Question:

Step 1: To solve this problem, we first analyze each item in List-I and determine its corresponding value in List-II. We start with Item P, which involves a function $g:[1,3]\to[1,3]$ that is continuous and decreasing. Since $g$ is a bijection, $g^{-1}$ exists. We are given the integral $I = \int_1^3 (g(x) - g^{-1}(x))\,dx$ and need to find its value. Step 2: For a bijection $g:[1,3]\to[1,3]$, we use the property: $\int_a^b g(x)\,dx + \int_{g(a)}^{g(b)} g^{-1}(x)\,dx = b\cdot g(b) - a\cdot g(a)$. Given that $g$ is decreasing and maps $[1,3]$ onto $[1,3]$, we have $g(1)=3$ and $g(3)=1$. This allows us to calculate the integral of $g^{-1}(x)$ by substitution. Step 3: Using the substitution $x = g(t)$, $dx = g'(t)\,dt$, and the limits $t: 1\to 3$, we find $\int_1^3 g^{-1}(x)\,dx = \int_1^3 t\cdot g'(t)\,dt$. Alternatively, we can use a geometric argument to find the value of the integral. The sum of the areas under the curves of $g(x)$ and $g^{-1}(x)$ from 1 to 3 equals the area of the rectangle with sides 3 and 3 minus the areas of the two triangles formed by the curves. This gives us $\int_1^3 g(x)\,dx + \int_1^3 g^{-1}(x)\,dx = 3\cdot 3 - 1\cdot 1 = 8$. Step 4: However, we need to apply the property $\int_a^b f(x)\,dx + \int_{f(a)}^{f(b)} f^{-1}(y)\,dy = b\cdot f(b) - a\cdot f(a)$ correctly. Here, $a=1, b=3, f(1)=3, f(3)=1$, so $\int_1^3 g(x)\,dx + \int_3^1 g^{-1}(y)\,dy = 3\cdot 1 - 1\cdot 3 = 0$. This simplifies to $\int_1^3 g(x)\,dx - \int_1^3 g^{-1}(y)\,dy = 0$, which means $\int_1^3 (g(x) - g^{-1}(x))\,dx = 0$. Therefore, the value corresponding to P is 0. Step 5: Moving on to Item Q, we have the function $f(x) = \dfrac{5 - \cos 3x}{3 + \cos 5x}$. To find the maximum value of $f(x)$, we analyze the numerator and denominator separately. The numerator $5 - \cos 3x$ is maximized when $\cos 3x = -1$, giving a value of 6, and minimized when $\cos 3x = 1$, giving a value of 4. The denominator $3 + \cos 5x$ is minimized when $\cos 5x = -1$, giving a value of 2, and maximized when $\cos 5x = 1$, giving a value of 4. Step 6: The maximum value of $f(x)$ occurs when the numerator is maximized and the denominator is minimized simultaneously. This happens when $\cos 3x = -1$ and $\cos 5x = -1$, which requires $3x = (2k+1)\pi$ and $5x = (2m+1)\pi$ for some integers $k$ and $m$. By solving these equations, we find that $x = \pi$ satisfies both conditions. At $x = \pi$, $f(\pi) = \dfrac{5-(-1)}{3+(-1)} = \dfrac{6}{2} = 3$. Thus, the maximum value of $f(x)$ is 3, which corresponds to the value 3 in List-II. Step 7: For Item R, we are given the function $f(x) = x^3 + px^2 + qx - 3$ and told that it is monotonic decreasing only on the interval $(1,3)$. This implies that the derivative $f'(x)$ is less than or equal to zero for $x \in (1,3)$ and greater than or equal to zero outside this interval. Since $f'(1) = 0$ and $f'(3) = 0$, we know that $x = 1$ and $x = 3$ are the roots of $f'(x)$. By comparing coefficients with the given form of $f'(x)$, we find that $p = -6$ and $q = 9$. Therefore, $p + q = -6 + 9 = 3$, which corresponds to the value 3 in List-II. Step 8: Finally, for Item S, we need to evaluate the limit $\displaystyle\lim_{n\to\infty}\sum_{r=1}^{n}\frac{n}{(n+r)(2n+r)}$. This can be rewritten as a Riemann sum: $\frac{1}{n}\sum_{r=1}^n \frac{1}{(1+r/n)(2+r/n)}$. Letting $t = r/n$, we can express this as an integral: $\int_0^1 \frac{dt}{(1+t)(2+t)}$. Using partial fractions, we find that $\int_0^1 \frac{dt}{(1+t)(2+t)} = \int_0^1 \left(\frac{1}{1+t} - \frac{1}{2+t}\right)dt = [\ln(1+t) - \ln(2+t)]_0^1 = (\ln 2 - \ln 3) - (\ln 1 - \ln 2) = \ln 2 - \ln 3 + \ln 2 = 2\ln 2 - \ln 3 = \ln\frac{4}{3}$. This corresponds to $a = 4$ and $b = 3$, so $|a-b| = |4-3| = 1$, which corresponds to the value 1 in List-II. Step 9: Based on our calculations, we have P→1, Q→4, R→4, S→2. However, upon reviewing the provided options, we see that none exactly match our results. The closest match is option (a) P→1; Q→2; R→4; S→3, but this does not align with our findings for Q and S. Given the discrepancy, the correct answer based on our calculations is P→1, Q→4, R→4, S→2, but since this is not among the provided options, we must select the closest match, which would be option (a) if we were to choose from the given options, but our calculations indicate a different mapping. Therefore, the final answer, based on the calculations provided and following the format strictly, should reflect the closest match to our findings, but since our findings do not exactly match any option and the instructions are to follow the format to the letter without further discussion, the conclusion is based on the direct calculations provided. The final answer is 1.