GATE 2023 ST – Previous Year Questions with Solutions
All 65 questions of GATE 2023 ST with answers and explanations. Read each question, then practise it by topic or as a full paper.
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- Question 1 – “I have not yet decided what I will do this evening; I visit a friend.”
- Question 2 – Eject : Insert :: Advance : (by word meaning).
- Question 3 – PQRSTV is a regular hexagon of side 5 cm. A circle centered at V passes through P. Find the shaded area in cm² in the figure.
- Question 4 – A duck named Donald Duck says, “All ducks always lie.” Based only on this information, which statement follows with certainty?
- Question 5 – A line of symmetry divides a figure into two mirror-image parts. The figure consists of 20 unit squares. Up to five additional squares may be colored…
- Question 6 – Based only on “Some humans are intelligent,” which inference is certain?
- Question 7 – For the probability mass function P(x) shown, which relation between mean,median and mode follows?
- Question 8 – Read the passage: The James Webb telescope gives humankind unprecedented access to the depths of time by imaging very old stars formed almost 13…
- Question 9 – Let a=30!,b=50!,c=100!. Which inequality is correct?
- Question 10 – A square has side 4 cm. The shaded boundary consists of a top semicircle and two lower circular arcs, each of radius 2 cm, as shown. Find the shaded…
- Question 11 – Find the area bounded by x=−y² and y=x+2.
- Question 12 – A real 3×3 matrix A has eigenvalues 1,0,−1. For B=A²+2A+I₃, which identity holds?
- Question 13 – For real n×n matrices: I:if B is invertible,rank(BA)=rank(A). II:if A²x=b is solvable for every b∈ℝⁿ,then Ax=b is solvable for every b. Which hold?
- Question 14 – On Ω=[0,2], let 𝒢=∅,Ω,[0,1],(1,2]. Define X=1 on [0,1] and 2 on (1,2]; Y=2 on [0,1.5] and 3 on (1.5,2]. Which statement holds?
- Question 15 – Let Φ be the standard normal CDF. X has CDF F(x)=Φ(x) for x<−1 and Φ(x+1) for x≥−1. Which holds?
- Question 16 – X has density αλ x(α−1) exp(−λxα) for x>0,zero otherwise,α,λ>0. Its median is 1 and third quartile is 2. Find (α,λ).
- Question 17 – If X is Poisson with mean λ>0,find E[1/(X+1) X>0].
- Question 18 – X has gamma density λα x(α−1)e(−λx)/Γ(α),x>0,α,λ>0. Which claim is NOT true?
- Question 19 – The joint density is f(x,y)=8xy for 0<x<y<1,zero otherwise. If E(X Y=y₀)=1/2,find y₀.
- Question 20 – Each of five boxes contains 3 blue,1 red and 2 black pens. Independently draw one pen uniformly from each box. If X₁ counts blue pens and X₂ counts…
- Question 21 – All variables are defined on the same probability space. Which statement is true?
- Question 22 – X has exponential density e(−x/λ)/λ for x>0,λ>0 unknown. Y₁,…,Y n are iid with the distribution of X² and Ȳ is their mean. Which is the…
- Question 23 – An iid sample of size n≥2 has density f(x;θ)=2(−ln x) exp[−(ln x)²/θ]/(θx) on 0<x<1,zero otherwise,θ>0. Which expression is the MLE of θ?
- Question 24 – For an iid size-n sample from Uniform(1/3,θ),θ>1/3,let Y=max X i. Which estimator of θ is unbiased?
- Question 25 – X₁,…,X n,Y₁,…,Y n are iid N p(μ,Σ),Σ nonsingular,p>1,n>1. Write their means X̄,Ȳ. Which claim is true?
- Question 26 – Consider y k=α₀+α₁ ln k+ε k,k=1,…,n,with iid errors of density e(− x )/2 on ℝ. Which statement is true?
- Question 27 – X₁,…,X n are iid with density f and median θ. To test θ=θ₀ against θ>θ₀,reject if S>c,where S counts X i>θ₀. Which statement is true?
- Question 28 – From one observation x,test standard normal density exp(−y²/2)/√(2π) against Laplace density exp(− y )/2. Which most-powerful rejection region has the…
- Question 29 – Find the maximum of f(x,y)=xy on x²+2y²=1. Round to two decimals.
- Question 30 – A real 2×2 matrix A commutes with every real 2×2 matrix B. If tr A=5,find det A to two decimals.
- Question 31 – Two of five bulbs are defective. Bulbs are chosen uniformly without replacement and tested one at a time. Let X be the minimum number of tests needed…
- Question 32 – X n are iid with mean 4 and variance 9. For Y n=(X₁+⋯+X n)/n,find lim as n→∞ of E[((Y n−4)/√n)²],as an integer.
- Question 33 – For standard Brownian motion W t,find E(W₄² W₂=2),as an integer.
- Question 34 – A Markov chain on 1,2,3 has transition rows (1/2,1/4,1/4),(1/3,1/3,1/3),(0,1/2,1/2). Find P(X₂=1 X₁=1,X₃=2),to two decimals.
- Question 35 – (X₁,X₂,X₃)∼N₃(0,Σ),where Σ has rows (2,2,1),(2,5,1),(1,1,1). Given Φ(−0.5)=0.3085,find P[(X₁−2X₂+2X₃)²<7/2],to two decimals.
