DeepSeek: on the 100-problem benchmark

Evaluate \(\underset{n \rightarrow \infty}{\mathrm{lim}}\frac{\overset{n}{\underset{k =1}{\sum}}\sqrt{1+\frac{k}{n}}}{n}\).

Analysis: runs 1, 3, and 6 agree at \(0.884488434\), while run 2 gives \(0.884381\), run 4 gives \(0.934497\), and run 5 gives \(\frac{\sqrt{2}}{2} + \frac{1}{2}\ln(1+\sqrt{2}) \approx 1.1478\).

Find the inflection points, as ordered pairs, of \(f \! \left(x \right) = x^{4}-4 x^{3}\).

Analysis: Run 2 includes an extra pair \((-\infty, 0)\) in addition to \((0, 0)\) and \((2, -16)\).

Find the inflection points, as ordered pairs, and intervals of concavity of \(f \! \left(x \right) = x^{4}-8 x^{2}+3\).

Analysis: run 1 reports the inflection \(y\)-value as \(-77/9\), run 3 as \(-29/9\), and run 2, run 4, run 6 as \(-13/9\); run 1, run 2, and run 6 list \((0,3)\) as an additional inflection point while run 4 and run 3 do not; and run 6 further subdivides the concavity intervals at \(x=0\), giving concave down on \((-\infty,-2/\sqrt{3})\cup(0,2/\sqrt{3})\) and concave up on \((-2/\sqrt{3},0)\cup(2/\sqrt{3},\infty)\), whereas the others keep \((-2/\sqrt{3},2/\sqrt{3})\) as a single concave-down interval.

Apply 3 iterations of Gauss-Seidel to the system \(10 x_{1}-x_{2}+2 x_{3} = 6\), \(-x_{1}+11 x_{2}-x_{3} = 25\), \(2 x_{1}-x_{2}+10 x_{3} = -11\), starting from \([x_{1}, x_{2}, x_{3}] = [0, 0, 0]\). Give \([x_{1}, x_{2}, x_{3}]\) after iteration 3, each to 6 decimal places.

Analysis: Run 1 gives \([0.500000, 2.626263, -1.897747]\), run 2 gives \([0.500000, 2.833333, -2.016667]\), run 3 gives \([0.523809, 2.355372, -1.486238]\), run 4 gives \([0.524287, 2.425531, -1.164914]\), run 5 gives \([0.500000, 2.833333, -1.966667]\), and run 6 gives \([0.524286, 2.426667, -1.184524]\), and at the requested six-decimal precision no two of these triples agree on all three components.

Assuming \(0<a\), compute the curvature \(\kappa \! \left(t \right)\) and torsion \(\tau \! \left(t \right)\) of the helix \(r \! \left(t \right) = \langle a \cos \! \left(t \right), a \sin \! \left(t \right), b t \rangle\).

Analysis: run 1 gives the curvature \(\kappa = \frac{|a|}{a^2+b^2}\), runs 2, 3, 5, and 6 give the signed torsion \(\tau = \frac{b}{a^2+b^2}\), and run 4 gives the absolute torsion \(\frac{|b|}{a^2+b^2}\).

For \(f \! \left(x \right) = {\mathrm e}^{x}\) and Taylor expansion centered at 0, find the smallest \(n\) such that the Lagrange remainder satisfies \({| \textit{R\_n} \! \left(x \right)|}< 1.0\times 10^{-6}\) for all \(x\) in \([0, 0.5]\).

Analysis: The answer by run 3 gives \(n=20\), the answer by run 1 gives \(n=100\), and the answer by run 6 gives \(0.000000\).

Find the envelope of the one-parameter family of lines \(y = m x +\frac{1}{m}\) for \(0<m\), and identify the resulting curve.

Analysis: run 4's expression still contains the parameter \(m\) and is not the envelope curve, run 5 gives the hyperbola \(4xy=1\), and run 1 gives the parabola \(y^2=4x\).

Decompose into partial fractions and evaluate \(\int \frac{3 x^{2}+2 x +1}{\left(x -1\right) \left(x +2\right)^{2}}d x\).

