1971 Mathematics Entrance Exam Options

50

Faculty of Computational Mathematics and Cybernetics

1. Solve system of equations
   $$\begin{cases} 4 \lg \sqrt{x} + 2^{y+1} = 3, \\ 24 \lg x + 2 \cdot 4^{y+1} = 37. \end{cases}$$

2. Find all solutions to the inequality
   $$\sqrt{3 \cos 2x} < \sqrt{2} \cos x,$$
   satisfying conditions $|x| < \pi$.

3. In the field of work there are tractor brigades containing the same number of caterpillar tractors and the same number of wheeled tractors, and in each brigade the number of all tractors is less than 9. If in each brigade the number of wheeled tractors is increased by 3 times, and crawler tractors by 2 times, then the total number of wheeled tractors in all brigades will be 27 more than the total number of caterpillar tractors, and in each brigade the number of all tractors will exceed 20. Determine the number of teams working in the field and the number of tracked and wheeled tractors in each team.

4. In a right triangle ABC, the length of the hypotenuse AB is equal to c, and the angle at vertex A is equal to α. On the extension of the hypotenuse AB, point B is taken as point M, and on the extension of leg AC, point N is taken as point C in such a way that BM=CN. Find the length of the common chord of two circles circumscribed about triangles ABC and AMN.

5. The base of the SABCD prim is a rectangle, the angle between the diagonals of which is equal to α, and α < π/3. The height of the pedestal passes through the point of intersection of the diagonals of the base and is equal to h. Given a triangular base having the same vertex S. Its base is a triangle, one vertex of which lies on the middle of the larger side of the rectangle ABCD, and the other two on its diagonals, and the projection of the vertex S onto the plane of the base lies inside this triangle. Find the volume of a triangular base if it is known that its lateral faces are equal.

Department of General Geological Faculty

1. Find all solutions to a system of equations
   $$\begin{cases} \cos 2x + \sin y = 2 \cos^2 30^\circ, \\ 2 \cos 2x - \sin y = \sin 540^\circ. \end{cases}$$

2. The automobile plant produces passenger cars and trucks. On the first day, 100 more trucks were produced than cars. On the second day, 150 more passenger cars were produced than on the first day, and 50 more trucks than on the first day. How many cars and how many trucks were produced on the first day, if on the second day 1.2 times more cars were produced than on the first?

3. For what values ​​of a are the roots of the quadratic equation
   $$x^2 - (2a+1)x + 2a = 0$$
   equal to each other?

4. Two circles of radius r touch each other. In addition, each of them touches from the outside the third circle of radius R at points A and B, respectively. Determine the radius r if AB=12 cm, R=8 cm.

5. Find all values ​​of x for which the inequality is true
   $$\log_4(2x+3) > \log_9 27.$$
   Department of Geophysics, Faculty of Geology

6. Solve system of equations
   $$\begin{cases} 0,5 \cdot 3^{x+1} + 7,5 \cdot 5^{y-1} = 12, \\ 2 \cdot 9^x + 5^y = 23. \end{cases}$$

7. Find all values ​​of x for which the inequality is true
   $$\frac{4-\sqrt{x+1}}{1-\sqrt{x+1}} \leq 0.$$

8. Solve the equation
   $$\sqrt{1-\cos 2x} = \sqrt{2} \sin x \left( \cos x - \frac{3}{2} \right).$$

9. Positive numbers a₁, a₂, a₃, a₄, a₅ in this order form a geometric progression. The sum of the logarithms of the numbers a₁, a₃, a₄, a₅ to base 3 is 9. Determine the numbers a₁, a₂, a₃, a₄, a₅ if log₃ a₄ is twice as large as log₃ a₂.

10. In triangle ABC on the sides AB = √3 cm, BC = 4 cm, AC = √7 cm, the median BD is checked. The inscribed circles of triangles ABD and BDC touch BD at points M and N, respectively. Determine the length of the segment MN.

