GATE 2020 ST – Previous Year Questions with Solutions
All 65 questions of GATE 2020 ST with answers and explanations. Read each question, then practise it by topic or as a full paper.
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- Question 1 – The Rajiv Gandhi Khel Ratna Award was conferred Mary Kom, a six-time world boxing champion, recently in a ceremony the Rashtrapati Bhawan in New…
- Question 2 – Despite a string of poor performances, the chances of K. L. Rahul’s selection in the team are .
- Question 3 – Select the word fitting the analogy: Cover : Uncover :: Associate : .
- Question 4 – Floods damaged summer-sown kharif crops. Officials believe winter-sown rabi output can recover the lost production so the country reaches…
- Question 5 – Subtract the sum of the first n odd positive integers from the sum of the first2n positive integers.
- Question 6 – Repo rate is the rate at which RBI lends to commercial banks; reverse repo is the rate at which RBI borrows from them. Under the usual lending-cost…
- Question 7 – P,Q,R,S,T,U,V,W sit around a circular table. S is opposite W; U is second right of R; T is third left of R; V neighbours S. Which must be true?
- Question 8 – Delhi and Agra are233 km apart. Car P leaves Delhi; Q leaves Agra toward Delhi one hour later. They meet75 min after Q starts. P travels10 km/h faster…
- Question 9 – A4×4 matrix has zero diagonal and mij=−mji. How many independent elements fully specify it?
- Question 10 – The chart gives annual profit percentages for companies P and Q during2013–2018. Both invest the same fixed amount each year. Find the ratio of total…
- Question 11 – M is a nonzero3×3 idempotent matrix and I the identity. Which statement is FALSE?
- Question 12 – Over the real field, let V=\(a,b,c) C3:a+ b=0,\ b+ c=0\. Which is a basis?
- Question 13 – On S=(x,y):x²−y²=4, let f(x,y)=6x+y². Which statement is true?
- Question 14 – For (x,y)≠(0,0), f=(4x2+9y2)/(4x2+9y2+64-8); set f(0,0)=c. Which is true?
- Question 15 – X has MGF M X(t)=1/6+et/3+e2t/3+e3t/6. Find P(X≤2).
- Question 16 – In Y ijk=+ i+ j+ ijk, i=1,2;j=1,2,3;k=1,2,3. Errors are iid N(0,σ²), Σα=Σβ=0, σ>0. SSE is residual sum of squares. Let χ²ν,a denote the(1−a) quantile.…
- Question 17 – X₁,…,X₂₀ are iid with density e- x-2 /3/6 on R. What is the distribution of W=(2/3) i=120 X i-2 ?
- Question 18 – A sample X₁,…,X₁₀ has Weibull density 3 x2e- x3 for x>0 (zero otherwise), θ>0. Let χ²ν,a be the(1−a) quantile. Which is a90% confidence interval for…
- Question 19 – A sample of size n≥2 comes from Uniform[−θ,θ], θ>0. Which statistic is minimal sufficient for θ?
- Question 20 – X₁,…,Xn are iid N(θ,2θ²), n≥2 and θ>0. Which statement is TRUE?
- Question 21 – N(t) is a Poisson process of rate2. Given N(3)=1, find the expected first arrival time.
- Question 22 – For Y i= 0+ 1x i2+ i, i=1,…,n≥2, find the OLS slope estimate. Sums run over i.
- Question 23 – X₁,…,Xn are iid Np(0,Σ), Σ positive definite and1≤p≤n−1. Define X̄=n⁻¹ΣXi and (n-1)S=(X i- X)(X i- X)T. Find the distribution of T²=nX̄ᵀS⁻¹X̄.
- Question 24 – A two-way fixed-effects ANOVA has no interaction and one observation per cell, with5 row levels and4 columns. Find residual degrees of freedom.
- Question 25 – A sample of size n≥2 has density θ⁻¹e(−x/θ), x>0, with θ∈1,2. Test H₀:θ=1 against H₁:θ=2. Which statement is true?
- Question 26 – X has characteristic function φ(t)=sin(t)cos(t)/t for t≠0 and φ(0)=1. Find P( X ≤3/2) (two decimals).
- Question 27 – X∼N₄(0,Σ), with =pmatrix1&.7&.6&.4\\.7&1&.5&.8\\.6&.5&1&.7\\.4&.8&.7&1pmatrix. Find P(X₁+X₂+X₃+X₄>0) (one decimal).
- Question 28 – A Markov chain on0,1 has P=pmatrix1/2&1/2\\1/3&2/3pmatrix and P(X₀=0)=1/3. Find27E[X₂] (two decimals).
- Question 29 – Independent events E,F,G have probabilities1/2,1/3,1/4. If p is the probability that at least two occur, find12p (one decimal).
- Question 30 – Let p(x,y,z)=P(X=x,Y=y,Z=z). The joint probability mass is alignedp(x,y,z)&=10!x!y!z!k!\,a,\&=(.2)x(.3)y(.4)z(.1)k.aligned Here k=10-x-y-z, and x,y,z…
- Question 31 – How many standard(reduced)4×4 Latin squares exist?
- Question 32 – A centred4-vector has covariance matrix with diagonal1 and every off-diagonal entry0.5. Find the proportion of total variation explained by its first…
- Question 33 – X₁,…,Xn have density e(−(x−2θ)) for x>2θ, zero otherwise, with θ>0 and n≥2. Let X(1)=min Xi. Find E[-1(X (1)-1/n) X 1-X 2=2].
