Let's go through each question individually.

Q23

Problem: Ball $A$ is dropped from the top of a building, and at the same time, ball $B$ is thrown upwards from the ground. When they collide, they move in opposite directions, and the speed of $A$ is twice the speed of $B$. The collision occurs at $\frac{\alpha}{\beta}$ of the building's height. Find $\alpha + \beta$.

Solution Outline:

1. Let the height of the building be $H$.
2. The motion of each ball can be described using the equations of motion under gravity.
3. Define $v_A$ and $v_B$ as the velocities of balls $A$ and $B$ at the point of collision.
4. Use the given condition $v_A = 2v_B$ and the fact that they collide at a certain height $\frac{\alpha}{\beta} H$.
5. Set up equations based on these conditions and solve for $\alpha + \beta$.

Q24

Problem: A particle moves in a straight line with constant acceleration $a$. The displacements of the particle from the origin at $t = t_1$, $t = t_2$, and $t = t_3$ are $s_1$, $s_2$, and $s_3$ respectively. If $t_1$, $t_2$, and $t_3$ are in arithmetic progression with a common difference $d$ and displacements are in geometric progression, find the value of $n$ (any number) if $a = \frac{\left(\sqrt{s_3} - \sqrt{s_1}\right)^2}{nd^2}.$

Solution Outline:

1. Use the conditions of arithmetic and geometric progression for the times and displacements.
2. Apply equations of motion to relate $s_1$, $s_2$, and $s_3$ to the acceleration $a$.
3. Use the formula given for $a$ and solve for $n$.

Q25

Problem: A particle with instantaneous velocity $\vec{v} = (2\hat{i} + 3\hat{j} + x\hat{k}) \, \text{m/s}$ and instantaneous acceleration $\vec{a} = 3\hat{i} - 4.5\hat{j} - 6\hat{k} \, \text{m/s}^2$. If the path of the particle is a straight line, find the magnitude of $x$ such that it is $\frac{5}{4}$.

Solution Outline:

1. For the particle to move in a straight line, the velocity and acceleration vectors must be parallel.
2. This implies that $\vec{v} \times \vec{a} = 0$.
3. Set up the cross product condition, solve for $x$, and then calculate the magnitude of $x$ as specified.

Would you like me to provide detailed solutions for each question?

Here are some related questions for further practice:

1. How do we calculate the relative velocity between two objects moving in opposite directions?
2. What conditions must be met for two moving objects to collide at a certain point?
3. How does the arithmetic progression of time intervals affect the displacement of an accelerating object?
4. How can we determine if two vectors are parallel based on their components?
5. What are the steps to solve problems involving motion with constant acceleration in multiple dimensions?

Tip: Always analyze the conditions of parallelism or perpendicularity when dealing with vector motion questions.