Moscow Aviation Institute (National Research University)

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______________________________________
VARIANT 227

1 What is the smallest solution of inequality √4 − 𝑥 + 𝑥 + 2 ≥ 0?

1) −5,2 2) −5 3) −4 4) 0
2 Find the sum of solutions of the equation 81 ∙ 3 𝑥+2
+3 𝑥+1
−32𝑥+4
− 27 = 0.

1) −1 2) 1 3) 5 4) −3
3 Calculate the value of cos −1 (1,5𝜋
+ arctg4) cos(arctg4) .

1) −1 2) 1 3) 0,25 4) −0,25
4 Calculate the value of log 30 2, if known, that log 30 3 = 𝑎 , log 30 5 = 𝑏.

1) 1−𝑎−𝑏 2) 𝑏−𝑎 3) 1+𝑎−𝑏 4) −(𝑎 + 𝑏)

5 𝐴𝐵 = 6 and 𝐵𝐶 = 12 are sides of triangle 𝐴𝐵𝐶. The sum of heights drawn to those sides is equal to
the doubled length of the third height. What is the length of the side 𝐴𝐶?

1) 8 2) 7 3) 9 4) 8,2
6 Find the sum of solutions of the equation sin 2𝑥 + 2 ctg 𝑥 = 3, that belong to the interval (0; 1,5𝜋).

1) 2𝜋 2) 1,5𝜋 3) 1,25𝜋 4) 1,75𝜋

7 Find all values of the parameter 𝑎 for which the system of equations
𝑥 2 + 𝑦 2 − 2𝑥 + 2𝑦 = 𝑎 − 2,
{
𝑥 2 − 2𝑥 − 𝑦 = 0
has only one solution.

1) [−0,25; 0] 2) 4 3) 0,5 4) 0
8 Find the smallest solution of inequality log 1 (3 − 8𝑥) ≥ 1.
9𝑥
Answer:

9 The heights 𝐴𝐴1 and 𝐵𝐵1 of acute isosceles triangle 𝐴𝐵𝐶(𝐴𝐶 = 𝐵𝐶) intersect at a point 𝑂. Find the
angle 𝐴𝐵𝐶, if 𝐶𝑂 = 𝐴𝐵.
Answer:

10 𝑆𝐴𝐵𝐶𝐷 is a regular tetragonal pyramid with a base 𝐴𝐵𝐶𝐷. Find the distance between 𝐵𝐶 and 𝑆𝐷, if
𝐴𝐵 = 12, 𝑆𝐶 = 10
Answer:
You can provide your own version of the answer to any problem.

2020
Chairman of the examination committee for mathematics ________________________
 Moscow Aviation Institute (National Research University)

«Approved by»
Prorector of MAI,
______________________________________
VARIANT 221

1 Find product of solutions of the equation

3𝑥 2𝑥 5
2
− 2 = .
3𝑥 + 6 − 𝑥 𝑥 + 2𝑥 − 6 6

1) 24 2) 48 3) 60 4) 36
2 4
Calculate the 32(𝑥1 + 𝑥2 ), where 𝑥1 , 𝑥2 − are the roots of the equation 5 √𝑥 − 2√𝑥 − 3 = 0.

1) 180 2) 130 3) 162 4) 194

3 Calculate the value of cos −1 (1,5𝜋
− arctg4) cos(arctg4) .

1) −0,25 2) 4 3) 0,25 4) −4
4 Calculate the value of log 4√2(0,175), if known, that log 2 196 = 𝑢 , log 2 56 = 𝑣.

1) 1 − 2𝑢 − 2𝑣 2) 5𝑢 − 6𝑣 − 4 3) 1 − 2𝑢 + 𝑣 4) 𝑣−𝑢
5 𝐴𝐵 = 6 and 𝐵𝐶 = 12 are sides of triangle 𝐴𝐵𝐶. The sum of heights drawn to those sides is equal
to the doubled length of the third height. What is the length of the side 𝐴𝐶?

1) 10 2) 6 3) 8 4) 9
6 Find sum of solutions of the equation 2 ctg 𝑥 + 1 = sin 2𝑥, that belong to the interval (−2𝜋; −0,5𝜋).

