The 2026 paper, solved
70 maths questions. Filter by unit, then tap a question for the answer. There is no official key yet, so 13 answers are marked to double-check, and 6 look like bonus marks because the question is faulty.
Q315-dim operator, min poly (x-2)^3(x-1): Jordan block structure, generalized eigenspace dim, rank of (T-2I)^2Linear algebraA
Answer: 3,3
Min poly (x-2)^3(x-1): gen. eigenspace of 2 has dim d=3 or 4. d=3 (one 3-block, eigenvalue 1 has mult 2): rank (T-2I)^2 = 1 + 2 = 3. d=4 gives (4,2), not an option; (4,1) impossible.
Sure
Q32Symmetric degenerate kernel of rank 2: max number of nonzero eigenvalues, eigenfunction orthogonalityMethodsD
Answer: 2, orthogonal
Rank-2 symmetric degenerate kernel has at most 2 nonzero eigenvalues; eigenfunctions of a symmetric kernel for distinct eigenvalues are orthogonal.
Sure
Q33F_n to f uniformly, each f_n uniformly continuous on metric space X: what follows for fAnalysisA
Uniform limit of uniformly continuous functions is uniformly continuous (3-epsilon argument), on any metric space.
Sure
Q34Count something between antipodal points of a sphere using global geometric arguments (1, 2, finite, infinite)Topology and DGD
Answer: Infinite
Antipodal points on S^2 are joined by infinitely many great semicircles (one per meridian).
Sure
Q35LP constraints x+2y<=6, 2x+y<=6; objective ax+by optimal on edge joining two vertices; find ratio a:bOR and numerical · Likely bonus marksA
Answer: 1:2
Vertices (0,0),(3,0),(2,2),(0,3). Alternate optima need objective parallel to an edge: x+2y=6 edge gives 1:2, 2x+y=6 edge gives 2:1. Both edges have integral (assignment-polytope-like) endpoints, so the clause does not discriminate. Picked 1:2.
Not sure
Defective: by the x↔y symmetry of the region, 1:2 (A) and 2:1 (B) both fit. Both solves picked A.
Q36Is Z5[x]/(x^2+2) a field, and its orderAlgebraA
Answer: field of order 25
x^2+2 has no root mod 5 (squares mod 5: 0,1,4; -2=3), so irreducible; Z5[x]/(x^2+2) is a field of 25 elements.
Sure
Q37N=2^a 3^b has 36 divisors, a>b, n congruent 2 mod 5: possible (a,b)Number theoryB
Answer: 8,3
(a+1)(b+1)=36, a>b, 2^a 3^b = 2 mod 5. (8,3): 9*4=36, 256*27 = 1*2 = 2 mod 5. (17,1): 2*3=1 mod 5 fails; (5,5),(2,11) violate a>b.
Sure
Q38Gram-Schmidt on (1,1,0),(1,0,1),(0,1,1); trace of projection onto first two vectorsLinear algebraA
Answer: 2
Projection onto a 2-dim subspace has trace = rank = 2.
Sure
Q39Y(x)=1+lambda*integral of x t y(t) dt on [0,1]: nonzero eigenvalue via consistency conditionMethodsD
Answer: lambda=3
y=1+lambda*c*x with c=int t y dt; c=1/2+lambda*c/3 so c(1-lambda/3)=1/2; singular (characteristic value) at lambda=3.
Sure
Q40Canonical transformation preserves symplectic structure: which quantity is invariantMechanicsD
Answer: phase space area
Canonical transformations preserve the symplectic form, hence phase-space volume/area (Liouville, Poincare invariant).
Sure
Q41(y-px)^2=p^2(1+p^2): nature of singular solution relative to general solutionODE and PDEB
Answer: envelope of general solution
Equation is Clairaut: y = px +/- p*sqrt(1+p^2). Singular solution exists (from x = -f'(p)) and is the envelope of the family of lines.
Sure
Q42Hausdorff second countable regular manifold: number of compatible metrics (0,1,finite,infinite)Topology and DGD
Answer: Infinite
Hausdorff, 2nd countable, regular implies metrizable (Urysohn); if d is compatible so is c*d for any c>0 (also min(d,1) etc.), so infinitely many.
