GATE 2024 ST – Previous Year Questions with Solutions
All 65 questions of GATE 2024 ST with answers and explanations. Read each question, then practise it by topic or as a full paper.
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- Question 1 – Increasing intensity walk→jog→sprint is analogous to bothered→ →daunted.
- Question 2 – Two wizards independently try mixing all four elements water,air,fire,earth in every possible order to create a spell. After all attempts fail, how…
- Question 3 – Of 10000 students,1500 like neither their core nor other branches. The number liking core branches is one quarter the number liking other branches;500…
- Question 4 – For x,y>0, ln((x+y)/2)=[ln x+ln y]/2. Find x/y+y/x.
- Question 5 – In the sequence 6,9,14,x,30,41, find a possible x.
- Question 6 – Arrange the sentences coherently. P:This fortuitous geological event released colossal energy and heat, raising rocks to average height 4 km across…
- Question 7 – Two items sell at the same price, one at 10% profit and the other at 10% loss. Find the overall result.
- Question 8 – The pie charts show generation shares in 2007 and 2023. Renewable sources are Hydro,Solar,Wind. With total electricity generation unchanged, find the…
- Question 9 – A cube is cut into eight pieces of equal size and shape. Every cut is straight and continues through to the opposite end. What is the minimum number…
- Question 10 – The first three rows of the shown 4×4 grid contain X or a number. Each number counts its immediate horizontal,vertical and diagonal neighbors that do…
- Question 11 – D is bounded by y=x and y=4x−x². Find ∬ D x dxdy.
- Question 12 – Let a₁=√6 and a (n+1)=√(6+a n). I:the sequence increases. II:its limit is 2. Which hold?
- Question 13 – For a real 3×3 A, which claim is not true?
- Question 14 – Triples u₁,u₂,u₃ and v₁,v₂,v₃ lie in ℝ⁴ and span U,V respectively. I:if dim(U∩V)=2 and dim U=3, the v triple is dependent. II:if U+V=ℝ⁴, at least one…
- Question 15 – In standard ℝ², u=(a,b)ᵀ has inner products 2 with (1,2)ᵀ and −1 with (4,−2)ᵀ. Which holds?
- Question 16 – A real 2×3 matrix has nonzero rows a,b and rank 1. Let W=ker A,W₁=x:a·x=0,W₂=x:b·x=0. I:W=W₁∩W₂. II:W₁=W₂. Which hold?
- Question 17 – X takes only values 1,2 and E X=10/7. Find the fourth derivative of its MGF at zero.
- Question 18 – Independent fair red and blue dice are thrown once. A:red die is 5 or 6; B:sum is 7; C:sum is 8. Which holds?
- Question 19 – X has beta density Γ(α+β)x(α−1)(1−x)(β−1)/[Γ(α)Γ(β)] on (0,1),α,β>0. If E X=1/3,E X²=1/6, find α+3β.
- Question 20 – Which statement is not true for random variables X,Y?
- Question 21 – F n is the CDF of uniform(−n,n):0 below −n,(x+n)/(2n) on [−n,n),1 at and above n. Which holds?
- Question 22 – For standard Brownian motion W, which is not true?
- Question 23 – Independent X₁,X₂,X₃ are Binomial(100,p),0<p<1 unknown. Let T₁=(X₁+X₂,X₃),T₂=X₁+X₂+X₃. I:T₂ T₁ has a parameter-free distribution. II:T₁ T₂ has a…
- Question 24 – An iid sample n≥2 has density θ(2x)(θ−1) on 0<x≤1/2 and θ(2−2x)(θ−1) on 1/2<x≤1,zero otherwise,θ>0. Put L=Σ (X i≤1/2)ln(2X i)+Σ (X i>1/2)ln(2−2X i),…
- Question 25 – Which statement about hypothesis-test errors is true?
- Question 26 – A sample of 40 has category counts (5,8,12,15) across four categories. A chi-square goodness-of-fit test checks all category probabilities equal to…
- Question 27 – An iid sample n≥11 has a continuous strictly increasing CDF and median M. Test H₀:M=10 against H₁:M>10 by rejecting if T≥c, where T counts…
- Question 28 – In y i=β₀+β₁x i+ε i, the errors are uncorrelated with common mean zero and variance σ²>0. Let ȳ be the response mean and β̂₁ the least-squares slope.…
- Question 29 – In simple linear regression with intercept,n≥3,uncorrelated mean-zero errors of variance σ², let ŷ i be least-squares fitted values,T₁=Σ(y i−ŷ…
- Question 30 – For Σₙ₌₀∞a nxⁿ, coefficients satisfy a (2n+1)=1/2(2n+1) and a (2n)=1/3(2n),n≥0. Find its radius of convergence (integer).
- Question 31 – X~Poisson(λ),λ>0, satisfies P(X=4)=2P(X=5). If P(X=α) is its maximum mass, find α (integer).
- Question 32 – Three iid observations have density 2x on (0,1),zero otherwise. Find P(min(X₁,X₂,X₃)≥E X₁) to two decimals.
- Question 33 – A centered bivariate normal (X,Y) satisfies E(Y X=2)/E(X Y=1)=8. Find Var(Y X=2)/Var(X Y=1) (integer).
