{"id": "ed65d411952414e55c0a5fe7", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Use the two graphs to find $\\lim_{x\\to0^-}g(f(x))$ and $\\lim_{x\\to0^+}g(f(x))$. Does the two-sided limit exist? The marked open circles are excluded values.\n", "options": null, "answer": "The one-sided limits are $2$ and $3$, respectively. The two-sided limit does not exist.", "solution": "From the inner graph, $f(x)\\to1^-$ as $x\\to0^-$ and $f(x)\\to1^+$ as $x\\to0^+$. The outer graph has left limit $2$ and right limit $3$ at $1$. The isolated values $f(0)$ and $g(1)$ do not determine these limits.", "has_image": true} {"id": "5b1b1dbf95bb6c6230af1a90", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The red line is tangent to $y=f(x)$ at $P$ and passes through $Q$. Starting Newton's method at the horizontal coordinate of $P$, find the next iterate.\n", "options": null, "answer": "$x_1=2$.", "solution": "The image gives $x_0=3$, $f(3)=2$, and tangent slope $f\\prime(3)=(2-(-2))/(3-1)=2$. Thus $x_1=x_0-f(x_0)/f\\prime(x_0)=3-2/2=2$.", "has_image": true} {"id": "a3450a0a2b19150f4d8a26e5", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The arrows show the velocity and acceleration of a regular planar trajectory at $P$. Find its curvature and the center and radius of its osculating circle at that instant.\n", "options": null, "answer": "$\\kappa=1/5$, radius $5$, center $(-3,5)$.", "solution": "The speed is $5$ and $\\det(v,a)=25>0$. Hence $\\kappa=|\\det(v,a)|/\\|v\\|^3=25/125=1/5$. The left unit normal is $(-4,3)/5$. The center is $P+5(-4,3)/5=(-3,5)$.", "has_image": true} {"id": "30349c9c24c8b2f78464f679", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Let $\\Gamma$ be the entire blue path from $A$ to $C$. Find $\\int_\\Gamma y\\,ds$ and its mean vertical coordinate with respect to arc length. The first piece is the indicated circular arc; the second is a straight segment.\n", "options": null, "answer": "$\\int_\\Gamma y\\,ds=14$; arc-length mean $14/(\\pi+5)$.", "solution": "On the quarter circle, set $y=2\\sin\\theta$, $ds=2\\,d\\theta$ for $0\\le\\theta\\le\\pi/2$, giving integral $4$. The straight piece has length $5$ and linearly varying height from $0$ to $4$, giving integral $5(0+4)/2=10$. The total length is $\\pi+5$, so the mean is $14/(\\pi+5)$.", "has_image": true} {"id": "2081fca58e997c057d8a2faf", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Assume $f$ is locally invertible at the marked point and has the derivative data displayed in the figure. Compute $(f^{-1})\\prime\\prime(3)$.\n", "options": null, "answer": "$(f^{-1})\\prime\\prime(3)=-3/4$.", "solution": "Differentiate $f(f^{-1}(u))=u$ twice. At $u=3$, the resulting identity is $(f^{-1})\\prime\\prime(3)=-f\\prime\\prime(2)/(f\\prime(2))^3=-6/8=-3/4$.", "has_image": true} {"id": "fb0800e70f87acc7e039ece4", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Let $f$ be positive on $(0,\\infty)$. The entire graph of $h(s)=\\log f(e^s)$ consists of the two straight rays shown. Determine whether $\\int_0^\\infty f(x)\\,dx$ converges, and evaluate it if it does.\n", "options": null, "answer": "The integral converges and equals $4$.", "solution": "The ray slopes are $-1/2$ and $-3/2$, and both meet the origin. Therefore $f(x)=x^{-1/2}$ on $(0,1]$ and $f(x)=x^{-3/2}$ on $[1,\\infty)$. Both improper integrals converge, with values $2$ and $2$.", "has_image": true} {"id": "6061408c896bbe6ed41f54db", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The shaded region is revolved about the dashed vertical line. Find the exact volume of the resulting solid. The two bounding graphs and the rotation axis are labeled in the picture.\n", "options": null, "answer": "$V=\\pi/2$.", "solution": "The curves meet at $x=0$ and $x=1$. A shell at $x$ has radius $2-x$ and height $x-x^2$. Consequently $$V=2\\pi\\int_0^1(2-x)(x-x^2)\\,dx=\\frac{\\pi}{2}.$$", "has_image": true} {"id": "e2dc9fb0de054216ec636cd1", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "A lamina of constant area density occupies the shaded part of the rectangle after the indicated semicircular notch is removed. Find its centroid.\n", "options": null, "answer": "$\\bar x=2$, $\\displaystyle\\bar y=\\frac{112-9\\pi}{3(24-\\pi)}$.", "solution": "Symmetry gives $\\bar x=2$. The rectangle has area $12$ and centroid height $3/2$. The removed semicircle has area $\\pi/2$ and centroid height $3-4/(3\\pi)$. Subtract its first moment and area: $$\\bar y=\\frac{18-(\\pi/2)(3-4/(3\\pi))}{12-\\pi/2}=\\frac{112-9\\pi}{3(24-\\pi)}.$$", "has_image": true} {"id": "7ae806d14fd56725c1862204", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "For an arbitrary continuous function $F$, express $\\iint_D F(x,y)\\,dA$ in the order $dy\\,dx$, where $D$ is the shaded region. Your bounds must describe the whole region without overcounting.\n", "options": null, "answer": "$$\\int_0^1\\int_0^{\\sqrt{x}}F(x,y)\\,dy\\,dx+\\int_1^2\\int_0^{2-x}F(x,y)\\,dy\\,dx.$$", "solution": "Horizontal boundaries are replaced by vertical slices. The curved upper boundary $y=\\sqrt{x}$ is active until the intersection $(1,1)$; the straight upper boundary $y=2-x$ is active afterward. Both slices start at $y=0$.", "has_image": true} {"id": "4c5d1f594f7e2161ef647ffd", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Use $u=(3x-y)/5$ and $v=(-x+2y)/5$ to evaluate $$\\iint_D e^{(3x-y)/5}\\,dx\\,dy,$$ where $D$ is the pictured parallelogram.\n", "options": null, "answer": "$5(e-1)$.", "solution": "The inverse transformation is $x=2u+v$, $y=u+3v$, with absolute Jacobian $5$. The four displayed vertices map to the corners of $[0,1]^2$. Thus the integral becomes $5\\int_0^1\\int_0^1e^u\\,dv\\,du=5(e-1)$.", "has_image": true} {"id": "7d8122cf3449f8ea10937b89", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Find the surface area of the portion of the plane $z=2x+y$ whose vertical projection onto the horizontal coordinate plane is the shaded disk.\n", "options": null, "answer": "$4\\pi\\sqrt6$.", "solution": "For a graph $z=g(x,y)$ the area element is $\\sqrt{1+g_x^2+g_y^2}\\,dA$. Here it equals $\\sqrt6\\,dA$. The pictured disk has radius $2$ and area $4\\pi$, giving $4\\pi\\sqrt6$.", "has_image": true} {"id": "4dfda6914498b96e57bca439", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "For $F(x,y,z)=(x,y,z)$, compute the outward flux through the five solid faces shown. The dashed top boundary does not represent an included face; the axis labels give the cube dimensions.\n", "options": null, "answer": "Outward flux $=2$.", "solution": "The full unit cube has divergence $\\nabla\\cdot F=3$, so its closed-surface flux is $3$. The omitted top face has outward normal $(0,0,1)$ and flux $\\iint z\\,dA=1$. Subtracting gives $3-1=2$.", "has_image": true} {"id": "ddd2217a2d7af907a5503ba9", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Evaluate $$\\int_C\\left(-\\frac{y}{x^2+y^2}\\,dx+\\frac{x}{x^2+y^2}\\,dy\\right)$$ along the directed spiral from $A$ to $B$. Follow the entire drawn path, including its full turn.\n", "options": null, "answer": "$5\\pi/2$.", "solution": "The differential form is the change in a continuously followed polar angle. The arrows show one complete counterclockwise turn followed by a quarter-turn, so the unwrapped angle change is $2\\pi+\\pi/2=5\\pi/2$. Endpoint principal angles alone would omit the full turn.", "has_image": true} {"id": "a898b1da55e806cf3f602c62", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "For $F=(-y,z,x)$, evaluate $\\oint_C F\\cdot dr$ around the spatial triangle. Its arrow order is $A\\to B\\to C\\to A$.\n", "options": null, "answer": "$6$.", "solution": "The curl is $\\nabla\\times F=(-1,-1,1)$. The oriented vector area is $\\frac12(B-A)\\times(C-A)=(0,-3,3)$. Stokes' theorem therefore gives $(-1,-1,1)\\cdot(0,-3,3)=6$.", "has_image": true} {"id": "a57a28149af178c4ba0082a7", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The blue curve is the intersection of the two constraint surfaces labeled in the diagram. For $F(x,y,z)=xy+z^2$, find every constrained stationary point, its two Lagrange multipliers in $\\nabla F=\\lambda\\nabla g+\\mu\\nabla h$, and its constrained classification. Justify completeness and the classifications by restricting the quadratic form to the constraint plane.\n", "options": null, "answer": "Maxima: $(1,1,-2)$ and $(-1,-1,2)$, with $F=5$, $\\lambda=5/6$, and respective $\\mu=-2/3,2/3$. Minima: $(\\sqrt3,-\\sqrt3,0)$ and $(-\\sqrt3,\\sqrt3,0)$, with $F=-3$, $\\lambda=-1/2$, $\\mu=0$. There are no other stationary points.", "solution": "An orthonormal basis of the pictured plane is $u=(1,-1,0)/\\sqrt2$ and $v=(1,1,-2)/\\sqrt6$. Write a feasible point as $au+bv$. The spherical constraint becomes $a^2+b^2=6$, while $F=-a^2/2+5b^2/6$. The two distinct restricted eigenvalues imply that the only stationary directions are the two basis axes, giving the four points above. On the feasible circle, transfer of squared amplitude toward the larger eigenvalue strictly increases $F$, which proves the minimum and maximum classifications by second variation. Finally solve $(y,x,2z)=2\\lambda(x,y,z)+\\mu(1,1,1)$ at each point to obtain the listed multipliers. The two constraint gradients are independent everywhere on the circle.", "has_image": true} {"id": "589599b0fa836ae57bbad667", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The continuous function $f$ is exactly the piecewise-affine graph shown. Compute its second derivative as a distribution on $\\mathbb R$. Then use the distributional differentiation rule, rather than direct integration of both affine pieces, to obtain its Fourier transform under $\\widehat f(\\omega)=\\int_{\\mathbb R}f(x)e^{-i\\omega x}\\,dx$, including the value at zero frequency.