| 1 | Limits & Series | Evaluate \(\underset{n \rightarrow \infty}{\mathrm{lim}}\frac{\overset{n}{\underset{k =1}{\sum}}\sqrt{1+\frac{k}{n}}}{n}\). |
| 2 | Limits & Series | Evaluate \(\underset{x \rightarrow 0}{\mathrm{lim}}\frac{\sin \! \left(x \right)-x +\frac{x^{3}}{6}}{x^{5}}\). |
| 3 | Differential Calculus | Find the inflection points, as ordered pairs, of \(f \! \left(x \right) = x^{4}-4 x^{3}\). |
| 4 | Differential Calculus | Find the inflection points, as ordered pairs, and intervals of concavity of \(f \! \left(x \right) = x^{4}-8 x^{2}+3\). |
| 5 | Numerical Methods | 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. |
| 6 | Differential Geometry & Vector Calculus | Assuming \(0<a\), compute the curvature \(\kappa \! \left(t \right)\) and torsion \(\tau \! \left(t \right)\) of the helix \(r \! \left(t \right) = \langle a \cos \! \left(t \right), a \sin \! \left(t \right), b t \rangle\). |
| 7 | Numerical Methods | 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 \([0, 0.5]\). |
| 8 | Differential Geometry & Vector Calculus | Find the envelope of the one-parameter family of lines \(y = m x +\frac{1}{m}\) for \(0<m\), and identify the resulting curve. |
| 9 | Limits & Series | Find the closed form of the infinite sum \(\overset{\infty}{\underset{n =1}{\sum}}\frac{H_{n}}{n^{2}}\), where \(H_{n} = \overset{n}{\underset{k =1}{\sum}}\frac{1}{k}\) is the n-th harmonic number. Express the result in terms of the Riemann zeta function. |
| 10 | Integration & Special Integrals | Evaluate \(\int_{0}^{1}\ln \! \left(x \right) \ln \! \left(1-x \right)d x\) in closed form. |
| 11 | Limits & Series | Find the closed form of the infinite sum \(\overset{\infty}{\underset{n =1}{\sum}}\frac{\cos \! \left(n \right)}{n^{2}}\) (the argument is in radians, not degrees). |
| 12 | Integration & Special Integrals | Decompose into partial fractions and evaluate \(\int \frac{3 x^{2}+2 x +1}{\left(x -1\right) \left(x +2\right)^{2}}d x\). |
| 13 | Integration & Special Integrals | Evaluate \(\int_{0}^{1}\frac{\arctan \! \left(x \right)}{x}d x\) in closed form. |
| 14 | Integration & Special Integrals | Find the total area enclosed by both loops of the lemniscate \(r^{2} = 4 \cos \! \left(2 \theta \right)\). |
| 15 | Integration & Special Integrals | Evaluate \(\int_{0}^{1}\frac{\ln \! \left(x^{2}+1\right)}{x^{2}+1}d x\) in closed form. Express the result using \(\pi\), \(\ln \! \left(2\right)\), and Catalan's constant \(G\). |
| 16 | Integration & Special Integrals | Evaluate \(\int_{0}^{1}\frac{\ln \! \left(x \right)}{\sqrt{-x^{2}+1}}d x\) in closed form. |
| 17 | Multivariable Calculus | Find the critical points of \(f \! \left(x , y\right) = x^{3}+y^{2}-3 x\) and classify them. |
| 18 | Multivariable Calculus | Locate and classify the critical points of the surface defined by \(f \! \left(x , y\right) = x^{3}+y^{3}-3 x y\). |
| 19 | Multivariable Calculus | Find the maximum rate of change of \(f \! \left(x , y\right) = x^{2} {\mathrm e}^{-y}\) at \([2, 0]\) and the direction in which it occurs. |
| 20 | Multivariable Calculus | Find the extreme values of \(f \! \left(x , y\right) = x^{2}+2 y^{2}\) subject to the constraint \(x +y = 3\). |
| 21 | Integration & Special Integrals | Reverse the order of integration and evaluate \(\int_{0}^{1}\int_{x}^{1}\sin \! \left(y^{2}\right)d y d x\). |
| 22 | Integration & Special Integrals | 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 \(r = 3\). |
| 23 | Integration & Special Integrals | Evaluate \(\int_{0}^{1}\int_{0}^{\sqrt{-x^{2}+1}}\left(x^{2}+y^{2}\right)d y d x\). |
| 24 | Integration & Special Integrals | Find the volume of the solid bounded above by \(z = -x^{2}-y^{2}+4\) and below by \(z = x^{2}+y^{2}\). |
