GATE 2026 ST – Previous Year Questions with Solutions
All 65 questions of GATE 2026 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 antonym of protagonist is .
- Question 2 – The two shown four-tile patterns may be used any number of times and in any orientation, without overlaps. Which shown new pattern cannot be…
- Question 3 – A knockout badminton tournament begins with 64 players. Each game has no tie and eliminates its loser. Every player continues until defeated or left…
- Question 4 – A student must enroll for 60–70 credits. Exactly 15 core credits and 20 project credits are compulsory; at least 10 specialization credits are…
- Question 5 – “When the teacher is in the room, all students stand silently.” If true, which is not necessarily true?
- Question 6 – Combinatorics studies counting, for example arrangements of N distinct objects in M circular spaces. Such counting is sometimes used to model physical…
- Question 7 – Panel I shows a structure’s front and top views. Which 3D structure in Panel II has both views?
- Question 8 – For positive S,K define H K(S)=max(S−K,0), where max(a,b)=a for a>b and b otherwise. Express the plotted N(S).
- Question 9 – In the six-round Olympic javelin final, each athlete’s best throw determines rank. Neeraj Chopra won gold, Jakub Vadlejch silver, Vitezlav Vesely…
- Question 10 – A fair die marked 1–6 is rolled twice. Find the probability the second number is an integer multiple of the first.
- Question 11 – Let p(t)=t(t−1)(t−2) and F(x)= 1/2x1/p(t)dt,1/2≤x<1. Which holds?
- Question 12 – Differentiable f:(−1,1)→ℝ satisfies: I:if f(0)≥0 and f′(x)>0 at every nonnegative zero, then f(x)>0 for x>0. II:if f(0)≤0 and f′(x)>0 at every…
- Question 13 – In ℝ⁴ let W₁=x:x₁+x₂+x₃+x₄=0,W₂=x:x₁+2x₂+3x₃+4x₄=0. Find dim(W₁∩W₂).
- Question 14 – Independent discrete X,Y have M X(t)=(1+2e(−t)+3e(2t))/6 and M (X+Y)(t)=(2+eᵗ+3e(3t))/6. Which holds?
- Question 15 – Identically distributed X,Y have variance σ²∈(0,∞). Their correlation coefficient is which expression?
- Question 16 – Independent identically distributed normal X,Y satisfy P(X+2Y≤3)=P(2X−Y≥4). Find E(X).
- Question 17 – X has density 4x²e(−2x) for x>0 and zero otherwise. If Y=ln X, which holds?
- Question 18 – A homogeneous Markov chain on 1,…,7 has transition matrix P=pmatrix1/3&0&2/3&0&0&0&0\\0&1/3&0&1/3&0&1/3&0\\1/2&0&1/2&0&0&0&0\\0&1/2&0&1/4&0&1/4&0\\1/2…
- Question 19 – A sample X₁,X₂ has density f α(x)=αx(α−1)e(−xα),x>0, zero otherwise,α>0 unknown. Let X (1)=min(X₁,X₂),X (2)=max(X₁,X₂). Which statement is correct?
- Question 20 – X₁,X₂ are independent N(θ,1),θ∈ℝ. Which listed statistic attains the Cramér–Rao lower bound for estimating its own expected value?
- Question 21 – For testing H₀ against H₁ at level α∈(0,1), which statement holds?
- Question 22 – Let iid continuous pairs (X i,Y i) have joint CDF F (X,Y)(x,y)=F X(x)F Y(y). For distinct sample pairs i,j find P[(X i−X j)(Y i−Y j)>0].
- Question 23 – For an iid sample X₁,…,X n,n≥2, let Z n(x) be the proportion of observations ≤x. Which hold?
- Question 24 – Evaluate nn 1-1/(2n)1+1/(2n)et2-1dt (integer).
- Question 25 – The system x+y+z=1,2x+y+3z=6,3x+2y+kz=k+1 has no solution. Find k (integer).
- Question 26 – In ℝ⁴ let U=x:x₁+x₂+x₄=0,x₃=0. Find dim U⊥ (integer).
- Question 27 – For n≥7, X has probabilities P(X=i)=2P(X=i−1),i=2,…,n, summing to 1 over i=1,…,n. Find (2ⁿ−1)P(X=7) (integer).
- Question 28 – X,Y have joint density x+y on 0<x<1,0<y<1, zero otherwise. Find 72[Var(X)+Var(Y)] (integer).
- Question 29 – Independent X₁,X₂,X₃ have densities ke(−kx),x>0, for k=1,2,3 respectively. For Y=min(X₁,X₂,X₃), find E(3Y²−Y) (integer).
- Question 30 – Continuous X has CDF F. Let Y=−3ln F(X). Find E(Y) (integer).
- Question 31 – For standard Brownian motion W, let Z₁=W(1)+W(2),Z₂=W(2)+W(3). If ρ is their correlation, find 10ρ to two decimals.
- Question 32 – A gamma sample has density λαx(α−1)e(−λx)/Γ(α),x>0, with unknown α,λ>0. The observed averages of x i and x i² are 2 and 5. Find the method-of-moments…
- Question 33 – A sample of size 10 has density 2(x−μ)e(−(x−μ)²) for x>μ, zero otherwise. Its observed minimum is 1. Using min X i−μ as pivot, a 95% confidence…
- Question 34 – X is uniform on (θ,θ+1),θ∈ℝ. For testing H₀:θ≤0 against H₁:θ>1, let β be the power of a level-0.05 UMP test. Find 225β (integer).
