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What is 125+278?

A) 393
B) 403
C) 413
D) 423
E) 433
F) 443

What is 142−87?

A) 45
B) 55
C) 65
D) 75
E) 95
F) 105

What is 17×23?

A) 391
B) 393
C) 395
D) 397
E) 399
F) 401

What is 196÷14?

A) 12
B) 13
C) 14
D) 15
E) 16
F) 17

What is the value of   7³?

A) 329
B) 343
C) 351
D) 357
E) 361
F) 369

What is the cube root of −125?

A) -25
B)  5
C) -5
D) 25
E) -3
F) -2

What is the fourth root of 256?

A) 2
B) 3
C) 4
D) 5
E) 6
F) 7

What is the three quarters of 136?

A) 34
B) 68
C) 2
D) 102
E) 96
F) 204 

What is 25% of 80?

A) 15
B) 18
C) 20
D) 22
E) 25
F) 30

What is √64 −2×3+5?

A) 23
B) 7
C) 9
D) 10
E) 11
F) 12

Solve for x: 2x + 5 = 15

A) x = 3
B) x = 4
C) x = 5
D) x = 6
E) x =-5
F) x =-6

What is the area of a trapezoid with bases 10 and 6, and height 4?

A) 15
B) 28
C) 30
D) 32
E) 64
F) 24

What is 3² + 4²?

A) 5
B) 16
C) 20
D) 25
E) 34
F) 36

If a triangle has angles of 30° and 70°, what is the third angle?

A) 40°
B) 100°
C) 80°
D) 50°
E) 60°
F) 110°

What is the greatest common divisor of 24, 36, 60?

A) 4
B) 6
C) 8
D) 12
E) 18
F) 24

Simplify: 3(x + 2) - 2x

A) x + 2
B) x + 4
C) x + 6
D) x + 8
E) 2x + 6
F) 3x + 6

What is the least common multiple of 12, 18, 30?

A) 6
B) 90 
C) 120
D) 150
E) 180
F) 210

Find the median of: 1,-1,2,-2,3,-3

A) 1
B) 2 
C) 0
D) -2
E) -1
F) 3

Find a mode of the following data set:
1,-3,7,2,-5,-1,0,2

A) 1
B) 0
C) 5
D) 2
E) -5
F) 0.5

What is the perimeter of a square with side length 9?

A) 18
B) 27
C) 36
D) 45
E) 54
F) 81

Solve: log₂(1/8) = ?

A) 1
B) 2
C) 3
D) -3
E) -2
F) -1

What is sin(750°)?

A) 0
B) 0.5
C) 0.707
D) 0.866
E) 1
F) 1.414

Factor: x² - 5x + 6

A) (x-1)(x-6)
B) (x-2)(x-3)
C) (x-2)(x-4)
D) (x-1)(x-5)
E) (x+2)(x+3)
F) (x-3)(3x-2)

What is the derivative of x³?

A) x²
B) 2x²
C) 3x²
D) 4x²
E) x³
F) 3x³

Find the roots of x² - 4x + 3 = 0

A) x = 1, 2
B) x = 1, 3
C) x = 2, 3
D) x = -1, -3
E) x = -1, 3
F) x = 0, 4

What is cos(π/4)?

A) 0
B) 1/2
C) 1/√2
D) √3/2
E) 1
F) -1/2

Evaluate: ∑(k=1 to 5) (k+2)

A) 10
B) 12
C) 15
D) 18
E) 20
F) 25

What is the equation of a line with slope 2 passing through (1, 3)?

A) y = 2x + 1
B) y = 2x - 1
C) y = 2x + 3
D) y = x + 2
E) y = 3x + 1
F) y = 2x + 5

Find the determinant of [[1,2,3],[2,3,-1],[5,2,1]]

A) -42 
B) 36
C) -32
D) -28
E) 24
F) -20

What is lim(x→0) sin(x)/x?

A) 0
B) 1
C) π
D) ∞
E) -1
F) undefined

Find the value of arcsin(x)+arccos (x) for  x ∈ [-1, 1].

A) 0
B) π/4 
C) π/2
D) π 
E) 2π
F) undefined

What is the integral of 1/sin(x)?

A) -ln|sin(x)| + C
B) ln|tan(x/2)| + C
C) -ln|cos(x)| + C
D) -cot(x) + C
E) tan(x) + C
F) sec(x) + C

Find the partial derivative ∂/∂x of f(x,y) = x²y + xy².

