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(3) Find the formula of un, as a function of n

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Cergy-Pontoise University, L1, year 2010–2011

Mathematics MS1 course (T. Banica), final exam, 11/01/11 15:00–17:30, documents and electronic devices forbidden

1. Consider the sequence u0 = 1, un+1 = 3un+ 2.

(1) Compute u1, u2, u3, u4, u5.

(2) Show that un>10, for any n >2.

(3) Find the formula of un, as a function of n.

(4) Does the sequence xn= 1/un converge?

2. Consider points A = (1,1), B = (2,3), C = (4,5).

(1) AreA, B, C on the same line?

(2) Find the distance from (0,0) to the line AC.

(3) Compute the angles of the triangleABC. (4) Compute the area of the triangle ABC.

3. Let K = sin(3x).

(1) Express K as a function of sin(x),cos(x).

(2) Express K as a function of sin(x).

(3) Express K as a function of cos(x). Discussion.

(4) Express K as a function of tan(x/2).

4. Let P be the plane x+y+ 2z+ 3 = 0 . (1) Does P pass through the point (2,−3,1)?

(2) Find the distance from (0,0,0) to the plane P.

(3) Find the intersection of P with the plane x+ 2y+z = 0.

(4) Find the intersection of P with the sphere centered at (2,2,2), of radius 1.

5. Consider the matrix A=

6 −3

4 −1

. (1) Find the characteristic polynomial ofA.

(2) Find the eigenvalues and eigenvectors of A.

(3) Express v = (23) as a combination of eigenvalues ofA.

(4) Compute explicitely (give the numeric answer) the vectorA5v.

=⇒justify all the answers

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