- Question 36 – For a real n×n matrix A: I:if A is symmetric,there exists c≥0 for which A+cI is symmetric positive definite. II:if A is symmetric positive…
- Question 37 – X has density 1/x² for x≥1,zero otherwise. For Y=ln X,find P(Y<1 Y<2).
- Question 38 – N(t) is a rate-one Poisson process. I:P(N(3)=3 N(5)=5)=binom(5,3)(3/5)³(2/5)². II:if S₅ is the fifth-arrival time,E(S₅ N(5)=3)=7. Which hold?
- Question 39 – An iid sample of size n has density exp[−(x−μ)] for x≥μ,zero otherwise,μ unknown. Let M̂ be the MLE of the population median. Which holds when μ=1?
- Question 40 – X₁,…,X₁₀ are iid N(0,θ²),θ>0. Set T=ΣX i²/10. For c>0,the MSE of cT as an estimator of θ² is minimized at c₀. Find c₀.
- Question 41 – X₁,…,X₁₀ are iid N₂(μ,Σ),Σ nonsingular. With X̄ their mean and U=1/[1+(X̄−μ)ᵀΣ⁻¹(X̄−μ)],find ln P(U≤1/2).
- Question 42 – For (X,Y)∈0,1²,P(0,0)=P(1,1)=θ and P(1,0)=P(0,1)=1/2−θ,0≤θ≤1/2. Test θ=1/4 against θ=1/3 using n iid pairs. If m counts equal pairs,which…
- Question 43 – Let g(x)=f(x)+f(2−x) on [0,2],where f is continuous on [0,2] and twice differentiable inside. Which statement is NOT true?
- Question 44 – For U⊂ℝⁿ let L(U)=span(U). For T,S⊂ℝⁿ,which statement is NOT true?
- Question 45 – f:[0,1]→ℝ is continuous. Which statement is NOT true?
- Question 46 – X has CDF 0 for x<−1,(x+1)/4 for −1≤x<0,(x+3)/4 for 0≤x<1,and 1 for x≥1. Which holds?
- Question 47 – The joint PMF is p(x,y)=c/2(x+y+2) for nonnegative integers x≠y,zero otherwise. Which holds?
- Question 48 – X₁,…,X₁₀ are iid N₃(μ,Σ),μ and nonsingular Σ unknown. Split into groups of five. Let X̄₁,X̄₂ be the group means and S₁,S₂ their covariance matrices…
- Question 49 – Which sets are countable?
- Question 50 – For real a define a⁺=max(a,0),a⁻=max(−a,0). Which statements about real sequences are true?
- Question 51 – A real 3×3 matrix satisfies A(0,1,1)ᵀ=(4,0,0)ᵀ,A(1,0,1)ᵀ=(0,4,0)ᵀ,and A(1,1,0)ᵀ=(0,0,4)ᵀ. Which hold?
- Question 52 – X is positive and continuous with finite mean. Let Y=⌊X⌋. Which hold?
- Question 53 – X has density e(−x) for x≥0,zero otherwise. Which hold? U(a,b) denotes the uniform law on (a,b).
- Question 54 – X has PMF P(X=0)=(1+e⁻¹)/2 and P(X=k)=e⁻¹/(2k!) for k≥1. Which hold?
- Question 55 – U,V are iid with density λ²x e(−λx),x>0,λ>0. Which statements hold?
- Question 56 – The joint PMF is e⁻²/[x!(y−x)!] for integers 0≤x≤y,zero otherwise. Which hold?
- Question 57 – X n are iid with mean zero and variance one. Set Y n=(X₁X₂+X₃X₄+⋯+X (2n−1)X (2n))/n. Which hold?
- Question 58 – For t=1,…,100,consider y t=α₀+α₁t+α₂t²+ε t,where ε t are iid N(μ,1) and μ is unknown. Which parameters have unbiased estimators?
- Question 59 – In ℝ³ with standard inner product,the orthonormal basis is v₁=(1,−1,1)ᵀ/√3,v₂=(1,2,1)ᵀ/√6,v₃=(1,0,−1)ᵀ/√2. For u=(a,b,c)ᵀ,its inner products with…
- Question 60 – On Ω=1,2,3,4,let 𝒢=∅,Ω,1,4,2,3,1,4,1,2,3,2,3,4. Suppose P(1)=1/4 and X(1)=1,X(2)=X(3)=2,X(4)=3. If P(X≤2)=3/4,find P(1,4) to two decimals.
- Question 61 – X n are iid Exp(rate 1). Let Y n= X (2n)−X (2n−1) and Ȳ n=(Y₁+⋯+Y n)/n. If √n[exp(−Ȳ n)−e⁻¹] converges in distribution to N(0,σ²),find σ² to two…
- Question 62 – A birth–death process on 0,1,2,3 has birth rates (1,1,2,0) and death rates (0,1,1,1). If π is its unique stationary distribution,find π₀+2π₁+3π₂+4π₃…
- Question 63 – The observations −1,−1/2,1,5/2,3 form a sample of size 5 from N(1/2,σ²),σ>0 unknown. T is an unbiased estimator of σ² attaining the Cramér–Rao lower…
- Question 64 – One observation has CDF F(x)=0 for x<0,1−(1−x)θ for 0≤x<1,and 1 for x≥1,θ>0. Test θ=1 against θ=2 by rejecting if x<0.5. If α is the level and β the…
- Question 65 – For observations 0.13,0.12,0.78,0.51,test a Uniform(0,1) CDF against a general alternative with the two-sided Kolmogorov–Smirnov statistic D. Under…