GATE 2022 ST – Previous Year Questions with Solutions
All 65 questions of GATE 2022 ST with answers and explanations. Read each question, then practise it by topic or as a full paper.
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- Question 1 – Mr. X speaks Japanese Chinese.
- Question 2 – Money is divided among P,Q,R,S in the ratio 5:2:4:3. R gets ₹1000 more than S. Find Q’s share in rupees.
- Question 3 – A trapezium has vertices P,Q,R,S in anticlockwise order,PQ∥SR,and side lengths PQ=11,QR=4,RS=6,SP=3 cm. Find the shortest distance between its…
- Question 4 – The figure is a 4×4 grid of unit squares with the unshaded cell representing a hole. How many squares can be formed using the unit cells without a…
- Question 5 – The diagram shows an art gallery with opaque boundary walls. A stationary guard has a 360° view. Which set lists all marked locations P,Q,R,S from…
- Question 6 – Passage: Mosquitoes threaten human health. Chemical control may have undesired consequences. Florida authorities have used genetically modified…
- Question 7 – Find the simultaneous solution of 2x−1>7 and 2x−9<1.
- Question 8 – P=(0,1),Q=(0,−3),R=(−2,−1),S=(2,−1) are the vertices of a quadrilateral. Find its enclosed area.
- Question 9 – Exactly one of P,Q,R,S,T copied in an exam. P says “R copied”;Q says “S copied”;R says “P did not copy”;S says “Only one of us is telling the truth”;T…
- Question 10 – In the square shown,P,Q,R,S mark its corners and T its center. X is a 180° rotation about diagonal SQ;Y a 180° rotation about diagonal PR;Z a 90°…
- Question 11 – A real 2×2 matrix M satisfies (I+M)⁻¹=I−αM with α≠0 and tr M=3. Find α.
- Question 12 – A linear pure-death process on 0,…,N has death rates μ i=5i,N≥1. Write p i(t)=P(X(t)=i). Which forward equations and initial conditions are correct?
- Question 13 – S² is the sample variance with divisor n−1 from n>1 iid normal observations with unknown μ and σ²>0. I:S² is unbiased for σ² and S is unbiased for σ.…
- Question 14 – Define f(x,y)=x²y/(x²+y²) off (0,0),and f(0,0)=0. Which holds on ℝ²?
- Question 15 – For iid observations with CDF F and empirical CDF F n,let D n=sup x F n(x)−F(x) and T n=√nD n. I:the law of T n is the same for all continuous F. II:D…
- Question 16 – Two Markov transition matrices have rows P₁:(1,0,0),(1/3,1/2,1/6),(0,0,1);P₂:(1/6,1/3,1/2),(1/4,0,3/4),(0,1,0). Which holds?
- Question 17 – X₁,…,X₂₀ are iid N₆(μ,Σ),Σ nonsingular,μ and Σ unknown. Let X̄=ΣX i/20 and S=Σ(X i−X̄)(X i−X̄)ᵀ/19. I:19S∼Wishart₆(19,Σ). II:(X₃−μ)ᵀS⁻¹(X₃−μ)∼χ²₆.…
- Question 18 – An iid sample of size 18 has density 2x exp(−x²/θ)/θ on x>0,zero otherwise. Let T=Σx i²,and let χ² (α,ν) denote the upper-tail α quantile. Which is a…
- Question 19 – For an iid sample of size n from f(x;θ),which statement is NOT true?
- Question 20 – Six iid Bernoulli(θ) observations are used to test θ=1/2 against θ>1/2 via Y=ΣX i. Observed Y=4. Find the p-value.
- Question 21 – Positive numbers a n satisfy 8/a (n+1)=7/a n+a n²/343,n≥1,a₁=3,and a n<7 for n≥2. I:a n is increasing. II:a n converges to a value in [3,7]. Which…
- Question 22 – A square matrix M of order n satisfies M²=0 and nullity(M)=6. Find the maximum possible n as an integer.