\n", "options": null, "answer": "$$D^2f=\\delta_0-2\\delta_1+\\delta_2.$$ For $\\omega\\ne0$, $$\\widehat f(\\omega)=\\frac{2e^{-i\\omega}(1-\\cos\\omega)}{\\omega^2},\\qquad\\widehat f(0)=1.$$", "solution": "The slopes read from the graph are $0,1,-1,0$. Because $f$ is continuous, its first distributional derivative has no atoms. The jumps of its piecewise-constant first derivative are $1,-2,1$, giving the stated second derivative. Fourier transformation yields $-\\omega^2\\widehat f=1-2e^{-i\\omega}+e^{-2i\\omega}$. Factor the right-hand side as $-2e^{-i\\omega}(1-\\cos\\omega)$ to obtain the formula. Its removable limit at zero is one, agreeing with the area under the graph.", "has_image": true} {"id": "ec8c6ffd6ce2c4fb0285a818", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The blue map $g:[0,1]\\to[0,1]$ is a quadratic determined by its labeled zeros and vertex. For the iteration $x_{n+1}=g(x_n)$, find every orbit of prime period two and determine its linear stability. Also classify the nonzero fixed point. Derive the period-two equation and distinguish its roots from fixed points.\n", "options": null, "answer": "There is exactly one prime period-two orbit: $$\\left\\{\\frac{21-\\sqrt{21}}{32},\\frac{21+\\sqrt{21}}{32}\\right\\}.$$ Its multiplier is $4/25$, so it is attracting. The nonzero fixed point is $11/16$, with multiplier $-6/5$, so it is repelling.", "solution": "The plot determines $g(x)=rx(1-x)$ with $r=16/5$. Factor the iterate equation as $$g(g(x))-x=(g(x)-x)\\left[r^2x^2-r(r+1)x+(r+1)\\right].$$ The quadratic factor has roots $(21\\pm\\sqrt{21})/32$, neither fixed, and substitution shows that they exchange under $g$. Their product multiplier is $g\\prime(a)g\\prime(b)=-r^2+2r+4=4/25$. The fixed point $(r-1)/r$ instead has multiplier $2-r=-6/5$. The absolute multipliers give the stated stability conclusions.", "has_image": true} {"id": "e1c4e43ee6634f398d14172d", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "A function is analytic throughout the complex plane except at the red singularities, each of which is nonremovable. Find the radius of convergence of its Taylor expansion about the blue center.\n", "options": null, "answer": "$R=2$.", "solution": "The distances from the center $1$ to the singularities are $2$, $4$, and $\\sqrt{17}$. The first nonremovable singularity encountered is at distance $2$, so the Taylor radius is $2$.", "has_image": true} {"id": "60a8f8a2ace2f29a3f4d23ff", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "For a series $\\sum_{n=1}^{\\infty}a_n$, the graph gives the two subsequential limits of $|a_n|^{1/n}$. Decide whether the series can converge and justify your conclusion. The dashed limits are exact.\n", "options": null, "answer": "The series diverges; $\\limsup |a_n|^{1/n}=3/2>1$.", "solution": "The odd subsequence approaches $3/2$, while the even subsequence approaches $1/2$. Thus the limsup in the root test is $3/2$. In particular, along sufficiently large odd indices the absolute terms grow rather than approach zero, so the series cannot converge.", "has_image": true} {"id": "95268192ac92a89ab64eb3c4", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "A function is defined on a punctured neighborhood of the origin. Its exact values on the two highlighted paths are given in the picture. Can it have a limit at the origin? Prove your answer using only information that is actually supplied.\n", "options": null, "answer": "No; the limit does not exist.", "solution": "Along the upper parabola the function has constant value $1/2$, so any full limit would have to equal $1/2$. Along the lower parabola it has constant value $-1/2$, forcing a different limit. Both paths approach the puncture, so this contradiction proves nonexistence. No assumption about the function away from these paths is needed.", "has_image": true} {"id": "e0ef04bda0fd96af4227ba8b", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "For each integer $n\\ge2$, let $f_n$ on $[0,1]$ be the triangular function specified by the labeled vertices and the zero segment in the picture. Find its pointwise limit. Is convergence uniform?\n", "options": null, "answer": "The pointwise limit is $0$, but convergence is not uniform.", "solution": "At $x=0$ every value is zero. For each fixed $x>0$, eventually $2/n", "options": null, "answer": "$\\chi=0$; the surface is nonorientable and is a Klein bottle.", "solution": "All corners become one vertex, the two edge pairs give two edges, and the square gives one face. Hence $\\chi=1-2+1=0$. One gluing pair reverses the transverse orientation while the other does not, preventing a consistent surface orientation. The resulting nonorientable closed surface of Euler characteristic zero is the Klein bottle.", "has_image": true} {"id": "ddcc67d238206db6beb207ea", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The diagram specifies a linear operator $A$ on the plane through its action on two independent invariant directions. Compute $A^{10}w$ for the marked vector $w$, without multiplying the matrix repeatedly.\n", "options": null, "answer": "$A^{10}w=(59050,59048)^T$.", "solution": "The picture gives $w=v_1+v_2$, with eigenvalues $3$ and $-1$. Thus $A^{10}w=3^{10}v_1+(-1)^{10}v_2=59049(1,1)+(1,-1)=(59050,59048)$.", "has_image": true} {"id": "ecd80c632c43a475e20f3032", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The nonzero chain vectors in the picture form a basis of the vector space, and every arrow denotes application of the displayed operator $N$. Determine the minimal polynomial of $A$, the rank of $N$, and $\\dim\\ker N^2$.\n", "options": null, "answer": "$m_A(t)=(t-2)^3$, $\\operatorname{rank}N=3$, and $\\dim\\ker N^2=4$.", "solution": "The two chains have lengths $3$ and $2$, so the Jordan blocks for eigenvalue $2$ have those sizes. The longest chain gives the minimal-polynomial exponent $3$. Each block contributes its size minus one to the rank, giving $2+1=3$. Each contributes the smaller of its size and $2$ to $\\dim\\ker N^2$, giving $2+2=4$.", "has_image": true} {"id": "542b8110693d26c97aa1df2f", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "On $L^2([0,2],w(x)\\,dx)$ use the positive weight in the figure and the inner product $\\langle f,g\\rangle_w=\\int_0^2w(x)f(x)g(x)\\,dx$. Find the orthogonal projection of $x^2$ onto $V=\\operatorname{span}\\{1,x\\}$ and the exact squared approximation error. Establish existence and uniqueness using the Gram matrix and verify the residual orthogonality equations.\n", "options": null, "answer": "$$P_V(x^2)=-\\frac{67}{78}+\\frac{28}{13}x,\\qquad\\|x^2-P_V(x^2)\\|_w^2=\\frac{193}{585}.$$", "solution": "The pictured weight gives moments $m_0=4$, $m_1=5$, $m_2=22/3$, $m_3=23/2$, $m_4=94/5$. For $p=a+bx$, residual orthogonality is $$\\begin{pmatrix}4&5\\\\5&22/3\\end{pmatrix}\\begin{pmatrix}a\\\\b\\end{pmatrix}=\\begin{pmatrix}22/3\\\\23/2\\end{pmatrix}.$$ The Gram determinant is $13/3>0$, proving uniqueness. Solving gives the stated coefficients. Finally $\\|x^2-p\\|_w^2=m_4-(a,b)(m_2,m_3)^T=193/585$. The normal equations are precisely the two required orthogonality identities.", "has_image": true} {"id": "3d9e3dd609a1d59dc468bd29", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The three labeled vectors are the edges meeting at one vertex of the pictured parallelepiped. Find its volume and decide whether the ordered edge triple is positively oriented in the standard coordinate system.\n", "options": null, "answer": "Volume $3$; the ordered triple is negatively oriented.", "solution": "The determinant with the displayed vectors as columns is $$\\det\\begin{pmatrix}1&0&1\\\\0&2&1\\\\1&1&0\\end{pmatrix}=-3.$$ Its absolute value gives the volume, and its sign gives the orientation.", "has_image": true} {"id": "1b8d16caf8318f2f9ce354a1", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "An invertible real linear map sends the unit disk to the displayed ellipse. Find its Euclidean operator norm, the Euclidean operator norm of its inverse, and the absolute value of its determinant. Can the determinant sign be inferred from these two filled regions alone?\n", "options": null, "answer": "$\\|A\\|_2=4$, $\\|A^{-1}\\|_2=1/2$, and $|\\det A|=8$. The sign is not determined.", "solution": "The ellipse semiaxes are the singular values, namely $4$ and $2$. The largest gives $\\|A\\|_2$, the reciprocal of the smallest gives the inverse norm, and their product is $|\\det A|$. Precomposing with a reflection changes the determinant sign but leaves the image of the disk unchanged.", "has_image": true} {"id": "741c6c2b7c58c748e7ac53ec", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Let $B$ be the vertex-edge incidence matrix of the directed network, with $-1$ at an edge's tail and $+1$ at its head. Find $\\dim\\ker B$ and one basis for the space of flows conserving mass at every vertex. Use the pictured edge order.