GATE 2023 MA – Previous Year Questions with Solutions
All 65 questions of GATE 2023 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 village was nestled in a green spot, the ocean and the hills.
- Question 2 – Disagree : Protest :: Agree : (by word meaning).
- Question 3 – A frabjous number is a three-digit number with all digits odd and no two adjacent digits equal. For example, 137 qualifies but 133 does not. How many…
- Question 4 – Which statement must be true about the mean and median of scores of all candidates appearing for GATE 2023?
- Question 5 – Ovals mark horizontal cross-sections at different heights h on the hill shown. Which option P, Q, R or S depicts its top view?
- Question 6 – Residency is a famous housing complex with many well-established individuals among its residents. A recent survey revealed that all residents well…
- Question 7 – Ankita climbs five stairs from the ground, moving either one or two stairs up each time and never down. F(N) is the number of ways to reach stair N;…
- Question 8 – Passage: DNA information is used to synthesize proteins necessary for life. DNA consists of nucleotides Adenine (A), Thymine (T), Cytosine (C), and…
- Question 9 – Which figure P,Q,R,S represents f(x)= x+2 - x-1 ?
- Question 10 – An opaque cylinder is suspended in a parallel light beam and casts a shadow on a screen perpendicular to the beam. It may be reoriented arbitrarily.…
- Question 11 – For f(x,y)=x²−(3/2)xy² and g(x,y)=4x⁴−5x²y+y² on ℝ², P:f has a saddle at (0,0); Q:g has a saddle there. Choose truth values.
- Question 12 – In usual ℝ³ let X= x + y + z ∈ℚ, Y=xyz=1, Z=x²+y²+z²=1, W=xyz=0. Which statement is correct?
- Question 13 – Let P(x)=1+e2 ix+2e3 ix. Evaluate NN-1 k=0N-1P(k2).
- Question 14 – A linear T:ℝ³→ℝ³ satisfies T(1,0,0)=(0,1,1),T(1,1,0)=(1,0,1),T(1,1,1)=(1,1,2). Which holds?
- Question 15 – On z <1 define f(z)=z-25z3+z5/5!-z7/7!+z9/9!-z11/11!. P:f has three zeros counting multiplicity in the unit disk. Q:f has one zero in 1/2< z <1.…
- Question 16 – Let N⊆ℝ be non-Lebesgue-measurable. P:the irrational points of N form a measurable set. Q:the boundary of N has positive Lebesgue outer measure.…
- Question 17 – Measurable E k⊆[0,1] have measure 1/k². Define E= n k nE k and F= n k nE k. P:m(E)=0; Q:m(F)=0. Choose truth values.
- Question 18 – In usual ℝ², X=(x,x sin(1/x)):0<x≤1∪(0×ℝ) and Y=(x,sin(1/x)):0<x≤1∪(0×ℝ). P:X is connected. Q:Y is connected. Choose truth values.
- Question 19 – M=pmatrix4&-3\\1&0pmatrix. P:M⁸+M¹² is diagonalizable. Q:M⁷+M⁹ is diagonalizable. Choose truth values.
- Question 20 – In C[0,1] with supremum metric let X=f:‖f‖∞≥1/2. Set f₁(x)=x and f₂(x)=1−x. P:f₁ lies in int X. Q:f₂ lies in int X. Choose truth values.
- Question 21 – Let ρ₁ be usual distance and ρ₂ the discrete metric on ℝ. X=n∈ℕ:n≥3,Y=n+1/n:n∈ℕ. Define f=2 on X and f=3 on Y. P:f:(X∪Y,ρ₁)→(ℝ,ρ₁) is uniformly…
- Question 22 – T:ℝ⁴→ℝ⁴ has kernel 4x₁+3x₂+2x₃+x₄=0 and rank(T−3I)=3. Find its minimal polynomial.
- Question 23 – In real C[0,1] with supremum norm let X=f:f(0)=f(1)=0, S X=f∈X:‖f‖∞=1 and F(f)=∫₀¹f(t)dt. How many elements are in f∈X:F(f)=‖F‖∩S X?