A) 2xy + y²
B) x² + 2xy
C) 2x + y
D) xy + y²
E) 2xy + 2y²
F) x² + y²

What is the Taylor series expansion of e^x around x=0 up to x²?

A) 1 + x
B) 1 + x + x²
C) 1 + x + x²/2
D) 1 + x/2 + x²/2
E) x + x²/2
F) 1 + 2x + x²

Find the eigenvalues of [[3, 1], [2, 2]].

A) λ = 1, 2
B) λ = 1, 4
C) λ = 3, 4
D) λ = 0, 5
E) λ = 2, 3
F) λ = 2, 4

What is the Laplace transform of ℒ{e^(at)}(s)?

A) 1/(s-a)
B) 1/(s+a)
C) a/(s-a)
D) s/(s-a)
E) 1/(s²-a²)
F) (s-a)/(s+a)

In the complex plane, what is |1 + i|?

A) 1
B) √2
C) 2
D) √3
E) 1/2
F) 2√2

What is the curl of the vector field F = (y, -x, 0)?

A) (0, 0, -2)
B) (0, 0, 2)
C) (0, 0, 0)
D) (0, 0, -1/2)
E) (1, 1, -2)
F) (1,-1, 0)

Find the residue of f(z) = 1/(z²(z-1)) at z = 0.

A) -1
B) 0
C) 1
D) 2
E) -2
F) 1/2

What is the order of the cyclic group Z/12Z?

A) 6
B) 8
C) 10
D) 12
E) 24
F) 144

In a principal ideal domain, every non-zero ideal is generated by how many elements?

A) 0
B) 1
C) 2
D) n
E) infinitely many
F) at most 2

What is the dimension of the kernel of the linear map T: R³ → R² given by T(x,y,z) = (x+y, y+z)?

A) 0
B) 1
C) 2
D) 3
E) 4
F) 6

In the Sobolev space H¹(Ω), where Ω ⊂ Rⁿ is a bounded domain, what is the norm of a function u?

A) ||u||_{L²}
B) ||u||_{L²} + ||∇u||_{L²}
C) (||u||²_{L²} + ||∇u||²_{L²})^{1/2}
D) max(||u||_{L²}, ||∇u||_{L²})
E) ||∇u||_{L²}
F) ||u||_{L∞}

What is the Euler characteristic of a torus?

A) -2
B) -1
C) 0
D) 1
E) 2
F) 4

In the hyperbolic plane, consider a triangle with angles α,β,γ. Which of the following statements is true?

A) α+β+γ=180°
B) α+β+γ<180°
C) α+β+γ>180°
D) α+β+γ=360°
E) α+β+γ=90°
F)  α+β+γ=270°

What is the fundamental group of S² (the 2-sphere)?

A) Z
B) Z²
C) Z/2Z
D) {e}
E) R
F) S¹

If the Ricci tensor of a Riemannian manifold is proportional to the metric, which of the following must hold?

A) M is flat
B) M is Einstein
C) M has constant positive curvature
D) M is a torus
E) The Riemann tensor vanishes

What is the Hausdorff dimension of the middle-third Cantor set?

A) 0
B) log(2)/log(3)
C) 1/2
D) 2/3
E) 1
F) log(3)/log(2)

In algebraic topology, what is the degree of the map f: S¹ → S¹ given by f(z) = z³?

A) 0
B) 1
C) 2
D) 3
E) -3
F) 9

Let T=I - K be a Fredholm operator on a Hilbert space, where K is compact. What is the index of T?

A) -dim(ker K)
B) 0
C) 1
D) dim(ker T)
E) ∞
F) undefined

In the theory of Lie algebras, what is the complex dimension of sl(2,C)?

A) 2
B) 3
C) 4
D) 6
E) 8
F) 9

What is the first Pontryagin class of complex projective plane CP²?

A) 0
B) 3h²
C) 4h²
D) 6h²
E) 9h²
F) 12h²

In topological K-theory, to what group K⁰(S²) is  isomorphic?

A) Z
B) Z²
C) Z⊕Z
D) 0
E) Z/2Z
F) Z³

What is the spectral radius of the matrix [[0, 1, 0], [0, 0, 1], [1, 0, 0]]?

