1971 Mathematics Entrance Exam Options

56

Option 1

1. The volume of a rectangular cone is $V$. A paramid is inscribed in a cone, at the base of which lies an equilateral triangle with an angle between the sides equal to $\alpha$. Find the volume of the paramid.

2. Without solving the equation
   $$x^2 - 2x - 15 = 0,$$
   calculate the sum of squares and the sum of cubes of its roots.

3. Solve the equation:
   $$\mathrm{tg} 2x \cdot \cos 3x + \sin 3x + \sqrt{2} \sin 5x = 0.$$

4. Solve inequality:
   $$(3x - 1)\log_2 x > 0.$$

Option 2

1. A regular paramid is inscribed in a ball, the side edge of which makes an angle $\alpha$ with the plane of the base. The volume of the paramid is $V$. Find the volume of the ball.

2. Solve the system of equations:
   $$\begin{cases} 10^3 - \lg(x - y) = 250, \\ \sqrt{x - y} + \frac{1}{2} \sqrt{x + y} = \frac{26 - y}{\sqrt{x - y}}. \end{cases}$$

3. Solve the equation:
   $$(1 + \cos 4x) \sin 2x = \cos^2 2x.$$

4. Solve inequality:
   $$\frac{x + 2}{5^{3-x}} > 25.$$

Kuibyshev Polytechnic Institute named after V.V. Kuibyshev

Faculty of Automation and Information and Measurement Technology

1. A plane is drawn through the vertex of a regular triangular paramid and the middle of the two sides of the base. Determine the area of ​​the section and the volume of the parts of the paramid into which it is divided by the section, knowing the side of the base $\alpha$ and the angle $\alpha$ formed by the plane of the section with the base.

2. The cost of 70 copies of the first volume and 60 copies of the second volume is 230 rubles. In fact, for all these books, 191 rubles fell, since a discount was made: on the first volume - by 15%, and on the second volume - by 20%. Find the original price of each volume.

3. Solve the equation:
   $$9^x - 2^{x + \frac{1}{2}} = 2^{x + \frac{7}{2}} - 3^{2x - 1}.$$

4. Solve inequality:
   $$(\log_{\sin x} 2)^2 < \log_{\sin x} (4 \sin^3 x).$$

Faculty of Electrical Engineering

1. A cone is inscribed into a ball of radius $R$, forming a cone that makes an angle $\alpha$ with the plane of the base. Calculate the volume of the cone.

2. Find three numbers that form a geometric progression, the sum of which is 56. It is known that if you add the corresponding 7, 12 and 9 to these numbers, you get three numbers that form an arithmetic-progressive.

3. Solve the equation:
   $$\lg \sqrt{x + 3} = 1 - 0.5 \lg (x + 24).$$

4. Find the domain of a function:
   $$y = \frac{2}{\sqrt{4 \sin x \cos x - 1}}.$$

Moscow Institute of Petrochemical and Gas Industry named after I. M. Gubkin

Option 1

1. Two workers were tasked with producing a batch of identical parts. After the first one worked for 7 hours and the second one for 4 hours, it turned out that they had completed 5/9 of the entire job. After working together for another 4 hours, they determined that they had 1/18 of the total work left to complete. In how many hours could each worker, working separately, complete the entire job?

2. Solve the equation:
   $$\sqrt[3]{x - 1} \sqrt[3]{3^{10x + 5}} = 3x - 9 \sqrt[3]{27^{3x - 7}}.$$