GATE 2021 MA – Previous Year Questions with Solutions
All 65 questions of GATE 2021 MA with answers and explanations. Read each question, then practise it by topic or as a full paper.
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- Question 1 – The ratio of boys to girls in a class is 7:3. Which is an acceptable total number of students?
- Question 2 – A polygon is convex if for every pair of its points P,Q, the segment PQ lies completely inside or on the polygon. Which shown polygon is not convex?
- Question 3 – Consider: (i) “Everybody in the class is prepared for the exam.” (ii) “Babu invited Danish to his home because he enjoys playing chess.” Which…
- Question 4 – The circular paper is folded along the shown lines and directions, punched in its final folded state and unfolded in reverse order. Which result is…
- Question 5 – is to surgery as writer is to . Which pair maintains the logical relation?
- Question 6 – Two rectangular paper sheets M,N each measure 6 cm×1 cm. M is rolled into an open cylinder by joining its short edges. N is cut into equal square…
- Question 7 – Item P costs ₹5400 and has marked price ₹5860. Item Q has profit 25% and marked price ₹10000. Their cost ratio P:Q is 3:4. Discount is marked price…
- Question 8 – Five bags each contain an identical set of ten distinct chocolates. One chocolate is picked uniformly and independently from each bag. Find the…
- Question 9 – All bacteria are microorganisms. All pathogens are microorganisms. Conclusions: I some pathogens are bacteria; II all pathogens are not bacteria.…
- Question 10 – Some suggest anti-obesity measures (AOM) such as displaying calorie information on restaurant menus. These measures sidestep the core causes of…
- Question 11 – A is real 3×4 and B real 4×3, with AB nonsingular. P:nullity A=0. Q:BA is nonsingular. Choose truth values.
- Question 12 – A nonconstant entire f(z)=u(x,y)+iv(x,y),z=x+iy, defines g₁=u x−iu y and g₂=v x+iv y. Which holds?
- Question 13 – A Möbius map T(z)=(az+b)/(cz+d),ad−bc≠0, sends 0,−i,∞ to 10,5−5i,5+5i respectively. Find T(Re z<0).
- Question 14 – R is the reduced row echelon form of a real 4×4 A, and its third column is (0,1,0,0)ᵀ. P:if (α,β,γ,0)ᵀ solves Ax=0, then γ=0. Q:rank[A b]=rank[R b]…
- Question 15 – For y″+λy=0,0<x<π,λ>0, with y(0)=0 and y(π)−y′(π)=0, which are the eigenvalues?
- Question 16 – For differentiable f:ℝ→ℝ, u(x,y)=xy+f(x²−y²) satisfies which equation?
- Question 17 – For u tt=u xx on x∈ℝ,t>0, u(x,0)=0 and u t(x,0)=4xe(−x²), find u(5,5).
- Question 18 – For x (n+1)=φ(x n),φ(x)=3+(x−3)³ on (2.5,3.5), and x₀=3.25, find the convergence order.
- Question 19 – For an orthonormal basis (e n) of a complex Hilbert space H: P:there is a bounded linear f:H→ℂ with f(e n)=1/n. Q:there is a bounded linear g:H→ℂ with…
- Question 20 – Let f:(−π/2,π/2)→ℝ be f(x)=π/2+x−arctan x. P: f(x)−f(y) < x−y for all x,y in the domain. Q:f has a fixed point. Choose truth values.
- Question 21 – On (0,1): P:d₁(x,y)= log(x/y) is a metric. Q:d₂(x,y)= x + y for x≠y and zero for x=y is a metric. Choose truth values.
- Question 22 – A twice continuously differentiable f:ℝ³→ℝ satisfies div(∇f)=6. On the unit sphere S let n be the unit outward normal. Find ∫ S∇f·n dS.
- Question 23 – P:every compact metrizable topological space is separable. Q:every Hausdorff topology on a finite set is metrizable. Choose truth values.
- Question 24 – On ℝ let T₁=U:0∉U or U=ℝ,T₂=U:0∈U or U=∅,T₃=T₁∩T₂. Find the closure of 1 in (ℝ,T₃).
- Question 25 – Differentiable f:ℝ²→ℝ has directional derivatives at zero D u f=√5 and D v f=√2, where u=(1/√5,2/√5),v=(1/√2,−1/√2). Find f x(0,0)+f y(0,0).
- Question 26 – Γ is the counterclockwise boundary of the square with vertices (0,0),(2,0),(2,2),(0,2). Evaluate ∮ Γ(1−y²)dx+xdy.
- Question 27 – Find the number of Sylow 5-subgroups of S₅.
- Question 28 – Let I=(x²+x+1) in R=ℤ₃[x]. Find the number of units in R/I.
- Question 29 – T:ℝ³→ℝ³ is linear and T(1,1,1)=(1,−1,1),T²(1,1,1)=(1,1,1),T²(1,1,2)=(1,1,1). Find rank T.
- Question 30 – For x²y″−4xy′+6y=0,x>0, with y(2)=0,y′(2)=4, find y(4).
- Question 31 – For f(x)=x⁴+2x³−11x²−12x+36, Newton iteration from x₀=2.1 finds root α=2. What is its convergence order?
- Question 32 – A cubic p(x)=α+β(x+2)+γ(x+2)(x+1)+δ(x+2)(x+1)x interpolates values at x=(−2,−1,0,1,2) of (2,−1,8,5,−34). Find α+β+γ+δ.
- Question 33 – For max 2x₁+3x₂ subject to 2x₁+x₂≤6,−x₁+x₂≤1,x₁+x₂≤3,x₁,x₂≥0, find the optimal value of its dual.
