.5995 Show that for any real a and b , we always have : cos a − b = cos a · cos b + sin a · sin b E.6082 In the plane provided with a refer-ence frame O ; I ; J orthonormal, any circle admits a stan-dard form of the form : x 2 + y 2 + a · x + b · y + c = 0 où a;b;c ∈ R 4. Probabilities E.5996 Let X be a random variable. For any real numbers a and b , we have : a E a ·X + b = a · E X + b b V a ·X = a 2 · V X 5. Binomial law E.5997 Establish that for any natural num-ber n and for any integer k such that 0 k<n , we have : n k +1 + n k = n +1 k +1 6. Unclassified financial years E.6080 Consider the plane provided with a O ; I ; J orthonormal coordinate system. Let ( d ) be a straight line admitting the director vector −→ u ( ¸ ; ˛ ) . Then the straight line ( d ) admits a standard form of the form : ˛ · x − ¸ · y + c = 0 https://chingmath.fr chapExoCorrec/5991 sacados/5991 chapExoCorrec/5992 sacados/5992 chapExoCorrec/5993 sacados/5993 chapExoCorrec/5994 sacados/5994 chapExoCorrec/5995 sacados/5995 chapExoCorrec/6082 sacados/6082 chapExoCorrec/5996 sacados/5996 chapExoCorrec/5997 sacados/5997 chapExoCorrec/6080 sacados/6080
HABHAB¸AB×AH¸−AB×AHABCHABCH E.6083 Consider the plane provided with a O ; I ; J orthonormal coordinate system. Let A ( x A ; y A ) , B ( x B ; y B ) , C ( x C ; y C ) be three points in the plane. The following numbers all have the same value : a x B − x A x C − x A + y B − y A y C − y A b AB × AC × cos −−→ AB ; −→ AC c 1 2 · −−→ AB + −→ AC 2 − −−→ AB 2 − −→ AC 2 d Noting H the orthogonal project of the point C on the line ( AB ) , consider the number ¸ defined according to the two cases : https://chingmath.fr chapExoCorrec/6083 sacados/6083 HABHAB¸AB×AH¸−AB×AHABCHABCH