A) 0
B) 1
C) √2
D) 2
E) √3/2
F) ∞

In the context of C*-algebras, what is the norm of a self-adjoint element a?

A) ||a|| = sup{|λ| : λ ∈ σ(a)}
B) ||a|| = tr(a)
C) ||a|| = rank(a)
D) ||a|| = det(a)
E) ||a|| = 1
F) ||a|| = ||a*a||^{1/2}

Let M ⊂ R^3 be a smooth surface, and let p ∈ M be a point where the Gauss curvature K(p) > 0. Which of the following statements must hold?

A) The surface is locally flat at p.
B) The surface is locally convex at p.
C) The surface is saddle-shaped at p.
D) The mean curvature at p is zero.
E) The surface has negative Gauss curvature somewhere near p.
F) The tangent plane at p intersects the surface only at p.

In symplectic geometry, what is the dimension of a Lagrangian submanifold of a 2n-dimensional symplectic manifold?

A) n/2
B) n
C) n+1
D) 2n-1
E) 2n
F) 4n

What is the Grothendieck group K₀ of the category of finite-dimensional vector spaces over a field?

A) 0
B) Z
C) Z²
D) N
E) Q
F) R

In the theory of operator algebras, what is the GNS (Gelfand–Naimark–Segal) construction associated with?

A) States on C*-algebras
B) Projections
C) Unitary operators
D) Compact operators
E) Trace class operators
F) Self-adjoint operators

What is the first Chern number of the complex projective line CP¹?

A) -2
B) -1
C) 0
D) 1
E) 2
F) 4

To which group is K₁(Z) isomorphic?

A) 0
B) Z/2Z
C) Z
D) Z²
E) {±1}
F) Q

What is the Todd genus of a complex 2-dimensional surface with Chern numbers c₁² = 9 and c₂ = 3?

A) 0
B) 1
C) 2
D) 3
E) 4
F) 6

In the context of étale cohomology, what is H²_{ét}(Spec k, μₙ) for k algebraically closed?

A) 0
B) Z/nZ
C) Z
D) k*
E) μₙ(k)
F) Br(k)[n]

According to the BSD conjecture, what is the L-function value L(E, 1) for an elliptic curve E/Q with analytic rank 0 and real period Ω = 2, where c_p are Tamagawa numbers?

A) 0
B) |Sha(E)|·∏c_p / |E(Q)_{tors}|²
C) 2·|Sha(E)|·∏c_p / |E(Q)_{tors}|²
D) ∞
E) 1
F) undefined

In derived algebraic geometry over C, what is the homotopy type of the derived moduli stack of perfect complexes on a smooth projective curve X?

A) BGL_n
B) K(Z, 2)
C) Maps(X, BGL_∞⁺)
D) Ω^∞ Σ^∞ X₊
E) X × BGL_∞
F) point

What is the motivic cohomology group H²_{M}(Spec k, Z(2)) for a field k containing all roots of unity?

A) 0
B) KM₂(k)
C) k*
D) Br(k)
E) H²_{ét}(k, μ_∞^⊗2)
F) k* ⊗ k*

What is the Euler characteristic of a compact, orientable surface M of genus  g in terms of the Gauss-Bonnet theorem?

A) 2 - 2g
B) 2g - 2
C) g 
D) 0
E) 2g
F) 1 - g

What is the orbifold Euler characteristic of the moduli space M_{g,n} of genus g curves with n marked points over C, when 2g-2+n > 0?

A) (-1)^{n-1} B_{2g+n-2}/(2g+n-2)!
B) 0
C) 1
D) (-1)^{3g-3+n}
E) B_{2n}/(2n)!
F) undefined

In higher topos theory, what is the  ∞-grupoid equivalent to the ∞-topos of sheaves on a contractible topological space with respect to the usual topology?

A) BG for some group G
B) ∞-Gpd
C) Set
D) point
E) K(Z,n) for some n
F) S^n for some n

What is the stable homotopy group π_{-1}^s?

A) 0
B) Z
C) Z/2Z
D) Z/24Z
E) undefined
F) Q

In the theory of vertex operator algebras, what is the central charge of the moonshine module V^♮?

A) 0
B) 1/2
C) 24
D) 26
E) 240
F) ∞

In spherical geometry, if a triangle has angles α, β, γ on a sphere of radius R, what is the formula for its area?

