GATE 2025 ST – Previous Year Questions with Solutions
All 65 questions of GATE 2025 ST with answers and explanations. Read each question, then practise it by topic or as a full paper.
Practise this paper in The GATE Grind →
- Question 1 – Even though I had planned to go skiing with my friends, I had to at the last moment because of an injury. Select the most appropriate phrase.
- Question 2 – The President, along with the Council of Ministers, to visit India next week. Select the most appropriate option.
- Question 3 – Electricity costs ₹7 per kWh. A 40 W desk light runs 10 hours each night for 180 days. Find its energy cost, and the percentage increase if it runs…
- Question 4 – In the shown block pattern, find entries (i),(ii),(iii),(iv), respectively.
- Question 5 – A bag contains Violet V,Yellow Y,Red R,Green G balls. Their counts satisfy Y+2V=50,V+G=50,Y+R=50,V+2R=50. Which shown pie chart represents the bag?
- Question 6 – From E.V. Lucas’s “A Funeral”: “His life was divided between the books, his friends, and long walks. A solitary man, he worked at all hours without…
- Question 7 – For the shown letter clock, O*=O Q S Z P R T and X*=X Z P W Y O Q. Which is P*?
- Question 8 – A five-digit number PQRST uses each of 1,2,3,4,5 once and satisfies P<Q,S>P>T,R<T. Find P.
- Question 9 – Variety P potatoes cost ₹800 per 5 kg; Q costs ₹800 per 4 kg. A mixture sells at ₹192/kg for 8% profit. Find the weight ratio P:Q.
- Question 10 – Villages P,Q,R have PQ=13 km,QR=14 km,RP=15 km as shown. A straight road joins Q,R. Find the minimum length of a road connecting P to road QR (km).
- Question 11 – Differentiable f:[0,∞)→[0,∞) has f(0)=0,f(x)>0 for x>0, and f(x)2= 0x[f(t)2+f(t)]dt. Which range contains f(2)?
- Question 12 – Which statement about countability is correct?
- Question 13 – Let S=(x,y,z)∈ℝ³∖0:z=−(x+y). Define S⊥=v∈ℝ³:v·s=0 for every s∈S. Which holds?
- Question 14 – For X~Poisson(ln2), find E[e((ln3)X)].
- Question 15 – (X₁,X₂,X₃) is multinomial with 100 trials and probabilities (3/10,1/10,3/5). Find E(X₂ X₃=40).
- Question 16 – The X n are iid with density 1/[π(1+x²)] on ℝ. Define Y n=1/2+arctan(X n)/π. Which convergence in probability holds?
- Question 17 – N(t) is a Poisson process of rate 2. Let X=N(6)−N(1),Y=N(5)−N(3),W=N(6)−N(5),Z=N(3)−N(1). Which holds?
- Question 18 – T is complete sufficient for a family of distributions, U is sufficient, and T≥0 almost surely under every member. Which is not necessarily true?
- Question 19 – For iid X₁,X₂~N(θ,1),θ∈ℝ, let φ be the size-.05 likelihood-ratio test of H₀:θ=0 against H₁:θ≠0. Which holds?
- Question 20 – X has density f₀=1 on [0,1] or f₁=1 on [1,2], zero elsewhere. Let φ be a most powerful level-.05 test of f₀ against f₁. Which is necessary?
- Question 21 – For simple densities f₀,f₁, let φ be a most powerful level-.05 test based on X. Which is not necessarily correct?
- Question 22 – For iid X n with CDF F, let F n be the empirical CDF. At a fixed real x, which holds?
- Question 23 – A bivariate normal (X,Y) has means 3,4, variances 25,100 and covariance 50ρ,−1<ρ<1. If E(Y X=5)=4.32, find ρ.
- Question 24 – For data (x i,y i) with Σx i²>0, β̂ minimizes Σ(y i−βx i)². Put v i=y i−x i,u i=2x i, and let γ̂ minimize Σ(v i−γu i)². If β̂=10, find γ̂.
- Question 25 – Evaluate I=2 01 01 y2[(1+xy)]dx\,dy (integer).
- Question 26 – P=pmatrix1&2\\-1&4pmatrix and Q=P³−2P²−4P+13I₂. Find det Q (integer).
- Question 27 – T:ℝ³→ℝ³ sends (x₁,x₂,x₃) to (3x₁+5x₂+x₃,x₃,2x₁+2x₃). Find rank T (integer).
- Question 28 – X has CDF F with lim (h→0−)F(3+h)=1/4 and F(3)=3/4. Find 16P(X=3) (integer).
- Question 29 – For X~Binomial(2,1/3), find 18E(X²) (integer).
- Question 30 – X is 10-dimensional N₁₀(0,I). Let Y=ln√(XᵀX) and M Y(t)=E(e(tY)),t>−10. Find M Y(2) (integer).
- Question 31 – For standard Brownian motion W, find E[(W(2)+W(3))²] (integer).
- Question 32 – Observed Bernoulli sample values are 0,1,1,1,0, with parameter θ∈(0,.7]. Find the MLE of θ to two decimals.
- Question 33 – Five observations are iid N(θ,6),θ∈ℝ. Let c(θ) be the Cramér–Rao variance bound for unbiased estimation of θ. Find 15inf (θ∈ℝ)c(θ) (integer).