\n", "options": null, "answer": "$\\dim\\ker B=2$; one basis is $(1,1,1,0)^T,(0,0,1,1)^T$.", "solution": "The connected graph has three vertices and four edges, so its incidence rank is $2$ and its nullity is $4-2=2$. The first displayed basis vector circulates around the triangle; the second circulates around the two oppositely directed edges between $A$ and $C$. Both conserve mass, and they are independent.", "has_image": true} {"id": "46980e00c4e473a8139bc9d5", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "A real homogeneous quadratic form on the plane has the labeled values and level sets shown. Determine its positive and negative indices, and give an invertible linear coordinate change reducing it to a difference of squares.\n", "options": null, "answer": "The inertia is $(1,1,0)$; $u=x$, $v=\\sqrt2\\,y$ gives $q=u^2-v^2$.", "solution": "Write $q=ax^2+2bxy+cy^2$. The three exact values yield $a=1$, $c=-2$, and $b=0$. Thus there is one positive and one negative squared direction and no null direction. Rescaling the vertical variable produces the stated canonical form.", "has_image": true} {"id": "d1188f00eae64047c775560b", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Interpret the diagram as a matrix: an edge between $r_i$ and $c_j$ carries entry $a_{ij}$, and an absent edge means zero. Compute the determinant by identifying the nonzero permutation products and their signs.\n", "options": null, "answer": "$\\det A=58$.", "solution": "The only nonzero perfect matchings are the identity matching, with product $1\\cdot3\\cdot6=18$, and the cyclic matching $(r_1c_2,r_2c_3,r_3c_1)$, with product $2\\cdot4\\cdot5=40$. Both permutations are even, so the determinant is $18+40=58$.", "has_image": true} {"id": "ac7c9f493b39f5785b2d68bf", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "On the pictured triangular element $T$, let $N_i$ be the affine nodal basis satisfying $N_i(P_j)=\\delta_{ij}$. For the diffusion tensor $D=\\begin{pmatrix}2&1\\\\1&3\\end{pmatrix}$ and constant source $f=6$, derive the element stiffness matrix $K_{ij}=\\int_T(\\nabla N_i)^TD\\nabla N_j\\,dA$ and load vector $b_i=\\int_TfN_i\\,dA$. Explain why the local stiffness matrix is singular although the diffusion tensor is positive definite.\n", "options": null, "answer": "$$K=\\begin{pmatrix}9/2&-1&-7/2\\\\-1&1/2&1/2\\\\-7/2&1/2&3\\end{pmatrix},\\qquad b=\\begin{pmatrix}2\\\\2\\\\2\\end{pmatrix}.$$ Its rank is $2$, with kernel $\\operatorname{span}\\{(1,1,1)^T\\}$.", "solution": "The triangle has area one. Its nodal basis is $N_1=1-x/2-y$, $N_2=x/2$, $N_3=y$, so the gradient columns are $(-1/2,-1)^T$, $(1/2,0)^T$, $(0,1)^T$. With these columns in $G$, constant gradients give $K=G^TDG$. Each basis function integrates to one third of the area, yielding $b_i=2$. Since $D$ is positive definite, zero element energy means a zero spatial gradient, hence a constant finite-element function. Its nodal coefficient vector is a multiple of $(1,1,1)^T$, proving exactly the stated nullspace.", "has_image": true} {"id": "f231d9169e1b23b6ea6a50f1", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The unit axes in the figure define right-handed rotations $R_a(\\varepsilon)=\\exp(\\varepsilon\\widehat a)$ and $R_b(\\varepsilon)=\\exp(\\varepsilon\\widehat b)$, where $\\widehat a v=a\\times v$. For $C(\\varepsilon)=R_a(\\varepsilon)R_b(\\varepsilon)R_a(-\\varepsilon)R_b(-\\varepsilon)$, compute $\\lim_{\\varepsilon\\to0}(C(\\varepsilon)-I)/\\varepsilon^2$. Give the leading rotation axis and angle, and justify the product expansion through second order.\n", "options": null, "answer": "$$\\lim_{\\varepsilon\\to0}\\frac{C(\\varepsilon)-I}{\\varepsilon^2}=\\frac12\\begin{pmatrix}0&-1&-1\\\\1&0&-1\\\\1&1&0\\end{pmatrix}.$$ The leading axis is $(1,-1,1)/\\sqrt3$ and the leading angle is $(\\sqrt3/2)\\varepsilon^2$.", "solution": "Set $A=\\widehat a$, $B=\\widehat b$. Multiplying $e^{\\varepsilon A}=I+\\varepsilon A+\\varepsilon^2A^2/2+O(\\varepsilon^3)$ and the corresponding other three expansions cancels the linear terms and leaves $C=I+\\varepsilon^2(AB-BA)+O(\\varepsilon^3)$. The cross-product identity gives $[\\widehat a,\\widehat b]=\\widehat{a\\times b}$. From the pictured axes, $a\\times b=(1,-1,1)/2$, whose norm is $\\sqrt3/2$. Its unit direction and scaled norm give the leading axis and angle.", "has_image": true} {"id": "9c891ab30ed8642c6a01165a", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The first three arrows are the columns of $A$; the red arrow is $b$. Decide whether $Ax=b$ is solvable. Give a nonzero vector $w$ satisfying $w^TA=0$ that certifies your conclusion.\n", "options": null, "answer": "No solution. A certificate is $w=(-1,-1,1)^T$, for which $w^Tb=-1$.", "solution": "Every displayed column lies in the plane $z=x+y$. The vector $w=(-1,-1,1)$ is perpendicular to that plane, so $w^TA=0$. But $w^Tb=-1\\ne0$, which would contradict $w^TAx=w^Tb$ if a solution existed.", "has_image": true} {"id": "f39eedf9b6a3cba18508eb34", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The graph and its branch labels define a proper convex function on the whole real line. Compute its Fenchel conjugate $f^*(p)=\\sup_{x\\in\\mathbb R}\\{px-f(x)\\}$ for every real $p$, and determine the entire maximizing set for each $p$. Find $\\partial f(0)$ and verify the biconjugacy identity $f^{**}=f$ by exhibiting a Fenchel--Young equality pair for every $x$.\n", "options": null, "answer": "$$f^*(p)=\\begin{cases}\\frac{p^2}{2},&p<0,\\\\ 0,&0\\leq p\\leq1,\\\\ \\frac{(p-1)^2}{4},&p>1.\\end{cases}$$ The unique maximizer is $x=p$ for $p<0$, $x=0$ for $0\\leq p\\leq1$, and $x=(p-1)/2$ for $p>1$. Also $\\partial f(0)=[0,1]$ and $f^{**}=f$.", "solution": "The left and right derivatives at the origin are zero and one, giving the subdifferential interval. On the negative half-line, the stationary condition for $px-x^2/2$ is $x=p$, admissible only for $p<0$. On the positive half-line, it is $x=(p-1)/2$, admissible only for $p>1$. For intermediate slopes the subgradient condition forces the cusp $x=0$. Substitution gives the three conjugate branches; strict convexity of $f$ makes each maximizing set a singleton. For $x<0$ choose $p=x$, for $x>0$ choose $p=2x+1$, and for $x=0$ choose any $p\\in[0,1]$. Each choice satisfies $f(x)+f^*(p)=px$. The universal Fenchel--Young inequality supplies the opposite bound, proving $f^{**}=f$.", "has_image": true} {"id": "04eaf29c395032d2753c9f17", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Let $A=uv^T$, where the two vectors are read from the figure. Compute its Moore--Penrose pseudoinverse.\n", "options": null, "answer": "$A^+=\\frac1{25}\\begin{pmatrix}2&4\\\\-1&-2\\end{pmatrix}$.", "solution": "For nonzero column vectors, $(uv^T)^+=vu^T/(\\|u\\|^2\\|v\\|^2)$. Both squared norms equal $5$, so the denominator is $25$. Substitution produces the displayed matrix. The two triple-product identities and symmetry of $AA^+$ and $A^+A$ verify the Moore--Penrose conditions.", "has_image": true} {"id": "29cf0413bda9541277b1c0c0", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Find the natural cubic spline interpolating the marked data. Give both polynomial pieces and its bending energy $J(s)=\\int(s\\prime\\prime)^2\\,dx$ over the displayed interval. Prove that it uniquely minimizes this energy among all $H^2$ functions interpolating those data; a polynomial fit alone is not a sufficient justification.\n", "options": null, "answer": "$$s(x)=\\begin{cases}(5x-x^3)/4,&0\\leq x\\leq1,\\\\ (3-x)-(3-x)^3/8,&1\\leq x\\leq3.\\end{cases}\\qquad J(s)=9/4.$$ It is the unique minimum-energy interpolant.", "solution": "Let $M_i$ denote the second derivative at each knot. The natural endpoint conditions give $M_0=M_2=0$. With step lengths $1$ and $2$, the interior spline equation is $6M_1=6[(-1)/2-1]=-9$, so $M_1=-3/2$. Standard integration of the linearly interpolated second derivative gives the stated pieces, which agree through their second derivatives. Their squared second derivatives integrate to $3/4+3/2=9/4$. For any other admissible interpolant write $g=s+h$, where $h$ vanishes at all knots. Piecewise integration by parts twice shows $\\int s\\prime\\prime h\\prime\\prime=0$: natural conditions remove endpoint derivative terms, continuity of $s\\prime\\prime$ cancels the interior derivative terms, and the knot values of $h$ remove the remaining terms. Thus $J(g)=J(s)+J(h)\\geq J(s)$. Equality forces $h$ to be affine, and its zero knot values force $h=0$.", "has_image": true} {"id": "2db58c4773e1a720bf148db7", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The arrows specify a permutation of the labeled symbols. Let $P$ be its permutation matrix. Find the smallest positive integer $k$ such that $P^k=I$ and determine $\\det P$.