Analysis: The answers use different coefficients on \(\ln|x-1|\), \(\ln|x+2|\), and the \(1/(x+2)\) term, so they represent distinct antiderivatives.

Find the critical points of \(f \! \left(x , y\right) = x^{3}+y^{2}-3 x\) and classify them.

Analysis: The first group classifies \((0,0)\) as a saddle, \((1,0)\) as a local minimum, and \((-1,0)\) as a local maximum; run 2 lists the points \((0,0)\), \((3,0)\), and \((-1,0)\) with different classifications; run 6 omits \((0,0)\) and labels \((-1,0)\) as a saddle.

Locate and classify the critical points of the surface defined by \(f \! \left(x , y\right) = x^{3}+y^{3}-3 x y\).

Analysis: The deepseek responses split into: one form giving both critical locations with classifications \((0,0)\) saddle and \((1,1)\) local minimum (with or without explicit \(z\)-values), another giving just the two surface points \((0,0,0)\) and \((1,1,-1)\) without classification, one ambiguous/atypical form \((-1,1,\text{local min};\,0,0,\text{saddle})\), and one single-entry form \([0,0,-1]\).

Find the extreme values of \(f \! \left(x , y\right) = x^{2}+2 y^{2}\) subject to the constraint \(x +y = 3\).

Analysis: The answer from run 5 reports only the minimum value \(-6\), while the answers from run 3 and run 1 also include a second value (\(6\) and \(12\) respectively) as a purported maximum.

Transform to cylindrical coordinates and compute \({\textcolor{gray}{\int}}_{\!\!\!0}^{2 \pi}{\textcolor{gray}{\int}}_{\!\!\!0}^{3}{\textcolor{gray}{\int}}_{\!\!\!0}^{4}r \textcolor{gray}{d}z \textcolor{gray}{d}r \textcolor{gray}{d}\theta\), where \(E\) is the solid bounded by \(z = 0\), \(z = 4\), and \(r = 3\).

Analysis: run 2 gives \(9\pi\), run 1 gives \(108\pi\), run 4 gives \(72\pi\), while run 5 and run 6 both give \(36\pi\).

Evaluate \(\int_{0}^{1}\int_{0}^{\sqrt{-x^{2}+1}}\left(x^{2}+y^{2}\right)d y d x\).

Analysis: Run 5's answer \(-\frac{1}{3}\) is negative, while run 1's answer \(0.5235987756 \approx \frac{\pi}{6}\) is positive, so they are not equivalent.

Find the determinant of the 4x4 matrix \(A = \left[\begin{array}{cccc} 2 & 1 & 0 & 1 \\ 1 & 3 & 1 & 0 \\ 0 & 1 & 2 & 1 \\ 1 & 0 & 1 & 3 \end{array}\right]\).

Analysis: The four answers give four different numerical values for the determinant: run 3 gives \(42\), run 4 gives \(-5\), run 1 gives \(12\), and run 6 gives \(20\).

Find the reduced row echelon form of \(A = \left[\begin{array}{cccc} 1 & 2 & -1 & 3 \\ 2 & 4 & 1 & -1 \\ 3 & 7 & 0 & 2 \end{array}\right]\).

Analysis: The answers by run 1 and run 3 agree on the last row \([0,0,1,-2]\), while run 6 gives \([0,1,0,-7],[0,0,1,4]\), differing in the second and third rows.

Find one valid singular value decomposition of the matrix \(\left[\begin{array}{cc} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{array}\right]\).

Analysis: Runs 4 and 1 share the same \(U\), \(\Sigma\), and \(V\); run 2 differs by a sign flip in column 2 of \(U\) without a matching flip in \(V\); run 5 presents an entirely different left-singular-vector matrix \(U\).

Diagonalize \(A = \left[\begin{array}{cc} 4 & 1 \\ 2 & 3 \end{array}\right]\) as \(A = D\), giving \(P\) and \(D\) explicitly.

Analysis: The matrices \(P\) differ: in run 1 the columns are \((1,2)\) and \((1,-1)\), while in run 3 the columns are \((1,1)\) and \((-1/2,1)\), which are not scalar multiples within each eigenspace of \(D=\operatorname{diag}(5,2)\).