1) −1,5𝜋 2) −1,25𝜋 3) −1,75𝜋 4) −0,75𝜋

7 Find all values of the parameter 𝑎 for which the system of equations
𝑥 2 + 𝑦 2 − 2𝑥 + 2𝑦 = 𝑎 − 2,
{
𝑥 2 − 2𝑥 − 𝑦 = 0
does not have a solution.

1) 𝑎 ≤ −0,5 2) 𝑎 ≤ −0,25 3) 𝑎<0 4) 𝑎 < −0,5

8 Find the biggest solution of inequality
log 1 (3 − 8𝑥) ≤ 1.
9𝑥

Answer:
9 The heights 𝐴𝐴1 and 𝐵𝐵1 of acute isosceles triangle 𝐴𝐵𝐶(𝐴𝐶 = 𝐵𝐶) intersect at a point 𝑂. Find the
angle 𝐴𝑂𝐵, if 𝐶𝑂 = 𝐴𝐵.
Answer:
10 𝑆𝐴𝐵𝐶𝐷 is a regular tetragonal pyramid with a base 𝐴𝐵𝐶𝐷. Find the distance between 𝐵𝐶 and 𝑆𝐷, if
𝐴𝐵 = 6, 𝑆𝐶 = 5.
Answer:
You can provide your own version of the answer to any problem.

2020
Chairman of the examination committee for mathematics ________________________
 Moscow Aviation Institute (National Research University)

«Approved by»
Prorector of MAI,
______________________________________
VARIANT 207

1 What is the smallest positive solution of inequality 4𝑡 + 2𝑡 −1 − 9 ≤ 0?

1) 0,5 2) 0,25 3) 0,025 4) 2

2 Solve the equation 2√𝑥 + 5𝑥 − 2 = 0.

1) 0,08(6 − √11) 2) 0,06(6 − √11) 3) 0,08(8 − √11) 4) 0,06(8 − √11)

3 Calculate the value of cos −1 (1,5𝜋 + arctg4) cos(arctg4) .

1) −1 2) 1 3) 0,25 4) −0,25
4 Calculate the value of lg 0,175, if known, that lg 7 = 𝑎, lg 2 = 𝑏.

1) 𝑎+2𝑏−1 2) 𝑎 − 2𝑏 − 1 3) 𝑎 − 2𝑏 + 1 4) 2𝑎 − 𝑏 − 1
5 𝐴𝐵 = 6 and 𝐵𝐶 = 3 are sides of triangle 𝐴𝐵𝐶. The doubled sum of heights drawn to those sides is
equal to the length of the third height. What is the length of the side 𝐴𝐶?

1) 2,4 2) 3,2 3) 3 4) 4
6 Find the sum of solutions of the equation sin 2𝑥 + 2 ctg 𝑥 = 3, that belong to the interval (0; 1,5𝜋).

1) 2𝜋 2) 1,5𝜋 3) 1,25𝜋 4) 1,75𝜋

7 Find the smallest solution of inequality log 1 (3 − 8𝑥) ≥ 1.
9𝑥

1) 1 2) 1 3) 1 4) 1
8 3 48 24

8 Find all values of the parameter 𝑎 for which the system of equations
𝑥 2 + 𝑦 2 − 2𝑥 + 2𝑦 = 𝑎 − 2,
{
𝑥 2 − 2𝑥 − 𝑦 = 0
has only one solution.

Answer:
9 The heights 𝐴𝐴1 and 𝐵𝐵1 of acute isosceles triangle 𝐴𝐵𝐶(𝐴𝐶 = 𝐵𝐶) intersect at a point 𝑂. Find the
angle 𝐴𝐵𝐶, if 𝐶𝑂 = 𝐴𝐵.

Answer:
10 𝑆𝐴𝐵𝐶𝐷 is a regular tetragonal pyramid with a base 𝐴𝐵𝐶𝐷. Find the distance between 𝐵𝐶 and 𝑆𝐷, if
𝐴𝐵 = 12, 𝑆𝐶 = 10

Answer:
You can provide your own version of the answer to any problem.

2020
Chairman of the examination committee for mathematics ________________________