The 2025 paper, solved
70 maths questions. Filter by unit, then tap a question for the answer. Every answer matches the commission’s official key of 10 May 2025. 1 question looks faulty and may be given as bonus marks.
Q31Cyclic group of prime order is simple; Lagrange's theoremAlgebraA
A true (prime order => only trivial subgroups => simple). R is Lagrange's theorem and is exactly why A holds.
Matches the official key
Q32HN/N ≅ ? (second isomorphism theorem)AlgebraA
Second isomorphism theorem: HN/N ≅ H/(H∩N).
Matches the official key
Q33Number of generators of cyclic group of order 8AlgebraC
Generators of C8 = phi(8) = 4.
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Q34Elements of A3AlgebraD
A3 = {I, (1 2 3), (1 3 2)}.
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Q35Order of a given permutation in S4AlgebraB
1->1, 2->3, 3->4, 4->2 is the 3-cycle (2 3 4); order 3.
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Q36Degree of splitting field of x⁴+x²+1 over QAlgebraA
x^4+x^2+1 = (x^2+x+1)(x^2-x+1); all roots are primitive 3rd/6th roots of unity, in Q(w); degree 2.
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Q37M maximal iff R/M is a fieldAlgebraD
M maximal iff R/M is a field (commutative ring with 1).
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Q38T self-adjoint iff ⟨Tx,x⟩ realAnalysisC
T self-adjoint iff <Tx,x> real for all x (complex Hilbert space).
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Q39Order: l² Hilbert → Banach → normed → topologicalAnalysisB
Implication chain: l2 Hilbert (II) => Banach (III) => normed (IV) => topological (I): II, III, IV, I. Reverse order (B) is the other reading.
Matches the official key
Official key (10.05.2025) says B: the order is weakest to strongest. My first solve said C (strongest to weakest).
Q40Xy/√(x²+y²): continuity/differentiability at originAnalysisD
|f| <= r/2 so continuous; f_x(0,0)=f_y(0,0)=0; along y=x, f/r = 1/2 != 0 so not differentiable.
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Q41Interval of uniform convergence of a series of functionsAnalysisD
On [1,inf), 1/(1+n^2 x) <= 1/n^2, Weierstrass M-test. On (0,c) not uniform (tail unbounded as x->0).
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Q42Ordering of series/convergence resultsAnalysisC
Chain of implications: partial sums converge (II) => series converges (III) => a_n->0 (IV) => {a_n} convergent hence Cauchy (I).
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Q43Max of sin x sin y sin(x+y)AnalysisB
Max of sin x sin y sin(x+y) is (sqrt3/2)^3 = 3sqrt3/8 at x=y=pi/3.
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Q44Divergence of a series ∑uₙ with reasonAnalysisA
u_n = 1/((2/3)^n+1) -> 1 != 0, so series diverges; R is the divergence (nth term) test.
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Q45Domain of uniform convergenceAnalysisD
f_n = n x e^{-n x^2}; sup on [0,a] is sqrt(n/2)e^{-1/2} -> inf (no uniform). On [1,2], f_n <= 2n e^{-n} -> 0, uniform. Exponent read as -n x^2.
Matches the official key
exponent in the printed f_n is small; read as -n x^2. If it were -x^2 the sequence would not converge pointwise at all, so D stands either way among the options.
Q46Nullity of T: V3→V2Linear algebraA
T(a1,a2,a3)=(a2,a3); kernel = {(a1,0,0)}, nullity 1.
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Q47Rank of given matrix ALinear algebraB
Row reduction (checked in python): rank 3.
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Q48Eigenvalues of given matrixLinear algebraB
Expand along row 2: (1-L)((1-L)^2-2)=0 => 1, 1+sqrt2, 1-sqrt2.
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Q49Dimension of subspace W1Linear algebraD
One independent linear constraint on R^5: dim 4.
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Q50Count of non-singular linear maps ℝ⁵→…Linear algebraA
Non-singular (invertible) maps need equal dimensions; R^5 -> R^3 cannot be, so 0.
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Q51Given T(e1),T(e2),T(e3): one-one/onto?Linear algebraD
T(e3)=T(e1)+T(e2), so T is singular: neither one-one nor onto.
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Q52Distance between orthogonal unit vectors (√2)Linear algebraB
|a-b|^2 = 1+1-2(0) = 2, distance sqrt2.