A) R^2(α + β + γ - π)
B) α + β + γ - π
C) R(α + β + γ)
D) R^2(αβγ)
E) πR^2
F) αβγ

In hyperbolic geometry, which map sends the interior of the Poincaré disk model to the upper half-plane model?


A) Identity map
B) Inversion in the unit circle followed by a translation
C) Möbius transformation
D) Reflection across the real axis
E) Rotation about the origin
F) Scaling by 2

What is the quantum dimension of the Jones-Wenzl projector JW_n in the Temperley-Lieb category at generic parameter q?

A) [n+1]_q
B) [n]_q!
C) 1
D) q^{n(n-1)/2}
E) ∏_{i=1}^n (q^i - q^{-i})/(q - q^{-1})
F) 0

If ω is a closed 1-form on a simply connected manifold M, which of the following is true?

A) ω is harmonic
B) ω is exact only if M is compact
C) dω ≠ 0
D) ω vanishes identically
E) ω = df for some function f
F) None of the above

What is the order of the Monster sporadic simple group?

A) 2^{46}·3^{20}·5^9·7^6·11^2·13^3·17·19·23·29·31·41·47·59·71
B) 2^{10}·3^7·5^3·7·11·13·17·19·23
C) 2^{41}·3^{13}·5^6·7^2·11·13·17·19·23·31·47
D) 2^{43}·3^{20}·5^9·7^6·11^2·13^3·17·19·23·29·31·41·47·59·71
E) 2^{48}·3^{20}·5^9·7^6·11^2·13^3·17·19·23·29·31·41·47·59·71
F) 2^{24}·3^{10}·5^6·7^3·11·13·17·23·29

In the theory of L-functions, what is the order of vanishing of L(s, χ) at s=1 for a primitive Dirichlet character χ of conductor q > 1 with  χ(-1) = 1?

A) 0
B) 1
C) -1 always
D) 1/2 always
E) depends on q
F) undefined

What is the Gromov-Witten invariant ⟨pt⟩_{0,1,d} for CP² when d = 1? What is the genus-0 Gromov-Witten invariant ⟨pt, pt⟩₀,₂,₁ for ℂP² with two point constraints and degree d = 1?

A) 0
B) 1
C) 2
D) 3
E) 4
F) 5

What is the homotopy type of the space of ∞-functors Fun(BG, BH) between the classifying ∞-groupoids BG and BH for finite groups G and H?

A) Map(BG, BH)
B) ∐_{[φ]} B(C_H(φ(G)))
C) BH^G
D) point
E) K(Hom(G,H), 0)
F) empty

Let ω be a 2-form on a 4-dimensional oriented Riemannian manifold M. If *ω = ω, which of the following is true? 

A) ω is exact
B) ω is closed
C) ω is self-dual 
D) ω vanishes identically 
E) dω ≠ 0 
F) None of the above 

In the context of anabelian geometry, what determines a hyperbolic curve X over a number field K up to isomorphism over K?

A) H¹(X, O_X)
B) π₁^{ét}(X) with its Galois action
C) The Jacobian J(X)
D) The canonical bundle K_X
E) H²(X, Z)
F) The genus g(X)

 What is the Seiberg-Witten invariant of the smooth, closed 4-manifold CP² # 9(-CP²) for its canonical Spin^c structure in the standard chamber?

A) 0
B) 1
C) -1
D) 2
E) 9
F) ∞

In chromatic homotopy theory, what is the height of the formal group law over F_p associated with Morava K-theory K(n)?

A) 0
B) 1
C) n
D) p
E) p^n
F) ∞

Let T be a torsion tensor on a manifold M. If T is totally antisymmetric, which of the following holds for all vector fields X, Y, Z? 

A) T(X,Y,Z) = T(Y,X,Z) 
B) T(X,Y,Z) = T(Z,Y,X) 
C) T(X,Y,Z) = 0 
D) T(X,Y,Z) =- T(X,Z,Y) 
E) T is symmetric in its first two arguments 
F) None of the above

For the quantum group U_q(sl_2), where q is not a root of unity, what is the dimension of the irreducible representation V(λ) with highest weight λ = n, where n is a non-negative integer?

A) n
B) n+1
C) 2n
D) n!
E) 2^n
F) (n+1)²

What is the motivic sphere spectrum S^{0,0} in the stable motivic homotopy category over a base field k?