Sure
Q43N=pqr distinct primes: number of square-free divisorsNumber theoryC
Answer: 8
Square-free divisors of pqr: 2^3 = 8.
Sure
Q44F_n(x)=nx/(1+n^3x^2)-type series on [0,inf): uniform convergence and interchange with integralAnalysisD
f_n = nx/(1+n^2x^2): sum_n f_n(x) ~ sum 1/(nx) diverges for x>0, so the series is not uniformly convergent (not even pointwise); term-by-term integration over [0,inf) invalid (int f_n diverges, sup f_n=1/2 not -> 0).
Fairly sure
Two solves agree. B is arguable only in the sense that both sides are infinite.
Q45BVP converted via Green's function to symmetric Fredholm equation; smallest lambda where resolvent kernel failsMethodsA
Answer: pi^2
EL: y''=lambda*y, Dirichlet Green's function of -d^2/dx^2 has characteristic values n^2 pi^2; smallest is pi^2 (sign convention aside, only positive options).
Fairly sure
sign of lambda (true value -pi^2 under one convention); chose smallest positive characteristic value pi^2.
Q46Non-orthogonal eigenvectors under dot product: what ensures orthogonality (mass matrix)MechanicsA
Answer: Mass matrix
Small oscillations: V a = w^2 T a; eigenvectors are orthogonal w.r.t. the mass (kinetic) matrix T, not the plain dot product (that needs T = identity).
Fairly sure
could be read as "identity matrix" (C) if the question means standard dot product works only when T = I; chose mass matrix.
Q47Phi: Z[x] to Z6, f maps to f(1) mod 6: determine kernelAlgebraB
Answer: 6, x-1
ker = {f : f(1) = 0 mod 6} = (6, x-1) (not principal).
Sure
Q48X=1003*997, find x mod 13Number theory · Likely bonus marksB
Answer: 3
1003 = 2, 997 = 9 mod 13, product 18 = 5 mod 13 (also 10^6 - 9 = 1 - 9 = 5). Options are 1,3,12,9, none is 5. Guess 3 (would arise from slip 10^6 = -1).
Not sure
Misprint: 1003×997 ≡ 5 (mod 13), which is not an option. 3 is what the same product gives mod 11, so the options may belong to a different modulus.
Q49Curve with k=1, tau=0: dimension of minimal ambient manifold containing itTopology and DGC
Answer: 2
k=1, tau=0 constant: plane circle; minimal ambient dimension 2.
Sure
Q50{A,H}=0 implies what about AMechanicsB
Answer: A conserved
dA/dt = {A,H} + dA/dt(explicit) = 0 for time-independent A.
Sure
Q51U_tt - c^2 u_xx transforms to canonical form using which methodODE and PDEB
Answer: characteristic variables
Hyperbolic u_tt - c^2 u_xx=0: xi=x-ct, eta=x+ct gives u_xi_eta=0.
Sure
Q52Count elements of F27 satisfying x^n=1 (exponent OCR-uncertain)AlgebraD
Answer: 13
F27* cyclic of order 26 = 2*13; number of x with x^13=1 is gcd(13,26)=13.
Sure
Q53Evaluate (2/97)Number theoryA
Answer: 1
97 = 1 mod 8, so (2/97) = +1.
Sure
Q54Every sequence has a Cauchy subsequence: which of compact/totally bounded/complete must holdTopology and DGB
Answer: totally bounded only
Every sequence has a Cauchy subsequence iff X totally bounded; completeness/compactness not implied (e.g. (0,1)).
Sure
Q55H=p^2/2m+V(q): condition on motion for action-angle variablesMechanicsC
Answer: periodic, bounded
Action-angle variables exist for bounded periodic (libration/rotation) motion.
Sure
Q56Group of order p^3 with center of order p: number of conjugacy classesAlgebra · Likely bonus marksB
Answer: p^2
|G|=p^3, |Z|=p: non-central classes have size p, count (p^3-p)/p = p^2-1, total p^2+p-1 (D4: 5). Not among options (p+2, p^2, p+1, p^3); p^2 is nearest.
