GATE 2025 MA – Previous Year Questions with Solutions
All 65 questions of GATE 2025 MA with answers and explanations. Read each question, then practise it by topic or as a full paper.
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- Question 1 – Ravi had younger brother who taught at university. He was widely regarded as honorable man. Choose the correct articles.
- Question 2 – The CEO’s workforce reduction was considered myopic because it sacrificed long-term stability for short-term gains. Choose a replacement for “myopic”…
- Question 3 – A class average was 30.8, with one mark incorrectly entered as 24 instead of 42. Correcting it raises the average to 31.4. How many students are…
- Question 4 – P is Q’s brother; S is Q’s daughter; T is S’s sister; R is Q’s mother. Statements: (1) R is S’s grandmother; (2) P is uncle to S and T; (3) R has only…
- Question 5 – According to the representative campus map, which statement is correct?
- Question 6 – G.K. Chesterton, in “A Piece of Chalk”, describes putting brown paper and chalks in his pocket. He reflects that pocket objects are primeval and…
- Question 7 – In the representative diagram QR ∥ ST. Their perpendicular separation is half the distance from P to QR. Find area(PST)/area(SQRT).
- Question 8 – A fair six-sided die is rolled three times. Find probability of exactly one 6.
- Question 9 – The square paper is folded along dotted lines as in I–III, then cut as in IV. Which pattern results on unfolding?
- Question 10 – A shop has four distinct ice-cream flavors. Any number of scoops per flavor is allowed and purchase order is irrelevant. How many purchases contain…
- Question 11 – S is the set of unit vectors w∈ℝ³ for which A=(1,-3,2)TwT is diagonalizable. Which statement holds?
- Question 12 – Let Y(s) be the Laplace transform of Bessel J₀(x), with transforms of J₀ and its first two derivatives assumed to exist. Which relation holds for s>0?
- Question 13 – For x³+4x²−10=0 in (1,3/2), consider S1: x k+1=10/(4+x k) converges for every initial x₀ in that interval. S2: x k+1=1210-x k3 diverges for some such…
- Question 14 – Maximize Z=2x₁+4x₂+4x₃−3x₄ with αx₁+x₂+x₃=4, x₁+βx₂+x₄=8, x≥0. S1: for α=2,β=1, (x₁,x₂) is an optimal basis. S2: for α=1,β=4, (x₃,x₂) is an optimal…
- Question 15 – In ℝ⁴ define S by x₁²+x₂²+x₃²−x₄²=0, T by the same expression=1, U by it=−1. Which statement holds?
- Question 16 – For X′=MX with M any real skew-symmetric 6×6 matrix, when is the origin stable?
- Question 17 – X=f∈C[0,1]:f(0)=f(1)=0 with sup norm; Y=C[0,1] with L² norm. On their closed unit balls U X,U Y, define Tf=Sf=∫₀¹f(t)dt. S1: sup over U X of Tf is…
- Question 18 – g(x,y)=f(x,y)e(2x+3y), where f is a nonzero C¹ homogeneous function of degree four on ℝ². On which listed set does xg x+yg y=0 hold?
- Question 19 – Classify (1+x2)u xx+2x(1-y2)u xy+(1-y2)u yy+xu x+(1-y2)u y=0.
- Question 20 – u t=u xx on 0<x<π,t>0, with u(0,t)=u(π,t)=0 and u(x,0)=sin4x cos3x. Find u(π/4,t).
- Question 21 – F:ℝ²→ℝ² is F(x,y)=(x3-3xy2-3x,3x2y-y3-3y). Where is the inverse function theorem applicable?
- Question 22 – f(x₁,x₂)=(x₁−x₂)² and g(x₁,x₂)=2x₁²+2x₂²−x₁x₂. S1: f⁻¹(K) is compact for every compact K⊂ℝ. S2: g⁻¹(K) is compact for every such K. Choose the truth…
- Question 23 – For which invertible M is the difference of characteristic polynomials p M(x)−p (M⁻¹)(x) constant?
