GATE 2022 MA – Previous Year Questions with Solutions
All 65 questions of GATE 2022 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 – As you grow older, an injury to your may take longer to .
- Question 2 – In a 500 m race P and Q have speeds in ratio 3:4. Q starts when P has already covered 140 m. How far apart are they when P wins? (m)
- Question 3 – School bells P,Q,R ring every 20,30,50 minutes respectively. All ring at noon. When do they next ring together?
- Question 4 – Some bottles are cups. All cups are knives. Conclusions: I some bottles are knives; II some knives are cups; III all cups are bottles; IV all knives…
- Question 5 – The shown disc has identical front and rear patterns. It is flipped once about one of fixed axes 1–1,2–2,3–3, each equally likely. What is the…
- Question 6 – Altruism is concern for others’ wellbeing. It has been shown to be motivated more by social bonding, familiarity and identification with a group.…
- Question 7 – Two identical fair dice have faces Q,R,S,T,U,V. They are thrown independently once. What is the probability both outcomes use only letters from Q,U,V,…
- Question 8 – An item is 10% cheaper at online store S than M. S adds ₹150 delivery, while M charges none. Buying from S saves ₹100. Find the item price at S,…
- Question 9 – Letters P,Q,R,S,T,U occupy the vertices of a regular convex hexagon. RS is longer than PQ and perpendicular to PQ. RU is parallel to TQ. Which…
- Question 10 – An ant starts at the bottom-left point P of the shown square grid and moves along grid lines toward the top-right corner, strictly decreasing its…
- Question 11 – M∈ℂ³ˣ³ has characteristic polynomial λ³+αλ²+βλ−1, with α+β≠0. Which identity holds?
- Question 12 – P:if M∈ℝᵐˣⁿ,m>n≥2 and rank M=n, then Mx=0 has only x=0. Q:if nonzero E∈ℝⁿˣⁿ satisfies E³=0, then I+E² is singular. Choose truth values.
- Question 13 – Let u=e2x[3x2y3y-3x2y3y]. Its harmonic conjugate v satisfies v(0,0)=2. For z=x+iy and f=u+iv, find 4+2if(iπ).
- Question 14 – Evaluate C z100/(z101+1)\,dz, where C is z =2 counterclockwise.
- Question 15 – The unit sphere of a real normed space contains distinct x,y and their entire joining segment. Which holds?
- Question 16 – Let (e k) be an orthonormal Hilbert-space basis, f k=e k+e (k+1), and g j= n=1j(-1)n+1e n. Find k=1 g j,f k 2.
- Question 17 – In usual ℝ² let A=x²+y²≤1,B=(x−2)²+y²≤1, M=A∪B and N=int A∪int B. Which holds?
- Question 18 – For x n=x (n−1)/2+1/x (n−1),n≥1, which is true?
- Question 19 – For real y₀, the initial value problem y′=cos(xy),x∈ℝ,y(0)=y₀ has how many solutions?
- Question 20 – Eigenfunctions at distinct eigenvalues of y″−3y′=λy,0<x<π, with y(0)=y(π)=0 are orthogonal with which weight?
- Question 21 – For u t−u xx=0,0<x<2,t>0, with u(x,0)=0,u(0,t)=1,u(2,t)=3, find the steady-state value at x=1.
- Question 22 – Let T₁ be the usual subspace topology on [0,1] and T₂ any topology there. P:if T₁⊊T₂, then ([0,1],T₂) is not compact. Q:if T₂⊊T₁, then ([0,1],T₂) is…
- Question 23 – Let p:[0,1]→0,1 be the indicator of [1/2,1], with the usual interval topology and its induced quotient topology on 0,1. Which holds?
- Question 24 – Give Y=∏ₙ₌₁∞ℝ its product topology and Δ=(x,x,x,…):x∈ℝ its subspace topology. Which holds?
- Question 25 – For Ax=b, A=pmatrix3&1&1\\1&4&1\\2&0&3pmatrix and b=(2,3,4)ᵀ, select all true statements.
- Question 26 – How many non-isomorphic abelian groups have order 2²·3³·5⁴?
- Question 27 – How many subgroups does a cyclic group of order 12 have?
- Question 28 – Find the radius of convergence of n03n+1z2n to two decimals.
- Question 29 – How many zeros, counting multiplicity, does 2z⁷−7z⁵+2z³−z+1 have in z <1?
- Question 30 – P is a degree-5 polynomial and x₀,…,x₆ are distinct in [2,3]. Let = i=06P(x i) j i(x i-x j)-1. Find α²−α+1.
- Question 31 – Find the maximum of 49−x²−y² on x+3y=10.
- Question 32 – For x,y≠0, f(x,y)=x²+xy+y²+1/x+1/y has a local minimum at (a,b). Find a³+b³ to two decimals.
- Question 33 – If x²φ″+xφ′+x²φ=0,x>0 has a Frobenius solution =xr n0a nxn with a₀≠0, find r²+1.
- Question 34 – Bessel functions satisfy J (α−1)(x)+J (α+1)(x)=(2α/x)J α(x). Find (πJ (3/2)(π))².
- Question 35 – Transform 7u xx+16u xy+4u yy=0 using ξ=y−2x,η=7y−2x into Au ξξ+Bu ξη+Cu ηη=0. Find (B²−4AC)/12³.