Find the minimum-norm least squares solution to \(A x = b\) for \(A = \left[\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right]\) and \(b = \left[\begin{array}{c} 1 \\ 1 \end{array}\right]\).

Analysis: The five answers are five distinct vectors in \(\mathbb{R}^3\): run 1 gives \((-1/3, 1/6, 2/3)\), run 3 gives \((-1/3, 2/3, 5/3)\), run 2 gives \((1/3, 1/3, 1/3)\), run 5 gives \((5/6, -1/3, 1/6)\), and run 6 gives \((-1/3, -1/6, 0)\).

Find an LU decomposition \(A = L U\) of \(A = \left[\begin{array}{cc} 4 & 3 \\ 6 & 3 \end{array}\right]\), with \(L\) unit lower triangular.

Analysis: Run 3 gives the product \(L \cdot U\) as a single matrix expression, while run 1 gives the pair \((L, U)\) as two separate matrices.

Solve the Riccati equation \(\frac{d}{d x}y \! \left(x \right) = 1+x^{2}-2 x y \! \left(x \right)+y \! \left(x \right)^{2}\).

Analysis: The answers by run 2 and run 1 give the general solution \(y = x + 1/(C - x)\) (equivalent up to the sign/name of the constant), while run 3's \(1/(x+1)\) and run 5's \(1\) are distinct expressions not equivalent to that family or to each other.

Solve the Riccati equation \(\frac{d}{d x}y \! \left(x \right) = y \! \left(x \right)^{2}+x^{2}\) in closed form.

Analysis: run 2 provides a closed-form solution in terms of Weierstrass \(\wp\) functions, while run 1 asserts that no closed-form solution exists.

Solve \(\frac{d}{d x}y \! \left(x \right) = \frac{x \left(x^{2}-x -1-2 x^{2} y \! \left(x \right)+2 x^{4}\right)}{\left(x^{2}-y \! \left(x \right)\right) \left(x +1\right)}\), expressing the implicit general solution.

Analysis: The six answers present different implicit relations in \(x\) and \(y\), each built from a distinct mixture of polynomial monomials, \(\ln(x+1)\), \(\ln(x^2-y)\) (or its variants), and in run 5 a \(\sqrt{-2x^4+2x^2y+2x^2-2y+1}\) factor; computing \(-F_x/F_y\) at common sample points gives a different value for every answer, so no two define the same solution family.

For \(0<L\), find the eigenvalues and nontrivial eigenfunctions of the boundary value problem \(\frac{d^{2}}{d x^{2}}y \! \left(x \right)+\lambda y \! \left(x \right) = 0\), \(y \! \left(0\right) = 0\), \(y \! \left(L \right) = 0\).

Analysis: The answer from run 1 includes an extraneous \(\lambda = 0\) eigenvalue, while the answers from run 4 and run 2 give only the nontrivial spectrum \(\lambda_n = n^2\pi^2/L^2\) with eigenfunctions \(\sin(n\pi x/L)\).

Solve Bessel's equation \(x^{2} \left(\frac{d^{2}}{d x^{2}}y \! \left(x \right)\right)+x \left(\frac{d}{d x}y \! \left(x \right)\right)+\left(x^{2}-1\right) y \! \left(x \right) = 0\).

Analysis: The answers by run 4 and run 3 give the general solution \(C_1 J_1(x) + C_2 Y_1(x)\) in terms of Bessel functions of order 1, while the answer by run 1 gives \(\arccos(x)\), which is a different function.

Solve \(\frac{d^{2}}{d x^{2}}y \! \left(x \right) = \frac{x^{2} \left(\frac{d}{d x}y \! \left(x \right)\right)^{2}-2 x y \! \left(x \right) \left(\frac{d}{d x}y \! \left(x \right)\right)+y \! \left(x \right)^{2}}{x^{2}}\).

Analysis: The first group gives the general solution \(y = C_1 x + C_2 x \ln(x)\), while run 6's answer \(y = C_1 x \exp(C_2/x)\) is a different family of functions.

