Question:

<div class="solution"> <p><strong>Step 1:</strong> Let's denote the common roots of the two equations as \(r_1\) and \(r_2\), and the third root of the first equation as \(r_3\). According to Vieta's formulas for the first equation \(x^3 - 5x^2 + 7x - a = 0\), we have \(r_1 + r_2 + r_3 = 5\), \(r_1r_2 + r_2r_3 + r_3r_1 = 7\), and \(r_1r_2r_3 = a\).</p> <p><strong>Step 2:</strong> For the second equation \(x^3 - 8x + b = 0\), since \(r_1\) and \(r_2\) are also roots, we have \(r_1^3 - 8r_1 + b = 0\) and \(r_2^3 - 8r_2 + b = 0\). Subtracting these two equations gives us \(r_1^3 - r_2^3 - 8(r_1 - r_2) = 0\), which simplifies to \((r_1 - r_2)(r_1^2 + r_1r_2 + r_2^2 - 8) = 0\). Since \(r_1 \neq r_2\), we must have \(r_1^2 + r_1r_2 + r_2^2 = 8\).</p> <p><strong>Step 3:</strong> From the equation \(r_1^2 + r_1r_2 + r_2^2 = 8\) and the Vieta's formula \(r_1r_2 + r_2r_3 + r_3r_1 = 7\), we can find a relationship between \(r_3\) and the other roots. However, to directly assess the given options, let's examine each option individually to see if any can be directly linked to the information we can derive from the equations and their roots.</p> <p><strong>Step 4:</strong> Option (a) suggests \(\log_4(a^3 + b^2 - 1) = 2\), which implies \(a^3 + b^2 - 1 = 4^2 = 16\). Without explicit values of \(a\) and \(b\), we must consider if this relationship can be derived or inferred from the given equations and their roots.</p> <p><strong>Step 5:</strong> For option (b), the integral \(\displaystyle\int_0^{\pi} \ln(\sin ax)\,dx = \dfrac{-\pi}{2}\ln 2\) does not directly relate to the roots of the given equations or their coefficients \(a\) and \(b\) without further context or a specific value for \(a\).</p> <p><strong>Step 6:</strong> Option (c) involves the limit \(\displaystyle\lim_{x \to 0}\left(\left[\dfrac{a^2 \sin x}{x}\right] + \left[\dfrac{b^2 \tan x}{x}\right]\right)\) being equal to 12. This option requires knowledge of \(a\) and \(b\) to assess its validity, as the limit's value depends on the behavior of \(\sin x / x\) and \(\tan x / x\) as \(x\) approaches 0, which both approach 1, and the greatest integer function \([\cdot]\) of \(a^2\) and \(b^2\).</p> <p><strong>Step 7:</strong> Option (d) suggests \(\tan^{-1}(\tan(a+b)) = 6 - 2\pi\). Given the periodic nature of \(\tan x\) and \(\tan^{-1} x\), for \(\tan^{-1}(\tan(a+b))\) to equal \(6 - 2\pi\), \(a + b\) must be such that it falls within a specific range that corresponds to this value, considering the principal value of \(\tan^{-1} x\) and the periodicity of \(\tan x\).</p> <p><strong>Answer:</strong> Given the complexity and the information provided, let's focus on what can be directly inferred or calculated. The question asks which of the statements is/are correct, but without explicit calculations or derivations for \(a\) and \(b\) based on the common roots, we must consider the principles of algebra and calculus that apply. However, my