- Question 24 – Balanced transportation has cost rows S1:(2,6,20,11), S2:(12,7,4,10), S3:(8,14,16,11), supplies (α+10,α+λ+10,5) and demands (α+5,10,λ+5,α+λ), α,λ>0.…
- Question 25 – S1=x≥0:2x₁+x₂≤4,x₁+2x₂≤5; S2=x≥0:2x₁−x₂≤5,x₁+2x₂≤5. Which statements hold?
- Question 26 – For (x,y)≠(0,0), f=(x2-y2)/(x2+y2)+x[1/(x2+y2)]. S1: lim (x→0) lim (y→0) f exists; S2: lim (y→0) lim (x→0) f exists; S3: joint limit at origin exists.…
- Question 27 – A real 7×7 M has characteristic polynomial (x−1)α(x−2)β(x−3)², α>β. Each of M−I, M−2I, M−3I has rank 5. Find m M(5), where m M is the minimal…
- Question 28 – Legendre P n satisfies (1−x²)y″−2xy′+n(n+1)y=0, with P n(1)=1. Find P₁₁′(1) (integer).
- Question 29 – At P=(1,1,√2), a unit vector a is tangent to the intersection 2x²+3y²−z²=3 and x²+y²=z². For f=x²+2y²−2√11 z, find the absolute directional derivative…
- Question 30 – Find the volume common to cylinders x²+y²≤4 and x²+z²≤4, to two decimals.
- Question 31 – W is the real vector space of all bounded real-valued solutions on ℝ of y⁽⁴⁾+2y″+y=0. Find dim W (integer).
- Question 32 – F=(y−z)i+(z−x)j+(x−y)k. S is the outward upper hemisphere x²+y²+(z−1)²=9, 1≤z≤4. Find -1 S( F) n\,dS (integer).
- Question 33 – I=(2 i)-1 C z/[1-(z3)]\,dz, where C is the positively oriented diamond Re z + Im z =1. Find 120I (integer).
- Question 34 – Quadrature ∫₋₁¹f(x)dx=αf(−1)+βf(1)+γf′(−1)+δf′(1) is exact through degree three. Find 9(α²+β²+γ²+δ²) (integer).
- Question 35 – Use Euler with h=0.5 for y′=sin(π(x+y)), y(0)=0. Find approximate y(1.5)+2y(1) (integer).
- Question 36 – For Ax=b with nonzero diagonal D, D-1A=pmatrix1&1&-2\\3&1&2\\1&1&1pmatrix and D⁻¹b=(4,4,1)ᵀ. S1: Jacobi from x⁽⁰⁾=(1,1,1)ᵀ gives x⁽¹⁾=(5,−1,−1)ᵀ. S2:…
- Question 37 – x²y″+7xy′+9y=x⁻³ ln x for x>0, with y(1)=y′(1)=0. Find y(e).
- Question 38 – Independent solutions of (1+x²)y″−xy′+cos²x y=0 satisfy y₁(0)=3,y₁′(0)=−1,y₂(0)=−5,y₂′(0)=2. Let W=y₁y₂′−y₂y₁′. Find W(1/2).
- Question 39 – C is the intersection z²=x²+y² and 4x+z=7. A point P on C has minimum distance to the xy-plane. Find its distance from the origin.
- Question 40 – u tt−9u xx=0 on ℝ,t>0, with u(x,0)=eˣ and u t(x,0)=sinx. Find u(π/2,π/6).
- Question 41 – A Möbius transformation sends 0,1/2,1 to −3,∞,2. It maps the circle centered at 1 with radius k to line αx+βy+γ=0. Find [2k(α+β)+γ]/[α+β−2kγ].
- Question 42 – Let U=Im z>0, D= z <1. Over all bijective analytic f:U→D with f(i)=0, find sup f(4i) .