A) Spec(Z)
B) P¹
C) 1_SH
D) A¹
E) G_m
F) pt

In arithmetic dynamics, for a rational map φ: P¹ → P¹ of degree d ≥ 2 defined over Q, what is the canonical height ĥ_φ(P) for a preperiodic point P?

A) 0
B) 1
C) log d
D) 1/d
E) ∞
F) undefined

 What is the E₂^{1,1}-term of the Adams spectral sequence computing the 2-primary stable homotopy groups π_*(S) of the classical sphere spectrum at prime 2?

A) 0
B) Z/2Z
C) Z/4Z
D) Z
E) Z²
F) undefined

In the Langlands program, what is the dimension of the space of cusp forms S_k(Γ₀(N)) of weight k ≥ 2 for Γ₀(1) = SL₂(Z)?

A) 0 if k odd
B) ⌊k/12⌋ if k ≡ 2 (mod 12)
C) ⌊k/12⌋ + 1 if k ≡ 0 (mod 12)
D) All of the above
E) k-1
F) ∞

In condensed mathematics, what is the cohomological dimension of Z_p as a condensed abelian group in the pro-étale site?	

A) 0
B) 1
C) 2
D) p
E) ∞
F) undefined

What is the K-theoretic Donaldson-Thomas invariant of the resolved conifold in degree 0?

A) 1
B) -1
C) 0
D) ∞
E) MacMahon function M(q)
F) 1/(1-q)

 In the context of the BSD conjecture, if an elliptic curve E/Q has rank 0 and L(E,1) ≠ 0, what is the order of the Tate-Shafarevich group |Sha(E/Q)|, where Ω is the real period and c_p are the Tamagawa numbers?

A) 1
B) L(E,1)·|E(Q)_{tors}|²/(Ω·∏c_p)
C) 0
D) |E(Q)_{tors}|
E) ∞
F) L'(E,1)

What is the leading coefficient of the Conway polynomial ∇_K(z) for the unknot K?

A) 0
B) 1
C) -1
D) 2
E) z
F) undefined

 In higher algebra, what is the ∞-categorical dimension of the derived category D(A) of a differential graded algebra A over a field k?

A) 0
B) 1
C) dim(A)
D) ∞
E) undefined
F) depends on A

For a reductive algebraic group G over a p-adic field k, what is the order of the group of connected components π₀(G(k)) = G(k)/G(k)⁰, where G_der is the derived group?

A) Always finite
B) Always infinite
C) 1 if G is simply connected
D) |π₁(G_der)|
E) p^n for some n
F) 0

In tropical geometry, what is the tropical determinant of the matrix [[a,b],[c,d]] in the max-plus algebra?

A) max(a+d, b+c)
B) min(a+d, b+c)
C) a+d+b+c
D) max(a,b,c,d)
E) ad-bc
F) max(a,d)-max(b,c)

Let R be the Riemann curvature tensor on a Riemannian manifold (M,g). Which of the following properties holds for all vector fields X,Y,Z,W? 

A) R(X,Y,Z,W) = -R(Y,X,Z,W) 
B) R(X,Y,Z,W) = R(Z,W,X,Y) 
C) R(X,Y,Z,W) + R(Y,Z,X,W) + R(Z,X,Y,W) = 0 
D) R(X,Y,Z,W) = g(R(X,Y)Z, W) E) All of the above 
F) None of the above

In non-commutative geometry, what is the Hochschild dimension of the algebra C^∞(M) of smooth functions on a compact manifold M of dimension n, over C?

A) 0
B) n
C) 2n
D) ∞
E) 1
F) undefined

Let Γ^k_{ij} be the Christoffel symbols of the Levi-Civita connection of a Riemannian manifold (M,g). Which of the following statements is true? 

A) Γ^k_{ij} = Γ^k_{ji} 
B) Γ^k_{ij} = -Γ^k_{ji} 
C) Γ^k_{ij} = 0 in all coordinates 
D) Γ^k_{ij} is a tensor 
E) Γ^k_{ij} = ∂g_{ij}/∂x^k 
F) None of the above 

"Let X,Y be vector fields on a smooth manifold M, and let f be a smooth function. What is the expression for the Lie bracket [X,fY]? 

A) f[X,Y]
B) f[X,Y] + (Xf)Y
C) f[X,Y] + (Yf)X
D) (Xf)Y - (Yf)X
E) f[X,Y]-(Yf)X
F) None of the above"