Key Concepts and Formulas Used:

1. Sum of Coefficients of a Polynomial: For any polynomial $P(x)$, the sum of its coefficients is obtained by substituting $x=1$ into the polynomial, i.e., $P(1)$.
2. L'Hôpital's Rule: If $\lim_{x \rightarrow a} \frac{f(x)}{g(x)}$ is of the indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x \rightarrow a} \frac{f(x)}{g(x)} = \lim_{x \rightarrow a} \frac{f'(x)}{g'(x)}$, provided the latter limit exists.
3. Fundamental Theorem of Calculus (Part 1): If $F(x) = \int_a^x f(t) dt$, then $F'(x) = f(x)$.
4. Standard Limit: $\lim_{x \rightarrow 0} \frac{\log(1+x)}{x} = 1$.
5. Common Roots of Quadratic Equations:
   • If two quadratic equations, $A_1 x^2 + B_1 x + C_1 = 0$ and $A_2 x^2 + B_2 x + C_2 = 0$, have a common root, then a specific condition involving their coefficients can be derived.
   • Crucial Point for Real Coefficients: If two quadratic equations with real coefficients share a common non-real (complex) root, then they must also share the conjugate of that root. This implies that both equations have both roots in common, which means their coefficients must be proportional: $\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2}$.

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Step-by-Step Solution:

Part 1: Finding the value of 'a'

We are given that $a$ is the sum of all coefficients in the expansion of $P(x) = \left(1-2 x+2 x^2\right)^{2023}\left(3-4 x^2+2 x^3\right)^{2024}$.

1. Apply the concept of sum of coefficients: To find the sum of coefficients of any polynomial, we substitute $x=1$ into the polynomial expression. This is because when $x=1$, all powers of $x$ become $1$, and thus each term reduces to its coefficient. $a = P(1) = \left(1-2(1)+2(1)^2\right)^{2023}\left(3-4(1)^2+2(1)^3\right)^{2024}$

2. Calculate the value: $a = (1-2+2)^{2023}(3-4+2)^{2024}$ $a = (1)^{2023}(1)^{2024}$ $a = 1 \times 1$ $a = 1$

   Therefore, the value of $a$ is $\mathbf{1}$.

Part 2: Finding the value of 'b'

We need to evaluate the limit b=\lim _\limits{x \rightarrow 0}\left(\frac{\int_0^x \frac{\log (1+t)}{t^{2024}+1} d t}{x^2}\right).

1. Check the indeterminate form: As $x \rightarrow 0$:

   • The numerator approaches $\int_0^0 \frac{\log (1+t)}{t^{2024}+1} d t = 0$.
   • The denominator $x^2$ approaches $0$. Since the limit is of the form $\frac{0}{0}$, we can apply L'Hôpital's Rule.

2. Apply L'Hôpital's Rule: We differentiate the numerator and the denominator with respect to $x$.

   • Differentiate the numerator: Using the Fundamental Theorem of Calculus, $\frac{d}{dx} \int_0^x f(t) dt = f(x)$. So, $\frac{d}{dx} \left(\int_0^x \frac{\log (1+t)}{t^{2024}+1} d t\right) = \frac{\log (1+x)}{x^{2024}+1}$.
   • Differentiate the denominator: $\frac{d}{dx} (x^2) = 2x$.

   Now, the limit becomes: b = \lim _\limits{x \rightarrow 0} \frac{\frac{\log (1+x)}{x^{2024}+1}}{2x} b = \lim _\limits{x \rightarrow 0} \frac{\log (1+x)}{2x(1+x^{2024})}

3. Evaluate the simplified limit: We can rewrite the expression to use the standard limit $\lim_{x \rightarrow 0} \frac{\log (1+x)}{x} = 1$: b = \lim _\limits{x \rightarrow 0} \left( \frac{\log (1+x)}{x} \cdot \frac{1}{2(1+x^{2024})} \right) As $x \rightarrow 0$:

   • $\frac{\log (1+x)}{x} \rightarrow 1$
   • $\frac{1}{2(1+x^{2024})} \rightarrow \frac{1}{2(1+0^{2024})} = \frac{1}{2(1+0)} = \frac{1}{2}$

   Therefore, $b = 1 \times \frac{1}{2}$ $b = \frac{1}{2}$

   The value of $b$ is $\mathbf{\frac{1}{2}}$.

   • Tip: Always verify the indeterminate form before applying L'Hôpital's Rule. Differentiating correctly, especially for integrals, is crucial. Remembering standard limits can often simplify calculations and avoid repeated applications of L'Hôpital's Rule.

Part 3: Using the common root condition

We are given two quadratic equations: $c x^2+d x+e=0$ and $2 b x^2+a x+4=0$. We have found $a=1$ and $b=\frac{1}{2}$. Substitute these values into the second equation: $2 \left(\frac{1}{2}\right) x^2 + (1) x + 4 = 0$ $x^2 + x + 4 = 0$

Now we have two equations that share a common root:

1. $c x^2+d x+e=0$

2. $x^2+x+4=0$

3. Analyze the nature of roots for the known equation: Let's find the discriminant of the equation $x^2+x+4=0$. For a quadratic equation $Ax^2 + Bx + C = 0$, the discriminant is $\Delta = B^2 - 4AC$. Here, $A=1$, $B=1$, $C=4$. $\Delta = (1)^2 - 4(1)(4) = 1 - 16 = -15$ Since $\Delta < 0$, the roots of the equation $x^2+x+4=0$ are non-real (complex) and are conjugates of each other.

4. Apply the common root condition for complex roots: When two quadratic equations with real coefficients have a common non-real root, it implies that they must have both roots in common. This is because complex roots always appear in conjugate pairs for polynomials with real coefficients. If one complex root is common, its conjugate must also be common. If both roots are common, then the two quadratic equations must be identical up to a constant multiplicative factor. In other words, their corresponding coefficients must be proportional.

   Therefore, for the equations $c x^2+d x+e=0$ and $1 x^2+1 x+4=0$ to have common roots, their coefficients must be proportional: $\frac{c}{1} = \frac{d}{1} = \frac{e}{4}$

   • Common Mistake: Directly equating coefficients without checking the nature of the roots can be incorrect if the common root is real. However, for non-real common roots in equations with real coefficients, proportionality holds. This is a subtle but important distinction.

Part 4: Determining the ratio d:c:e

From the proportionality derived above: $\frac{c}{1} = \frac{d}{1} = \frac{e}{4}$

This directly implies the ratio of the coefficients $c:d:e$. $c:d:e = 1:1:4$

The question asks for the ratio $\mathrm{d}: \mathrm{c}: \mathrm{e}$. From $c:d:e = 1:1:4$, we can state that:

• $d$ is proportional to $1$
• $c$ is proportional to $1$
• $e$ is proportional to $4$

Therefore, $\mathrm{d}: \mathrm{c}: \mathrm{e} = 1:1:4$.

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Summary and Key Takeaway:

This problem beautifully integrates concepts from Binomial Theorem, Limits (L'Hôpital's Rule and Standard Limits), and Properties of Quadratic Equations. The key takeaways are:

• The sum of coefficients is found by substituting $x=1$.
• L'Hôpital's Rule is a powerful tool for indeterminate limits, often combined with the Fundamental Theorem of Calculus for integral expressions.
• When quadratic equations with real coefficients share a complex root, they must share both roots, leading to proportional coefficients. This specific condition is crucial for solving problems involving common roots. The final answer is $\mathrm{d}: \mathrm{c}: \mathrm{e} = 1:1:4$.