Let's analyze and solve each of the propositions presented in the image. Here's a step-by-step breakdown:

Exercise 1:

We need to determine which of the propositions $(P_1)$ to $(P_7)$ are true or false and justify them.

1. Proposition $P_1$:

$(\forall x \in \mathbb{R}) : -6x^2 + 5x - 1 \leq 0$

Let's solve the quadratic inequality: The expression is $-6x^2 + 5x - 1$. We find the roots of the equation $-6x^2 + 5x - 1 = 0$ using the discriminant method: $b^2 - 4ac = 5^2 - 4(-6)(-1) = 25 - 24 = 1$ The roots are $x = 1$ and $x = \frac{1}{6}$.

For $x \in \left[\frac{1}{6}, 1\right]$, $-6x^2 + 5x - 1$ will be less than or equal to 0. However, for other $x$, the expression will be positive. Thus, $P_1$ is false.

2. Proposition $P_2$:

$(\exists! x \in \mathbb{R}) : x^3 - 2\sqrt{2} = 0$ Solving, $x^3 = 2\sqrt{2}$, so $x = \sqrt[3]{2\sqrt{2}}$. There exists exactly one real solution to this equation. Hence, $P_2$ is true.

3. Proposition $P_3$:

$(\forall m \in \mathbb{R}^*)(\exists x \in \mathbb{R}) : x^2 - mx + m^2 = 0$

The discriminant of this quadratic equation is $m^2 - 4m^2 = -3m^2$. For $m \neq 0$, this discriminant is always negative, implying no real solutions. Therefore, $P_3$ is false.

4. Proposition $P_4$:

$(\forall m \in \mathbb{R})(\exists x \in \mathbb{R}) : x^2 - 3mx + 2m^2 + 1 = 0$

The discriminant here is $(3m)^2 - 4(1)(2m^2 + 1) = 9m^2 - 8m^2 - 4 = m^2 - 4$. For all $m \neq \pm 2$, there are real solutions. When $m = 2$ or $m = -2$, it has a double root. Hence, $P_4$ is true.

5. Proposition $P_5$:

$(\exists m \in \mathbb{R})(\exists x \in \mathbb{R}) : x^2 - 3mx + 2m^2 + 1 = 0$ As verified previously, for most values of $m$, this has solutions, so $P_5$ is true.

6. Proposition $P_6$:

(a) Check $m = 6$, then $x^2 + x + 5 = 6$, which gives $x^2 + x - 1 = 0$. The discriminant is $1 + 4 = 5 > 0$, so $P_6$ is true.

For $m = -1$, $x^2 + x + 5 = -1$ leads to a contradiction, so $P_6$ is false.

7. Proposition $P_7$:

Checking, you will find the propositions work for any $m > 0$.

Do you want a more detailed explanation of any part? Here are some related questions:

1. What is the definition of discriminants and how do they determine the nature of solutions for quadratic equations?
2. Can you explain why $P_1$ is false by graphing the function $-6x^2 + 5x - 1$?
3. Why do discriminants differ for equations in $P_4$ and $P_5$ cases?
4. What other real solutions exist for $x^3 = 2\sqrt{2}$?
5. How can uniqueness be verified in mathematical proofs like $P_2$?

Tip: Always check your roots using a discriminant!