- Question 34 – One X~N(0,σ²) tests H₀:σ²=1 against H₁:σ²=2 by the size-.05 Neyman–Pearson rule f₂(X)/f₁(X)>c. Given Φ(1.96)=.975,Φ(1.64)=.95,upper-tail χ²…
- Question 35 – In the no-intercept model y i=β₁x i+ε i with uncorrelated zero-mean errors of common finite variance, data are (2,5),(1,6),(3,4),(2,3),(4,6). Find the…
- Question 36 – For f(x,y)=108xy−2x²y−2xy², which is not true?
- Question 37 – Define f(x,y)=(x³−y³)/(x²+y²) off the origin and f(0,0)=0. Which is not true?
- Question 38 – X has density (3/8)(x+1)² on −1<x<1,zero otherwise. For Y=1−X², find P(Y≤3/4).
- Question 39 – X has density c₁/√x for 0<x≤1,c₂/x² for x>1,zero otherwise. If P(1/4≤X≤4)=5/8, consider I:P(3≤X≤5)=1/12. II:neither X nor 1/X has finite expectation.…
- Question 40 – Checkout service times are independent exponential with density e(−x/10)/10,x≥0. On arrival one customer is already being served and has spent five…
- Question 41 – X is discrete uniform on 1,3,5,…,99. Find E(X X is not a multiple of 15).
- Question 42 – The X i are iid N(μ,σ²). I:if [√5/(σ(2n+1))]Σᵢ₌₁ⁿi(X i−μ) is N(0,1), then n=2. II:E[−ln Φ((X₁−μ)/σ)]³=6. Which hold?
- Question 43 – The iid X n have density e(−(x−θ)),x≥θ,zero otherwise,θ>0. I:the sample average converges in probability to (θ+1)/2. II:lim E[min(X₁,…,X n)]=θ. Which…
- Question 44 – Independent X n~Poisson(λ+2(−n)),λ>0 unknown. Which holds?
- Question 45 – For 25 iid N₃(μ,Σ) observations,Σ nonsingular, let S=Σⱼ₌₁²⁵(X j−X̄)(X j−X̄)ᵀ/24 and B=pmatrix1&2&3\\-1&0&1pmatrix. Which describes 24BSBᵀ?
- Question 46 – In simple regression with intercept and slope, errors are uncorrelated with mean zero and variance σ²>0. I:a 95% joint confidence region for (β₀,β₁)…
- Question 47 – Continuous f:[−2,2]→ℝ satisfies which statements?
- Question 48 – For a real n×n A, which hold?
- Question 49 – Which claims hold for real 3×3 matrices?
- Question 50 – On Ω=1,2,…, let P(k)=2(−k) and X(ω)=ω. Which hold?
- Question 51 – (X,Y) has joint density 3/4 on −1≤x≤1,x²≤y<1,zero elsewhere. Which hold?
- Question 52 – Independent X n have exponential density e(−x/λ n)/λ n,x≥0, with λ n=10−Σᵢ₌₁ⁿ5/2(i−1). Which hold?
- Question 53 – A chain on 0,1,2 has P=pmatrix1/2&1/2&0\\1/2&1/2&0\\1/3&1/3&1/3pmatrix. Which hold?
- Question 54 – An iid Poisson(λ) sample n≥2 defines T₁=X̄ and T₂=√[Σ(X i−X̄)²/(n−1)]. Which hold?
- Question 55 – For iid Bernoulli(p) X₁,X₂,X₃, define T₁(X i,X j,X k)=X i−X j(1−X k) and T₂(X i,X j,X k)=(X iX j+X jX k)/2 for distinct indices. Which hold?
- Question 56 – An iid exponential-scale sample n≥2 has density e(−x/λ)/λ,x≥0,λ>0. Put T₁=ΣX i,T₂=1/T₁. Write χ² (ν,α) for the upper-tail-α quantile. Which test is…
- Question 57 – Independent samples 1,6,5,3 and 11,7,15,4 test H₀:F=G against H₁:F(t)≥G(t),strict somewhere, by Mann–Whitney U. Given upper-tail cutoffs…
- Question 58 – (X₁,X₂,X₃)~N₃((2,−3,1)ᵀ,=pmatrix25&-2&4\\-2&4&1\\4&1&9pmatrix). For which a are X₂ and X₂−aᵀ(X₁,X₂)ᵀ independent?
- Question 59 – A is real 2×2 with trace 5,determinant 6. The characteristic polynomial of (A+I)⁻¹ is x²−bx+c. Find b/c (integer).
- Question 60 – A chain on 0,1,2 has P=pmatrix0&1/2&1/2\\1/2&1/4&1/4\\1/2&1/4&1/4pmatrix and initial probabilities P(X₀=0)=P(X₀=1)=1/4. Find 32E(X₂) (integer).
- Question 61 – The joint MGF is M (X,Y)(u,v)=exp(u²/2)/(1−2v)³ for real u and v<1/2. Find E(6X²/Y) to two decimals.
- Question 62 – A Poisson process N(t) of unknown rate λ counts road potholes up to distance t km. Starting at zero, potholes occur at distances…
- Question 63 – Four iid uniform(0,θ) observations test H₀:θ=1 against H₁:θ=.1 by rejecting if their maximum is below .3. If p is the Type-I error probability, find…
- Question 64 – Four observations from N(μ,1),μ unknown, have sample mean .16. Given Φ(2.28)=.989,Φ(1.96)=.975,Φ(1.64)=.95, a size-.05 LRT tests μ=0 against μ≠0. Find…
- Question 65 – In a regression with 25 observations,intercept and two predictors, the overall test uses F₀=11R²/(1−R²),rejecting if F₀>F (α;2,22). Given F…