- Question 23 – A unit vector u∈ℝ⁴ is orthogonal to S=x:x₁+x₂+x₃+x₄=0. Let v=(1,−2,1,1)ᵀ. If u and v−αu are orthogonal,find α² to two decimals.
- Question 24 – B(t) is standard Brownian motion. If P[B(2)+2B(3)>1]=1−Φ(1/√α),α>0,find α as an integer.
- Question 25 – X,Y are independent exponential variables with E X²=1/2 and E Y²=2/9. Find P(X<2Y) to two decimals.
- Question 26 – X has PMF P(X=x)=(3/4)(x−1)/4,x=1,2,… . Evaluate Σ from n=0 to ∞ of P(n<X≤n+3),to two decimals.
- Question 27 – X i are iid N(1,4),and S n=ΣX i². Find lim as n→∞ of [Var(S n)/n−(E(S n)/n)²],as an integer.
- Question 28 – Calls arrive as a rate-one Poisson process per minute. Choose five two-minute intervals. Let p be the probability each interval has at most one call.…
- Question 29 – X has density c(x−⌊x⌋) on 0<x<3,zero elsewhere. Find P(X∈[1/2,2]),to two decimals.
- Question 30 – X has MGF M(t),mean 1/2 and variance 1/4. Y has MGF H(t)=[(3/4)e(2t)+1/4]M(t) near zero. Find Var Y as an integer.
- Question 31 – X i are iid with density x(−5/6)e(−x/8)/[√2 Γ(1/6)] on x>0. Let X̄ n be their mean. If √n[X̄ n(3−X̄ n)−20/9]→N(0,σ²) in distribution,find σ² to two…
- Question 32 – X(t) is a Poisson process with P(X(t)=n)=e(−4t)(4t)n/n!. If C(t₁,t₂) is its covariance function,find C(5,3),as an integer.
- Question 33 – Four observations 6,5,3,6 come from density 2(θ−x)/θ² on 0<x<θ,zero elsewhere. Find the method-of-moments estimate of θ as an integer.
- Question 34 – If a company pays a quarterly dividend,the next is paid with probability .7. If it stops a dividend,the next is not paid with probability .5. Find the…
- Question 35 – Eight iid observations have density x/8 on 0<x<4,zero elsewhere. If α=Var[F₈(2)] for their empirical CDF,find 128α as an integer.
- Question 36 – A symmetric 3×3 matrix M has eigenvalues −1,1,2 and corresponding unit eigenvectors u,v,w. If Mx=u+2(v+w) and M²y=u−(v+2w),find x+y ².
- Question 37 – Which series converge conditionally? S₁=Σ (n=0)∞(−1)n n/(n²+4); S₂=Σ (n=0)∞(−1)n(√(n²+1)−n).
- Question 38 – Four independent observations (3,6)ᵀ,(4,4)ᵀ,(5,7)ᵀ,(4,7)ᵀ come from a bivariate normal population. If μ̂,Σ̂ are their MLEs,find Σ̂μ̂.
- Question 39 – X∼N₃(μ,Σ),μ=(2,−3,2)ᵀ,Σ has rows (4,−1,1),(−1,2,a),(1,a,2). The partial correlation of X₂ and X₃ given X₁ is 5/7. Find a.
- Question 40 – The line y=αx,α≥√2,divides R=(x,y):0≤x≤√y,0≤y≤2 into equal areas. Find α.
- Question 41 – The joint density of (X,Y,Z) is (2x+3y+z)/3 on the unit cube,zero elsewhere. Which point lies on the regression surface of X on (Y,Z)?
- Question 42 – One observation has density 2x/θ² on 0<x<θ. Using pivot X/θ,which interval is an 80% confidence interval for θ when x=10? Limits are rounded to three…
- Question 43 – X₁,…,X₇ are iid N(0,θ),θ>0. Set K=(X₁²+X₂²)/Σ (i=1)7X i². I:K and ΣX i² are independent. II:7K/2∼F (2,7). III:E K²=8/63. Which hold?