Not sure
Misprint: the true count is p²+p−1 (5, 11, 29 for p = 2, 3, 5, checked by brute force). p² is the nearest option.
Q57Eigenvalues of y''+lambda y=0, y(0)=y(pi)=0ODE and PDEC
Answer: n^2
y=sin(nx), lambda=n^2, n=1,2,...
Sure
Q58Surface z=x^2+y^2: Gaussian curvature at origin via fundamental formsTopology and DGC
Answer: 4
K=(f_xx f_yy - f_xy^2)/(1+f_x^2+f_y^2)^2 = 4 at origin.
Sure
Q59Gauss-Seidel on diagonally dominant PDE-discretised system: convergence rate vs JacobiOR and numericalB
Answer: Faster
For diagonally dominant systems GS typically converges faster than Jacobi (rho_GS = rho_J^2 for consistently ordered matrices).
Fairly sure
"faster" is the usual textbook claim, not universal for all diagonally dominant systems.
Q60F of bounded variation and in L^2: strongest valid conclusion on Fourier coefficients a_nAnalysisB
BV implies a_n = O(1/n); f in L2 gives sum |a_n|^2 < inf. A fails (sawtooth a_n ~ 1/n), C fails, D fails for discontinuous BV.
Sure
Q61Sphere in inviscid flow, add slight viscosity (Navier-Stokes): qualitative drag correctionMechanicsB
Answer: becomes finite
Inviscid: d'Alembert paradox, drag 0. Small viscosity (Navier-Stokes) gives finite drag (Stokes drag 6 pi mu a U at low Re).
Sure
Q62Curve (t,t^2,t^3): curvature at t=0 via Frenet formulasTopology and DGC
Answer: 2
gamma'=(1,0,0), gamma''=(0,2,0) at t=0: k = |gamma' x gamma''|/|gamma'|^3 = 2.
Sure
Q63N<=50 maximising divisor count: which nNumber theory · Likely bonus marksA
Answer: 30
Square-free n<50 with max divisors (8): both 30 and 42 qualify (36, 48 not square-free). Two correct; picked 30.
Not sure
Defective: 30 and 42 are both square-free with 8 divisors, so A and D are both correct.
Q64G acts on itself by conjugation, all classes of size 1: G is abelian/trivial/simple/cyclicAlgebraA
Answer: abelian
All conjugacy classes of size 1 iff G abelian (not necessarily trivial, cyclic or simple).
Sure
Q65Primal has alternate optima, dual region non-degenerate: implication for dualOR and numerical · Likely bonus marksA
Answer: dual unique solution
Primal alt. optima iff dual degenerate at optimum, so hypothesis (non-degenerate dual) is contradictory; dual is feasible and bounded (C, D out). Between unique/multiple, picked unique.
Not sure
The premises contradict each other: primal alternate optima force a degenerate dual. Both solves picked A.