- Question 34 – Minimize 2x₁−5x₂ subject to 2x₁+3x₂+s₁=12,−x₁+x₂+s₂=1,−x₁+2x₂+s₃=3, all five variables nonnegative. If (x₁,2,s₁,s₂,s₃)ᵀ is a basic feasible solution,…
- Question 35 – In a complex Hilbert space, ⟨u,v⟩=2. Evaluate (2)-1 02\ u+eitv\ 2eitdt.
- Question 36 – In R=a+b√(−17):a,b∈ℤ, P:2+√(−17) is irreducible. Q:2+√(−17) is prime. Choose truth values.
- Question 37 – For u xx+4u xy+(x²+4y²)u yy=sin(x+y), P:the PDE is parabolic on x²/4+y²=1. Q:it is hyperbolic inside that ellipse. Choose truth values.
- Question 38 – Solve xu x+u y=1 with u(x,0)=−x²,x>0. Find u(2,1).
- Question 39 – For y″+ay′+by=f(t),a,b>0,a≠b,a²−4b=0, with y(0)=y′(0)=0, which Laplace-transform solution holds?
- Question 40 – For y″+2αy′+β²y=0 with α>β>0, classify the critical point.
- Question 41 – Explicit Euler solves y′=−10y,t>0,y(0)=1 via y (n+1)=y n+h f(t n,y n). Which step range ensures y n→0?
- Question 42 – Maximize c₁x₁+c₂x₂ under three inequalities a i₁x₁+a i₂x₂≤b i (i=1,2,3),x≥0. A feasible (p,q) has c₁p+c₂q=6, and all feasible objectives lie in…
- Question 43 – In real L²[−1,1] with its usual norm, let M=f:∫₋₁¹f=0 and f(x)=x². Find inf (g∈M)‖f−g‖.
- Question 44 – On C[0,1] with supremum norm define (Tf)(x)=∫₀ˣx f(t)dt. Let R(T) be its range. P:T is bounded linear. Q:T⁻¹:R(T)→C[0,1] exists and is bounded. Choose…
- Question 45 – Let ℓ¹ have its usual norm and X=x∈ℓ¹:Σₙ₌₁∞n x(n) <∞. Define T:X→ℓ¹ by (Tx)(n)=nx(n), using on X the inherited ℓ¹ norm. Which holds?
- Question 46 – On [0,1], f n(x)=nx³e(−nx). P:(f n) is equicontinuous. Q:Σₙ₌₁∞f n does not converge uniformly. Choose truth values.
- Question 47 – Define f(x,y)=√(x²+y²) sin(y²/x) for x≠0 and zero for x=0. P:f is continuous but not differentiable at (0,0). Q:every unit-direction derivative exists…
- Question 48 – V is bounded by y=x²+z² and y=4. Evaluate V15x2+z2\,dV.
- Question 49 – For f(x,y)=4xy−2x²−y⁴, which description holds?
- Question 50 – The equation xy−z log y+e(xz)=1 defines y=f(x,z) near (x,y,z)=(0,1,1). Find ∇f(0,1).
- Question 51 – On ℝ let T₁ be the upper-limit topology with basis (a,b], T₂ the cofinite topology and T₃ the usual topology with basis (a,b). Which inclusion order…
- Question 52 – Let X₁ be ℝ with upper-limit basis (a,b], and X₂ be ℝ with the cofinite topology. Which holds?
- Question 53 – For an inner product ⟨·,·⟩ on ℝⁿ, P: ⟨u,v⟩ ≤(⟨u,u⟩+⟨v,v⟩)/2 for all u,v. Q:if ⟨u,v⟩=⟨2u,−v⟩ for every v, then u=0. Choose truth values.
- Question 54 – G has order 5⁴ and its center has 5² elements. Find the number of conjugacy classes.
- Question 55 – F is a finite field whose nonzero multiplicative group contains a subgroup of order 17. Find the smallest possible F .
- Question 56 – R=z=x+iy:0<x<1,−11π<y<11π, with positively oriented boundary Γ. Evaluate (2 i)-1 ez/(ez-2)\,dz.
- Question 57 – On z <2π, f(z)=3z²/(1−cos z) for z≠0 and f(0)=6. If f=Σₙ₌₀∞a nzⁿ, find 6a₂.
- Question 58 – Count the zeros with multiplicity of P(z)=3z⁵+2iz²+7iz+1 in 1< z <7.
- Question 59 – A has characteristic polynomial det(xI−A)=x⁴(x−1)²(x−2)³. If rank(A²)<rank(A³)=rank(A⁴), find the geometric multiplicity of eigenvalue zero.
- Question 60 – A power-series solution y=Σₖ₌₀∞a kxᵏ,a₀≠0, satisfies y″−24x²y=0. Find a₄/a₀.
- Question 61 – Suppose u(x,t)=Ae(−t)sin x solves u t=u xx on 0<x<π,t>0, with zero endpoint values and initial data u(x,0)=60 for 0<x≤π/2 and 40 for π/2<x<π. Find πA.
- Question 62 – V is the real vector space of polynomials p=a₀+a₁x+a₂x². Define T(p)=[p(0)−p(1)]+[p(0)+p(1)]x+p(0)x². Find the sum of its eigenvalues.
- Question 63 – The quadrature ∫₀²x f(x)dx≈αf(0)+βf(1)+γf(2) is exact for every polynomial f of degree at most 2. Find 2β−γ.
- Question 64 – For x∈(0,1), let n be the number of zeros immediately after the decimal point before the first nonzero decimal digit. Define f(x)=0 for rational x and…
- Question 65 – The point x̄=(11/3,2/3,0)ᵀ is optimal for max 4x₁+x₂−3x₃ subject to 2x₁+4x₂+ax₃≤10,x₁−x₂+bx₃≤3,2x₁+3x₂+5x₃≤11,x≥0, where a,b∈ℝ. If (p,q,r)ᵀ is dual…