\n", "options": null, "answer": "$k=6$ and $\\det P=-1$.", "solution": "The picture has disjoint cycles of lengths $3$ and $2$. The order is their least common multiple, $6$. Their signs multiply to $(-1)^{3-1}(-1)^{2-1}=-1$.", "has_image": true} {"id": "74abfe10da960cc0dfb07da1", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The diagram is the complete phase line for a continuously differentiable autonomous equation $y\\prime=F(y)$. Classify each equilibrium as stable or unstable and find the forward limit of the solution starting at $y(0)=1$.\n", "options": null, "answer": "The equilibria $-1$ and $2$ are asymptotically stable; $0$ is unstable. The indicated solution tends to $2$.", "solution": "Arrows point toward $-1$ from both sides and toward $2$ from both sides, establishing their local asymptotic stability. They point away from $0$, so it is unstable. A solution starting between $0$ and $2$ increases, cannot cross an equilibrium by uniqueness, and must approach $2$.", "has_image": true} {"id": "f7bd8cc80ec8ebb3dad8443e", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "For the right-hand side in the graph, consider $y\\prime=F(y)$ with $y(0)=0$ on nonnegative time. Does this initial-value problem have a unique solution? Exhibit two distinct solutions and explain the relevant uniqueness hypothesis.\n", "options": null, "answer": "It is not unique. Examples are $y(t)=0$ and $y(t)=t^2/4$ for $t\\ge0$.", "solution": "Both functions satisfy the initial condition and the differential equation. More generally, a solution may stay at zero until any time $c\\ge0$ and then follow $(t-c)^2/4$. The pictured cusp is continuous but not locally Lipschitz in $y$ at zero, so the standard local uniqueness theorem does not apply.", "has_image": true} {"id": "c39ad01e84dfffed20228959", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The picture gives both invariant directions and their eigenvalues for a planar linear system. Characterize all initial points whose trajectories converge to the origin as time tends to infinity. Classify the equilibrium.\n", "options": null, "answer": "Exactly the points with $x_0+y_0=0$ converge to the origin. The origin is a saddle.", "solution": "Decompose an initial point into components along $(1,1)$ and $(1,-1)$. The first component grows as $e^{2t}$, while the second decays as $e^{-t}$. Convergence is therefore equivalent to zero coefficient in the first direction, namely $x_0+y_0=0$. Opposite eigenvalue signs give a saddle.", "has_image": true} {"id": "ce787641cf900947c7b27bf2", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Consider the nonzero eigenfunctions of $-\\phi\\prime\\prime=\\lambda\\phi$ on the rod, with the boundary conditions and length shown. Derive the equation for positive eigenvalues. Determine exactly when a zero eigenvalue occurs and characterize the existence and number of negative eigenvalues as the real parameter $\\alpha$ varies. Justify the threshold with both the boundary equation and a Rayleigh-quotient argument.\n", "options": null, "answer": "Positive eigenvalues are $\\lambda=k^2$ with $k>0$ satisfying $k\\cos(2k)+\\alpha\\sin(2k)=0$. A zero eigenvalue occurs exactly at $\\alpha=-1/2$, with $\\phi(x)=Cx$. Exactly one negative eigenvalue exists if $\\alpha<-1/2$, and none exists otherwise. It is $-\\kappa^2$, where $\\kappa>0$ uniquely solves $\\kappa\\coth(2\\kappa)=-\\alpha$.", "solution": "The left condition selects $\\sin(kx)$ for positive eigenvalues, $x$ for zero, and $\\sinh(\\kappa x)$ for negative eigenvalues. Substitution at the other endpoint gives the stated three equations. The function $\\kappa\\coth(2\\kappa)$ increases strictly from $1/2$ to infinity, so the negative equation has exactly the asserted number of solutions. Integration by parts gives the quadratic form $\\int_0^2|\\phi\\prime|^2\\,dx+\\alpha|\\phi(2)|^2$. The trace estimate $|\\phi(2)|^2\\leq2\\int_0^2|\\phi\\prime|^2\\,dx$ makes this nonnegative when $\\alpha\\geq-1/2$; for nonnegative $\\alpha$ this is immediate, while for negative $\\alpha$ the trace bound gives the result. Below the threshold the test function $\\phi=x$ has form value $2+4\\alpha<0$. This matches the boundary-equation classification.", "has_image": true} {"id": "642aea44a971d1adeadc85f1", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The pictured curve solves $y\\prime=ry(1-y/K)$ with positive constants $r,K$. Use the exact marked data and asymptote to find $r$ and the first time the population reaches $4$.\n", "options": null, "answer": "$r=\\log2$ and the requested time is $t=2$.", "solution": "The asymptote gives $K=6$. The initial point gives $y(t)=6/(1+2e^{-rt})$. Substitution of the second point yields $3=6/(1+2e^{-r})$, so $e^{-r}=1/2$. Setting $y=4$ then gives $2e^{-rt}=1/2$, hence $t=2$.", "has_image": true} {"id": "b8cda8255e7f4cd2786960a3", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The system satisfies $y\\prime+2y=u(t)$ with $y(0)=0$, where the forcing is shown. Find the Laplace transform $Y(s)$ and a time-domain expression using Heaviside functions.\n", "options": null, "answer": "$$Y(s)=\\frac{3(e^{-s}-e^{-3s})}{s(s+2)}.$$ $$y(t)=\\frac32\\big[(1-e^{-2(t-1)})H(t-1)-(1-e^{-2(t-3)})H(t-3)\\big].$$", "solution": "The plot gives $u(t)=3[H(t-1)-H(t-3)]$. Thus $U(s)=3(e^{-s}-e^{-3s})/s$ and $(s+2)Y(s)=U(s)$. The inverse of $1/(s(s+2))$ is $(1-e^{-2t})/2$; applying the two delays gives the response.", "has_image": true} {"id": "c45ca9aab8abd2a2f21ad343", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "For $y\\prime=-2y$, use the implicit trapezoidal method starting at the plotted initial point and taking the two time steps shown. Compute both new approximations.\n", "options": null, "answer": "$y_1=1/3$ and $y_2=1/9$.", "solution": "The method is $y_{n+1}=y_n+(h/2)(-2y_n-2y_{n+1})$, so $(1+h)y_{n+1}=(1-h)y_n$. The diagram gives $h=1/2$ and $y_0=1$, hence each step multiplies the value by $1/3$.", "has_image": true} {"id": "dbd9460e0e18c8fe17b9271f", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The rod satisfies $u_t=u_{xx}$ on $0", "options": null, "answer": "$u(x,t)\\sim (8/\\pi^2)e^{-t}\\sin x$ for interior fixed $x$.", "solution": "The separated modes are $e^{-n^2t}\\sin(nx)$. The first coefficient is $$b_1=\\frac2\\pi\\left[\\int_0^{\\pi/2}\\frac{2x}{\\pi}\\sin x\\,dx+\\int_{\\pi/2}^{\\pi}\\left(2-\\frac{2x}{\\pi}\\right)\\sin x\\,dx\\right]=\\frac8{\\pi^2}.$$ It is nonzero and decays slower than every higher mode.", "has_image": true} {"id": "964db9b25b8ca3bc59ccb191", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The initial data for $u_t+2u_x=0$ on the real line are shown, including the endpoint conventions. Give the profile at $t=1$ and evaluate $u(7/2,1)$.\n", "options": null, "answer": "$u(x,1)=2$ on $[2,3)$, $u(x,1)=-1$ on $[3,4)$, and $0$ elsewhere. In particular $u(7/2,1)=-1$.", "solution": "Characteristics have speed $2$, so $u(x,t)=u(x-2t,0)$. Translate the whole pictured initial profile two units to the right. The argument at the requested point is $7/2-2=3/2$, which lies on the negative plateau.", "has_image": true} {"id": "007040de075e9199171be1ba", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The graph gives all local equilibria of a scalar parameter-dependent differential equation. Identify the bifurcation, state the number of stable equilibria on each side of its critical parameter, and give one simple scalar equation realizing this diagram.\n", "options": null, "answer": "A saddle-node bifurcation occurs at $\\lambda=0$. There are no local equilibria for $\\lambda<0$ and one stable equilibrium for $\\lambda>0$. A realization is $y\\prime=y^2-\\lambda$.", "solution": "The two branches are born together at the turning point. For the proposed equation, equilibria satisfy $y^2=\\lambda$. At the negative branch the derivative with respect to $y$ is $2y<0$, while at the positive branch it is positive. This matches both the locations and the stability styles.", "has_image": true} {"id": "0387f898f6a497d500390e19", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Evaluate $$\\oint_C\\frac{e^z}{z^2(z-3)}\\,dz,$$ where $C$ is the directed circle in the picture. The red marks show the poles.\n", "options": null, "answer": "$-8\\pi i/9$.", "solution": "Only the double pole at the origin is enclosed. Its residue is $$\\left.\\frac{d}{dz}\\frac{e^z}{z-3}\\right|_{z=0}=-\\frac13-\\frac19=-\\frac49.$$ The residue theorem multiplies this by $2\\pi i$, giving $-8\\pi i/9$.", "has_image": true} {"id": "cd6534b73d42ad2b120ed94c", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "A meromorphic function has no zeros or poles on a positively oriented simple boundary $C$. Inside $C$ it has poles of total multiplicity $3$. The figure is the complete image walk $f(C)$ in the value plane. How many zeros lie inside $C$, counted with multiplicity?