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We are given the following information in the question:

- A particle \( P \) has a mass of 2 kg.
- There are two forces acting on \( P \): 
  \[
  \mathbf{F_1} = (2\mathbf{i} + 4\mathbf{j}) \, \text{N} \quad \text{and} \quad \mathbf{F_2} = (c\mathbf{i} - 2\mathbf{j}) \, \text{N}
  \]
  where \( c \) is a constant.
- The magnitude of the acceleration of \( P \) is \( \sqrt{5} \, \text{ms}^{-2} \).

We need to find the two possible values of \( c \).

### Step 1: Calculate the net force
The net force \( \mathbf{F} \) acting on the particle is the sum of \( \mathbf{F_1} \) and \( \mathbf{F_2} \):
\[
\mathbf{F} = \mathbf{F_1} + \mathbf{F_2} = (2 + c)\mathbf{i} + (4 - 2)\mathbf{j}
\]
This simplifies to:
\[
\mathbf{F} = (2 + c)\mathbf{i} + 2\mathbf{j}
\]

### Step 2: Use Newton's second law
Newton's second law states:
\[
\mathbf{F} = m \mathbf{a}
\]
where \( m = 2 \, \text{kg} \) is the mass and \( \mathbf{a} \) is the acceleration.

Since the magnitude of the acceleration is given as \( |\mathbf{a}| = \sqrt{5} \, \text{ms}^{-2} \), we can use this in our calculations. The magnitude of the net force is:
\[
|\mathbf{F}| = m |\mathbf{a}| = 2 \times \sqrt{5} = 2\sqrt{5} \, \text{N}
\]

### Step 3: Find the magnitude of the net force
The magnitude of the net force \( \mathbf{F} \) is:
\[
|\mathbf{F}| = \sqrt{(2 + c)^2 + 2^2}
\]
Simplifying the expression inside the square root:
\[
|\mathbf{F}| = \sqrt{(2 + c)^2 + 4}
\]

### Step 4: Set up the equation
Now, equate the magnitudes of the net force:
\[
\sqrt{(2 + c)^2 + 4} = 2\sqrt{5}
\]
Square both sides to remove the square root:
\[
(2 + c)^2 + 4 = 20
\]
Simplify:
\[
(2 + c)^2 = 16
\]

### Step 5: Solve for \( c \)
Take the square root of both sides:
\[
2 + c = \pm 4
\]
Thus, the two possible values of \( c \) are:
\[
c = 4 - 2 = 2 \quad \text{or} \quad c = -4 - 2 = -6
\]

### Final Answer:
The two possible values of \( c \) are \( c = 2 \) and \( c = -6 \).

---

Do you have any questions, or would you like further clarification on any part of this solution?

Here are 5 related questions to deepen your understanding:
1. How does Newton's second law relate force and acceleration in general?
2. Why do we square both sides of the equation when solving for \( c \)?
3. Can you explain why vector components are added separately (i.e., \( \mathbf{i} \) and \( \mathbf{j} \) components)?
4. What happens to the motion of the particle if only one of the forces was applied?
5. How would the values of \( c \) change if the mass of the particle were different?

**Tip**: When solving vector problems, always keep track of each component separately and treat them as independent dimensions.

A particle P is moving on a smooth horizontal surface under the action of two forces. Given that:
- The mass of P is 2 kg.
- The two forces are (2i + 4j) N and (ci - 2j) N, where c is a constant.
- The magnitude of the acceleration of P is √5 ms⁻².
Find the two possible values of c.

Learn how to determine the two possible values of c when a particle of mass 2 kg is acted upon by two forces and has an acceleration of √5 ms⁻². This detailed solution involves vector addition, Newton's Second Law, and solving algebraic equations. Suitable for Grades 11-12.

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