GATE 2021 ST – Previous Year Questions with Solutions
All 65 questions of GATE 2021 ST with answers and explanations. Read each question, then practise it by topic or as a full paper.
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- Question 1 – A city’s current population is 11,02,500 and has increased by 5% per year. What was its population two years ago?
- Question 2 – Positive integers p,q satisfy p/q+q/p=3. Find p²/q²+q²/p².
- Question 3 – Find the least number of squares to add so that the line P–Q becomes a line of symmetry of the shown arrangement.
- Question 4 – Nostalgia is to anticipation as is to . Which pair maintains the relation?
- Question 5 – Which sentences are grammatically correct? (i)I woke up from sleep. (ii)I woked up from sleep. (iii)I was woken up from sleep. (iv)I was wokened up…
- Question 6 – Statements:All purple are green;all black are green. Conclusions:I:Some black are purple. II:No black is purple. Which option is logically correct?
- Question 7 – Passage:Computers are ubiquitous and improve efficiency in almost all fields,from agriculture to space exploration. AI is a hot topic and enables…
- Question 8 – A square sheet of side 1 is cut along a main diagonal. One resulting triangle is revolved about its short edge to make a solid cone. Find its volume…
- Question 9 – The bar chart shows daily exercise minutes for students X and Y during one week. On how many days did one student exercise at least 10% longer than…
- Question 10 – Corners are cut from an equilateral triangle to produce the regular convex hexagon shown. Find the ratio of hexagon area to original triangle area.
- Question 11 – X is a nonconstant positive random variable with E X=9. Which holds?
- Question 12 – For standard Brownian motion W(t),find Var[W(1)W(2)].
- Question 13 – An iid sample of size n≥2 has density exp[−(x−θ)/θ]/θ for x>θ,zero otherwise,θ>0. Find the method-of-moments estimator of θ.
- Question 14 – A realization of n≥2 iid N(μ,σ²) observations is given,μ∈ℝ,σ>0. P:a 95% confidence interval for μ is unique when σ is known. Q:a 95% confidence…
- Question 15 – X₁,…,X n are iid N(0,σ²),n≥2. Let T₁=ΣX i²,T₂=X̄² and let 𝒮 be the joint density family for σ>0. Write χ² (ν,α) for the (1−α) quantile. To test σ=1…
- Question 16 – A sample of 60 binary pairs gives counts n₁₁=10,n₁₀=20,n₀₁=20,n₀₀=10. Test independence of X,Y by Pearson’s chi-square test. Which statement gives the…
- Question 17 – (X,Y) is bivariate normal with mean zero,unit marginal variances and correlation ρ∈(−1,1). Let Φ ρ(0,0)=P(X≤0,Y≤0). Find Kendall’s τ.
- Question 18 – In Y i=β₀+β₁x i+ε i,i=1,…,n,n≥3,the errors are iid with mean zero and variance σ². Let β̂₀,β̂₁ be OLS estimators,x̄=Σx i/n,S₁=Σ(x i−x̄)²,S₂=Σy i(x…
- Question 19 – Independent vectors X₁,X₂,X₃∼N₄(0,Σ₁) and Y₁,…,Y₄∼N₄(0,Σ₂) have positive definite covariance matrices. Let X=[X₁ X₂ X₃],Y=[Y₁ Y₂ Y₃ Y₄],and…
- Question 20 – Find n(2n+n2n2(n/2))1/(2n-n(1/n)).
- Question 21 – Let I=4 1/21 1-x2x1x2+y2\,dy\,dx. Find eI+,to two decimals.
- Question 22 – A has rows (0,0,1),(1,0,0),(0,1,0). Find the nullity of 5A(I₃+A+A²).
- Question 23 – A real 2×2 matrix has eigenvalues 1,−1 with corresponding eigenvectors (√3/2,1/2)ᵀ and (−1/2,√3/2)ᵀ. If A²⁰²¹ has entries a,b,c,d,find a+b+c+d,to two…
- Question 24 – Events A,B satisfy P(B)=3/4 and P(A∪Bᶜ)=1/2. If A and B are independent,find P(A),to two decimals.
