GATE 2026 MA – Previous Year Questions with Solutions
All 65 questions of GATE 2026 MA with answers and explanations. Read each question, then practise it by topic or as a full paper.
Practise this paper in The GATE Grind →
- Question 1 – Suresh said, “I did it yesterday.” Choose the correct indirect speech.
- Question 2 – Which tile continues the shown sequence?
- Question 3 – The gallery walkways are the lines shown, and black dots are junctions. A guard at a junction watches every walkway incident to it. Find the minimum…
- Question 4 – June 2 is a Thursday. What weekday is July 3 in the same year?
- Question 5 – H denotes a head-up coin and T a tail-up coin. Six coins begin HHHTTT and must become TTTHHH, as shown. A step swaps two adjacent coins without…
- Question 6 – Exacerbate : Mitigate :: . Choose the pair with the same relationship.
- Question 7 – Fold Panel I along dashed lines into a cube, with shading outside. Which Panel II cubes can result?
- Question 8 – Patients with high cholesterol also have high BP. Some high-BP patients have diabetes, but nobody has both cholesterol and diabetes. There are 75…
- Question 9 – Four people P,Q,R,S have distinct ages. P says “I am younger than S.” Q says “I am neither youngest nor oldest.” R says “P is older than me.” Who is…
- Question 10 – Three externally tangent circles C₁,C₂,C₃ have centers O₁,O₂,O₃ and radii r₁ = 2 cm, r₂ = 1 cm, r₃. The shown angle O₁O₃O₂ is 90°. Find r₃ in cm.
- Question 11 – Let M₃(ℝ) be all real 3×3 matrices and T₃(ℝ) its upper-triangular subspace, with usual operations. To which listed real vector space is M₃(ℝ)/T₃(ℝ)…
- Question 12 – Reverse the integration order in I= 01 0 ydx\,dy+ 12 y-11dx\,dy.
- Question 13 – Let C be the Cantor set. On [0,1], define f(x) = x²⁰²⁶ on C; cos(πx) on [0,1/2]∖C; sin(πx) on [1/2,1]∖C. Find its Lebesgue integral.
- Question 14 – A group G has order 595. Which statement is true?
- Question 15 – On the real line, u t=ku xx, t > 0, k ∈ ℝ, with u(x,0)=e-a x , a > 0. For Fourier convention u(w,t)= -u(x,t)eiwxdx, find the transform.
- Question 16 – f: R2 R2 is f(x,y)=(ex y,ex y). Which statement is true?
- Question 17 – f: C C is f(z)= z 2-5 z+2. At which listed point is f complex differentiable?
- Question 18 – Let D = z ∈ ℂ: z < 1. An analytic map f:D→D has f(0) = 0. Which is a possible f′(0)?
- Question 19 – X = C¹([0,1],ℝ). Define p 1(x)= x(t) , p 2(x)= x(t) , and p₃ = p₁+p₂. Which statement is true?
- Question 20 – Let d₁,d₂,d₃ be metrics on a set X with at least two elements. Which formula does NOT define a metric?
- Question 21 – P is real symmetric positive definite, 3×3. Which statements about QTPQ are true?
- Question 22 – f(x,y)=xx2+y2/ x for x ≠ 0 and 0 for x = 0. Which statements hold at (0,0)?
- Question 23 – Over ℤ₁₁, let p(x) = x²+x+4. Which statements are true?
- Question 24 – Give ℝ the cofinite topology =\A R:Ac is finite\\\. Which statements hold?
- Question 25 – For distinct a,b > 0, consider af xx+bf yy=8f. Which pairs (ξ,η) make f=e2 x+ y a solution?
- Question 26 – Assign tasks X,Y,Z to agents P,Q,R, one each. Cost rows (X,Y,Z) are P:(4,2,8), Q:(2,3,7), R:(3,1,6). Which statements hold for the optimum?
- Question 27 – For 2x(x-2)2y+3xy+(x-2)y=0, which statements about x = 2 are true?
- Question 28 – T reflects ℝ³ through a two-dimensional subspace. P is its matrix in the standard basis. Find 2 trace(P) − 3 det(P) (integer).
- Question 29 – Find the degree over ℤ₃ of the field extension Z 3[x]/ x3-x+1 (integer).
- Question 30 – For y′ = x+y, y(0) = 1, use modified Euler with h = 0.05. Find the second approximation to y(h) to two decimals.
- Question 31 – = n 01n2+( ex)n7n2+x8\,dx. Find 14α (integer).
- Question 32 – On ℝ³ with ℓ¹ norm, T has matrix pmatrix1&1&3\\2&2&-2\\1&3&-3pmatrix. Find its operator norm (integer).
- Question 33 – Maximize z = 10x₁+20x₂, subject to x₁ ≤ 36, x₂ ≥ 42, x₁+x₂ ≥ 48, 5x₁+x₂ ≤ 150, x₁,x₂ ≥ 0. Find maximum z (integer).
- Question 34 – How many distinct topologies on 1,2,3 have exactly four open sets? (integer)
- Question 35 – The series n=0 a n(z-2)n converges at z = 5 and diverges at z = −1. Find its radius of convergence (integer).