- Question 24 – A continuous map is proper when inverse images of compact sets are compact. Using usual/subspace/product topologies, which map is proper?
- Question 25 – Minimize 3x₁+4x₂ subject to x₁−x₂≤1,x₁+x₂≥3,x₁,x₂≥0. Find the optimal value.
- Question 26 – For c>0, u xx−u tt/c²=0 on ℝ×(0,∞), with u(x,0)=sin x and u t(x,0)=cos x. The dependence interval for P=(3,2) has endpoints Q,R on the x-axis. If…
- Question 27 – Near zero, z/(1-z-z2)= n=0 a nzn. Find a₆+a₅.
- Question 28 – Let p(x)=x³−2x+2. The polynomial q of degree at most 4 interpolates q(−2)=p(−2),q(−1)=p(−1),q(0)=2.5,q(1)=p(1),q(3)=p(3). Find q⁽⁴⁾(0).
- Question 29 – For fixed real c, the trapezoidal-rule value of ∫₀²(9x²−5cx⁴)dx equals the exact integral. Find c to two decimals.
- Question 30 – If 14 -xx (x2+y2)-1dy\,dx= -/4/4 r-1dr\,d, find α.
- Question 31 – The portion of z=2x+2y−100 inside x²+y²=1 has surface area απ. Find α.
- Question 32 – On real L²[−1,1], F(f)=∫₋₁¹f(x)x²dx. With the usual L² norm, find 5‖F‖².
- Question 33 – Solve y″+4y=t for 0≤t≤2 and =2 for t>2, with y(0)=y′(0)=0. If α=y(π/2), find (4/π)α to two decimals.
- Question 34 – In ℝ⁴ with the standard inner product, M=x:x₁=x₃. Find dim M⊥.
- Question 35 – Let M=pmatrix3&-1&-2\\0&2&4\\0&0&1pmatrix. If 6M⁻¹=M²−6M+αI, find α.
- Question 36 – In GL₂(ℂ), let S=pmatrix0&-1\\1&0pmatrix, T₁=iI,T₂=diag(i,−i), T 3=pmatrix0&1\\1&0pmatrix and G j=⟨S,T j⟩. Write Z(G) for the center. Which holds?
- Question 37 – On real ℓ² with usual norm, S(x₁,x₂,…)=(0,x₁,x₂,…) and T(x₁,x₂,…)=(0,x₂,0,x₄,…). Which holds?
- Question 38 – Let c₀₀ be finitely supported real sequences with supremum norm. Define F(x) n=(n+1/n)x n and G(x) n=x n/(n+1/n). Which holds?
- Question 39 – For xu x+yu y=u with data u(t,t)=f(t),t>0, consider: P:if f(t)=2t+1 there is a unique local solution near the initial curve. Q:if f(t)=2t−1 there are…
- Question 40 – For Mx=b, M=pmatrix2&-1\\-4&3pmatrix and b=(−2,5)ᵀ. P:if M=LU with lower triangular L having unit diagonal, trace U=3. Q:Jacobi iterates converge to…
- Question 41 – Let φ,ψ be independent solutions of y″+(2−cos x)y=0 on ℝ. Suppose α<β,φ(α)=φ(β)=0 and φ≠0 on (α,β). P:φ′(α)φ′(β)>0. Q:φ(x)ψ(x)≠0 throughout (α,β).…
- Question 42 – On the unit disk f(z)=Σₙ₌₀∞a nzⁿ, where a₀=a₁=1 and a n=1/2(2n) for n≥2. P:f is one-to-one in some neighborhood of every disk point. Q:f( z ≤1/2) is…
- Question 43 – Let Ω⊆ℂ be open and connected and contain U= z ≤1/2. Let J consist of analytic f on Ω with sup (z,w∈U) f(z)−f(w) =1. P:some f∈J has f′(0) ≥2. Q:every…
- Question 44 – On ℝ let τ=U⊆ℝ:U=∅ or 1∈U. Which statements hold?
