TASKS of MATHEMATICS for economic applications AA. 2012/13
Intermediate Test December 2012
I M 1) Compute 3 , using trigonometric form for complex numbers.
" 3 " 3
) '
%
I M 2) Given the matrices œ and œ , verify if
" $ % # # #
# ! % # # "
" $ % ' ' $
they have the same eigenvalues, and then check if they are similar matrices.
I M 3) Given the linear map 0 À Ä ß 0 œ † , with œ , if the
" " " !
! " " "
" # # 5
‘% ‘$ — —
dimension of the Kernel is equal to , find a basis for the Image and a basis for the Kernel of# the linear map.
I M 4) The linear system has solutions. Find all the solutions
B C #D œ "
#B C D œ "
#B 7C 5D œ "
∞"
of the linear system.
I M 5) Given the matrix œ check if, varying the parameter , the matrix can5
" ! !
! 5 "
! " "
admit a multiple eigenvalue.
I Winter Exam Session 2013
I M 1) After finding the roots , , , D" D# D$ D% of the equation D #D #D )D ) œ !% $ # , then calculate $ D † D † D † D" # $ %.
I M 2) The matrix œ is similar to the matrix by means of the matrix
" # "
# " "
" ! "
(similarity transformation) œ . Determine the matrix and its eigenvalues.
" ! "
! " "
! ! "
I M 3) Given the matrix œ check if, varying the parameter , the matrix7
$ " "
! # !
7 5 $
can admit multiple eigenvalues, and then, for such values, check, varying the parameter , if5 the matrix is a diagonalisable one.
I M 4) Given the linear map 0 À Ä ß 0 œ † , with œ , if the
" " ! !
! " " !
! ! " "
5 ! ! "
‘% ‘% — —
dimension of the Kernel is maximum, find the image of the vector "ß "ß "ß ".
II M 1) The system of equations is satisfied at a uni-
0 Bß Cß D œ D #BC œ !
1 Bß Cß D œ B C %B C D œ !# # #
que point P œ "ß Cß D ; determine that point and check which implicit function can be so defined; then calculate the first order derivatives of such function.
II M 2) Given the function 0 Bß C œ BC " B C # determine its stationary points and then check their type.
II M 3) Solve the problem .
Max/min s.t.:
0 Bß C œ B C B C Ÿ "
C #B
# #
# #
II M 4) Given the function 0 Bß C œ , that it is not differentia- BC
B C Bß C Á !ß !
! Bß C œ !ß !
# #
ble at the point !ß ! , using the definition check for the directions for which the directional@ derivative W@0 !ß ! exists.
II Winter Exam Session 2013
I M 1) After finding the roots , , , of the equation D D D D" # $ % D D D *D "! œ !% $ # , then calculate .% D D D D" # $ %
I M 2) Check for existence and number of solutions, on varying parameters and , for the7 5 linear system of equations . When the system has solutions, find
B C #D œ "
#B $C D œ "
B 'C (D œ 7
&C &D œ 5 them.
I M 3) Vector — œ B ß B " # has coordinates "ß # in the base #ß " à "ß #. Determine its coordinates in the base $ß # à "ß " .
I M 4) Determine an orthogonal matrix that diagonalizes œ .
! " "
" ! "
" " !
II M 1) Given 0 Bß C œ B $BC $ , and unit vector of @ "ß " , find the points B ß C! ! at which it results W@0 B ß C ! !œ ! and W@ß@# 0 B ß C ! !œ !.
II M 2) Given the equation 0 Bß Cß D œ B C C D BCD œ ! $ $ $ and the point P! œ "ß !ß " that satisfies it, determine the equation of the tangent plane to the surface of the implicit function defined with such equation.
II M 3) Solve the problem Max/min .
s.t.:
0 Bß Cß D œ BCD B C D œ &
We recommend using the simplest procedure.
II M 4) Solve the problem .
Max/min s.t.:
0 Bß C œ B BC C B " C Ÿ "
B #C Ÿ !
#
I Additional Exam Session 2013 I M 1) Compute " 3#! " 3 "#.
I M 2) Find a basis for ‘# consisting of eigenvectors of , where is a matrix similar to
œ % " œ $ "
# " & #
with the similarity transformation matrix.
I M 3) Given the matrix œ , check, varying the parameters and , the5 7
" ! ! 5
! " ! 5 7 ! " ! 7 ! ! "
maximum possible dimension for the Kernel of the linear map 0 À‘% Ä‘%ß 0 — œ —† , and then check, for such and , if vector 5 7 —œ "ß "ß "ß " may belong to the Kernel of such linear map.
I M 4) Check if the matrix œ may be diagonalised with an orthogonal matrix.
# # "
! " !
" # #
II M 1) Verify that with the equation 0 Bß C œ B B C / œ ! % # C it is always possible to define an implicit function C œ C B , at any point P that satisfies it. If P! ! œ "ß ! , compute C "w and C "ww .
II M 2) Given the function 0 Bß C œ B/ C/ C B, check if there exist directions such that@ W@0 !ß ! œ ! and W#@ß@0 !ß ! œ ! .
II M 3) Check if the function 0 Bß C œ B † C B is differentiable at the point !ß ! .
II M 4) Solve the problem .
Max/min s.t.:
0 Bß C œ B C B %C Ÿ % B )C Ÿ %
# #
#
I Summer Exam Session 2013
I M 1) After finding the roots , , of the equation D" D# D$ D $D %D ) œ !$ # , then calculate .$ D † D † D" # $
I M 2) The matrix œ admits the eigenvalue - œ ". Check if such mat-
# # "
" # 5
$ ! #
rix is a diagonalizable one.