For the Fourier-Bessel expansion \(f \! \left(r \right) = -r^{2}+1 = \overset{\infty}{\underset{n =1}{\sum}}c_{n} J_{0}\! \left(\alpha_{n} r \right)\) on \([0, 1]\), where \(\alpha_{n}\) is the n-th positive zero of \(J_{0}\! \left(x \right)\), give the closed form for \(c_{n}\) and the numeric values of \(c_{1}\), \(c_{2}\), \(c_{3}\) to 4 decimal places.

Analysis: As complete answers, all six differ: runs 1, 3, and 6 share one closed form but report different numeric triples, runs 2 and 5 share the other closed form but differ numerically, and run 4 gives only the closed form with no reported values of \(c_1, c_2, c_3\).

Find the harmonic conjugate of \(u \! \left(x , y\right) = x^{2}-y^{2}\).

Analysis: Run 1's answer \(x^2 - y^2 + 2ixy\) includes the imaginary unit \(i\), making it a complex-valued expression (namely \((x+iy)^2\)), while run 6's answer \(x^2 - y^2 + 2xy\) is purely real and lacks the factor of \(i\).

Evaluate \(\int \frac{{\mathrm e}^{z}}{\left(z -1\right)^{3}}d z\) over the contour \({| z |} = 3\).

Analysis: Run 2 gives \(0\) while run 1 gives \(-2\pi i\, e\), which are distinct complex numbers.

Find the Laurent series expansion of \(f \! \left(z \right) = \frac{1}{z \left(z -2\right)}\) about \(z = 0\) for \(0<{| z |}<2\).

Analysis: run 3 gives the closed-form \(\frac{1}{2(z-2)} - \frac{1}{2z}\) and run 4 gives the full Laurent series with explicit continuation, both representing the same infinite expansion; run 2 truncates the series at \(z^4/64\), while run 1 includes one additional term up to \(z^5/128\).

Find the residue of \(f \! \left(z \right) = \frac{\sin \! \left(z \right)}{z^{4}}\) at \(z = 0\).

Analysis: run 1 and run 4 give \(0\), while run 2 gives \(-1/6\).

Evaluate \(\int_{0}^{2 \pi}\frac{1}{2+\cos \! \left(\theta \right)}d \theta\).

Analysis: The answer from run 4 is \(\frac{\pi}{\sqrt{3}} \approx 1.8138\), while the answer from run 1 is \(\frac{1}{\sqrt{3}} \approx 0.5774\), differing by a factor of \(\pi\).

Find the resultant of \(p = x^{3}-2\) and \(q = x^{2}-2\) with respect to \(x\).

Analysis: The answers \(-2\) and \(-4\) are distinct numerical values, so run 5 and run 1 are not mathematically equivalent.

Solve the nonlinear system \(x^{3}+y^{3} = 35\), \(x +y = 5\) over the reals.

Analysis: Run 2 and run 1 list all four real solutions \((5,0), (0,5), (2,3), (3,2)\), while run 4 lists only \((5,0)\).

Compute a lexicographic Gröbner basis (with \(y <x\)) of the ideal \(\langle x^{2}+y^{2}-4, x y -1\rangle\) and use it to find all complex solutions of the system.

Analysis: No two full answers match: they present different polynomial sets, different claimed Gröbner bases, or only solution points, and the last answer lists only the four incorrect points \((\pm\sqrt{3},\pm\sqrt{3})\) with no basis.

Evaluate \(\int_{0}^{\frac{\pi}{2}}\ln \! \left(\sin \! \left(x \right)\right)^{2}d x\) in closed form.

Analysis: The run 3 answer evaluates to \(\pi^3/12 + \pi(\ln 2)^2\), while the others give \(-\pi^3/24 + \pi(\ln 2)^2/2\).

Express \(4 \sin \! \left(x \right) \sin \! \left(2 x \right) \sin \! \left(4 x \right)\) as a sum of cosines.

Analysis: The answers from run 1, run 2, and run 4 give \(\cos(x) - \cos(3x) - \cos(5x) + \cos(7x)\), while run 6's answer multiplies every term by \(4\).