- Question 44 – I:For a variable with density e(− x )/2 on ℝ,there exist iid X₁,X₂ such that X₁−X₂ has this law. II:For a variable uniform on (−2,2),there exist iid…
- Question 45 – X n are independent exponential variables of mean 1/2. “i.o.” means infinitely often. Which hold?
- Question 46 – X n has probabilities n/(n+1) at 0 and 1/(n+1) at n. Let X=0 almost surely. Which hold?
- Question 47 – M is symmetric 3×3 with eigenvalues 1,2,3. N has real eigenvalues and MN+NᵀM=3I. Which cannot be eigenvalues of N?
- Question 48 – A real 3×2 matrix M has SVD M=USVᵀ,where U,V are orthogonal and S=[[√3,0,0],[0,1,0]]ᵀ. Which hold?
- Question 49 – For a>0,suppose P(aX/(2π)∈ℤ)=1. Let φ X(t) be its characteristic function. Which hold?
- Question 50 – Which real functions are uniformly continuous on [0,∞)?
- Question 51 – Independent size-seven samples are population 1:(18,20,16,20,17,18,14) and population 2:(17,18,14,20,14,13,16). For the two-sided 5% Mann–Whitney test…
- Question 52 – Fit Y=β₀+β₁X₁+β₂X₂+β₃X₃+ε with independent N(0,σ²) errors to 52 observations. Regression sum of squares is 18.6,total sum of squares 79.23. Find the…
- Question 53 – Three iid observations have density 2x on (0,1),zero elsewhere. Let p be the probability the maximum is at least twice the minimum. Find p to three…
- Question 54 – Fit Y=β₀+β₁X+ε with independent N(0,σ²) errors to x=(0,1,2,3,4),y=(3,4,5,6,7). Let Ŷ₀ be the least-squares fitted mean at X=6. If Var Ŷ₀=cσ²,find c to…
- Question 55 – The Poisson(θ) sample is 0,1,1,2,0. If α is the minimum-variance unbiased estimate of P(X≤1),find 5⁴α as an integer.
- Question 56 – For n paired observations used in Spearman rank correlation,x i are distinct for i≥2 and x₁=x₂. Given Σd i²=19.5,where d i=rank(x i)−rank(y i),find…
- Question 57 – At each repeated food experiment,70% of cats previously eating A choose A,and 50% previously eating B choose A. Initially 40% ate A in experiment 1.…
- Question 58 – An iid size-five sample from density 3x²/θ³ on 0<x<θ is 3,6,4,7,5. Find the MLE of the 1/8 quantile,to one decimal.
- Question 59 – A gamma density is x⁴e(−x/β)/(24β⁵),x>0,β>0. For β=2,find the Cramér–Rao lower bound for variance of an unbiased estimator of β² from n=8…
- Question 60 – Four iid Bernoulli(θ) observations test θ=1/4 against θ>1/4 by rejecting when ΣX i>2. Let α be type-I error and γ(θ) the power. Find 16α+7γ(1/2),as an…
- Question 61 – For iid N(μ,2²) observations,test μ=10 against μ=12 by rejecting for large X̄. Given Φ(1.645)=.95 and Φ(2.33)=.99,find the smallest integer sample…
- Question 62 – Y₁<⋯<Y n are order statistics of an iid sample from a continuous distribution symmetric about its mean μ. Find the smallest n with P(Y₁<μ<Y n)≥.99.
- Question 63 – A point P=(x,y,z) lies on z=xy+5 and x+y+z=1 and is nearest to the origin. Find its distance from the origin as an integer.
- Question 64 – X∼Uniform(0,4),and Y X=x∼Uniform(0,x²/4). Find E Y²,to three decimals.
- Question 65 – X=(X₁,X₂,X₃)ᵀ∼N₃(μ,Σ),μ=(3,2,4)ᵀ,and Σ has rows (4,2,1),(2,3,0),(1,0,2). Find E(X₁ X₂=4,X₃=7),as an integer.