Solutions must include all necessary comments and calculations.
In the following problems assume:
- UD is the unit digit, and TD is the tens digit of the Student ID;
- c = -1
- d is the remainder of dividing UD by 3, then increased by 1.
- p is the remainder of dividing UD + TD by 3, then increased by 2.
- q is the remainder of dividing UD by 4, then increased by 3.
Duration: 2h
Problem 1. Compute and mark on the complex plane:
Points: 60
[3p] [sqrt[3]{8cosleft(frac{3dpi}{4}
ight)-8jsinleft(-frac{3dpi}{4}
ight)}]
[4p] [|z+(-d+j(d+1)+1)/(j-d)|<(|z-d+j|)]
[3p] the roots of (w(z)=z^{4}+(j-sqrt{3})^{4(d+2)})
Problem 2.
[10p] Find solution of a system of linear equations
[x_{1}+x_{2}+x_{3}+2x_{4}=(-1)]
[-cx_{1}+cx_{2}=c]
[cx_{1}+(c+1)x_{2}+(c+3)x_{3}+(2c+1)x_{4}=-c-5]
[-x_{2}+2x_{3}-x_{4}=(-5)]
with variables (x_{1}, x_{2}, x_{3}, x_{4}) in (R).
Problem 3. For a tuple of vectors (A=((3p, p, 3p), (2p+3, 1, p+3), (2p-1, 1, p+1))) in (R^{3}):
[3p] verify if A is a basis of (R^{3}),
[4p] verify if ((p-7, p-1, 2p-5)) in span A
[3p] find a basis and the dimension of span A
Problem 4. Find values:
[3p] [lim_{n o+infty}left(frac{qn-3}{1-qn}
ight)^{qn}]
[3p] [lim_{x o q}arctanleft(frac{q^{2}-2qx+q+x^{2}-x}{x-q}
ight)]
[4p] [lim_{n o+infty}sqrt[n]{(q+1)^{n}+n^{q}left(frac{q}{2}
ight)^{n}}]
Problem 5.
[10p] Find parameters a, b in (R) for which a function (f(x)=egin{cases} frac{ an(px)}{2x} & ext{for } x<0, \ aarcsinleft(cosleft(frac{4pi}{6}
ight)
ight) & ext{for } x=1, \ frac{sin(px)}{ln(1+px)}+b & ext{for } x>1 end{cases}) is continuous at (x=0).
Problem 6. For a function (f(x)=frac{x-2d}{sqrt{x^{2}+d^{2}}}):
[3p] find the left slant asymptote,
[5p] examine the monotonicity and find local extreme values,
[2p] verify if it is injective.
