Mathematics Exam + Solutions

Mathematics Exam 11/13/2024 ⏱ 60’


1) LIMITS Calculate the following limit

lim log $ √𝑥
!→# %
o 0
o −∞
!
o "
o +∞



2) VECTORS Determine the polar coordinates (𝜌, 𝛼) of 𝑢
(⃗ with A (0, –4)
#
o 𝜌 = 4 ;𝛼 = $
%
o 𝜌 = −4 ; 𝛼 = − $ 𝜋
o they are not defined
%
o 𝜌 = 4 ;𝛼 = $
𝜋



3) VECTORS a- Determine 𝑘 so that 𝑢 (2, 2𝑘) and 𝑣 4√3, 18 are orthogonal
b- Determine the angle 𝛼 between 𝑢 and the versor −𝚥(⃗(0, – 1)
#
o 𝑢 = 4√3, 18 ; 𝛼 = %
#
o 𝑢 = 41, −√38 ; 𝛼 = &
#
o 𝑢 = 42, −2√38 ; 𝛼 = &
#
o 𝑢 = 42, 2√38 ; 𝛼 = − &



4) DERIVATIVE Given the function 𝑓 (𝑥 ) = −10(𝑥 $ − 2), determine if there
is a maximum or a minimum in 𝑥 = 0
o the function has a minimum in 𝑥 = 0

, o the function has a maximum in 𝑥 = 0
o the function neither has a maximum nor a minimum in 𝑥 = 0
o the function is increasing in 𝑥 = 0


2 3
5) ALGEBRA Determine the inverse matrix 𝐴'! of A ? @
−1 1
!1 −3
o ?
( 1
@
2
5 −15
o ? @
5 10
! 5 0
o (? @
0 5
1 −3
o ? @
1 2


6) ALGEBRA Determine the real eigenvalues and eventual orthogonal
−3 0
eigenvectors of A ? @
0 3
o no, there are no real eigenvalues
o yes, there are two real eigenvalues, but no eigenvectors
o yes, there are two real eigenvalues with their respective eigenvectors,
but they are not necessarily orthogonal
o yes, there are two eigenvalues with their respective orthogonal
eigenvectors


7) ALGEBRA Given the following equations, determine 𝑚∥ and 𝑚* so that
the lines can be respectively parallel and orthogonal
2𝑥 + 𝑦 = 7
C
𝑚𝑥 − 𝑦 = −1


!
o 𝑚∥ = 2 ; 𝑚* = $
o 𝑚∥ = 2 ; 𝑚* = 1