- Question 25 – A fair die is rolled twice independently,giving X,Y. Find E[X+Y (X−Y)²=1].
- Question 26 – X has CDF 0 for x<1,a/2 for 1≤x<2,c/6 for 2≤x<3,and 1 for x≥3. Let A n=[1+1/n,3−1/n] and A=⋃ (n≥1)A n. Given P(X≤1)=1/2 and E X=5/3,find P(X∈A),to two…
- Question 27 – The kth order statistic from a size-eight Uniform(0,2) sample has density (7/32)x⁶(2−x) for 0<x<2,zero otherwise. Find k.
- Question 28 – For each α>0,independent Bernoulli variables X n(α) have P(X n(α)=1)=1/n(2α). Let S=α>0:X n(α)→0 almost surely. Find inf S,to two decimals.
- Question 29 – X n are iid Uniform(0,2). Set Z n=−ln[(∏ (i=1)n(2−X i))(1/n)]. Find its almost-sure limit,to two decimals.
- Question 30 – A Markov chain on 0,1 has transition rows (.25,.75),(.75,.25),and P(X₀=0)=P(X₀=1)=.5. Find Σ (k=1)100 E[(X (2k))(2k)].
- Question 31 – The observations 0,2 are a sample of size two from Binomial(2,p). Test p=1/2 against p≠1/2. Find the observed likelihood-ratio statistic,to two…
- Question 32 – X has density (3/13)(1−x)(9−x) on 0<x<1,zero otherwise. Find (4/3)E[X(X²−15X+27)],to two decimals.
- Question 33 – (Y,X₁,X₂) has mean (5,2,0)ᵀ and covariance rows (10,.5,−.5),(.5,7,1.5),(−.5,1.5,2). Find the multiple correlation coefficient between Y and its best…
- Question 34 – Three iid observations from a bivariate normal population with unknown mean μ and positive definite covariance Σ are (1,2)ᵀ,(4,−2)ᵀ,(−5,0)ᵀ. Find the…
- Question 35 – For Y i=α+βx i+ε i,i=1,2,3,fixed x i and iid errors of mean zero and finite variance,let α̂,β̂ be OLS estimates. Data are…
- Question 36 – Define f:ℝ→ℝ by f(x)=x³sin x if x=0 or x is irrational,and f(p/q)=1/q³ for nonzero reduced rational p/q,q∈ℕ. Which holds?
- Question 37 – For f:[0,∞)→ℝ,which statement is true?
- Question 38 – A differentiable f:ℝ→ℝ satisfies f(0)=0 and f′(x)+2f(x)>0 for all x. Which holds?
- Question 39 – Let M be all symmetric positive definite real 3×3 matrices,and S=A∈M:A⁵⁰−A⁴⁸/4=0. How many elements are in S?
- Question 40 – For a real 3×3 A with I+A invertible,let B=(I+A)⁻¹(I−A). Which statement is true?
- Question 41 – X is Poisson with E X²=110. Which claim is NOT true?
- Question 42 – X∼Uniform[−π/2,π/2]. Which statement is NOT true?
- Question 43 – On sample space ℕ,let P(n)=a n. Which statement is NOT true?
- Question 44 – The joint density is 4/(x+y)³ for x>1,y>1,zero elsewhere. Which claim is NOT true?
- Question 45 – X₁,X₂,X₃ are uncorrelated with common variance σ². Let Y₁=2X₁+X₂+X₃,Y₂=X₁+2X₂+X₃,Y₃=X₁+X₂+2X₃. P:the covariance-matrix eigenvalues of (Y₁,Y₂,Y₃) sum…
- Question 46 – A time-homogeneous discrete Markov chain has finite or countable state space S. Which statement is true?