Find the Fourier series expansion (first four nonzero terms) of the square wave \(f \! \left(x \right) = \mathrm{signum}\! \left(\sin \! \left(x \right)\right)\) on \([-\pi, \pi]\).

Analysis: The first group writes the series as the sum \(\frac{4}{\pi}\sin(x)+\frac{4}{3\pi}\sin(3x)+\frac{4}{5\pi}\sin(5x)+\frac{4}{7\pi}\sin(7x)\), while the second lists only the sequence of Fourier coefficients \(0,\frac{4}{\pi},0,\frac{4}{3\pi},0,\frac{4}{5\pi},0,\frac{4}{7\pi}\) without the accompanying \(\sin(nx)\) basis functions.

Find \(D_{7}\), the number of derangements of 7 elements.

Analysis: The number of derangements of 7 elements differs between run 2's value of \(133496\) and run 1's value of \(1854\).

Find the Stirling number of the second kind \(S \! \left(6, 3\right)\).

Analysis: The groups give four distinct numerical values: \(20\), \(90\), \(245\), and \(546\).

Find the number of binary strings of length 8 with no consecutive 1's.

Analysis: run 4's answer is \(55\), run 2's is \(34\), run 1's is \(42\), and run 5's is \(21\).

Find the number of labeled trees on 6 vertices.

Analysis: Run 2 gives \(105\) while run 1 gives \(720 = 6^{6-2}\) (Cayley's formula for labeled trees on \(6\) vertices), so the two values are not equal.

Find the number of spanning trees in the complete bipartite graph \(K_{2,3}\).

Analysis: The groups give the values \(6\), \(10\), and \(12\) for the number of spanning trees.

Find the number of ways to partition a 6-element set into 3 unordered subsets of size 2.

Analysis: The answer by run 4 is \(10\), the correct count obtained as \(\binom{6}{2}\binom{4}{2}\binom{2}{2}/3! = 15\); the answer by run 1 is \(120\); the answer by run 3 is \(3\); the answer by run 2 is \(5\).

For data \(X = [1, 2, 3, 4, 5]\) and \(Y = [ 2.1, 3.9, 4.8, 6.2, 7.0]\), fit a least squares regression line \(y = b x +a\) and compute the residual sum of squares.

Analysis: The groups give four distinct residual sum of squares values: \(0.06\), \(0.068\), \(0.14\), and \(0.028\).

Find the range and interquartile range of \([5, 12, 18, 23, 28, 35, 41]\) using Tukey's hinges, with the median excluded from the lower and upper halves.

Analysis: Run 1 gives range \(7\) and IQR \(18\), run 2 gives range \(35\) and IQR \(23\), and run 3 just repeats the data set.

For data \(\{1, 2, 3, 4, 5\}\), compute the least squares regression line \(y = b x +a\) and the coefficient of determination \(R^{2}\).

Analysis: The groups agree on the slope \(0.9\) but differ in intercept and \(R^2\): run 1 gives \((0.2, 0.9, 0.6429)\), run 3 gives \((0.3, 0.9, 0.5625)\), run 4 gives \((0.2, 0.9, 0.5625)\), run 5 gives the line \(0.2 + 0.9x\) with \(R^2 = 0.72\), and run 6 gives \((0.2, 0.9, 0.45)\).

For \(\mathit{X\,~\,Beta(\alpha=2,\,\beta=3)\,}\), find \(E_{X}\), \(\mathit{Var}_{X}\), and the mode.

Analysis: The mode differs between the two answers: run 2 gives \(1/3\) while run 3 gives \(1/4\), which is the correct mode \((\alpha-1)/(\alpha+\beta-2) = 1/3\) versus an incorrect value.

Find median and sample standard deviation (denominator \(n -1\)) of \([3, 7, 8, 12, 15]\).

Analysis: The medians differ: run 1 gives \(7\), run 5 gives \(5\), and run 6 gives \(9\).

Find the linear regression equation \(y = a x +b\) that best fits the points \([1, 2]\), \([2, 4]\), \([3, 5]\).