| 25 | Linear Algebra | Find the determinant of the 4x4 matrix \(A = \left[\begin{array}{cccc} 2 & 1 & 0 & 1 \\ 1 & 3 & 1 & 0 \\ 0 & 1 & 2 & 1 \\ 1 & 0 & 1 & 3 \end{array}\right]\). |
| 26 | Linear Algebra | Find the reduced row echelon form of \(A = \left[\begin{array}{cccc} 1 & 2 & -1 & 3 \\ 2 & 4 & 1 & -1 \\ 3 & 7 & 0 & 2 \end{array}\right]\). |
| 27 | Linear Algebra | Find the characteristic polynomial and eigenvalues of the companion matrix \(A = \left[\begin{array}{ccc} 0 & 0 & 2 \\ 1 & 0 & -1 \\ 0 & 1 & 2 \end{array}\right]\). |
| 28 | Linear Algebra | Find one valid QR decomposition of the matrix \(\left[\begin{array}{cc} 1 & 1 \\ 1 & 0 \\ 0 & 1 \end{array}\right]\). |
| 29 | Linear Algebra | Find one valid singular value decomposition of the matrix \(\left[\begin{array}{cc} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{array}\right]\). |
| 30 | Linear Algebra | Diagonalize \(A = \left[\begin{array}{cc} 4 & 1 \\ 2 & 3 \end{array}\right]\) as \(A = D\), giving \(P\) and \(D\) explicitly. |
| 31 | Linear Algebra | 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 \(b = \left[\begin{array}{c} 1 \\ 1 \end{array}\right]\). |
| 32 | Linear Algebra | 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. |
| 33 | ODEs & Special Functions | Solve the Riccati equation \(\frac{d}{d x}y \! \left(x \right) = 1+x^{2}-2 x y \! \left(x \right)+y \! \left(x \right)^{2}\). |
| 34 | ODEs & Special Functions | Solve the Riccati equation \(\frac{d}{d x}y \! \left(x \right) = y \! \left(x \right)^{2}+x^{2}\) in closed form. |
| 35 | ODEs & Special Functions | 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. |
| 36 | ODEs & Special Functions | For \(0<L\), find the eigenvalues and nontrivial eigenfunctions of the boundary value problem \(\frac{d^{2}}{d x^{2}}y \! \left(x \right)+\lambda y \! \left(x \right) = 0\), \(y \! \left(0\right) = 0\), \(y \! \left(L \right) = 0\). |
| 37 | ODEs & Special Functions | Solve Bessel's equation \(x^{2} \left(\frac{d^{2}}{d x^{2}}y \! \left(x \right)\right)+x \left(\frac{d}{d x}y \! \left(x \right)\right)+\left(x^{2}-1\right) y \! \left(x \right) = 0\). |
| 38 | ODEs & Special Functions | Solve \(\frac{d^{2}}{d x^{2}}y \! \left(x \right) = \frac{x^{2} \left(\frac{d}{d x}y \! \left(x \right)\right)^{2}-2 x y \! \left(x \right) \left(\frac{d}{d x}y \! \left(x \right)\right)+y \! \left(x \right)^{2}}{x^{2}}\). |
| 39 | ODEs & Special Functions | Determine the series solution for \(\frac{d^{2}}{d x^{2}}y \! \left(x \right)+x y \! \left(x \right) = 0\) (Airy's equation) around \(x = 0\) up to order 5. |
| 40 | ODEs & Special 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. |
| 41 | Complex Analysis | Find the harmonic conjugate of \(u \! \left(x , y\right) = x^{2}-y^{2}\). |
| 42 | Complex Analysis | Find all 6th roots of \(-64\) in polar form and identify which roots lie in the upper half-plane. |
| 43 | Complex Analysis | Evaluate \(\int \frac{{\mathrm e}^{z}}{\left(z -1\right)^{3}}d z\) over the contour \({| z |} = 3\). |
| 44 | Complex Analysis | Find the Laurent series expansion of \(f \! \left(z \right) = \frac{1}{z \left(z -2\right)}\) about \(z = 0\) for \(0<{| z |}<2\). |
| 45 | Complex Analysis | Find the residue of \(f \! \left(z \right) = \frac{\sin \! \left(z \right)}{z^{4}}\) at \(z = 0\). |
| 46 | Complex Analysis | Find the Laurent series of \(f \! \left(z \right) = \frac{1}{z^{2} \left(z -1\right)}\) around \(z = 0\) in both regions: (a) \(0<{| z |}<1\) and (b) \(1<{| z |}\), each with 5 non-zero terms. |
| 47 | Complex Analysis | Find the residue of \(f \! \left(z \right) = \frac{{\mathrm e}^{z}}{\left(z -\mathrm{I} \pi \right)^{2}}\) at \(z = \mathrm{I} \pi\). |
| 48 | Complex Analysis | Evaluate \(\int_{0}^{2 \pi}\frac{1}{2+\cos \! \left(\theta \right)}d \theta\). |
| 49 | Polynomial & Abstract Algebra | Find the resultant of \(p = x^{3}-2\) and \(q = x^{2}-2\) with respect to \(x\). |