I M 3) Given the linear map 0 À Ä ß 0 œ † , with œ " " 5 ,
! 7 "
‘$ ‘# — —
knowing that the vector "ß "ß # belongs to the Kernel, find the dimension of the Image of the map.
I M 4) Check for existence and number of solutions, on varying parameters and , for the7 5
linear system of equations .
B B œ $
#B B B B œ # B B 7B #B œ 5
" %
" # $ %
" # $ %
II M 1) Find maxima and minima for 0 Bß C œ $C #B in the quadrangle having as vertices the points , , !ß ! "ß ! "ß # and . !ß "
II M 2) Given 0 Bß C œ B BC $C # , find point T! at which W?0 T ! œ # and W@0 T ! œ # , where is the unit vector of # ? "ß " and is the unit vector of @ "ß " Þ
II M 3) Solve the problem .
Max/min s.t.:
0 Bß C œ B #C B #
% C Ÿ "
#
#
II M 4) Verify that with the equation 0 Bß C œ / BC cosB C œ ! , at the point "ß " , it is possible to define an implicit function C œ C B , and then compute C "w and C "ww .
II Summer Exam Session 2013
I M 1) Calculate $ .
"& $
# " $3 # " 3
I M 2) Check when the matrix œ , varying the parameter , is a diagona-5
! 5 !
" ! !
! ! 5
lizable one.
I M 3) Given a linear map 0 À Ä ß 0 œ † , with œ + , - , knowing
+ , -
‘$ ‘# — — " " "
# # #
that the image of the vector "ß "ß " is the vector "ß " and that the vector "ß !ß " belongs to the Kernel, check if the dimension of the Image of the map may be equal to ."
I M 4) Find the representation of the vector "ß #ß $ in the basis "ß !ß " à "ß "ß ! à "ß !ß ! . II M 1) Analyse stationary points for 0 Bß C œ B C 5 BC $ $ on varying the parameter .5 II M 2) Solve the problem Max/min
s.t.:
0 Bß C œ B C B %C Ÿ %# # # # Þ
II M 3) Given 0 Bß C œ BC C #, if W?0 "ß " œ ! and W?0 "ß " œ #, determine the unit vector ? œ cosαàsenα.
II M 4) Verify that with the system sen cos , at the
0 Bß Cß D œ B C D #B œ "
1 Bß Cß D œ /#BD /CBœ !
point "ß "ß #, it is possible to define an implicit function B Ä C B ß D B , and then compu- te the equation of the tangent line at B œ ".
I Autumn Exam Session 2013
I M 1) If D œ # cos 3sen and D œ % cos 3sen , calculate D † D .
& & & &
% %
" 1 1 # 1 1 $ " #
I M 2) Given the basis !ß "ß " à "ß "ß ! à "ß "ß " , find the coordinates of the vector
"ß #ß # in this basis.
I M 3) Check when the matrix œ , varying the parameter , is a diagonaliza-5
! " "
5 ! 5
! " "
ble one.
I M 4) Given the linear maps 0 À Ä ß 0 œ † , with œ and
" 5
" "
5 !
‘# ‘$ — —
1 À Ä ß 1 œ † œ " 5 " 5
! " "
‘$ ‘# — —, with , check, varying the parameter , when the dimentions of the Image and of the Kernel of the composite map 1 0 — À‘# Ä‘# are equal.
II M 1) Analyse stationary points for 0 Bß C œ B BC 5C # # on varying the parameter .5 II M 2) Solve the problem Max/min
s.t.:
0 Bß C œ B BC C B C Ÿ "# # # # Þ
II M 3) Given 0 Bß Cß D œ B CD # , if and are the unit vectors of ? @ "ß !ß " and !ß "ß ", compute .W#?ß@0 "ß "ß "
II M 4) Verify if with the equation 0 Bß C œ B C B C œ ! $ $ # # , at the points !ß ! or
"ß " , it is possible to define an implicit function C œ C B , and then compute and Cw Cww. II Autumn Exam Session 2013
I M 1) Once you have found the three roots of the equation B #B %B ) œ !$ # , verify that they cannot be the three third order roots of a complex number.
I M 2) After verifying that œ # " and œ $ $ are similar matrices,
" # ) (
check how many are the matrices that realize the similarity.
I M 3) Check for existence and number of solutions, on varying parameters , and , for7 5 2
the linear system of equations .
B #B #B B œ "
#B $B B B œ # B *B 7B 5B œ 2
" # $ %
" # $ %
" # $ %
I M 4) In a linear map 0 À‘$ Ä‘$ß 0 — œ —† , the two vectors —" œ "ß !ß ! and
—# œ !ß "ß " form a basis for the Kernel of such map, and the vector ˜" œ $ß $ß ' is the image of the vector —$ œ "ß #ß " . Find the matrix and check if is a diagonali- zable matrix.
II M 1) Given the function 0 Bß C œ , that it is not differentiable at B
B C B C Á !
! B C œ !
#
the point !ß ! , using the definition check for the directions for which the directional deri-@ vative W@0 !ß ! exists.
II M 2) Verify that with the system , at the point , it is
0 Bß Cß D œ BCD / œ !
1 Bß Cß D œ BCD / œ ! "ß "ß "
BC CD
possible to define an implicit function B Ä C B ß D B , and then compute the equation of the tangent line at B œ ".
II M 3) Solve the problem .
Max/min s.t.:
0 Bß C œ B C B #B C " Ÿ ! B #B C " Ÿ !
#
#
II M 4) Check if the problem Max/min may have solutions.
s.t.:
0 Bß Cß D œ BC D BD CD œ %