\n", "options": null, "answer": "$5$ zeros counted with multiplicity.", "solution": "Each of the two counterclockwise image loops encloses the origin, so the image walk has winding number $2$. By the argument principle, $N-P=2$. With $P=3$, the zero count is $N=5$.", "has_image": true} {"id": "96d3b3b27c8181a609064680", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "An analytic function $f=u+iv$ has the real-part gradient shown. Determine the complex derivative $f\\prime(P)$ and the greatest directional derivative of $u$ over unit directions at $P$.\n", "options": null, "answer": "$f\\prime(P)=2+3i$; the greatest directional derivative is $\\sqrt{13}$.", "solution": "The Cauchy--Riemann equations give $f\\prime=u_x-iu_y=2+3i$. The greatest unit-direction derivative of the real part is the gradient norm, $\\sqrt{2^2+(-3)^2}=\\sqrt{13}$.", "has_image": true} {"id": "b56f3b98c6a924be4800de5b", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "A Schwarz--Christoffel map $F$ takes the upper half-plane conformally onto the interior of the polygon in the diagram. Its real prevertices $a_1<\\cdots", "options": null, "answer": "$$F\\prime(z)=C\\,\\frac{(z-a_4)^{1/2}}{\\prod_{j\\ne4}(z-a_j)^{1/2}},\\qquad C\\ne0.$$ Its square has a simple zero at $a_4$ and simple poles at the other prevertices. Locally $F(z)-F(a_4)\\sim K(z-a_4)^{3/2}$ with $K\\ne0$.", "solution": "For an interior angle $\\beta_j\\pi$, the Schwarz--Christoffel derivative has factor $(z-a_j)^{\\beta_j-1}$. The five convex right angles have $\\beta_j=1/2$, while the pictured concave corner has $\\beta_4=3/2$. These give the derivative formula, and squaring makes every local order integral. Integrating the local half-power at $a_4$ produces exponent $3/2$ for the map itself. That exponent expands the upper half-plane angle into the reentrant angle rather than the convex right angles. The exponents sum to $-2$, consistent with infinity being a regular boundary point.", "has_image": true} {"id": "b022593894dfc293b8c0115f", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Start with the logarithm value $\\log1=0$ and analytically continue it along the entire directed path shown, avoiding the origin. What value is obtained at the endpoint?\n", "options": null, "answer": "$3\\pi i$.", "solution": "Along a continuation path, the logarithm is $\\log|z|+i\\theta$ with a continuous unwrapped argument. The path makes one full counterclockwise turn and then a half-turn, so the endpoint argument is $3\\pi$. Its endpoint modulus is $1$, giving $3\\pi i$.", "has_image": true} {"id": "7db6e71594949d526684003a", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Use Rouche's theorem to count the zeros of $p(z)=z^5+3z+1$ inside the circle shown, including multiplicity.\n", "options": null, "answer": "All $5$ zeros lie inside.", "solution": "On the circle, $|z^5|=(3/2)^5=243/32$, whereas $|3z+1|\\le3(3/2)+1=11/2=176/32$. The strict inequality allows comparison with $z^5$, which has five zeros inside.", "has_image": true} {"id": "53231635504c896208c7836d", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Under $w=(z-1)/z$, find the image of the pictured boundary circle in the extended complex plane and the image of its open interior.\n", "options": null, "answer": "The boundary maps to $\\operatorname{Im}w=\\operatorname{Re}w$ together with infinity; the interior maps to $\\operatorname{Im}w>\\operatorname{Re}w$.", "solution": "The marked points map as $0\\mapsto\\infty$, $1\\mapsto0$, and $i\\mapsto1+i$. A generalized circle through infinity is a line, and these values determine the line of slope one. The center $(1+i)/2$ maps to $i$, which lies above that line, fixing the interior side.", "has_image": true} {"id": "b3acdd031ac15f414bbe189b", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Let $\\zeta_0,\\ldots,\\zeta_4$ be the five complex vertices in the picture. Compute $\\sum_{j=0}^4(2-\\zeta_j)^{-1}$ without separately evaluating the trigonometric coordinates.\n", "options": null, "answer": "$80/31$.", "solution": "The regular unit pentagon anchored at $1$ consists of the roots of $P(z)=z^5-1$. Its logarithmic derivative satisfies $P\\prime(z)/P(z)=\\sum_j1/(z-\\zeta_j)$. Evaluating at $z=2$ gives $5\\cdot2^4/(2^5-1)=80/31$.", "has_image": true} {"id": "7bd086ec765210aa937984fc", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The vector $(X,Y)$ is uniformly distributed over the union of the two shaded components. Define $U=X^2$ and $V=Y/X$. Derive the complete joint density of $(U,V)$, explicitly accounting for every inverse branch, and determine whether $U$ and $V$ are independent. Verify that the transformed density integrates to one.\n", "options": null, "answer": "$f_{U,V}(u,v)=1/6$ for $1", "options": null, "answer": "$c=3/2$ and the event probability is $1/16$.", "solution": "The support integral is $\\int_0^2\\int_0^{2-x}xy\\,dy\\,dx=2/3$, so $c=3/2$. On the smaller triangle the unnormalized integral is $\\int_0^1\\int_0^{1-x}xy\\,dy\\,dx=1/24$. Multiplication by $c$ gives $1/16$.", "has_image": true} {"id": "437b7a0a5bc5a2a48b59d406", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The three independent clocks have exponential waiting times with rates shown. Find the probability that clock $A$ rings first and clock $B$ rings second. Each clock is removed after it rings.\n", "options": null, "answer": "$3/40$.", "solution": "The chance that $A$ wins the initial race is $2/(2+3+5)=1/5$. By memorylessness, after that event the remaining clocks race with rates $3$ and $5$, so $B$ wins with probability $3/8$. Their product is $3/40$.", "has_image": true} {"id": "96634c287adde174059a18f4", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "A homogeneous Poisson process has unknown intensity $\\lambda$, with prior density $p(\\lambda)=\\lambda e^{-\\lambda}$ for $\\lambda>0$. Only the shaded time windows are observed, and the dots show every arrival in those windows. Derive the posterior density and the full posterior predictive mass function for the number $N$ of arrivals in a new unit-length interval. Also find the predictive mean and variance.\n", "options": null, "answer": "The posterior is Gamma with shape $6$ and rate $7$: $$p(\\lambda\\mid\\mathrm{data})=\\frac{7^6}{\\Gamma(6)}\\lambda^5e^{-7\\lambda}.$$ For $k\\geq0$, $$\\mathbb P(N=k\\mid\\mathrm{data})=\\binom{k+5}{5}\\left(\\frac78\\right)^6\\left(\\frac18\\right)^k.$$ Its mean is $6/7$ and variance $48/49$.", "solution": "The observed windows have lengths $2,1,3$ and contain a total of four arrivals. The unobserved gaps contribute no exposure, so the likelihood is proportional to $\\lambda^4e^{-6\\lambda}$. Combining with the prior yields the normalized Gamma density above. Integrating the Poisson mass $e^{-\\lambda}\\lambda^k/k!$ against that posterior gives the stated negative-binomial mass. By total expectation the predictive mean is $\\mathbb E[\\lambda]=6/7$. Total variance gives $\\mathbb E[\\lambda]+\\operatorname{Var}(\\lambda)=6/7+6/49=48/49$.", "has_image": true} {"id": "0c727f7b789487408715d2fc", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Start the Markov chain at state $1$. The double circles are absorbing states. Find the probability of eventual absorption at the rightmost state and the expected number of transitions until absorption.\n", "options": null, "answer": "Rightmost absorption probability $3/5$; expected transition count $14/5$.", "solution": "Let $h_j$ be the rightmost absorption probability. Then $h_1=(3/4)h_2$ and $h_2=(1/2)h_1+1/2$, yielding $h_1=3/5$. For expected absorption times, $e_1=1+(3/4)e_2$ and $e_2=1+(1/2)e_1$. Solving gives $e_1=14/5$.", "has_image": true} {"id": "04ea3174cc7268db42cef7fb", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "A homogeneous Poisson process is observed in the two colored windows. Conditional on exactly three events in their union, find the probability that exactly two occurred in $A$ and one in $B$. Events outside these windows are not counted.\n", "options": null, "answer": "$4/9$.", "solution": "Given the total count in disjoint windows, allocation is binomial with probabilities proportional to window lengths. Their lengths are $2$ and $1$, so the requested probability is $\\binom32(2/3)^2(1/3)=4/9$. The unobserved gap does not contribute to either count.", "has_image": true} {"id": "ea74d1664a9c48b285af7d27", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The graph is a cumulative distribution function; its nonhorizontal continuous piece is straight. Identify all atoms and their probabilities, and compute $\\mathbb E[X]$.\n", "options": null, "answer": "Atoms: $\\mathbb P(X=0)=1/5$ and $\\mathbb P(X=2)=2/5$. The mean is $1$.", "solution": "The jumps at the marked endpoints have sizes $1/5$ and $1-3/5=2/5$. The slope on $(0,1)$ is $2/5$, giving a continuous density there. Hence $\\mathbb E[X]=(2/5)\\int_0^1x\\,dx+2(2/5)=1$.", "has_image": true} {"id": "046abab2840aaa75c461e82e", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Let $X,Y$ be independent uniform random variables on $[0,1]$, and set $S=X+Y$. The highlighted diagonal specifies a threshold $s$. Find both $\\mathbb P(S\\le s)$ and the density $f_S(s)$ at that threshold.