- Question 43 – On a nonempty connected open Ω⊂ℂ, f is nonconstant. S1: if f is continuous and f² analytic, f is analytic. S2: if f² and f³ are analytic, f is…
- Question 44 – On the unit sphere S⊂ℝ³ identify points exactly when their third coordinates agree. Give the equivalence-class space the quotient topology. To which…
- Question 45 – Let W be the real span of all Legendre polynomials. X=C[−1,1] with sup norm, Y=C[−1,1] with L² norm. Where is W dense?
- Question 46 – X=ℝ and Y=[0,1], each with the usual distance. In which space is [0,1/4) open?
- Question 47 – K is algebraically closed and contains a finite field F. L consists of elements of K algebraic over F. S1: L is algebraically closed. S2: L is…
- Question 48 – T:M₂(ℝ)→M₂(ℝ) sends X to AXB, with A=pmatrix1&-2\\1&4pmatrix, B=pmatrix6&5\\-2&-1pmatrix. P is its matrix in the standard basis. Which statements…
- Question 49 – Maximize Z=3x₁+5x₂ with x₁+x₃=4, 2x₂+x₄=12, 3x₁+2x₂+x₅=18, x≥0. Optimal basis is (x₃,x₂,x₁), with inverse B-1=pmatrix&&-\\0&&0\\0&-&pmatrix. Let…
- Question 50 – For 0<α<1, Cα[0,1] is the Hölder Banach space with norm f(0) +sup (s≠t) f(t)−f(s) / t−s α. T:Cα→C[0,1] is inclusion into sup norm. Which statements…
- Question 51 – On ℝ,t>0, u t+3u x=u and v t+3v x=v², both with initial data cosx. Which statements hold?
- Question 52 – u t=2u xx on 0<x<1,t>0 with zero endpoint values and u(x,0)=2x(1−x). Which statements hold?
- Question 53 – f:ℝ²→ℝ² is (e2 x2 y,e2 x2 y). Which statements hold?
- Question 54 – An infinite orthonormal set x k lies in a real Hilbert space X. Let Y=spanx₁,…,x n and S nx= k=1n x,x k x k. Which statements hold?
- Question 55 – On [0,1], f₁(x)=x/2 and f n+1=f n-12(f n2-x). Which statements hold?
- Question 56 – For x∈(0,π), u n(x)=sin(nx)/√n. Which statements about the series hold?
- Question 57 – A nonconstant continuous f:S¹→ℝ is defined on the unit circle with usual topology. Which statements hold?
- Question 58 – X is uncountable with cocountable topology: ∅ is open and every set with finite or countable complement is open. Which statements hold?
- Question 59 – Rings are commutative associative with 1≠0 and unital homomorphisms. P1:R is a product of two nonzero rings. P2:R has nonzero idempotents r₁,r₂ with…
- Question 60 – E⊂F⊂K are nonalgebraic field extensions. α∈K∖F is algebraic over F. L is generated over E by coefficients of the monic minimal polynomial of α over F.…
- Question 61 – Under ⟨f,g⟩=∫₋₁¹f(x)g(x)dx, polynomial p(x)=α+βx²−30x⁴ is orthogonal to every polynomial of degree at most three. Find α+5β (integer).
- Question 62 – M is the symmetric matrix of Q=2x₁²+2x₂²+3x₃²+4x₁x₂+2x₁x₃+2x₂x₃. For any nonzero Y∈ℝ³, a n=YT(M+I)n+1Y/[YT(M+I)nY]. Find lim a n (integer).
- Question 63 – Distinct nonzero real α,β and a polynomial Q of degree <5 define f(z)=[6( z)-6(e2 z-Q(z))]/z6. The source says f “satisfies Morera’s theorem in ℂ∖0”.…
- Question 64 – In a group G with identity e, g≠e, g²=e, h≠e, h²≠e and ghg⁻¹=h². Find the least positive n with hⁿ=e (integer).
- Question 65 – On ℝ² let d₁ be Manhattan distance and d₂=d₁/(1+d₁). The d₁-open ball centered at zero with radius 1/7 equals the d₂-open ball with radius 1/α. Find α…