Radical equations and inequalities

9000024803

Level: 
A
Removing radical in an equation by squaring both sides may enrich the set of solutions of this equation and checking the solutions of the new equation in the original equation may be necessary. Identify a correct conclusion in the particular case of the following equation. \[ -\sqrt{x^{2 } - 2x + 1} = x \]
If we look for the solution in the set \(\mathbb{R}^{-}\), then squaring both sides of the equation gives an equivalent equation. The checking of the solution is not necessary.
If we look for the solution in the set \(\mathbb{R}^{+}\), then squaring both sides of the equation gives an equivalent equation. The checking of the solution is not necessary.
If we look for the solution in the set \(\mathbb{R}\), then squaring both sides of the equation gives an equivalent equation. The checking of the solution is not necessary.
None of the above.

9000024807

Level: 
C
A body hangs on a string of the length \(l_{1}\). The length \(l\) of the spring defines the period \(T\) of motion by the relation \[ T = 2\pi \sqrt{ \frac{l} {g}}, \] where \(g\) is a standard acceleration of gravity. We have to adjust the length of the string such that the period doubles. Find the new length of the string.
We elongate the string by \(3\cdot l_{1}\), i.e. \(l_{2} = l_{1} + 3l_{1}\).
The length doubles, i.e. \(l_{2} = 2l_{1}\).
The new length will be half of the original length, i.e. \(l_{2} = \frac{1} {2}l_1\).
We shorten the string by \(3\cdot l_{1}\), i.e. \(l_{2} = l_{1} - 3l_{1}\).

9000024808

Level: 
C
In the following list identify a true statement referring to the following equation. \[ \sqrt{4x^{2 } - \sqrt{8x + 5}} = 2x + 1 \]
The equation has a unique solution, this solution is a negative number.
The equation has two solutions, both solutions have an opposite sign.
The equation has a unique solution, this solution is a positive number.
The equation does not have a solution.

9000024805

Level: 
C
A falling body dropped at a velocity \(60\, \mathrm{m}\mathrm{s}^{-1}\). Find the initial height \(h\), if the relation between the velocity and the initial height \(h\) is \(v = \sqrt{2hg}\). Use \(g = 10\, \mathrm{m}\mathrm{s}^{-2}\) for acceleration of gravity.
The initial height is between \(150\, \mathrm{m}\) and \(200\, \mathrm{m}\).
The initial height is smaller than \(100\, \mathrm{m}\).
The initial height is between \(100\, \mathrm{m}\) and \(150\, \mathrm{m}\).
The initial height is bigger than \(200\, \mathrm{m}\).

9000024802

Level: 
A
Consider the equation \[ \sqrt{x^{2 } - 2x + 1} = x + 2 \] and the equation which arises from this equation by squaring both sides of the equation, i.e. the equation \[ \left (\sqrt{x^{2 } - 2x + 1}\right )^{2} = (x + 2)^{2}. \] Identify a true statement.
Both equations are equivalent only if \(x\geq - 2\).
Both equations are equivalent.
Both equations are equivalent only if \(x\leq - 2\).
None of the above.