\n", "options": null, "answer": "$\\mathbb P(S\\le3/2)=7/8$ and $f_S(3/2)=1/2$.", "solution": "The excluded upper-right triangle has legs $1/2$, hence area $1/8$. This gives the cumulative probability $7/8$. More generally, for $1", "options": null, "answer": "$\\Sigma=\\begin{pmatrix}5&4\\\\4&5\\end{pmatrix}$, correlation $4/5$, and inside probability $1-e^{-1/2}$.", "solution": "The covariance eigenvalues are the squared semiaxes, $9$ and $1$, with the first eigenvector at angle $\\pi/4$. Rotating this diagonal matrix gives the stated covariance, hence correlation $4/5$. The squared Mahalanobis distance has a chi-square law with two degrees of freedom, so its probability of being at most $1$ is $1-e^{-1/2}$.", "has_image": true} {"id": "e650d3d5f6efab898527eb98", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Independent lifetimes have an exponential distribution with rate $\\lambda$. The diagram shows each observed duration and whether it ends in failure or independent right censoring. Derive the likelihood up to factors not involving $\\lambda$ and find its maximum-likelihood estimate.\n", "options": null, "answer": "$L(\\lambda)\\propto\\lambda^3e^{-15\\lambda}$; $\\widehat\\lambda=1/5$.", "solution": "Each observed failure contributes $\\lambda e^{-\\lambda t}$ and each right-censored observation contributes its survival probability $e^{-\\lambda t}$. The image gives three failures and total exposure $15$. Differentiating $3\\log\\lambda-15\\lambda$ yields $3/\\lambda-15=0$. Its second derivative is negative.", "has_image": true} {"id": "c8ec5ed201d743a7b1cac82d", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The mosaic labels are observed counts, not percentages. Test independence of group and outcome by calculating Pearson's chi-square statistic and its degrees of freedom. No continuity correction is to be used.\n", "options": null, "answer": "$\\chi^2=20$ with $1$ degree of freedom.", "solution": "Both group totals and both outcome totals are $40$, while the grand total is $80$. Every expected count under independence is $40\\cdot40/80=20$. Thus the statistic is $4(10^2/20)=20$, with $(2-1)(2-1)=1$ degree of freedom.", "has_image": true} {"id": "babce12827ead49099398852", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Use the paired measurements to calculate the t statistic for testing zero mean change, with change defined as after minus before. Give the degrees of freedom.\n", "options": null, "answer": "$t=2\\sqrt{10}$ with $4$ degrees of freedom.", "solution": "The within-person differences are $2,1,3,2,2$. Their mean is $2$, and their sample variance is $[0+1+1+0+0]/4=1/2$. Therefore $t=2/\\sqrt{(1/2)/5}=2\\sqrt{10}$, with one fewer degree of freedom than the number of pairs.", "has_image": true} {"id": "21c71de36f180bc291583998", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "For an ordinary least-squares line with an intercept fitted to the plotted observations, identify the point of greatest leverage and calculate its exact leverage. Do not confuse leverage with a large vertical residual.\n", "options": null, "answer": "The point with $x=8$ has greatest leverage, $h=87/97$.", "solution": "Leverage is $h_i=1/n+(x_i-\\bar x)^2/S_{xx}$. The graph gives $n=5$, $\\bar x=14/5$, and $S_{xx}=194/5$. The furthest horizontal coordinate from the mean is $8$, giving $h=1/5+(26/5)^2/(194/5)=87/97$.", "has_image": true} {"id": "ec7bbde3cca8e7bb8ba698da", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "A Galton--Watson process begins with one ancestor and has the offspring law shown. Derive its extinction probability. Conditional on eventual extinction, determine the offspring law of the resulting branching process and the expected total progeny, including the ancestor. Justify the conditioning formula using independence of descendant subtrees.\n", "options": null, "answer": "The extinction probability is $q=1/2$. The conditioned offspring probabilities at $0,1,2$ are $(1/2,1/4,1/4)$, and the conditional expected total progeny is $4$.", "solution": "The offspring generating function is $G(z)=1/4+z/4+z^2/2$, with mean $G\\prime(1)=5/4>1$. Extinction probability is the smallest solution of $G(q)=q$ in $[0,1]$; the roots are $1/2$ and one. Given that the ancestor has $k$ children, all descendant subtrees become extinct with probability $q^k$. Bayes conditioning therefore changes the offspring law to $\\widetilde p_k=p_kq^k/q$ and its generating function to $\\widetilde G(z)=G(qz)/q$. This gives $(1/2,1/4,1/4)$ and mean $3/4$. Summing the expected generation sizes of the conditioned subcritical process gives $\\sum_{n\\geq0}(3/4)^n=4$.", "has_image": true} {"id": "c699b7dec299ca41900a5a4d", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The arrows are jump rates of a continuous-time Markov chain, and $\\Delta$ is absorbing. Starting at state $0$, let $\\tau$ be the first hitting time of $\\Delta$. Construct the transient generator and use the backward equations to derive $\\mathbb E_0[e^{-s\\tau}]$ for $s\\geq0$. Compute the exact mean and variance of $\\tau$, checking them against derivatives of the transform.\n", "options": null, "answer": "$$\\mathbb E_0[e^{-s\\tau}]=\\frac{8s+29}{s^3+10s^2+31s+29},\\qquad\\mathbb E_0[\\tau]=\\frac{23}{29},\\qquad\\operatorname{Var}_0(\\tau)=\\frac{317}{841}.$$", "solution": "In transient-state order $(0,1,2)$, the diagram gives $$T=\\begin{pmatrix}-3&2&1\\\\1&-4&0\\\\0&1&-3\\end{pmatrix},\\qquad q=\\begin{pmatrix}0\\\\3\\\\2\\end{pmatrix}.$$ With $\\phi_i(s)=\\mathbb E_i[e^{-s\\tau}]$, first-jump conditioning gives $(sI-T)\\phi=q$, whose first component is the stated rational function. For moments, solve $(-T)m=\\mathbf1$ and $(-T)m^{(2)}=2m$. These give $m_0=23/29$ and $m_0^{(2)}=846/841$, so the variance is $846/841-(23/29)^2=317/841$. The same values follow from $-\\phi_0\\prime(0)$ and $\\phi_0\\prime\\prime(0)$.", "has_image": true} {"id": "16de30d86054856c0a9e33db", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "A random spanning tree is sampled with probability proportional to the product of its edge weights. The picture defines two variable weights $u,v>0$. Use the weighted matrix-tree theorem to derive its partition function $Z(u,v)$. Then set $u=4$, $v=2$ and compute the covariance of the indicators for inclusion of edges $AC$ and $CE$. Derive the marginal and joint inclusion formulas by differentiating the partition function, rather than assuming the two edge events are independent.\n", "options": null, "answer": "$$Z(u,v)=23uv+53u+40v+65.$$ At the stated weights, the inclusion probabilities for $AC$, $CE$, and both are $396/541$, $264/541$, and $184/541$, respectively. The covariance is $-5000/292681$.", "solution": "Deleting the row and column of $E$ from the weighted Laplacian gives $$\\begin{pmatrix}u+2&-1&-u&0\\\\-1&4&-2&-1\\\\-u&-2&u+v+5&-3\\\\0&-1&-3&6\\end{pmatrix}.$$ Its determinant is the stated partition polynomial. Because every tree contains either zero or one copy of each special edge, differentiation yields $P(AC)=uZ_u/Z$, $P(CE)=vZ_v/Z$, and $P(AC,CE)=uvZ_{uv}/Z$. At the specified weights, $Z=541$, giving the displayed probabilities. Subtracting the product of the marginals from the joint probability gives $-5000/292681$, exhibiting negative dependence.", "has_image": true} {"id": "9c14076df61aff00fbf9d83f", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Assume $t>0$. For which values of $t$ is the highlighted edge set $\\{AB,AC,CD\\}$ the unique minimum spanning tree? Describe what occurs at the boundary value.\n", "options": null, "answer": "It is uniquely optimal for $0", "options": null, "answer": "The shortest path is $A\\to C\\to B\\to D$ of weight $1$. The stated Dijkstra procedure can report $3$.", "solution": "The desired route has weight $2-3+2=1$. Dijkstra finalizes $B$ first at tentative distance $1$ and records distance $3$ to $D$. It next processes $C$ at distance $2$, but the negative edge would lower the already-finalized distance to $B$ to $-1$. Without reopening $B$, that improvement is not propagated to $D$.", "has_image": true} {"id": "98d29b206e7272a9fae04eef", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Find the maximum flow from $s$ to $t$ in the directed network. Give a feasible flow attaining the value and a cut certifying that no larger value is possible. Labels are capacities.\n", "options": null, "answer": "Maximum flow $5$. One flow uses $f_{sa}=3$, $f_{sb}=2$, $f_{ab}=1$, $f_{at}=2$, $f_{bt}=3$.", "solution": "The listed flow respects every capacity and conserves flow at the intermediate vertices. Its total value is $3+2=5$. The cut separating $s$ from all other vertices has capacity $3+2=5$, so the max-flow min-cut theorem proves optimality.", "has_image": true} {"id": "a57d638943bbfad954c38dc2", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Find the chromatic polynomial of the graph and the number of proper colorings with three labeled colors. Adjacent vertices must receive different colors.\n", "options": null, "answer": "$P(k)=(k-1)[(k-1)^5-(k-1)]$; there are $60$ proper three-colorings.", "solution": "The core is a cycle of length five, whose chromatic polynomial is $(k-1)^5-(k-1)$. Once that cycle is colored, the pendant vertex has $k-1$ choices. At $k=3$, this gives $2(32-2)=60$.", "has_image": true} {"id": "e0732e65969e93bc9b3a656b", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "A postman must start and finish at the same vertex and traverse every edge of the pictured network at least once. Repeated edges are allowed. Find the minimum possible total route length.