- Question 36 – Solve y=pmatrix-3&4\\-2&3pmatrixy, y(0)=pmatrix1\\2pmatrix.
- Question 37 – Define x,y P=xTPy on ℝ³ with P = diag(1,9,16). For V = span(1,0,0)ᵀ,(1,1,1)ᵀ, choose an orthonormal basis.
- Question 38 – Evaluate (1-z)(3 z+5 z)1-z101\,dz, where Γ is z = 7/9, counterclockwise.
- Question 39 – On which domain does f(z)= n=15-n(nz) represent an analytic function?
- Question 40 – Let δ be the unit impulse (zero away from its point and total integral 1). Solve 2y+y+2y=(x-5), y(0) = y′(0) = 0. Which y(10) is correct?
- Question 41 – Which ring under usual addition and multiplication is a unique factorization domain?
- Question 42 – On ℓ∞ with supremum norm, T(x 1,x 2,)=(x 1,x 2/2,x 3/3,). Which statement holds?
- Question 43 – u tt=c2u xx on ℝ, t > 0,c > 0, with initial displacement 1 for x < 1 and zero otherwise, and initial velocity zero. Find maximum u(0,t) for t > 0.
- Question 44 – T is harmonic in 0 < x,y < 1, with T(x,0)=x, T(0,y)=y, T(x,1)=1+x, T(1,y)=1+y. Find T(1/2,1/3).
- Question 45 – For subsets E₁,E₂ of a normed space, define E₁+E₂ = x+y:x∈E₁,y∈E₂ and E₁×E₂ as their Cartesian product. Which statement is NOT true?
- Question 46 – Transportation cell costs are c₁₁=10,c₁₂=12,c₂₁=11,c₂₂=13. Multipliers satisfy α₁=x, α₂=x+1, β₁=y, β₂=y+2. Given x=4,c₁₃=19,β₃=y+5, find y and reduced…
- Question 47 – A map f:X→Y is continuous and bijective. Which condition guarantees it is a homeomorphism?
- Question 48 – Nonzero real 3×3 matrices P,Q are simultaneously upper triangularizable by the same nonsingular S. Let K=PQ−QP. Which statements hold?
- Question 49 – S= n=1\(x,y) R2:(x-n)2+y2=1/n2\ with the usual topology. Which statements hold?
- Question 50 – On D = z <1, set f(z)=z/(1-z) and g(z)=(1+z)/(1-z). S₁: f(D)⊆g(D); S₂: g(D)⊆f(D). Which statements hold?
- Question 51 – Let α,β be Frobenius indicial roots at zero for 2x2y-xy+(1-x2)y=0. Which statements hold?
- Question 52 – T: Q[x] Q[x] sends p(x) to p(x²). Which statements hold?
- Question 53 – For a normed linear space X and its continuous dual X′, which statements hold?
- Question 54 – Let G=ℤ₃₀⊕ℤ₁₂ under addition. Which statements hold?
- Question 55 – Let J=pmatrix2&1&0&0&0&0\\0&2&0&0&0&0\\0&0&2&1&0&0\\0&0&0&2&0&0\\0&0&0&0&3&1\\0&0&0&0&0&3pmatrix. Find geometric multiplicity of eigenvalue 2…
- Question 56 – = S F n\,dS, with F=(2x+3z) i+(xz-y) j+(y2+2z) k. S is the radius-9 sphere centered at (3,−1,2), with outward normal. Find α/(36π) (integer).
- Question 57 – Maximize z=xTpmatrix2&0&0\\0&2&1\\0&1&2pmatrixx over real vectors with xᵀx=1. Find the maximum (integer).
- Question 58 – P₃(ℝ) contains real polynomials of degree at most three. T:P₃(ℝ)→ℝ² sends p to (p(1),p′(1)). Find nullity (integer).
- Question 59 – Count zeros, with multiplicity, of z6+5z3+4z+11 in 1 < z < 3 (integer).
- Question 60 – On real L²[0,π], Tf(x)= x 0 f(t) t\,dt+ x 0 f(t) t\,dt. Find (4/π)‖T‖ (integer).
- Question 61 – u is harmonic in the open unit disk and equals x²−y² on its boundary. Find 4[u(1/2,0)-u(0,1/2)] (integer).
- Question 62 – X=5,6,7,8,9,10 has topology ∅,X,5,6,7,8,9,10. How many subsets are neither open nor closed? (integer)
- Question 63 – (4,0,2,β) is optimal for minimizing x₁+3x₂+2x₃−αx₄, with 4x₁+x₂+x₃=18, −3x₁+2x₃+x₄=2 and all variables nonnegative. For real α,β, find maximum 22(α+β)…
- Question 64 – On D=[0,2]×[0,2], let f(t)=⌈t⌉, the smallest integer at least t. Find D f(x+y)dx\,dy (integer).
- Question 65 – Apply Jacobi to pmatrix1&2&1\\0&2&2\\1&1&1pmatrixx=pmatrix2\\2\\2pmatrix starting x⁽⁰⁾=(0,0,0)ᵀ. Find x₁⁽²⁾+x₂⁽²⁾+x₃⁽²⁾ (integer).