Q66Y'=y^(1/3), y(0)=0: unique, none, infinitely many, or exactly two solutionsODE and PDEC
y'=y^(1/3), y(0)=0: y=0 and y=±(2x/3)^(3/2) all work, and can be glued at any x0 (non-Lipschitz at 0) -> infinitely many solutions
Sure
Q67Two-degree system with kinetic and potential coupling, after diagonalisation one eigenfrequency: meaningMechanicsB
A zero eigenfrequency means zero restoring force for that normal mode: free (rigid-body) motion mode
Sure
Q68Evaluate improper integral of sin x / x (type) using Fourier-transform reasoningMethodsD
Int_0^inf sin x / x dx = pi/2 (Dirichlet integral, via Fourier transform of rectangular pulse)
Sure
Q69Modified Euler unstable on stiff equation; condition for stability improvement with RK4OR and numericalD
RK4 has a bounded stability region (|h*lambda| < ~2.78) so stiff problems need small step size to stay stable
Sure
Q70F analytic and conformal at z0 implies f'(z0) nonzeroComplexA
Analytic and conformal at z0 iff f'(z0) != 0
Sure
Q71Torque-free rotation about intermediate axis (I1<I2<I3): long-term behaviourMechanicsB
Rotation about intermediate axis is unstable (Euler eqns linearised give exp. growth, sqrt of positive product) -> exponential divergence (tumbling)
Sure
Q72Number of generators of cyclic group of order 105AlgebraB
phi(105)=phi(3)phi(5)phi(7)=2*4*6=48 (checked)
Sure
Q73Evaluate (-1/p) for p=101Number theoryA
(-1/p)=(-1)^((p-1)/2); p=101, (p-1)/2=50 even -> 1 (checked)
Sure
Q74Classify PDE x^2 u_xx + 2xy u_xy + ... (elliptic/hyperbolic/parabolic/mixed)ODE and PDEC
A=x^2, B=xy, C=y^2: B^2-AC = x^2y^2 - x^2y^2 = 0 -> parabolic
Sure
Q75F,g independent in V*; T(x)=(f(x),g(x)): rank of T and dim kernelLinear algebraA
f,g independent -> T onto R^2, rank 2; nullity n-2
Sure
Q76Extremal of integral (y'^2+y^2) with constraint integral y=1 via Lagrange multiplier: formMethodsC
Euler-Lagrange: y''-y=-lambda/2 gives y=lambda/2 + c1 e^x + c2 e^-x; with free ends (natural BC y'(0)=y'(1)=0) c1=c2=0 so y=constant=1 (int y=1). Constant is the only option that can occur
Fairly sure
No boundary conditions are printed. C assumes natural boundary conditions.
Q77Number of zeros of z^5-2z+1 (OCR-uncertain) insideComplexA
z^5-2z+1=(z-1)(z^4+z^3+z^2+z-1). Roots (computed): 1, 0.5188, -1.2906, -0.114+-1.217i. Only 0.5188 is strictly inside |z|<1 (z=1 is on the circle). Rouche with -2z also gives 1
Sure
Q78Phase I objective zero, artificial variables in basis at zero level, dual feasible: implicationOR and numericalD
Phase I optimum 0 with artificials still basic at zero level, feasible original problem -> degenerate basic feasible solution
Sure
Q79(D^2-4D+4)z=0: form of complementary functionODE and PDEC
(D-2)^2 z=0 -> z=(A+Bx)e^(2x) (A,B arbitrary functions of y)
Sure
Q80Operator with rank(T)=2 and given eigenvalue data: find minimal polynomial, then trace and nullityLinear algebraA
T^2=T: min poly x^2-x; eigenvalues 1 (mult 2 = rank), 0 (mult 2). trace=2, nullity=4-2=2
Sure
Q81W=(z-i)/(z+i)-type map: image of real axisComplexA
z real: |z-i|=|z+i| so |w|=1: image is the unit circle
Sure
Q82Degenerate kernel gives singular linear system: condition for existence of solution (rank condition)MethodsB
Singular (degenerate-kernel) system is solvable iff consistency holds: rank A = rank [A|b] (Fredholm alternative). "Rank condition satisfied" is the only sensible option
Sure
Q83Hermitian A with A^2=I, 4-dim, equal multiplicities: eigenvalues, orthogonal decomposition, traceLinear algebraA
A^2=I, A Hermitian -> eigenvalues +-1; dim 4 with equal multiplicity (2 each) -> trace = 2-2 = 0
Sure
Q84Q=x^2+y^2+z^2+2xy+2yz+2zx: matrix form, diagonalise, rank and indexLinear algebraB
Q=(x+y+z)^2: matrix of all ones, eigenvalues 3,0,0 -> rank 1, positive index 1 -> (1,1)
Fairly sure
Rank 1. If "index" means the number of negative squares, the answer is A (1,0).
