1. The problem is to verify if the given table of nodes $x_k$ and weights $\lambda_k$ corresponds correctly to the Gauss-Chebyshev quadrature method for different values of $n$. 2. The Gauss-Chebyshev quadrature nodes for $n$ points are given by: $$x_k = \cos\left(\frac{2k-1}{2n}\pi\right), \quad k=1,2,\ldots,n$$ 3. The weights for Gauss-Chebyshev quadrature are all equal and given by: $$\lambda_k = \frac{\pi}{n}$$ 4. Check the table for each $n$: - For $n=2$, nodes are $\pm 0.7071068$ which is $\pm \frac{\sqrt{2}}{2}$, correct. - Weights are $1.5707963$ which is approximately $\frac{\pi}{2}$, correct. - For $n=3$, nodes are $0$ and $\pm 0.8660254$ which correspond to $\cos(\pi/6)=\sqrt{3}/2$ and $\cos(\pi/2)=0$, correct. - Weights are $1.0471976$ which is approximately $\frac{\pi}{3}$, correct. - For $n=4$, nodes are $\pm 0.3826834$ and $\pm 0.9238795$ which correspond to $\cos(3\pi/8)$ and $\cos(\pi/8)$, correct. - Weights are $0.7853982$ which is approximately $\frac{\pi}{4}$, correct. - For $n=5$, nodes are $0$, $\pm 0.5877853$, and $\pm 0.9510565$ which correspond to $\cos(\pi/2)$, $\cos(\pi/5)$, and $\cos(\pi/10)$, correct. - Weights are $0.6283185$ which is approximately $\frac{\pi}{5}$, correct. 5. Since all nodes and weights match the theoretical values for Gauss-Chebyshev quadrature, the table is correct. Final answer: The table is correct for the Gauss-Chebyshev method.