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Q53Rank and nullity of T: R²→R³Linear algebraD
Image spanned by (1,1,0),(1,-1,1): rank 2; nullity 2-2 = 0.
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Q54PI for RHS e^{2x} sin xODE and PDEC
PI = e^{2x} (1/(D^2+2D+5)) sin x = e^{2x}(4 sin x - 2 cos x)/20 = -(1/10)e^{2x}(cos x - 2 sin x).
Matches the official key
Q55PI of a linear PDEODE and PDEB
z = x^2 y: z_xx = 2y, z_y = x^2, difference 2y - x^2.
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Q56IVP has unique/no/infinite solutionsODE and PDEB
General solution y = C x^4 gives y(0)=0 != 2: no solution.
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Q57Singular solution of Clairaut-type equationODE and PDED
p-discriminant of p^3 - 4xy p + 8y^2: 256x^3y^3 - 1728y^4 = 0 => 27y = 4x^3; verified it satisfies the ODE.
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Q58General solution form of a given ODEODE and PDEB
Laplace equation: u = f(x+iy)+g(x-iy).
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Q59Match PDEs with elliptic/parabolic/hyperbolicODE and PDED
a heat = parabolic (II); b wave = hyperbolic (III); c Laplace = elliptic (I); d y u_x - x u_y = 0 => u=f(x^2+y^2) (IV).
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Q60Complete solution of z = f(p,q)ODE and PDEB
Clairaut form z = px+qy+f(p,q): complete integral z = ax+by+a^2+b^2.
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Q61Residue at z=0ComplexC
z cos(1/z) = z - 1/(2z) + ...; residue -1/2.
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Q62Image ofComplexA
z=(2w+5)/(4(w+1)); |z|=1 => u^2+v^2+u-3/4=0, centre (-1/2,0), radius 1.
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Q63∮ overComplexB
Residue at z=-1: (1/3!) d^3/dz^3 e^{2z} = 8e^{-2}/6; integral = 2 pi i * 8/(6e^2) = 8 pi i/(3e^2).
Matches the official key
Q64F(z) given u = eˣ(x cos y − y sin y)ComplexA
f = z e^z has real part e^x(x cos y - y sin y).
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Q65Value of a definite integral (contour method)ComplexB
Integral of d(theta)/(a+b sin theta) over 0..2pi = 2 pi/sqrt(a^2-b^2) = 2 pi/3.
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Q66P/q has pole of order m when q has zero of order mComplexA
q has zero of order m at z0 and p(z0)!=0, so p/q has a pole of order m; R explains A
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Q67Continuity equation, spherical symmetryMechanicsB
Spherical symmetry: d(rho)/dt + (1/r^2) d(rho q r^2)/dr = 0
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Q68DOF of rigid body with one point fixed (3)MechanicsC
Rigid body with one point fixed: 3 degrees of freedom (Euler angles)
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Q69Which Poisson-bracket identity failsMechanicsD
[q_r,p_s] = delta_rs, not 0; other three always hold (antisymmetry, Jacobi, [q,q]=0)
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Q70Continuity eq; incompressible ⇒ ∇·v=0MechanicsC
A is the continuity equation (true). R is false: incompressible means div v = 0, not curl v = 0
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Q71Coupled-oscillator Lagrangian, normal modesMechanicsD
p1=2q1'+q2', p2=2q2'+q1' so p1+p2=3(q1'+q2') (A ok); p1'=-2q1, p2'=-2q2 (B ok); adding EOM gives s''=-(2/3)s, s=q1+q2 (C ok); p1'/p2' = q1/q2, not q2/q1 (D false)
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Q72Relations between φ and ψ (Cauchy–Riemann)MechanicsD
Cauchy-Riemann: phi_x = psi_y, phi_y = -psi_x, so a and d
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Q73Hamiltonian from given LagrangianMechanicsC
p_x=2a x', p_y=2b y'; H = p_x^2/4a + p_y^2/4b + kxy
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Q74Extremal of a functional with end conditionsMethodsC
F has no y: y'/(x sqrt(1+y'^2)) = c gives x^2+(y-k)^2=R^2; through (1,0),(2,1): k=2, R^2=5
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Q75Solution of an integral equationMethodsA
L{t^n F(t)} = (-1)^n d^n f/ds^n
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Q76Solution of Fredholm integral equationMethodsA
c = int t phi = 1/2 + lambda c/3, c = 3/(2(3-lambda)); phi = 1 + 3 lambda x/(2(3-lambda))
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Q77Laplace transform of t cos tMethodsB
-d/ds[s/(s^2+1)] = (s^2-1)/(s^2+1)^2
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Q78Euler–Lagrange equation is 2nd-order ODE (type)MethodsC
E-L: F_y'y' y'' + F_y'y y' + F_y'x - F_y = 0 is second order, in general non-linear (depends on F)
Matches the official key
Both solves agree on C. B (linear) is the only real competitor.