| 50 | Polynomial & Abstract Algebra | Solve the polynomial equation \(x^{4}-2 x^{3}-3 x^{2}+4 x +4 = 0\) over the reals, finding all roots. |
| 51 | Polynomial & Abstract Algebra | Solve the nonlinear system \(x^{3}+y^{3} = 35\), \(x +y = 5\) over the reals. |
| 52 | Polynomial & Abstract Algebra | Find the minimal polynomial of \(\alpha = \sqrt{2}+\mathit{root}_{3}\! \left(3\right)\) over \(Q\). |
| 53 | Polynomial & Abstract Algebra | 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. |
| 54 | Polynomial & Abstract Algebra | Determine the Galois group of \(x^{4}-2\) over \(Q\). |
| 55 | Polynomial & Abstract Algebra | Find the discriminant of \(f \! \left(x \right) = x^{3}+p x +q\) in terms of \(p\) and \(q\), and state the condition on \([p, q]\) for \(f\) to have a repeated root. |
| 56 | Polynomial & Abstract Algebra | Express the elementary symmetric polynomial \(e_{3}\! \left(x_{1}, x_{2}, x_{3}\right)\) in terms of the power sums \(p_{k} = x_{1}^{k}+x_{2}^{k}+x_{3}^{k}\) for \([k = 1, 2, 3]\). |
| 57 | Trigonometry & Fourier Analysis | Evaluate \(\int_{0}^{\frac{\pi}{2}}\frac{x}{\tan \! \left(x \right)}d x\) in closed form. Express the result using \(\pi\) and \(\ln \! \left(2\right)\). |
| 58 | Trigonometry & Fourier Analysis | Evaluate \(\int_{0}^{\frac{\pi}{2}}\ln \! \left(\sin \! \left(x \right)\right)^{2}d x\) in closed form. |
| 59 | Trigonometry & Fourier Analysis | Evaluate \(\int_{0}^{\frac{\pi}{4}}\ln \! \left(1+\tan \! \left(x \right)\right)d x\) in closed form. Express the result using \(\pi\) and \(\ln \! \left(2\right)\). |
| 60 | Trigonometry & Fourier Analysis | Express \(4 \sin \! \left(x \right) \sin \! \left(2 x \right) \sin \! \left(4 x \right)\) as a sum of cosines. |
| 61 | Trigonometry & Fourier Analysis | Find a closed form for \(\overset{n}{\underset{k =0}{\sum}}\cos \! \left(k \theta \right)\), for \(\theta \neq 2 \pi m\) with \(m\) in \(Z\). |
| 62 | Trigonometry & Fourier Analysis | Express \(T_{7}\! \left(x \right)\), the Chebyshev polynomial of the first kind of degree 7, explicitly as a polynomial in \(x\). |
| 63 | Trigonometry & Fourier Analysis | 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 \([-\pi, \pi]\). |
| 64 | Trigonometry & Fourier Analysis | Express \(\sin \! \left(\frac{\pi}{5}\right)\) in radicals. |
| 65 | Combinatorics & Graph Theory | Find \(D_{7}\), the number of derangements of 7 elements. |
| 66 | Combinatorics & Graph Theory | Find the Stirling number of the second kind \(S \! \left(6, 3\right)\). |
| 67 | Combinatorics & Graph Theory | Find the number of binary strings of length 8 with no consecutive 1's. |
| 68 | Combinatorics & Graph Theory | Find the number of ways to partition the set \(\{1, 2, 3, 4\}\) into non-empty subsets. |
| 69 | Combinatorics & Graph Theory | Find the number of labeled trees on 6 vertices. |
| 70 | Combinatorics & Graph Theory | Find the number of spanning trees in the complete bipartite graph \(K_{2,3}\). |
| 71 | Combinatorics & Graph Theory | Find the chromatic polynomial of the complete graph on 4 vertices. |
| 72 | Combinatorics & Graph Theory | Find the number of ways to partition a 6-element set into 3 unordered subsets of size 2. |
| 73 | Statistics & Probability | 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. |
| 74 | Statistics & Probability | 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. |
| 75 | Statistics & Probability | For data \(\{1, 2, 3, 4, 5\}\), compute the least squares regression line \(y = b x +a\) and the coefficient of determination \(R^{2}\). |
| 76 | Statistics & Probability | Find the two-tailed p-value for \(t = 2.5\) with \(\mathit{df} = 20\) using the Student t distribution. |