- Question 34 – Three independent observed pairs are (1,3),(2,4),(7,8). Find their Spearman rank correlation to two decimals.
- Question 35 – A normal linear regression has n=25 observations, an intercept and four predictors, with unknown error variance. If R²=5/6, find adjusted R² to two…
- Question 36 – Let F be all real functions continuous on [a,b] and differentiable on (a,b). Which is correct?
- Question 37 – On U=(x,y):x+y≤2, find the minimum of f(x,y)=(x−1)⁴+(y−2)⁴.
- Question 38 – A 10×10 real P has diagonal entries 9/10 and off-diagonal entries −1/10. Find its rank.
- Question 39 – X has CDF 0 for x<0, α(1+2x²) for 0≤x<1 and 1 for x≥1. Its median is 1/√2. Find α.
- Question 40 – X>0 has lognormal density f(x)=( x2)-1[-12(( x-)/)2]. If ln(E(X²)/(E X)²)=4, find Var(ln X).
- Question 41 – For λ>0, joint masses are p (X,Y)(m,n)=λⁿe(−λ)/[2ⁿm!(n−m)!],m=0,…,n,n=0,1,…. Which holds?
- Question 42 – An iid sample n≥2 is N(−θ,θ), with θ>0 unknown. Which holds?
- Question 43 – X₁,X₂ are iid exponential of scale θ>0. For H₀:θ≤1 against H₁:θ>1, φ=1 (X₁>1). With S=X₁+X₂, which test has the same power function as φ?
- Question 44 – Independent X,Y₁,Y₂ have exponential rates 2,1,1 respectively. R i is the ascending rank of Y i among the three observations. Find E(R₁+R₂).
- Question 45 – Five iid bivariate N₂(0,I₂) vectors form rows of a 5×2 matrix X. Let W=XᵀX and Z=W₁₁+4W₁₂+4W₂₂. Find Var Z.
- Question 46 – For y i=α+βx i+ε i,i=1,…,24, iid N(0,σ²) errors, the centered sums are S xx=22.82,S yy=43.62,S xy=15.48. Find the F₁,₂₂ statistic testing β=0 against…
- Question 47 – For n≥1, x n=1+1/√2+1/√3+⋯+1/√n−2(√n−1). Which hold?
- Question 48 – Let O be all real 3×3 orthogonal matrices with determinant 1. Which hold?
- Question 49 – Independent standard normal X₁,X₂,X₃ define Y₁=X₁−X₂,Y₂=X₁+X₂−2X₃,Y₃=X₁+X₂+X₃. Which hold?
- Question 50 – Independent X n converge almost surely to zero. Which are necessarily true?
- Question 51 – A Markov chain on 1,2,3 has P=pmatrix0&1/2&1/2\\1/3&0&2/3\\2/5&3/5&0pmatrix and π=(18,24,25)/67. Which hold?
- Question 52 – An iid sample is uniform on (−θ/2,θ/2),θ>0. Which hold?
- Question 53 – X~N₃(0,diag(4,9,4)),α=(2,0,−1)ᵀ,β=(1,1,1)ᵀ. Which hold?
- Question 54 – For Y=Xβ+ε with ε~N n(0,I),X real n×p and λ>0, let β̂=(XᵀX+λI p)⁻¹XᵀY. Which hold?
- Question 55 – Find the number of saddle points of f(x,y)=x²y²+8x−4y (integer).
- Question 56 – A 5×5 P has zeros on the diagonal, ones above and minus ones below. Find the product of its five eigenvalues (integer).
- Question 57 – P=pmatrix2&1\\1&2pmatrix, Q=pmatrix1&1\\-2&4pmatrix. Find trace(P⁵+Q⁴) (integer).
- Question 58 – Independent X,Y,Z have MGFs M X(t)=(2+eᵗ)²/9,M Y(t)=exp(eᵗ−1),M Z(t)=exp[2(eᵗ−1)]. Find 10P(X>Y+Z) to two decimals.
- Question 59 – Service densities at pumps P₁,P₂ are λe(−λx) and λ²xe(−λx),x>0. Each pump is selected with probability 1/2. If the chance of service exceeding one…
- Question 60 – Independent X n take values ±1/2ⁿ, each with probability 1/2. If Σᵢ₌₁ⁿX i converges in distribution to U, find 6P(U≤2/3) (integer).
- Question 61 – Seven iid observations have density (λ³x²/2)e(−λx),x>0,λ>0 unknown. The MLE λ̂ has bias E(λ̂−λ)=αλ. Find 1/α (integer).
- Question 62 – X₁,X₂ are iid with density e(x−θ) for x≤θ, zero otherwise,θ∈ℝ. For H₀:θ≥0 against H₁:θ<0, let β(θ) be the power of a level-.09 UMP test. Find β(ln…
- Question 63 – X~Bin(3,θ),0<θ<1. To test H₀:1/4≤θ≤3/4 against the outside alternatives, reject for X=0 or 3. Find the test size to two decimals.
- Question 64 – X N 3(0,pmatrix1&.4&0\\.4&1&.6\\0&.6&1pmatrix). Find the partial correlation of X₁,X₂ given X₃ to two decimals.
- Question 65 – Bivariate normal X,Y have means 2,3,variances 16,25 and covariance 14. Find 2π[P(X>2,Y>3)−1/4] to two decimals.