Q79Max distance between curves on [0,1]AnalysisC
max of x - x^2 on [0,1] at x=1/2 gives 1/4
Matches the official key
Q80Nature of LPP solution (feasible/bounded/many)OR and numericalD
x1>=x2>=3+3x1 forces x1<=-1.5, contradicting x1>=0: infeasible, so none of the options
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Q81Which set in R² is not convexOR and numericalB
Annulus 1<=r^2<=4 is not convex (the others are a disc, a disc, a half-plane)
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Q82NR iteration formulaOR and numericalA
Newton-Raphson: x_{n+1} = x_n - f(x_n)/f'(x_n)
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Q83Max value of objective subject to constraintsOR and numericalC
Vertices: (0,0)->0, (3.5,0)->17.5, (7/3,5/3)->23.33, (0,4)->28 (feasible); max 28
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Q84LPP: none/unbounded/alternative/unique optimumOR and numericalB
Take x2=x1, x1->inf: feasible, z=x1->inf, so unbounded
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Q85Third divided difference of 1/x at a,b,c,dOR and numericalD
Third divided difference of 1/x = (-1)^3/(abcd) = -1/(abcd)
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Q86Interpolating polynomial from tableOR and numericalA
x^3+x^2-x+2 gives 2,3,12,147 at x=0,1,2,5
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Q87Rate of convergence of NR (quadratic)OR and numericalA
Newton-Raphson converges quadratically (order 2)
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Q88Twin-prime pairs between 51 and 100Number theoryB
Primes 51-100: 53,59,61,67,71,73,79,83,89,97; twin pairs (59,61),(71,73) = 2
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Q89Σ(180)Number theoryB
180=2^2*3^2*5: sigma = 7*13*6 = 546
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Q902^340 mod 341Number theoryD
341=11*31; 2^10=1 mod 11, 2^5=1 mod 31, 10 | 340, so 2^340 = 1 (checked in python)
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Q91Chronological order: Panchasiddhantika, Aryabhatiya, Brahmasphuta, TrisatikaNumber theoryB
Aryabhatiya (499) < Panchasiddhantika (c. 505-575) < Brahmasphutasiddhanta (628) < Trisatika (c. 750): I, IV, III, II
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Q921729, 1887, 6174, Bhaskara II birth yearNumber theoryB
Taxi-cab 1729, Ramanujan born 1887, Kaprekar constant 6174, Bhaskara II born 1114: a-III, b-I, c-II, d-IV
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Q93Match-list (content garbled in extraction)Number theoryD
(a,b)=III, (a,b]=IV, [a,b)=I, [a,b]=II
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Q94Which Tᵢ fails for metric spaces (incorrect statement)Topology and DG · Likely bonus marksD
Every metric space is T1, T2, T3 and T4, so all four statements are true; no option is incorrect. Guess D
Matches the official key
Defective: every metric space is T1, T2, T3 and T4, so no statement is incorrect. The official key still gives D.
Q95Totally bounded metric space is separableTopology and DGB
{a,b} and {b,c} are in the family but their intersection {b} is not, so not closed under intersection
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Q96(content garbled in extraction)Topology and DGA
Totally bounded metric space is separable (compact needs completeness too; "reducible" is meaningless)
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Q97Gaussian curvature of sphere radius r (1/r²)Topology and DGD
Sphere of radius r: K = 1/r^2
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Q98Curvature/torsion of r=(a cos u, a sin u, bu)Topology and DGB
Circular helix: torsion = b/(a^2+b^2), curvature a/(a^2+b^2)
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Q99Gaussian curvature of sphere radius aTopology and DGC
K = 1/a^2
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Q100Constant curvature and torsion ⇒ circular helixTopology and DGC
Constant curvature and torsion implies circular helix (standard theorem); "helix"/"cylindrical helix" are also true but weaker
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(C) is the standard answer. (A) cylindrical helix and (B) helix are also true statements (circular helix is a special case), but (C) is the most specific.