51

Faculty of Chemistry

1. An equilibrium trapezoid $ABCD$ is described around a circle of radius $r$. E and K are the points of contact of this circle with the sides of the trapezoid. The angle between the base $AB$ and the side $AD$ of the trapezoid is $60^\circ$. Prove that line $EK$ is parallel to line $AB$ and find the area of ​​trapezoid $ABEK$.

2. Solve the equation:

$$2 \log \sqrt{\frac{2}{3}(x-0,5)} + x = x^0, 5x^2 + \cos^4\left(\frac{\pi}{4}\right).$$

3. Three excavators were tasked with digging a pit: the 1st and 2nd - with a capacity of 800 $m^3$, and the 3rd - with a capacity of 400 $m^3$. The 1st and 2nd excavators together remove more soil per hour; the second excavator removes more soil than the 3rd. The 1st and 3rd excavators began working at the same time, and the 2nd at the moment when the 1st removed $300 m^3$ of soil. When the 3rd excavator had completed 2/3 of its work, the 2nd removed $100 m^3$ of soil. The 3rd excavator was the first to complete its task. How much soil had the 1st excavator removed by the time the 3rd had finished digging its pit?

4. Solve inequality:

$$1 - \sqrt{\frac{1}{2} + \log_3 \sin x - \log_3 \cos x} > 0.$$

5. A regular right triangular feature $ABCA'B'C'$ is described around a ball of radius $r$. Point $M$ is the middle of the side edge $BB'$, point $N$ is the middle of the side edge $CC'$. A right circular cylinder is inscribed into a ball so that its base lies in the $AMN$ plane. Find the volume of this cylinder.

Patrice Lumutba University of Other Lifestyles

Faculty of Physics and Mathematics

1. Point $A$ is located at a distance of $s$ km from point $B$. A pedestrian left from point $A$ to point $B$, and after $T$ hours a cyclist left in the same direction. The cyclist caught up with the pedestrian $p$ km from $A$, reached $B$ and immediately turned back. Having traveled $p$ km from $B$, the cyclist again met the pedestrian and, continuing his journey, returned to $A$ later than the pedestrian arrived at $B$. To what extent is the pedestrian’s arrival at $B$ equal to that of the cyclist’s return at $A$?

2. Solve the system of equations:

$$\begin{cases} 2u + v + w = 6, \\ 3u + 2v + w = 9, \\ 3u^3 + 2v^3 + w^3 = 27. \end{cases}$$

3. Solve inequality:

$$\cos x \cdot \cos 3x + \cos^2 2x < -\frac{1}{4}.$$

4. A ball of radius $R$ is high as a pyramid with a square base. However, the side edges of the pyramid are perpendicular to the plane of the base, and the largest side edge forms an angle $\alpha$ with it. Determine the lateral surface of the pyramid.

Faculty of Engineering

1. The distance between two cities is 600 km; a freight train travels 8 hours slower than a passenger train. If the speed of each train is increased by 10 km/h, then the passenger train will continue the same only 5 hours faster.

commodity. Determine the speed of each train.

2. Solve the equation:

$$2\cos 2x \cdot (\text{ctg } x - 1) = 1 + \text{ctg } x.$$

3. Solve the equation:

$$|x^2 - 2x - 4| = 3x - 2.$$

4. The height of the cylinder is equal to the height of the cone. The lateral surface of the cylinder is related to the lateral surface of the cone as 3:2. In addition, it is known that the angle made by the edge of the cone with the plane of the base is equal to $\alpha$. Find the ratio of the volume of the cylinder to the volume of the cone.

Faculty of Economics

1. One of the two machines takes three days longer to process a batch of parts than the other. How many days would it take to process this batch of parts by each machine separately, if it is known that if they worked together on these machines, 3 times more batch of parts would be processed in 20 days.

2. Solve the equation:

$$\frac{2^x + 10}{4} = \frac{9}{2^{x-2}}.$$

3. Solve the equation:

$$2 - 2\sin^2 3x - \tg^2 3x = 0.$$

4. Find the volume of a cone described around a regular quadrilateral whose sides are equal to $a$ and whose two angles are equal to $60^\circ$.