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Ex1/ 1/ For A, B ∈ Mn (R) be square matrices of order n, assumed to be invertible

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Ex1/ 1/ For A, B ∈ Mn (R) be square matrices of order n, assumed to be invertible. Simplify as much as possible, detailing all the steps, the following expression: [in the picture] 2/For C, D ∈ M2 (R) be square matrices of order 2. Say if the following statements are still true. Justify the answers.. Ex2/ 1/ By using the Gauss method and specifying the operations carried out at each stage, solve the homogeneous system AX = 0. 2/ Is matrix A invertible? Justify using your answer to the previous question. 3/ Show that S is a solution of the non-homogeneous system AX = B. 4/ Using your answers to the previous questions, give all the solutions of the non-homogeneous system AX = B. Do not use the Gauss method

Ex3/ We consider the function f (x) [in the picture] defined on R \ {1}. 1/ Calculate the following limits by justifying your answers 2/ Calculate the derivative f 'and then give the table of variation of f. The limit values will be indicated in question 1 on the table. 3/ Determine the largest value of a ∈ R + for which f is injective over the interval] 1, a]. Justify your answer. 4/ Denote by I =] 1, a] the interval of the previous question. Give the interval J for which f: I → J is a bijection. Justify your answer 5/ Let g: J → I be the reciprocal bijection of the bijection f: I → J. (a) Show that g (−1/2) = 3/2

(b) Is the bijection g differentiable at the point −1/2? If so, calculate g '(- 1/2) We will justify the answers. 6. Show that there is at least one solution of the equation ( in the picture ) in the interval [2, 5]. Justify your answer Ex4/ Let a ∈ N be a parameter, and x → f (x) a function three times differentiable in 0. We suppose that f has in 0 the following limited expansion: [ function in the picture] 1/Determine the values of f (0), f’(0) , f’’(0), f’’’(0) 2/ Write the expansions limited to order 3 of f (ax) and f (x)^2 in 0. 3/ Ex5/ Questions 1,2,3 are independent of each other 1/ (a) Find in exponential form the square roots of u = √3 + i. (b) Find in algebraic form the square roots of u. (c) Deduce the values of cos π / 12 and sin π / 12 2. (a) Calculate in algebraic form (2 + i) ^ 3 (b) Deduce the cubic roots of 2 + 11i in algebraic form. 3/3. Solve in C the equation z ^ 2 = z + 1/2. We will look for solutions in algebraic form.