- Question 36 – Which describes ℝ[X]/(X⁴+4)?
- Question 37 – For proper nonzero ideals J₁,J₂ of a nonzero commutative ring R, P:for every r₁,r₂∈R there is a unique r∈R with r−r₁∈J₁,r−r₂∈J₂. Q:J₁+J₂=R. Which…
- Question 38 – Continuous f:[−π,π]→ℝ satisfies f(x)>f(0)/2 for x <δ,0<δ<π. Put P (n,δ)(x)=(1+cos x−cosδ)ⁿ. Which integral necessarily tends to zero as n→∞?
- Question 39 – P:if n0a nxn converges at −3 and diverges at 6, then Σ(−1)ⁿa n converges. Q:the convergence interval of n=2(-1)nxn/(4n n) is [−4,4]. Choose truth…
- Question 40 – On [0,1] let f n(x)=x²/[x²+(1−nx)²]. Which holds?
- Question 41 – In ℚ with usual distance, E=p∈ℚ:2<p²<3. Which holds?
- Question 42 – On L²[−1,1] define (Tf)(x)=f(−x), and let M=ker(I−T). Find the distance from φ(t)=eᵗ to M.
- Question 43 – X,Y,Z are Banach spaces, T:X→Y linear, and S:Y→Z bounded linear and injective. If S∘T is bounded, which must hold?
- Question 44 – A quadratic interpolant through (−1,f(−1)),(0,f(0)),(2,f(2)) approximates f′(0)=−(2/3)f(−1)+αf(0)+βf(2), for smooth f on (−3,3). Find 1/(αβ).
- Question 45 – Find work of F=(x+y)i−(x²+y²)j along the upper unit semicircle from (1,0) to (−1,0).
- Question 46 – For u tt−u xx=0 on 0<x<π,t>0, u(x,0)=sin x+sin2x+sin3x,u t(x,0)=0 and u(0,t)=u(π,t)=0, find u(π/2,π).
- Question 47 – A linear T:ℝ²→ℝ² satisfies T(1,2)=(1,0),T(2,1)=(1,1). If T⁻¹(p,q)=(x,y), which holds?
- Question 48 – For max 5x₁+12x₂ subject to x₁+2x₂+x₃≤10,2x₁−x₂+3x₃=8 and x≥0, suppose y=(α,−1)ᵀ is dual feasible. Which must hold?
- Question 49 – Let K⊆ℂ be all elements algebraic over ℚ. Which hold?
- Question 50 – A Möbius T satisfies T(0)=α,T(α)=0,T(∞)=−α, where α=(−1+i)/√2. L is the line through zero of slope −1; C is the unit circle. Which hold?
- Question 51 – For a>0 define D a:L²(ℝ)→L²(ℝ) by (D af)(x)=a(−1/2)f(x/a) almost everywhere. Which hold?
- Question 52 – An orthonormal set in L²[−1,1] has φ n=C nP n, where P n is the degree-n Legendre polynomial. Which listed statements hold?
- Question 53 – Let T be the smallest topology on ℝ for which every singleton is closed. For X=(ℝ,T), which hold?
- Question 54 – On ℤ let T be generated by A (m,n)=m+nk:k∈ℤ,m,n∈ℤ,n≠0. Which hold?
- Question 55 – For the primal min cᵀx subject to Ax=b,x≥0, let x⁰,y⁰ be primal and dual feasible. Which hold?
- Question 56 – In ℝ³ set W₁=x:x₁+2x₂−x₃=0,W₂=x:2x₁+3x₃=0. Which hold?
- Question 57 – Three companies C₁,C₂,C₃ bid for three jobs J₁,J₂,J₃, each company taking one job. Cost rows are C₁:(10,12,8),C₂:(9,15,10),C₃:(15,10,9). Find the…
- Question 58 – For y′=f(x,y), a second-order Runge–Kutta method uses K₁=hf(x i,y i),K₂=hf(x i+αh,y i+βK₁), y (i+1)=y i+(K₁+3K₂)/4. Find αβ to two decimals.
- Question 59 – Find the surface area of z=x²+y² between planes z=0 and z=1, to one decimal.
- Question 60 – For f(x,y,z)=x+x cos z−y sin z+y, find the rate of change at P₀=(2,−1,0) toward P₁=(0,1,2).
- Question 61 – A harmonic u on 1<x<2,1<y<2 has u x(1,y)=y,u x(2,y)=5,u y(x,1)=αx²/7,u y(x,2)=x. If a solution exists, find α.
- Question 62 – For xu x+(x²+y)u y=u on ℝ², u(2,y)=y−4. Find u(1,2).
- Question 63 – Maximize 6x₁+5x₂ subject to 3x₁+2x₂≤12,−x₁+x₂≤1,x₁,x₂≥0. Find the optimum.
- Question 64 – Plants P₁,P₂,P₃ supply (15,25,10) units to markets M₁,…,M₄ demanding (10,10,10,20). Cost rows are P₁:(1,3,1,3),P₂:(2,2,4,1),P₃:(2,1,1,2). Find the…
- Question 65 – A real 3×3 M satisfies M²=2M+3I and det M=−9. Find trace M.