\n", "options": null, "answer": "$22$.", "solution": "All four vertices have odd degree. The original total edge length is $18$. The possible shortest-path pairings of the odd vertices cost $2+2=4$, $3+3=6$, or $4+4=8$. Duplicating the edges in the cheapest pairing makes every degree even, so an Euler circuit of length $18+4=22$ exists and is optimal.", "has_image": true} {"id": "ca4c15982f8d43be492df365", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The upward edges are the cover relations of a partial order. Find its width and the number of linear extensions. Give enough detail to make the extension count checkable.\n", "options": null, "answer": "Width $2$; exactly $5$ linear extensions.", "solution": "The incomparable pair $B,C$ gives a lower bound of two for the width. The chains $A", "options": null, "answer": "Maximum size $3$; for example $\\{a1,b2,d3\\}$. The Hall witness is $\\{a,b,c\\}$ with neighborhood $\\{1,2\\}$.", "solution": "The first three left vertices can collectively use only two right vertices, so at most two of them can be matched. The remaining left vertex adds at most one more. The displayed matching reaches three. Since the specified subset has three elements but only two neighbors, Hall's condition fails.", "has_image": true} {"id": "c1597e726bb0859bb9efcefb", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "How many binary words of length five are accepted by the pictured deterministic automaton? Derive a recurrence from its state structure rather than listing every word.\n", "options": null, "answer": "$13$ words.", "solution": "The automaton accepts exactly words with no consecutive ones. Such a word either ends in a zero after a valid shorter word, or ends in $01$ after a valid word two places shorter, with the initial short cases treated separately. Thus $a_n=a_{n-1}+a_{n-2}$, $a_0=1$, $a_1=2$, giving $1,2,3,5,8,13$.", "has_image": true} {"id": "8e21c6add27a21df769b7c4f", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Let $B_t$ be standard Brownian motion. The blue points in the diagram are the only observations beyond the deterministic starting value, and the two red vertical lines mark target times. Derive the joint conditional distribution of the Brownian values at those target times, in increasing time order. Determine whether they are conditionally independent. Show the covariance-block calculation, using $\\operatorname{Cov}(B_s,B_t)=\\min(s,t)$.\n", "options": null, "answer": "The conditional law is $$\\mathcal N\\!\\left(\\begin{pmatrix}1\\\\1/2\\end{pmatrix},\\begin{pmatrix}1/4&0\\\\0&1/2\\end{pmatrix}\\right).$$ The two target values are conditionally independent.", "solution": "Let $X=(B_{1/2},B_2)^T$ and $Y=(B_1,B_3)^T$, with observed value $y=(2,-1)^T$. Brownian covariance gives $$K_{XX}=\\begin{pmatrix}1/2&1/2\\\\1/2&2\\end{pmatrix},\\quad K_{YY}=\\begin{pmatrix}1&1\\\\1&3\\end{pmatrix},\\quad K_{XY}=\\begin{pmatrix}1/2&1/2\\\\1&2\\end{pmatrix}.$$ The Gaussian conditioning formulas yield mean $K_{XY}K_{YY}^{-1}y=(1,1/2)^T$ and covariance $K_{XX}-K_{XY}K_{YY}^{-1}K_{YX}=\\operatorname{diag}(1/4,1/2)$. In a joint Gaussian conditional law, zero cross-covariance implies independence. The two target times lie in distinct bridge segments separated by an observed time.", "has_image": true} {"id": "d8b2b8c4f0940a38782569e7", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The two polygon interiors are distinct two-cells. Identify every corner to one vertex and identify all edges with the same letter in the direction of their arrows. Orient each two-cell counterclockwise in its displayed polygon. Compute $H_0$, $H_1$, and $H_2$ of the resulting CW complex with integer coefficients. Write the cellular boundary matrix and use Smith normal form, or an equivalent integral-generator argument, to identify any torsion.\n", "options": null, "answer": "$H_0\\cong\\mathbb Z$, $H_1\\cong\\mathbb Z\\oplus\\mathbb Z/2\\mathbb Z$, and $H_2=0$.", "solution": "Read the oriented attaching words as $aab^{-1}$ and $bbbc^{-1}c^{-1}$. With edge order $(a,b,c)$ and the displayed face order, $$\\partial_2=\\begin{pmatrix}2&0\\\\-1&3\\\\0&-2\\end{pmatrix},\\qquad\\partial_1=0.$$ The columns are independent, so $H_2=0$. The greatest common divisor of all entries is one, while the greatest common divisor of the nonzero two-by-two minors $6,-4,2$ is two. Thus the Smith invariant factors are $1,2$, leaving one free generator and one order-two torsion generator in the cokernel. Equivalently, the relations give $b=2a$ and $2(c-3a)=0$. The single vertex gives connectedness and $H_0\\cong\\mathbb Z$.", "has_image": true} {"id": "8745d9f35b2d6e225dd529f0", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "For this plane embedding, determine the number of faces, including the unbounded face. Then give the multiset of vertex degrees of the plane dual.\n", "options": null, "answer": "There are $5$ faces. The dual degree multiset is $\\{4,3,3,3,3\\}$.", "solution": "The connected graph has $V=5$ and $E=8$, so Euler's formula gives $F=2-V+E=5$. Four bounded faces are triangles, while the unbounded face has a four-edge boundary. Each dual vertex has degree equal to the corresponding face-boundary length.", "has_image": true} {"id": "478227f6f1b75978ccfc4d6d", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The row player maximizes and the column player minimizes the payoff in the table. Find the equilibrium mixed strategy of each player and the value of the game.\n", "options": null, "answer": "Row probabilities $(1/3,2/3)$, column probabilities $(1/2,1/2)$, game value $1$.", "solution": "If the row player uses the first row with probability $p$, the two column payoffs are $3p$ and $2-3p$. Equalizing gives $p=1/3$ and value $1$. If the first column has probability $q$, the row payoffs are $4q-1$ and $2-2q$, giving $q=1/2$. Both probabilities are interior, so mutual indifference verifies equilibrium.", "has_image": true} {"id": "9870f181d6bdb13e682144b0", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The figure is a Cayley diagram of a finite group; the marked identity is $e$. Starting at any vertex, blue and red arrows are right multiplication by $r$ and $t$, respectively. Find the order of $rt$ and list the subgroup it generates using vertex labels.\n", "options": null, "answer": "The order is $3$, and the subgroup is $\\{e,4,2\\}$.", "solution": "To right multiply by $rt$, follow a blue edge and then a red edge. Starting at the identity, successive applications reach vertices $4$, $2$, and then $e$. No earlier return occurs, so the order is three and these are precisely the subgroup elements.", "has_image": true} {"id": "616360ca116d49cd8549c9b6", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Treat every drawn vertex and edge as a simplex, and include exactly the shaded triangles as two-dimensional simplices. Compute the Euler characteristic and the dimension of the first homology group over the real numbers.\n", "options": null, "answer": "$\\chi=0$ and $\\dim H_1=1$.", "solution": "There are six vertices, eight edges, and two filled faces, so $\\chi=6-8+2=0$. The two shaded triangles form a contractible disk. The unfilled triangular boundary is a single loop attached at one vertex. The complex is connected and has no two-dimensional homology, hence $0=1-\\dim H_1$, giving one.", "has_image": true} {"id": "f102e6d70aa2c8ae849dc9d7", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Let $f\\in C^2([1,2])$ satisfy the exact value and derivative bounds annotated in the figure; the displayed curve illustrates these conditions and no polynomial formula is assumed. Starting at the marked $x_0$, apply Newton iteration. Prove that all iterates stay in the interval and converge to the unique root $\\alpha$. Derive an explicit bound for $|x_n-\\alpha|$ and find the smallest $n$ certified by that bound to give error at most $10^{-12}$.\n", "options": null, "answer": "The iterates decrease to $\\alpha$ and remain in $[\\alpha,3/2]$. A valid error certificate is $$|x_n-\\alpha|\\leq2\\left(\\frac1{16}\\right)^{2^n}.$$ The smallest iteration index certified by this bound is $n=4$.", "solution": "The sign values and $f\\prime\\geq2$ give a unique root between the left endpoint and $x_0$. Convexity places the tangent zero at any point to the right of the root between that root and the point itself. Thus Newton iterates stay in the stated invariant interval, decrease, and converge to the root. The mean value theorem gives $e_0\\leq f(x_0)/2=1/8$. Taylor expansion about $x_n$, evaluated at the root, gives $$0\\leq e_{n+1}\\leq\\frac{\\sup f\\prime\\prime}{2\\inf f\\prime}e_n^2\\leq\\frac12e_n^2.