SELECTED EXAM QUESTIONS
AGH UST 2023
Question 1 [1p.] The slope of the tangent to the graph of the function
f (x) = −x3 + 3x2 − 7x − 2
at the point P = (1, −5) is equal to
A. −4 B. 4 C. −2 D. 2
Question 2 [1p.] The equation cos2 x = cos x in the interval [−π, 0]
A. has exactly one solution. B. has exactly two solutions.
C. has exactly three solutions. D. has no solutions.
Question 3 [1p.] The expression
1 1
− a
b
,
a−b b −1 a− a −1 b
for a, b ̸= 0 and |a| =
̸ |b|, is equal to
a4 +b4 a3 +b3 a3 +b3 a4 +b4
A. a3 −b3 B. a4 −b4 C. a2 −b2 D. a4 −b4 .
Question 4 [1p.] The limit
−7n3 + n2 + 2n
lim
n→∞ n3 + 3n2 + 4n5
is equal to
A. − 47 B. −7 C. 0 D. −∞
Question 5 [2p.] Assume that
ax + b
f (x) = ,
cx + d
where b = 11, d = 12 and cx + d ̸= 0. The point A = (6, −4) is the center of symmetry of the graph of
the function f . Compute the ratio ab . Enter the first three digits of the decimal expansion of the result.
ANSWER:
Question 6 [3p.] Prove that for all real numbers a, b, such that 4a2 + b2 ¬ 4 the inequality 2a + b ¬ 3
holds true.
Question 7 [3p.] For what values of the parameter k the domain of the function
p
f (x) = (k + 3)x2 + (k + 3)x + 2
is the set of real numbers?
log9 7
Question 8 [3p.] Consider the numbers a = log 8 + 3 log 5 and b = log7 49 . Calculate ab .
Question 9 [4p.] We randomly create the nine-digit number with different digits taken from the set
{1, 2, 3, 4, 5, 6, 7, 8, 9}. Calculate the probability of getting the odd number where the digits 5 and 7 are
adjacent.
Question 10 [4p.] Solve the equation
1 x+8
21 · 23 · 25 · . . . · 22x−1 = ·4 .
2
Question 11 [4p.] For what values of the parameter p the roots x1 , x2 of the equation
x2 − (p + 3)x + p = 0
satisfy the condition |x1 − x2 | < 3?
Question 12 [5p.] We draw five numbers consecutively without replacement from the numbers 1 to 20.
Find the probability that the second number drawn is divisible by 4 and the last number is divisible by 5.
Question 13 [5p.] Consider the function:
f : [−4, 4] ∋ x 7→ |3 − x| − |4 + 2x| + |6x|.
(a) Find its smallest and largest value.
(b) Solve the inequality
f (x) > 8.
Question 14 [6p.] In a regular triangular prism,
√
the sine of the angle between the diagonal of the side
face and the adjacent side face is equal to 2 5 3 . Calculate the lenght ratio of the edge of the base of the
prism to the height of the prism.
Question 15 [7p.] The rectangle with edges parallel to the axis OX and OY is inscribed in the figure
bounded by the parabola y = 81 x2 and the straight line y = 6. Find the coordinates of the rectangle’s
vertices with the maximal area.
