GATE 2024 MA – Previous Year Questions with Solutions
All 65 questions of GATE 2024 MA with answers and explanations. Read each question, then practise it by topic or as a full paper.
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- Question 1 – Increasing intensity drizzle → rain → downpour is analogous to → quarrel → feud.
- Question 2 – All heroes are winners; all winners are lucky. Inferences: I all lucky people are heroes; II some lucky people are heroes; III some winners are…
- Question 3 – A student should multiply positive real p by positive real q, but instead divides p by q. Percentage error is 80%. Select q.
- Question 4 – Divide the sum of the first twenty consecutive positive odd numbers by 20². Find the result.
- Question 5 – The girls:boys ratio in class VIII equals boys:girls in IX. Class totals are 450 and 360, with equal girl counts in both classes. Find girls per…
- Question 6 – Fill (i)–(iv): “Yoko Roi stands (i) as an author for standing (ii) as an honorary fellow, after she stood (iii) her writings that stand (iv) the…
- Question 7 – Seven identical cylindrical chalk sticks fit tightly in a cylindrical container as shown. Container and sticks have equal length. Find…
- Question 8 – The graph relates cardiovascular-disease mortality risk to daily steps. Which statement follows from the plotted data?
- Question 9 – Five equal cubes and one smaller cube are assembled in Figure A. View X gives Figure B. Which shown planar image is the view from Y?
- Question 10 – A cube is held with a body diagonal vertical. Find the least positive rotation about that axis leaving its appearance unchanged.
- Question 11 – An entire f satisfies f(z) =1 whenever Im z=0. Which statement is correct?
- Question 12 – C is the ellipse z−2 + z+2 =8 oriented counterclockwise. Evaluate C z2/(z2-2z+2)\,dz.
- Question 13 – For A⊆X in any topological space, write int, ∂ and overbar for interior, boundary and closure. Which inclusion need not hold?
- Question 14 – Evaluate the source limit 0-1 0 e-x/[3x+x2+x+4]dx.
- Question 15 – In the subring ℝ[X²,X³] of ℝ[X], I: the ring is a UFD; II: it is a PID. Choose truth values.
- Question 16 – For a group of order 2024, n p is its Sylow p-subgroup count. Which statement always holds?
- Question 17 – I: Every compact Hausdorff space is normal. II: Every metric space is normal. Choose truth values.
- Question 18 – On ℤ give the topology with basis S(a,b)=an+b:n∈ℤ, a≠0. I: Every S(a,b) is both open and closed. II: The only connected set containing x is x. Choose…
- Question 19 – A=pmatrix0&2\\2&0pmatrix. On M₂(ℂ), T(B)=AB. Find T’s characteristic polynomial.
- Question 20 – A is a complex normal matrix. I: real eigenvalues imply A is Hermitian. II: unit-modulus eigenvalues imply A is unitary. Choose truth values.
- Question 21 – A=pmatrix2&-1&1\\1&2&-1\\-1&1&2pmatrix. For (A+εI)x=b, ε>0, I: Jacobi converges for every start. II: Gauss–Seidel converges for every start. Choose…
- Question 22 – For y′=f(x,y), use y n=y (n−1)+ak₁+bk₂, with k₁=hf(x (n−1),y (n−1)), k₂=hf(x (n−1)+αh,y (n−1)+βk₁). Which (a,b,α,β) gives order two?
- Question 23 – u t−4u xx=0 on 0<x<1,t>0, zero endpoint values, u(x,0)=sinπx. Let g(t)=∫₀¹u(x,t)²dx. Choose its behavior.
- Question 24 – y₁,y₂ are different solutions of y+(ex)y=(ex+1) on [0,1]. Select the general solution, constants real.
- Question 25 – Minimize 5x₁+2x₂ subject to 2x₁+x₂≤2, x₁+x₂≥1, x₁,x₂≥0. Find optimum.
- Question 26 – f:ℝ⁴→ℝ is differentiable and homogeneous of degree 3, with f(p)=6 at p=(1,1/2,1/3,1/4). Find 12f x₁(p)+6f x₂(p)+4f x₃(p)+3f x₄(p) (integer).
- Question 27 – How many nonisomorphic finite groups have exactly three conjugacy classes? (integer)
- Question 28 – For x,y>0, f(x,y)=(x²−y²,2xy). Let g be its local inverse near f(2,1). Find det Dg at f(2,1), to two decimals.
- Question 29 – Find GL₂(F₃) , the number of invertible 2×2 matrices over the three-element field (integer).
- Question 30 – In ℝ⁴ with standard inner product, W₁=span(1,0,0,−1) and W₂=span(2,1,0,−1). Find dim(W₁⊥∩W₂⊥).
- Question 31 – How many group homomorphisms ℤ/4ℤ→S₄ exist? (integer)
- Question 32 – P f is the degree-at-most-two interpolant to f at a−h,a,a+h, h>0. Find the largest degree d for which every polynomial g of degree d satisfies g″(a)=P…
- Question 33 – P f is the quadratic interpolant of f(x)=x² x at −1,0,1. Find sup on [−1,1] of f−P f , to two decimals.