$$ Hence $e_n/2\\leq(e_0/2)^{2^n}\\leq16^{-2^n}$. At $n=3$ the bound is $2^{-31}>10^{-12}$, whereas at $n=4$ it is $2^{-63}<10^{-12}$.", "has_image": true} {"id": "f20e2407f83b2b45db793798", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Use normalized binary floating-point arithmetic with three significant bits, unbounded exponent range, and rounding to nearest with ties to even. The image gives the operands and neighboring representable numbers. Evaluate $\\operatorname{fl}(\\operatorname{fl}(a+b)+c)$ and $\\operatorname{fl}(a+\\operatorname{fl}(b+c))$.\n", "options": null, "answer": "The two results are $0$ and $1$, respectively.", "solution": "The exact intermediate sum $a+b=9$ is halfway between $8$ and $10$. Ties-to-even chooses $8$, so the first expression becomes $8-8=0$. In the other grouping, $b+c=-7$ is exactly representable, as is $8-7=1$.", "has_image": true} {"id": "d468733dcef9ab3b2b02014d", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Using every sample in the image, approximate the integral over the full sampled interval by the composite Simpson rule. State the grid spacing and the resulting approximation.\n", "options": null, "answer": "$h=1$ and the approximation is $23/3$.", "solution": "There are four equal subintervals. The Simpson weights are $1,4,2,4,1$, so $$\\int_0^4f(x)\\,dx\\approx\\frac13[1+4(2)+2(0)+4(3)+2]=\\frac{23}{3}.$$ No exact error can be inferred without additional smoothness information.", "has_image": true} {"id": "c622bd08639e0847867b535d", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Use the three horizontal sample locations to construct a formula for $f\\prime(0)$ that is exact for every polynomial of degree at most two. Then apply it to the pictured values.\n", "options": null, "answer": "$f\\prime(0)\\approx-\\frac23f(-1)+\\frac12f(0)+\\frac16f(2)$, giving $1/3$.", "solution": "Let the weights be $a,b,c$. Exactness for $1,x,x^2$ gives $a+b+c=0$, $-a+2c=1$, and $a+4c=0$. Solving yields $(-2/3,1/2,1/6)$. The plotted values give $-4/3+1/2+7/6=1/3$.", "has_image": true} {"id": "8de9670d5d6b47939dd5a346", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "For the interval shown, determine positive $d,w$ so that the symmetric rule $w[f(1-d)+f(1+d)]$ integrates every polynomial of degree at most three exactly.\n", "options": null, "answer": "$d=\\sqrt3$, $w=3$, so the rule is $3[f(1-\\sqrt3)+f(1+\\sqrt3)]$.", "solution": "Shift to the centered variable $u=x-1$, whose interval is $[-3,3]$. Exactness for the constant gives $2w=6$. Exactness for $u^2$ gives $2wd^2=\\int_{-3}^3u^2\\,du=18$, hence $d^2=3$. Symmetry automatically handles the two odd powers.", "has_image": true} {"id": "3384968fffbecb182b78cb8c", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "A gradient method moves from $x_0$ to $x_1$ as shown. The red arrow gives the exact gradient at the initial point. Determine the scalar step size in $x_1=x_0-\\alpha\\nabla f(x_0)$ and show that the step direction is a strict descent direction.\n", "options": null, "answer": "$\\alpha=1/4$; the directional inner product is $-5<0$.", "solution": "The displacement is $d=x_1-x_0=(-1,-1/2)=-(1/4)(4,2)$, so the step size is $1/4$. The first-order directional change is $\\nabla f(x_0)\\cdot d=4(-1)+2(-1/2)=-5$, certifying strict descent.", "has_image": true} {"id": "9d0e60db3a80d2cf1f73c981", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Minimize $f(x,y)=(x-3)^2+(y-3)^2$ over the shaded region. Give the minimizer, minimum value, and nonnegative KKT multipliers in the order of the constraint labels in the image.\n", "options": null, "answer": "Minimizer $(2,2)$, minimum $2$, and multipliers $(2,0,0)$.", "solution": "The nearest feasible point to $(3,3)$ is its perpendicular projection onto the active edge, namely $(2,2)$. The coordinate constraints are inactive. Stationarity is $(-2,-2)+\\lambda_1(1,1)=0$, giving $\\lambda_1=2$. Complementarity gives the other multipliers zero. Convexity makes these conditions sufficient.", "has_image": true} {"id": "36ffaed22f6cf01937198963", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The graph joins the marked vertices by straight segments. Decide whether the function is convex. Then find its greatest convex minorant on the displayed interval: the pointwise largest convex function lying below it.\n", "options": null, "answer": "It is not convex. The greatest convex minorant is $g(x)=2x$ for $0\\le x\\le2$ and $g(x)=4x-4$ for $2\\le x\\le3$.", "solution": "At the midpoint, $f(1)=3>[f(0)+f(2)]/2=2$, violating convexity. The lower convex hull joins $(0,0)$ to $(2,4)$ and then to $(3,8)$, with increasing slopes $2,4$. Any convex minorant is bounded above by the first chord on $[0,2]$ and by the original straight graph on $[2,3]$. The proposed function reaches both bounds.", "has_image": true} {"id": "02e60611cf927fbdfca12d73", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Apply forward Euler with constant positive step size to the linear system whose entire spectrum is shown. Determine the interval of step sizes for which every discrete mode decays asymptotically.\n", "options": null, "answer": "$0", "options": null, "answer": "Old solution $(1,1)$; new solution $(0,2)$; amplification $\\sqrt{80401}$.", "solution": "Subtracting the original equations gives $0.01y=0.01$, while after the perturbation it gives $0.01y=0.02$. The solution change has norm $\\sqrt2$, equal to the original solution norm, so its relative size is one. The original right-hand side has norm $\\sqrt{80401}/100$, and its change has norm $1/100$. The requested ratio is therefore $\\sqrt{80401}$.", "has_image": true} {"id": "8348509c83f97569509a5e2c", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Use the sampled points in the figure to estimate $\\pi$ by the circle-to-square area ratio. Also give the plug-in estimated standard error of this estimator under independent uniform sampling. Count every plotted mark once.\n", "options": null, "answer": "$\\widehat\\pi=16/5$; estimated standard error $4/(5\\sqrt5)$.", "solution": "There are twenty samples, sixteen inside the unit circle. Thus $\\widehat p=4/5$ and $\\widehat\\pi=4\\widehat p=16/5$. The indicator sample mean has estimated standard error $\\sqrt{\\widehat p(1-\\widehat p)/20}$; scaling by four gives $4/(5\\sqrt5)$.", "has_image": true} {"id": "1308d4c06b18222a02d99c48", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The patch is a spherical triangle whose vertices and great-circle boundaries are shown. Find its area and the integral of Gaussian curvature over it.\n", "options": null, "answer": "Area $2\\pi$; integrated Gaussian curvature $\\pi/2$.", "solution": "The vertex coordinates give sphere radius $2$, and the coordinate-plane great circles meet at right angles. The spherical excess is $3(\\pi/2)-\\pi=\\pi/2$. Area equals radius squared times excess, giving $4(\\pi/2)=2\\pi$. Since curvature is $1/4$, its integral is $\\pi/2$.", "has_image": true} {"id": "8bb88f0efd59b0b3fa37de67", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "Extend the pictured straight ramp periodically with period $2\\pi$, using the red endpoint values. At $x=\\pi$, to what value does its Fourier series converge? Also find its mean over one period.\n", "options": null, "answer": "The Fourier series converges to $1$ at the jump, and the mean is $1$.", "solution": "The left limit at the jump is $2$, while periodic continuation gives right limit $0$. Dirichlet's convergence theorem gives their average, $1$, regardless of the separately assigned value $5$. The mean of the linear ramp from height zero to height two is also one.", "has_image": true} {"id": "fc5e27cb855ec05c9754a641", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "On the standard oriented torus $\\mathbb R^2/\\mathbb Z^2$, two loops have the lifts shown, each starting at the origin. Find their algebraic intersection number in the order $(\\alpha,\\beta)$ and the smallest number of transverse intersections achievable within their homotopy classes.\n", "options": null, "answer": "Algebraic intersection $+5$; minimum transverse intersection count $5$.", "solution": "The lift displacements give homology vectors $(2,1)$ and $(1,3)$. Their oriented intersection is the determinant $2\\cdot3-1\\cdot1=5$. Any transverse representatives have at least its absolute value in geometric intersections. Straight torus geodesics after a generic translation attain that count.", "has_image": true} {"id": "193559c3f81b7d4ba3434506", "subject": "Mathematics", "level": "Undergraduate", "qtype": "Short answer", "question": "The blue graph is a quadratic with the exact marked values on the displayed interval. Among all affine functions $p(x)=a+bx$, find the one minimizing the largest absolute vertical error. Give the minimum error and prove optimality.\n", "options": null, "answer": "$p(x)=1/2$ with minimum uniform error $1/2$.", "solution": "The marked values determine $f(x)=x^2$ on $[-1,1]$. The candidate $p=1/2$ has errors $+1/2,-1/2,+1/2$ at the three ordered points $-1,0,1$, and nowhere exceeds that absolute error. The alternating extremal errors certify optimality for degree-one approximation. Directly, an error strictly below $1/2$ would force $p(-1)>1/2$, $p(1)>1/2$, but $p(0)<1/2$, contradicting affinity.", "has_image": true}