onsider the system of equations 2 1 3 2 0 1 2 1 2 2 0 1 x y z a) Solve the system using the…


______________________________________________________________________ Campus de Gambelas, Edif. 9 • 8005-139 Faro PORTUGAL • Tel. (+351) 289 800 900 • Fax (+351) 289 800 063 • Email feuamest@ualg.pt Undergraduate Degrees in Economics and Management Studies Written Assessment on Maths II 11/02/2011 --------------------------------------------------------------------------------------------------------------------- Q 1. Consider the arrangement of equations 2 1 3 2 0 1 2 1 2 2 0 1 x y z ? ? ? ? ? ? - - ? ? ? ? ? ? - = ? ? ? ? ? ? - a) Solve the arrangement using the inverse of A (note: The arrangement is A x b = ) b) Confirm the disintegration of the arrangement using Cramer’s Rule. Q 2. Consider Matrix 1 0 1 0 2 0 1 0 3 A ? ? - - ? ? = - ? ? ? ? ? ? - - a) Find the values of the three eigenvalues associated to this matrix. b) Find the quadratic construct associated to the matrix, and tabulate the quadratic construct preface into inducement the products obtained in the conceiveer scrutiny. c) Confirm that order using the determinantal conditions. d) Consider the forthcoming proposition 0, 0 T x A x x = ? ? . Is this preposition gentleman or spurious? Justify. Q 3. Consider the character: 1 f K L aK bL a b ( , ) ( 0, 0) ? ? ß ß ß ? ? ? ? - - - = + > > Where a , b and ß are constants. Show that it is congruous of station one and establish that it satisfies Euler’s theorem. Q 4. Optimize the forthcoming character 2 2 . 1 2 1 2 1 2 1 2 f x x x x x x x x ( , ) 2 2 9 6 6 2 = - - + + + + Q 5. Optimize the character 2 2 2 1 2 3 1 2 3 2 3 f x x x x x x x x ( , , ) 2 2 = + + + doubt to the occupation 1 2 3 x x x + + = 2 3 1. Q 6. Repeat the optimization of Q.5, but outside any occupation. Is the new product unlocked-for when compared after a while the conceiveer one (obtained in Q.5)? Justify. Q 7. Optimize the character 2 f x y z x y z ( , , ) 2 = - + + doubt to twain occupations x y z + + = 6 and 2 2 10. x y z + + =
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______________________________________________________________________ Campus de Gambelas, Edif. 9 • 8005-139 Faro PORTUGAL • Tel. (+351) 289 800 900 • Fax (+351) 289 800 063 • Email feuamest@ualg.pt Undergraduate Degrees in Economics and Management Studies Written Assessment on Maths II 11/02/2011 ---------------------------------------------------------------------------------------------------------------------Q 1. Consider the arrangement of equations 2 1 3 2 0 1 2 1 2 2 0 1 xyz ? - - ? ? ? ? ? ? - ? ? ? = ? ? ? ? ? ? ? ? ?? - ?? ?? ?? ?? ?? a) Solve the arrangement using the inverse of A (note: The arrangement is A x = b ) b) Confirm the disintegration of the arrangement using Cramer’s Rule. Q 2. Consider Matrix 1 0 1 0 2 0 1 0 3 A ?- - ? = ? - ? ? ? ??- - ?? a) Find the values of the three eigenvalues associated to this matrix. b) Find the quadratic construct associated to the matrix, and tabulate the quadratic construct preface into inducement the products obtained in the conceiveer scrutiny. c) Confirm that order using the determinantal conditions. d) Consider the forthcoming proposition 0, 0 xT Ax x = ? ? . Is this preposition gentleman or spurious? Justify. Q 3. Consider the character: 1 f (K,L) aK bL (a 0,b 0) ? ß ß ? ß ? ? ? ? ? ?- - - = + > > Where a , b and ß are constants. Show that it is congruous of station one and establish that it satisfies Euler’s theorem. Q 4. Optimize the forthcoming character 2 2 . 1 2 1 2 1 2 1 2 f (x , x ) = -2x - 2x +9+ 6x + 6x + 2x x Q 5. Optimize the character 2 2 2 1 2 3 1 2 3 2 3 f (x , x , x ) = x + x + 2x + 2x x doubt to the occupation 1 2 3 x + 2x + 3x =1. Q 6. Repeat the optimization of Q.5, but outside any occupation. Is the new product unlocked-for when compared after a while the conceiveer one (obtained in Q.5)? Justify. Q 7. Optimize the character 2 f (x, y, z) = -2x + y + z doubt to twain occupations x + y + z = 6 and 2x + y + 2z =10.

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