Find the magnitude of vector a + b, where a = [1, -3] and b = [5, 2].

$$a = \begin{bmatrix} 1 \\ -3 \end{bmatrix}$$ $$b = \begin{bmatrix} 5 \\ 2 \end{bmatrix}$$

To add vectors, we add their corresponding components.

$$a + b = \begin{bmatrix} 1 \\ -3 \end{bmatrix} + \begin{bmatrix} 5 \\ 2 \end{bmatrix} = \begin{bmatrix} 1 + 5 \\ -3 + 2 \end{bmatrix} = \begin{bmatrix} 6 \\ -1 \end{bmatrix}$$

For a vector $v = \begin{bmatrix} x \\ y \end{bmatrix}$, its magnitude $||v||$ is given by the formula $||v|| = \sqrt{x^2 + y^2}$.
Let $v = a + b = \begin{bmatrix} 6 \\ -1 \end{bmatrix}$.

$$||a + b|| = \sqrt{6^2 + (-1)^2}$$ $$||a + b|| = \sqrt{36 + 1}$$ $$||a + b|| = \sqrt{37}$$

a. $\sqrt{34}$
b. 5
c. 2
d. $\sqrt{37}$

The calculated magnitude matches option d.

The final answer is $\boxed{\text{d. } \sqrt{37}}$.