- Question 47 – Customers arrive as a rate-ten Poisson process. Independently each is male or female with probability 1/2. Let N(t) count all arrivals,M(t) males,and…
- Question 48 – X (1)<⋯<X (5) are order statistics of a size-five Uniform(0,θ) sample. P:3X (2) is unbiased for θ. Q:Var[E(2X (3) X (5))]≤Var(2X (3)). Which hold?
- Question 49 – X₁,…,X n are iid exponential with mean θ>0,n≥2. Let X (1)=min X i and T=ΣX i. Find E(X (1) T).
- Question 50 – For an iid size-n sample from Uniform(0,θ),θ>0,n≥2,let X (1),X (n) be the extremes. P:(X (1),X (n)) is complete. Q:the range X (n)−X (1) is ancillary.…
- Question 51 – X n are iid with CDF F. For real a<b,F(x)=0 on x≤a,0<F(x)<1 on a<x<b,and F(x)=1 on x≥b. Let S n be the empirical CDF. Which statement is NOT true?
- Question 52 – (X₁,X₂,X₃,X₄)∼N₄(μ,Σ),μ=(1,0,0,1)ᵀ and Σ has rows (1,.2,0,0),(.2,2,0,0),(0,0,2,.2),(0,0,.2,1). Which holds?
- Question 53 – Y∼N₈(0,I₈). The quadratic forms YᵀΣ₁Y and YᵀΣ₂Y are independent central chi-square variables with 3 and 4 degrees of freedom. P:Σ₁ and Σ₂ are…
- Question 54 – A centered bivariate normal density is f(x,y)=exp[(3/2)xy−(25/32)x²−2y²]/π on ℝ². Find 8E(XY).
- Question 55 – Let f(x,y)=8x²−2y. If M,m are its maximum and minimum on x²+y²=1,find M−m,to two decimals.
- Question 56 – Let A=[a u₁ u₂ u₃],B=[b u₁ u₂ u₃],C=[u₂ u₃ u₁ a+b] be 4×4 matrices of column vectors. If det A=6,det B=2,find det(A+B)−det C.
- Question 57 – X has MGF M(t)=(et−1)/[t(1−t)] for t<1,with the continuous value at t=0. Find P(X>1),to two decimals.
- Question 58 – X n are iid Uniform(0,3). Independently,Y has PMF P(Y=k)=1/[(e−1)k!],k=1,2,… . Find P(maxX₁,…,X Y≤1),to two decimals.
- Question 59 – X n are iid Exp(rate 1),and X (n)=maxX₁,…,X n. If X (n)−ln n converges in distribution to Z,find the median of Z,to two decimals.
- Question 60 – Visitors arrive at a park as a rate-one Poisson process. Each stays an independent Exp(rate 1) time,independent of arrivals. Currently 10 visitors are…
- Question 61 – The observations .90,.50,.01,.95 are iid with density [θ/(1−θ)]x[(2θ−1)/(1−θ)] on 0<x<1,where .5≤θ<1. Find the MLE of θ,to two decimals.
- Question 62 – A size-100 sample from N(μ,9) has mean 5.608. Given Φ(1.96)=.975 and Φ(1.64)=.95,find the p-value of the UMP unbiased test μ=5.02 against μ≠5.02,to…
- Question 63 – X takes 7,8,9,10. Under p₀ the probabilities are .90,.05,.04,.01;under p₁ they are .69,.10,.16,.05. Find the power of the most-powerful size-.05 test…
- Question 64 – X₁,…,X₁₀ are iid with density f θ(x)=f(x−θ),where f is symmetric about zero. For testing θ=1.2 against θ≠1.2,let T⁺ be the Wilcoxon signed-rank…
- Question 65 – Fit Y i=β₀+β₁x (1i)+⋯+β₂₂x (22i)+ε i,i=1,…,123,with iid N(0,σ²) errors. Regression sum of squares is 338.92,total sum of squares 522.30. Find 100…