| 77 | Statistics & Probability | Perform a two-tailed one-sample t-test at \(\alpha = 0.05\) for \(\mathit{H0}\) using the sample \([9, 10, 11, 12, 13]\) (assume normal population with unknown variance). |
| 78 | Statistics & Probability | For \(\mathit{X\,~\,Beta(alpha=2,\,beta=3)\,}\), find \(E_{X}\), \(\mathit{Var}_{X}\), and the mode. |
| 79 | Statistics & Probability | Find median and sample standard deviation (denominator \(n -1\)) of \([3, 7, 8, 12, 15]\). |
| 80 | Statistics & Probability | Find the linear regression equation \(y = a x +b\) that best fits the points \([1, 2]\), \([2, 4]\), \([3, 5]\). |
| 81 | Integration & Special Integrals | Evaluate \(\int_{0}^{1}\frac{\ln \! \left(x \right)^{2}}{x^{2}+1}d x\) in closed form. |
| 82 | Differential Geometry & Vector Calculus | Find the equation of the osculating circle to the curve \(y = x^{2}\) at the point \([1, 1]\). |
| 83 | Integration & Special Integrals | Evaluate \(\int_{0}^{1}\frac{\ln \! \left(1-x \right)^{2}}{x}d x\) in closed form. Express the result using \(\zeta \! \left(3\right)\). |
| 84 | Differential Geometry & Vector Calculus | Find the total arc length of the cardioid \(r = 1+\cos \! \left(\theta \right)\). |
| 85 | Integration & Special Integrals | Evaluate \(\int_{0}^{\infty}\frac{\ln \! \left(x \right)}{\left(x^{2}+1\right)^{2}}d x\) in closed form. |
| 86 | Differential Geometry & Vector Calculus | Find the Jacobian determinant of the transformation \(u = x^{2}-y^{2}\), \(v = 2 x y\). |
| 87 | Differential Geometry & Vector Calculus | Evaluate \(\int F d C\) where \(F = \langle z , x , y\rangle\) and \(C\) is the circle \(x^{2}+y^{2} = 1\), \(z = 1\), oriented counterclockwise when viewed from above. |
| 88 | Linear Algebra | Find the Jordan normal form of the matrix \(A = \left[\begin{array}{ccc} 2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{array}\right]\). |
| 89 | Linear Algebra | Project the vector \(\langle 2, 3, 4\rangle\) onto the subspace of \(R^{3}\) spanned by \(\langle 1, 1, 1\rangle\) and \(\langle 1, 0, 0\rangle\). |
| 90 | ODEs & Special Functions | Solve the IVP \(\frac{d^{2}}{d t^{2}}y \! \left(t \right)+2 \frac{d}{d t}y \! \left(t \right)+2 y \! \left(t \right) = \delta \! \left(t -\pi \right)\), \(y \! \left(0\right) = 0\), \(D\! \left(y \right)\! \left(0\right) = 0\). |
| 91 | ODEs & Special Functions | Solve the Riccati equation \(\frac{d}{d x}y \! \left(x \right) = y \! \left(x \right)^{2}+2 x y \! \left(x \right)+x^{2}-1\). |
| 92 | ODEs & Special Functions | Solve Laplace's equation \(\mathit{Laplacian} \! \left(u \! \left(r , \theta \right), \left[r , \theta \right]\right) = 0\) on the annulus \(1<r <2\) in polar coordinates, with boundary conditions \(u \! \left(1, \theta \right) = \cos \! \left(2 \theta \right)\) and \(u \! \left(2, \theta \right) = 0\). Give the explicit harmonic solution \(u \! \left(r , \theta \right)\). |
| 93 | Complex Analysis | 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. |
| 94 | Complex Analysis | Find the residue of \(f \! \left(z \right) = {\mathrm e}^{\frac{1}{z}}\) at \(z = 0\). |
| 95 | Complex Analysis | Evaluate \(\int_{0}^{2 \pi}\frac{\cos \! \left(\theta \right)}{5+4 \cos \! \left(\theta \right)}d \theta\). |
| 96 | Polynomial & Abstract Algebra | Factor the polynomial \(x^{8}+x^{4}+1\) completely over the complex numbers. |
| 97 | Combinatorics & Graph Theory | Find the number of distinct necklaces with 6 beads using 3 colors, where rotations are considered identical. |
| 98 | Limits & Series | 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. |
| 99 | Number Theory | Derive the minimal polynomial of \(\cos \! \left(\frac{2 \pi}{9}\right)\) over \(Q\) and give its numerical value to 6 decimal places. |
| 100 | ODEs & Special Functions | 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. |
