Sure, let's solve these problems one by one.

1. Evaluate the expression $(2 + \sqrt{3})^2 - 4(\sqrt{3} - 1)$.

   $(2 + \sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3}$

   $4(\sqrt{3} - 1) = 4\sqrt{3} - 4$

   Subtract the two results:

   $(7 + 4\sqrt{3}) - (4\sqrt{3} - 4) = 7 + 4 = 11$

   Answer: A) 11

2. Find the allowable values for the expression $\frac{x+7}{\sqrt{4-x}}$.

   The expression is valid when the denominator is positive:

   $\sqrt{4 - x} > 0 \Rightarrow 4 - x > 0 \Rightarrow x < 4$

   Answer: Г) $x < 4$

3. Find the roots of the equation $\sqrt{x^2 + 9} = \sqrt{13}$.

   Square both sides:

   $x^2 + 9 = 13$

   $x^2 = 4$

   $x = \pm 2$

   Answer: A) -2 and 2

4. The expression $\sin^2(90^\circ - \alpha) + \sin^2 \alpha - 1$ is identically equal to:

   Using identities: $\sin(90^\circ - \alpha) = \cos \alpha$

   $\cos^2 \alpha + \sin^2 \alpha = 1$

   $(\cos^2 \alpha + \sin^2 \alpha) - 1 = 0$

   Answer: В) 0

5. The sequence $\{a_n\}$ is defined by $a_1 = 2$ and $a_{n+1} = 2a_n - 1$. Find the sum of the first three terms.

   $a_1 = 2$

   $a_2 = 2 \times 2 - 1 = 3$

   $a_3 = 2 \times 3 - 1 = 5$

   Sum = $a_1 + a_2 + a_3 = 2 + 3 + 5 = 10$

   Answer: В) 10