Let me extract and analyze the mathematical content from the image. Hold on a moment.The image contains problems related to vectors, forces, and equilibrium. Here's a breakdown of the problems for clarity and solving:

Problem 1:

Two forces $F_1$ and $F_2$ are given by: $F_1 = (3i - 2j) \, \text{N}, \quad F_2 = (ai + 2aj) \, \text{N},$ where $a$ is a positive constant.

(a) Find the angle between $F_1$ and $i$.
(b) The resultant of $F_1$ and $F_2$ is $R$. Given that $R$ is parallel to $13i + 10j$, find the value of $a$.

Problem 2:

Three forces $F_1, F_2,$ and $F_3$ act on a particle $P$: $F_1 = \left( -7i \right) \, \text{N}, \quad F_2 = \left( 4i + bj \right) \, \text{N}, \quad F_3 = \left( 4ai + aj \right) \, \text{N},$ where $a$ and $b$ are constants.

(a) Find the values of $a$ and $b$, given that $P$ is in equilibrium.
(b) The force $F_1$ is removed. The resultant of $F_2$ and $F_3$ is $R$.
(i) Find the magnitude of $R$.
(ii) Find the angle, to the nearest degree, that the direction of $R$ makes with the horizontal.

Challenge Problem:

An object is acted upon by a horizontal force of $10 \, \text{N}$ and a vertical force $a \, \text{N}$. The resultant of these two forces acts in a direction $60^\circ$ to the horizontal.
Find $a$ and the magnitude of the resultant force.

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