1st Exam in Mathematics - 2021/2022 - 1st semester
Outcome no. 1: Mathematical logic (9 pts, 3 pts needed to pass)
1. (3 pts) Check whether the expression
(p ? q) ? (p ? q),
where p and q are sentences, is a tautology.
2. (3 pts) Determine the logic value of the expression
? ?x ? ? : (|x| > |x + 2|).
3. (3 pts) Write the negation of the sentence
?x ? ? : (|x| < 1) ? (|x + 1| ? 2).
Outcome no. 2: Algebra of complex numbers (9 pts, 3 pts needed to pass)
1. (3 pts) Find real and imaginary parts of (1-i)/(2+i).
2. (3 pts) Write z = 1 - ?3i in a polar form.
3. (3 pts) Find all complex numbers z's satisfying equation Re(z虏 - 2) = 啪.
Outcome no. 3: Matrix calculus (9 pts, 3 pts needed to pass)
1. (3 pts) Solve the equation X 路 [1 2; 2 1] = 3 路 [1 1; 3 3].
2. (3 pts) Calculate |1 2 0; 2 3 0; 0 1 7|.
3. (3 pts) Determine the rank of the matrix [1 2 0 2; 2 4 0 -1].
Outcome no. 4: Vector calculus and analytic geometry (9 pts, 3 pts needed to pass)
1. (3 pts) Find the equation of the line passing through points A(1, 2, 3) and B(0, -2, 5).
2. (3 pts) Find parametric equations of the line l passing through A(1, 2, 3) and perpendicular to the plane ? : y = 1.
3. (3 pts) Find the common point of the plane ? : y + z = 1 and the line l : x/1 = y/-2 = z/-1.
Outcome no. 5: Differential calculus (9 pts, 3 pts needed to pass)
1. (3 pts) Compute by definition f'(1) for f(x) = (x - 1)虏.
2. (3 pts) Calculate ( (cos x) / (x虏 + e^x + 1) )'.
3. (3 pts) Use l'Hospital's Rule to find lim_{x?1} (ln x) / (x虏 - 1).