- Question 34 – For positive x,y,z with xy+yz+zx=12, find maximum xyz to two decimals.
- Question 35 – Outward flux of F=(x³,y³,z³) through the unit sphere is απ. Find α to two decimals.
- Question 36 – D is the unit disk, H the upper half-plane. Over analytic f:D→H with f(0)=i/2, find sup f′(0) .
- Question 37 – On z =1, which function is a uniform limit of polynomials in z?
- Question 38 – Which condition suffices for f:[0,1]→ℝ to be Lebesgue measurable?
- Question 39 – On M₂(ℝ)≅ℝ⁴, g(A)=trace(A²). Classify the zero matrix.
- Question 40 – I: Some proper subgroup G of (ℚ,+) has finite quotient ℚ/G. II: Some subgroup G has quotient isomorphic to (ℤ,+). Choose truth values.
- Question 41 – X=ℝ/ℤ has the quotient topology from usual ℝ. I:X is compact. II:Removing any one point leaves X connected. Choose truth values.
- Question 42 – Five distinct unit vectors v₁,…,v₅ in ℝ⁷ have nonpositive integer pairwise inner products for unequal indices. N counts ordered pairs (r,s), including…
- Question 43 – For f(x)= x + x−1 + x−2 on [−1,2], which listed quadrature rule gives the exact integral?
- Question 44 – For y′=e(−y²)+1,y(0)=0, I:the IVP has a unique solution on ℝ. II:Every solution is bounded on its maximal existence interval. Choose truth values.
- Question 45 – A is a nondiagonalizable real 2×2 matrix with real eigenvalue λ and eigenvector v. Choose the general solution of y′=Ay, with arbitrary constants…
- Question 46 – On x,y>0, transform y²u xx−x²u yy+yu y=0 using η=x²/2, ξ=y²/2. It becomes (u -u )+(u +u )/(2)+(a u +b u )/(2)=0. Find (a,b).
- Question 47 – On finitely supported real sequences C₀₀ define f(x)=Σₙ₌₁∞x n/√n. I:f extends continuously linearly to ℓ¹. II:f extends continuously linearly to ℓ².…
- Question 48 – In real ℓ²(ℤ) with usual inner product, T sends x to y with y j=(x j+x (−j))/2. Which statements hold?
- Question 49 – X=ℝ² with norm x + y . On S=(x,0), f(x,0)=2x. A Hahn–Banach extension F agrees with f and preserves its norm. Which statements hold?
- Question 50 – The intervals [a,b),a<b form a topology basis on ℝ. Which statements hold?
- Question 51 – T,S:ℝ⁴→ℝ⁴ are nonzero and nonidentity; T²=T. Which statements hold?
- Question 52 – Let p₁<p₂ be the two fixed points of g(x)=eˣ−2, and x n=g(x (n−1)). Which statements hold?
- Question 53 – For y″+λy=0 on [0,π], y(0)=y′(0),y(π)=y′(π), select the listed eigenvalues.
- Question 54 – f(x,y)=(1−cos(x²/y²))√(x²+y²) for y≠0, and zero otherwise. Which statements hold at (0,0)?
- Question 55 – Let S=f n:n≥1⊂C[0,1], f n(x)=xe(−nx), with supremum metric. Which statements hold?
- Question 56 – T:ℝ⁴→ℝ⁴ has only eigenvalues 1,2, dim ker(T−I)=1 and rank(T−2I)=2. Which Jordan forms are possible?
- Question 57 – In real L²[−1,1], ‖f‖²=18/5, ∫f=2 and ∫xf=0. P₁ is the subspace of polynomials of degree at most one. Find inf (g∈P₁)‖f−g‖² to two decimals.
- Question 58 – Maximize 10x+6y−8z with 5x−2y+6z≤20, 10x+4y−6z≤30, x,y,z≥0. Find maximum to two decimals.
- Question 59 – K is the splitting field over ℚ of (X²−2)(X²−3). How many fields F satisfy ℚ⊆F⊆K? (integer)
- Question 60 – H⊆S₃ consists of permutations σ for which trace(A₁A₂A₃)=trace(A σ(1)A σ(2)A σ(3)) for every complex 2×2 triple. Find H (integer).
- Question 61 – A C¹ path runs from (0,2) to (3,0). For F(x,y)=(1−2y,1−2x), find ∫F·dr to two decimals.
- Question 62 – For u tt−u xx=0 on ℝ,t>0, u(x,0)=0 and initial velocity v(x)=x⁴(1−x)⁴ on 0<x<1, zero otherwise. Find α=inft>0:u(2,t)>0 to two decimals.
- Question 63 – The BVP x²y″−2xy′+2y=0 on [1,2], y(1)−y′(1)=1, y(2)−ky′(2)=4, has infinitely many solutions for which k? (two decimals)
- Question 64 – Find the global maximum of (x²+y²)e(2−x−y) for x,y≥0, to two decimals.
- Question 65 – In the upper half-disk 0<r<2,0<θ<π, u rr+r⁻¹u r+r⁻²u θθ=0, with u(r,0)=u(r,π)=0 for 0≤r≤2 and u(2,θ)=k sin2θ. Given u(1,π/4)=2, find k to two…