- Question 45 – Let R=p(x)∈ℚ[x]:p(0)∈ℤ. For a∈R write (a) for its generated ideal. Which statements hold?
- Question 46 – Let S₁=ℤ[x]/(2,x³) and S₂=ℤ₂[x]/(x²), with the displayed ideals generated by their entries. Which statements hold?
- Question 47 – Let f n:ℝ→ℝ equal n²(x−n) on [n,n+1/n²], and zero elsewhere. Write m for Lebesgue measure. Which statements hold as n→∞?
- Question 48 – Write χ E for the indicator of E. Let f(x)= n=1(2n/n3) [2-(n+1),2-n](x) on [0,1]. P:f∈Lᵖ[0,1] for every 1<p<2. Q:f∈L¹[0,1]. Select all correct…
- Question 49 – For x′=y,y′=x with x(0)=α,y(0)=β, which choices give x(t) + y(t) →0 as t→∞?
- Question 50 – For h>0 define D hf(a)=[αf(a−h)+βf(a)+γf(a+2h)]/(6h). If this equals f′(a) for every real polynomial of degree at most 2 and every real a, which hold?
- Question 51 – A twice continuously differentiable f on [a,b] has f′<0,f″<0 on (a,b), and f(ζ)=0 at ζ∈(a,b). Newton iteration is x (k+1)=x k−f(x k)/f′(x k). If x…
- Question 52 – Let f(x,y)=2x²y/(x²+y²) away from the origin and f(0,0)=0. Which statements hold?
- Question 53 – In C[0,1] with supremum metric, let f n(x)=xⁿ and P=f n:n∈ℕ. Which hold?
- Question 54 – Let G be abelian and Φ:G→(ℤ,+) a surjective homomorphism, with Φ(a)=1. P:for every g∈G there is n∈ℤ with gaⁿ∈ker Φ. Q:G=(ker Φ)⟨a⟩ and (ker Φ)∩⟨a⟩=e.…
- Question 55 – C is x²+y²=4,z=2 oriented counterclockwise viewed from above. If C( x+ex)dx+4x\,dy+ez2z\,dz =, find α.
- Question 56 – In real ℓ² let M=\x: n1x n/4n=0\. The orthogonal projection of (1,1/2,1/3,…) onto M⊥ is ( n11/(n4n))(1/4,1/42,1/43,). Find α.
- Question 57 – For a balanced transportation problem with destinations P,Q,R,S, the five cost rows and supplies are: source 1:(13,8,12,9),20; source…
- Question 58 – In S₈, σ is the product of a disjoint 4-cycle and 3-cycle. How many permutations are conjugate to σ?
- Question 59 – A is a real 3×3 matrix with det(A+iI)=0 and det A=3. Find trace(A²).
- Question 60 – A=(a ij) is real 3×3 with A(1,2,1)ᵀ=2(1,2,1)ᵀ, A(0,1,1)ᵀ=2(0,1,1)ᵀ and A(−1,1,0)ᵀ=4(−1,1,0)ᵀ. If m is its minimal-polynomial degree, find…
- Question 61 – On Ω=x²+y²<4, u xx+u yy=0 and u=1+2x² on ∂Ω. Find u(0,1).
- Question 62 – For k≥0, y k is a degree-k polynomial satisfying y k(1)=5 and (1−x²)y k″−2xy k′+k(k+1)y k=0. If 12 -11 k=1n(y k-y k-1)2dx- -11 k=1ny k2dx=24, find the…
- Question 63 – For 4(ln x)y″+3y′+y=0,x>1, let r₁,r₂ be the indicial roots at regular singular point x=1. Find r₁−r₂ to two decimals.
- Question 64 – For 0<x<π,t>0 solve u xx−u tt=sin x sin(2t), with u(x,0)=u t(x,0)=0 and u(0,t)=u(π,t)=0. Find u(π/2,3π/2) to two decimals.
- Question 65 – Maximize 3x₁+2x₂+5x₃ subject to x₁+2x₂+x₃≤44,x₁+2x₃≤48,x₁+4x₂≤52 and x₁,x₂,x₃≥0. Find the optimal value.