Analysis: run 1 gives slope \(1.5\) and intercept \(2/3\), run 4 gives \(2\) and \(0.5\), and run 2 gives \(1\) and \(4/3\).

Find the equation of the osculating circle to the curve \(y = x^{2}\) at the point \([1, 1]\).

Analysis: run 1 gives center \((3,1)\) with \(r^2=4\), run 5 gives center \((2,3)\) with \(r^2=3\), and run 2 gives center \((0,5/2)\) with \(r^2=5/4\).

Project the vector \(\langle 2, 3, 4\rangle\) onto the subspace of \(R^{3}\) spanned by \(\langle 1, 1, 1\rangle\) and \(\langle 1, 0, 0\rangle\).

Analysis: The answer by run 1 is \([3, 2, 2]\), the answer by run 2 is \([2, 3, 4]\) (the original vector unchanged), and the answer by run 6 is \([2, 2, 2]\).

Solve the Riccati equation \(\frac{d}{d x}y \! \left(x \right) = y \! \left(x \right)^{2}+2 x y \! \left(x \right)+x^{2}-1\).

Analysis: The answers from run 4 and run 1 give the general solution \(y(x) = -x + 1/(C + x)\), while run 5 gives only the constant \(-1\).

Evaluate \(\mathrm{I}\! \left(a \right) = \int_{0}^{\infty}\frac{\cos \! \left(a x \right)}{x^{4}+1}d x\) for \(0<a\) in closed form.

Analysis: Run 1 contains a factor \(\arccos(e^{-a/\sqrt{2}}\cos(a/\sqrt{2}))\) that depends on \(a\), whereas run 2 reduces via \(\sin(b+\pi/4)=(\sin b+\cos b)/\sqrt{2}\) to \(\frac{\pi}{2\sqrt{2}}e^{-a/\sqrt{2}}(\cos(a/\sqrt{2})+\sin(a/\sqrt{2}))\), so the two expressions have different functional dependence on \(a\).

Factor the polynomial \(x^{8}+x^{4}+1\) completely over the complex numbers.

Analysis: runs 1 and 6 are identical, runs 2 and 3 are identical, and runs 4 and 5 are identical; these three pairwise groups are mathematically different from one another.

Find the closed form of the infinite sum \(\overset{\infty}{\underset{n =1}{\sum}}\frac{H_{n}}{n^{3}}\), where \(H_{n} = \overset{n}{\underset{k =1}{\sum}}\frac{1}{k}\) is the n-th harmonic number.

Analysis: Run 1 gives a rational multiple of \(\pi^4\), while run 2 gives a combination of \(\pi^2 \zeta(3)\) and \(\zeta(5)\), which are not equal as real numbers.

Derive the minimal polynomial of \(\cos \! \left(\frac{2 \pi}{9}\right)\) over \(Q\) and give its numerical value to 6 decimal places.

Analysis: runs 3, 4, and 6 agree at \(-0.173648\), runs 1 and 2 agree at \(0.173648\), and run 5 gives \(0.766044\).

Solve the wave equation \(\frac{\partial^{2}}{\partial t^{2}}u \! \left(x , t\right) = c^{2} \left(\frac{\partial^{2}}{\partial x^{2}}u \! \left(x , t\right)\right)\) on \(0<x \boldsymbol{\land}x \boldsymbol{\land}x \boldsymbol{\land}x <\pi\) with \(0<c\), with boundary conditions \(u \! \left(0, t\right) = u \! \left(\pi , t\right) = 0\), initial displacement \(u \! \left(x , 0\right) = x \left(\pi -x \right)\), and initial velocity \(\left(\frac{\partial}{\partial t}u \! \left(x , t\right)\right)\! \left(x , 0\right) = 0\). Express the solution \(u \! \left(x , t\right)\) as a Fourier sine series with the coefficients given in closed form.

Analysis: Run 1's expression is missing the \(\frac{1}{\pi}\) factor and the summation over \(n\), while run 3 provides the complete series \(u(x,t) = \sum_{n=1}^{\infty} \frac{4(1-(-1)^n)}{\pi n^3}\sin(nx)\cos(cnt)\).