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Problem: Solve the inequality for all values of the parameter \( a \) : \[ \sqrt{|x+1|} \leq a-x \text {. } \]

17.9 USE problems

2.57 $

Problem: Find all the values of the parameter a, for which the function is even: \[ f(x)=|x-a+1|+|x+3 a| . \]

17.10 USE problems

2.57 $

Problem: Solve the system of equations: \[ \left\{\begin{array}{l} x-2 y+\frac{3}{x-2 y}=4 \\ \frac{y}{x-2 y-3}=5 \end{array}\right. \]

17.11 USE problems

0 $

Problem: Solve the system of equations: \[ \left\{\begin{array}{l} \sqrt{3 x-5}=\sqrt{3 y+7} \\ y^{2}+x=10 \end{array}\right. \]

17.12 USE problems

0 $

Problem: Solve the irrational equation: \[ \left(x^{2}-4\right) \sqrt{3+5 x-2 x^{2}}=0 \text {. } \]

17.13 USE problems

0 $

Problem: Solve the irrational equation: \[ \sqrt{x-1}+\sqrt{x+2}=\sqrt{x+34}-\sqrt{x+7} . \]

17.14 USE problems

0 $

Problem: Find all values of the parameter a, for which 1) the equation \( a x^{2}+x+17=0 \) has only one root; 2) the equation \( a x^{2}+(5-3 a) x-a=0 \) has two roots of different signs.

17.15 USE problems

0 $

Problem: Find all values of the parameter a, for which 1) the equation \( 4 x^{2}+4 x=a^{2}-1 \) has two different positive roots. 2) the equation \( (a-2) x^{2}+2(a-2) x+ \) \( +2=0 \) has no roots.

17.16 USE problems

0 $

Problem: Determine the number of roots of the equation for each value of the parameter a. \[ a x^{2}+(a+1)^{2} x+a+2=0 . \]

17.17 USE problems

0 $

Problem: Find all values of \( a \) for which the sum of the real roots' squares of the equation \[ x^{2}-a x+a-2=0 \text { is minimal. } \]

17.18 USE problems

0 $

Problem: The quadratic equation \( x^{2}+p x+q=0 \) has two different roots \( x_{1} \) and \( x_{2} \). Find \( p \) and \( q \), if the numbers of \( x_{1}+1 \) и \( x_{2}+1 \) are the roots of the equation \( x^{2}-p^{2} x+p q=0 \).

17.19 USE problems

2.57 $

Problem: In the cuboid \( A B C D A_{1} B_{1} C_{1} D_{1} \) the lengths of the edges are \( A D=12, A B=5, A A_{1}=8 \). Find the volume of the pyramid \( M B_{1} C_{1} D \), if \( M \) is a point on \( A A_{1} \) moreover \( A M=5 \).

17.21 USE problems

2.57 $

Problem: We have \( A B C \) right triangle, with the right angle \( C \). On the line, containing \( A D \) median, point \( E \) is taken, which is at a distance of 4 from the vertex \( A \). Find the area of the triangle \( B C E \), if \( B C=6 \), \( A C=4 \).

17.23 USE problems

3.08 $

Problem: 1) Appliances are connected to the power outlet, the total resistance of which is \( 60 \mathrm{Ohm} \). An electric heater is supposed to be connected to the outlet in parallel with them. Determine (in ohms) the smallest possible resistance of this electric heater, if it is known that in case of a parallel connection of two conductors with resistances \( R_{1} \) and \( R_{2} \) their total resistance is given by the formla \( R=\frac{R_{1} \cdot R_{2}}{R_{1}+R_{2}} \), and for the normal functioning of the electrical network the total resistance of it must be not less than \( 10 \mathrm{Ohm} \). 2) Find the biggest value of the function: \( y=17 x-5 \sin x+7 \) on the segment \( \left[-\frac{\pi}{2} ; 0\right] \). 3) The first pipe passes 1 liter less water per minute, than the second pipe. How many liters of water per minute passes through the second pipe, if it fills the reservoir with the volume of 624 liters 2 minutes faster, than the first pipe fills the reservoir with the volume of 650 liters?

17.25 USE problems

3.08 $

Problem: Solve for all values of the parameter: \[ a^{2}(x-1)-a(2 x-5)=3 x-6 \text {. } \]

17.26 USE problems

0 $

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