- Question 35 – (X₁,X₂,X₃)ᵀ is normal with mean (1,1,1)ᵀ and covariance =pmatrix2&1&1\\1&2&1\\1&1&2pmatrix. Find Var(X₁ X₂=1,X₃=−1) to two decimals.
- Question 36 – For f(x₁,x₂)=2x₁⁴+x₂²+x₂x₁², which holds?
- Question 37 – For x₁∈(0,4), define x (n+1)=2−√(4−x n). I:x n→0. II:x (n+1)/x n→1/4. Which hold?
- Question 38 – A is real n×n,n≥2. I:every complex eigenvalue’s conjugate is also an eigenvalue. II:for a nonreal eigenvalue λ=x+iy,y≠0 and eigenvector v, Re v and Im…
- Question 39 – Consider x+2y−z=a,x+y+3z=b,2x+3y+2z=c. I:a solution exists for every (a,b,c)∈ℝ³. II:for (a,b,c)=0 the solutions are (−7t,4t,t):t∈ℝ. Which hold?
- Question 40 – On Ω let the σ-algebra be Ω,∅,A,Aᶜ, with P(A)=P(Aᶜ)=1/2 and both sets nonempty. Let X=1 A,Y=1 (Aᶜ). Which holds?
- Question 41 – Independent iid geometric X,Y have P(X=x)=p(1−p)x,x=0,1,…,0<p<1. Which holds?
- Question 42 – X has support 0,1,2,… and P(X≥k+1 X≥k)=p for every k≥0,0<p<1. Which holds?
- Question 43 – An iid sample of size 3 has P θ(X i=1)=θ,P θ(X i=2)=1−θ,0<θ<1. Which statistic is not sufficient?
- Question 44 – An iid sample of size n≥2 is used to test H₀:density (2π)(−1/2)e(−x²/2) against H₁:density e(− x )/2. At level 0.05, the likelihood-ratio critical…
- Question 45 – An iid sample n>1 has f β(x)=βe(−x)(1−e(−x))(β−1),x>0,β>0. Test H₀:β=1 against H₁:β>1 at level α. Which holds?
- Question 46 – Five iid observations come from a continuous distribution with population median θ (1/2). Test H₀:θ (1/2)=2.5 against H₁:θ (1/2)>2.5. Which is a…
- Question 47 – Independent iid X,Y follow N p(μ,Σ),Σ positive definite. Which follows a chi-square law as stated?
- Question 48 – In Y i=β₀+β₁x i1+β₂x i2+β₃x i3+ε i,i=1,…,31, the errors are iid N(0,1). The 5% overall F-test rejects all three slopes zero. Upper-tail F critical…
- Question 49 – Define f(x,y)=(x³+y³)/√(x²+2y²) away from the origin and zero at the origin. Which hold?
- Question 50 – A=pmatrix 1&1&0\\0& 2&1\\0&0& 3pmatrix with real λ i. Which hold?
- Question 51 – A=pmatrix3&1&1\\1&0&2\\1&2&0pmatrix and S=x∈ℝ³:xᵀAx=1. Which hold?
- Question 52 – X has full support 1,2,3,…, E(X²)<∞ and CDF F. Which hold?
- Question 53 – For n≥1, P(X n=x)=[1/(5n)](1−1/(5n))x,x=0,1,…. Let Z n=X n/n converge in distribution to V. Which hold?
- Question 54 – Independent X (2k−1)~Bin(1,θ),X (2k)~Bin(1,1−θ),0<θ<1. The Y k are iid Poisson(λ),λ>0. Put S 2n= k=1n(X 2k-1-X 2k+1-2), W n= k=1nY k2 and σ…
- Question 55 – N(t) is a homogeneous Poisson process of rate 3 and T₁ its first arrival time. Which hold?
- Question 56 – An iid sample n≥2 has Laplace density e(− x−μ )/2,μ∈ℝ. Let μ̂ be an MLE and X̄ the sample mean. Which hold?
- Question 57 – From a single observation, test H₀:θ=1 against H₁:θ≠1 at level 0.05. For which density families does a UMP test exist?
- Question 58 – X~N p(μ,Σ),Σ positive definite, and real A satisfies AᵀA=I. Let Y=AX. Which hold?
- Question 59 – For X=(X₁,…,X p)ᵀ~N p(0,Σ),Σ positive definite, which hold?
- Question 60 – Y~N₃(0,I₃), A is the 3×3 matrix with every entry 1/3, and B=I₃−A. Which hold?
- Question 61 – On D=(x,y):0<x²+y²≤1, let I α=∬ D(x²+y²)(−α)dxdy,α≥0. Find N₀=supα≥0:I α<∞ (integer).
- Question 62 – For X~N(3,1), find E[X⁴(X−3)] (integer).
- Question 63 – Independent U₁,U₂,U₃ are exponential with respective rates 1,2,3. Let X=min(U₁,U₃),Y=min(U₂,U₃). Find P(X=Y) to two decimals.
- Question 64 – X₁,X₂ are iid uniform(0,θ),θ>0, with maximum T=X (2). Let ψ(θ)=P θ(X₁+X₂<1). A continuous δ:(0,∞)→ℝ makes δ(T) an unbiased estimator of ψ(θ) for every…
- Question 65 – A single X has density 2x/θ² on 0<x<θ, zero otherwise,θ>0. Test H₀:θ≤1 against H₁:θ>1 at level .05. Let β₁ be the size of a UMP test and β₂ its power…