Q85Value of Legendre symbol (29/43)Number theoryB
29=1 mod 4: (29/43)=(43/29)=(14/29)=(2/29)(7/29); 29=5 mod 8 so (2/29)=-1; (7/29)=(29/7)=(1/7)=1 -> -1 (checked: 29^21 = -1 mod 43)
Sure
Q86F analytic,ComplexA
f(z)/z^3 is entire and bounded by 1 (f has zero of order >=3 at 0) -> constant by Liouville: f=a z^3, |a|<=1
Sure
Q873x3 real symmetric with char poly (x-1)^2(x-3): diagonalizable?, quadratic form signatureLinear algebraC
Real symmetric -> diagonalizable; eigenvalues 1,1,3 all positive -> signature (3,0)
Sure
Q88Which gives higher accuracy for smooth functionsOR and numericalD
Hermite uses derivative data as well as values (osculatory, higher accuracy for smooth functions)
Sure
Q89Volterra kernel not of convolution type: correct way to apply Laplace transformMethodsB
Laplace transform needs convolution kernel; a variable transformation (e.g. t=e^s) can convert some non-convolution kernels (Mellin type) to convolution form. Other options (ignore kernel, convert to PDE, direct transform) are not valid approaches
Not sure
poorly posed; B (variable transformation) is a best guess.
Q90Ax=b, A is 3x4 rank 2, consistent: dimension of solution space, null spaceLinear algebraC
A is 3x4 rank 2: nullity 4-2=2; solution set = particular + 2-parameter null space, dimension 2
Sure
Q91Y'' + (1/x)y' type equation for x>0: evaluate W(x)*x (constant, x, x^2, 1/x) (OCR-uncertain)ODE and PDEA
Homogeneous y'+y/x=0 -> y1=c/x, Wronskian (Abel) W=c/x so W*x=constant
Sure
Q92Integral over [0,2pi] of 1/(2+sin theta) d theta (OCR-uncertain)ComplexA
Integrand = d/dtheta ln(2+sin theta); over full period integral = 0
Sure
Q93RCF has companion blocks of degrees 2 and 3: dimension, minimal and characteristic polynomialLinear algebraD
Companion blocks of degree 2 and 3 -> dimension 2+3=5
Sure
Q94Unbalanced transportation, non-degenerate BFS, MODI shows multiple optima: geometric implicationOR and numericalB
Multiple optima in transportation LP: optimal solutions lie on an edge (face) of the feasible polytope
Sure
Q95F analytic in simply connected domain, integral over every closed curve is 0, then f is ...ComplexD
Integrand is conj(f(z)) (bar visible in image). Green: closed integral of conj(f) dz = 2i * Double integral conj(f') dA = 0 for all C -> f'=0 -> f constant. (If bar were absent, answer would be trivially "analytic")
Sure
The overbar on f(z) is printed in both English and Hindi. Without it, every analytic f would work.
Q96Compact connected manifold: number of connected componentsTopology and DGD
Connected means exactly 1 component
Sure
Q97Y(x)=lambda * integral of G(x,t)y(t)dt: condition for nontrivial solution via operator theory (OCR-uncertain)MethodsD
Homogeneous Fredholm eqn y=lambda K y has non-trivial solution iff lambda is an eigenvalue i.e. Fredholm determinant D(lambda)=det(I-lambda K)=0
Sure
Q98A^3-3A^2+2A=0: eigenvalues, invertibility, rank of A(A-I)Linear algebraA
Char eq of 3x3 A: x(x-1)(x-2) so eigenvalues 0,1,2 (A singular, not invertible). A(A-I) has eigenvalues 0,0,2 -> rank 1
Sure
Q99Curve with constant nonzero curvature and torsion: circle/line/helix/ellipseTopology and DGC
Constant nonzero curvature and torsion -> circular helix (Frenet equations); circle has zero torsion, line zero curvature
Sure
Q100F=u+iv entire with u^2+v^2 = exp(2x+...) (OCR-uncertain) and f(0)=1: find fComplex · Likely bonus marksD
Option D is NOT visible in the photo (cut off). Math: |f'|^2 = (1/4)Laplacian(e^2x+y^2) = e^2x+1/2, but log|f'|^2 would have to be harmonic (f' zero-free), and it is not -> no entire f exists. A, B, C all fail (|e^z|^2=e^2x; |e^z+iz|^2 and |e^z+z|^2 have extra cross terms). Guess D = "no such function"
Not sure
Option D is cut off in the photo. No entire function satisfies the condition and A–C all fail, so D is right only if it reads "none of these". The second solve guessed A.