- Question 34 – For Yi = α + βxi + εi, i = 1,…,7, the xi are fixed and the errors are independent with mean zero and a common finite variance. OLS estimates are α̂…
- Question 35 – Observed data0,1,2,3 come from N(θ,5), with θ restricted to[2,∞). Find the maximum-likelihood estimate.
- Question 36 – For f(x,y)=x⁴−2x³y+16y+17 on R², which is true?
- Question 37 – Over C³, T(x,y,z)=(x,3y/2-z/2,y/2+3z/2). Which is true?
- Question 38 – For real a,b,c, let M=pmatrixa&ac&0\\1&c&0\&bc&1pmatrix. Which is true?
- Question 39 – A4×4 matrix has minimal polynomial(x−1)²(x−3)². Which statement is FALSE?
- Question 40 – For f(x,y)= y-2 x-1 , which is true?
- Question 41 – Which function is uniformly continuous on its stated domain?
- Question 42 – On[0,2π]³ the joint density is (1- x 1 x 2 x 3)/(83), zero elsewhere. Which is true?
- Question 43 – Two bivariate-normal populations have means(0,0),(1,1), common covariance pmatrix1&.5\\.5&1pmatrix, equal priors and equal misclassification costs.…
- Question 44 – X₁,…,Xn,n≥2, have density θ⁻¹exp[−(x−θ)/θ] for x>θ, θ>0. Which statement is true?
- Question 45 – (X,Y) is centred bivariate normal with variances1 and correlation−1<ρ<1. Find E[max(X,Y)].
- Question 46 – Independent X₁,…,X₁₀∼N₃(0,I₃). Let J₃ be all ones and T= i=110X iT(I 3-J 3/3)X i. Find its distribution.
- Question 47 – Independent X₁,X₂,X₃∼N₄(0,Σ), Σ positive definite. Let X be the3×4 matrix whose rows are Xᵢᵀ. Wm(n,V) denotes order-m Wishart with n degrees of…
- Question 48 – X∼N₃((1,2,3),Σ), where =pmatrix1&.5&0\\.5&1&0\\0&0&5pmatrix. Which pair is independent?
- Question 49 – In one-way ANOVA Yij=μ+τi+εij,i=1,2,3;j=1,2,3,4, errors are iid N(0,σ²),Στi=0. Test allτi=0 using MST/MSE. If observed ratio is104.45, what is the…
- Question 50 – For iid N(θ,1),n≥2, test H₀:θ∈[1,2] against θ<1 orθ>2. Which statement is true?
- Question 51 – A pure birth process has rates λn=2ⁿ,n≥0. Let T be time to grow from0 to5. Find256Var(T).
- Question 52 – A Markov chain on0,1,2 has P=pmatrix0&1&0\\.3&0&.7\\0&1&0pmatrix. Find nP(X 2n=2 X 0=2) (one decimal).
- Question 53 – For y>0, X Y=y has density y exp(−yx),x>0. Y has density y exp(−y),y>0. Find E[Y X=1] (one decimal).
- Question 54 – The joint MGF is M X,Y(s,t)=e2s2+t. Let p=P(X+2Y<1). Given Φ(0) = .5, Φ(.5) = .6915, Φ(1) = .8413 and Φ(1.5) = .9332, find 2p+1 to two decimal places.
- Question 55 – A Markov chain on0,1,2,3 has P=pmatrix1/2&1/2&0&0\\1/2&1/4&1/4&0\\0&0&1/4&3/4\\0&0&0&1pmatrix. Starting at1, let p be the probability of visiting0…
- Question 56 – Joint mass is P(X=x,Y=y)=xy(1/4)x for x=1,2,… and y=0,…,x, zero otherwise. Find Var(Y).
- Question 57 – X takes1,2,3,4,5. Under H₀ probabilities are(.10,.10,.10,.10,.60); under H₁ they are(.05,.06,.08,.09,.72). Find power of the most powerful level0.1…
- Question 58 – A centred trivariate normal has variances1 and all pairwise correlations1/3. Find partial correlation of X₂ and X₃ with X₁ fixed (two decimals).
- Question 59 – Four iid observations come from a continuous symmetric location density fθ(x)=f(x−θ), with f(−x)=f(x). Under H₀:θ=0, let T⁺ be the Wilcoxon positive…
- Question 60 – Fit an intercept-and-slope OLS regression to x=(−2,−1,0,1,2),y=(3,5,8,9,10). Predict y at x=5.
- Question 61 – On D=\(x,y,z):0 x,y,z1,\ x+y+z1\ let I= D(x+y)\,dx\,dy\,dz. Find84I.
- Question 62 – Joint CDF is0 if x<0 or y<0; (1-e-x)/4 if x≥0 and0≤y<1; and1−e(−x) if x≥0,y≥1. Find16Var(X)+32Var(Y).
- Question 63 – Iid Xᵢ have EX₁=0,EX₁²=1,EX₁⁴=3. Let Y n=n-1 X i2. If P(Y n+ n(Y n-1)/32)=(c), find c² (one decimal).
- Question 64 – Joint density is (81/4)x 12x 22x 32 on−1≤x₁≤x₂≤x₃≤1, zero otherwise. Find Var(X₁+X₂+X₃) (one decimal).
- Question 65 – Five iid observations have Laplace density f(x;θ)=exp(− x−θ )/2. Test θ=0 against θ<0 using Yi=1 if Xi>0,0 otherwise. Reject ifΣYi≤2. Find test…