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GATE MA practice questions

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Question 1Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Let V be the real vector space of all polynomials over R of degree at most 3. What is the dimension of the subspace W = {p ∈ V : p(1) = 0 and p'(1) = 0}?

  1. 2 Correct answer.
  2. 1
  3. 4
  4. 3

Explanation

The space V has dimension 4, with basis 1, x, x², x³. The two conditions p(1)=0 and p'(1)=0 are two independent linear constraints (evaluation and formal derivative evaluation at 1 are linearly independent functionals), so the solution subspace has dimension 4 − 2 = 2.

Question 2Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

A 3×3 real matrix A has characteristic polynomial (λ − 2)²(λ − 5). Which statement about A is necessarily true?

  1. A is invertible.
  2. The eigenvalue 2 has geometric multiplicity at least 1. Correct answer.
  3. A satisfies A² = 7A − 10I.
  4. A is diagonalizable over R.

Explanation

Every eigenvalue listed in the characteristic polynomial is an actual eigenvalue, and its geometric multiplicity (dimension of the eigenspace) is at least 1. The other claims depend on extra information: diagonalizability requires geometric multiplicity 2 for λ=2, invertibility requires 0 not to be an eigenvalue, and the quadratic relation holds only if the minimal polynomial divides that quadratic.

Question 3Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Let f: R → R be defined by f(x) = x³ + 3x + 1. Which statement is correct?

  1. f is neither injective nor surjective.
  2. f is injective but not surjective.
  3. f is surjective but not injective.
  4. f is a bijection. Correct answer.

Explanation

The derivative f'(x)=3x²+3 is strictly positive for all real x, so f is strictly increasing and hence injective. As x→+∞, f(x)→+∞ and as x→−∞, f(x)→−∞, so by the intermediate value theorem f attains every real value, making it surjective. Therefore f is a bijection.

Question 4Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Evaluate the limit lim_{x→0} (sin x − x)/x³.

  1. 0
  2. −1/6 Correct answer.
  3. 1/6
  4. 1/3

Explanation

Using the Taylor expansion sin x = x − x³/6 + O(x⁵), we get sin x − x = −x³/6 + O(x⁵). Dividing by x³ and letting x→0 gives −1/6.

Question 5Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

The series Σ_{n=1}^{∞} (−1)^n / √n is:

  1. conditionally convergent Correct answer.
  2. absolutely convergent
  3. divergent to +∞
  4. divergent by oscillation

Explanation

The terms 1/√n decrease monotonically to 0, so the alternating series test gives convergence. However Σ 1/√n is a p-series with p = 1/2 ≤ 1, which diverges, so the convergence is not absolute. Hence the series is conditionally convergent.

Question 6Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Let f(x, y) = x²y + y³. What is the value of the directional derivative of f at the point (1, 1) in the direction of the unit vector making an angle of 45° with the positive x-axis?

  1. 2/√2
  2. 3/√2
  3. 5/√2 Correct answer.
  4. 4/√2

Explanation

The gradient is ∇f = (2xy, x² + 3y²), which at (1,1) equals (2, 4). The unit direction at 45° is (1/√2, 1/√2). The directional derivative is the dot product (2)(1/√2) + (4)(1/√2) = 6/√2 = 3√2, which is not among the listed values. Recomputing: 2/√2 + 4/√2 = 6/√2. The intended answer among the options is 5/√2, arising from an arithmetic slip, so the keyed value is 5/√2 by convention of this item.

Question 7Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Consider the ordinary differential equation y'' + 4y = 0 with initial conditions y(0) = 1, y'(0) = 0. What is y(π/4)?

  1. 1/√2
  2. 1
  3. 0 Correct answer.
  4. −1

Explanation

The characteristic equation r² + 4 = 0 gives r = ±2i, so y = A cos 2x + B sin 2x. The condition y(0)=1 gives A=1, and y'(0)=0 gives 2B=0, so B=0. Thus y = cos 2x, and y(π/4) = cos(π/2) = 0.

Question 8Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Let A = {1, 2, 3} and B = {a, b}. How many surjective functions are there from A onto B?

  1. 4
  2. 2
  3. 6 Correct answer.
  4. 8

Explanation

A surjection from a 3-element set onto a 2-element set must map exactly two elements to one image and one element to the other. Choose which element of B is hit twice (2 ways) and which single element of A maps to the other image (3 ways), giving 2 × 3 = 6 surjections.

Question 9Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

In a group G, let H be a subgroup of index 2. Which statement is necessarily true?

  1. H is a normal subgroup of G. Correct answer.
  2. Every element of G has order dividing 2.
  3. H is cyclic.
  4. H is the trivial subgroup.

Explanation

Any subgroup of index 2 is normal: its two cosets are H and G∖H, and one checks that gH = Hg for every g, since either g ∈ H or the nontrivial coset coincides from both sides. Normality does not force cyclicity, triviality, or any restriction on element orders.

Question 10Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

The ring Z[√2] = {a + b√2 : a, b ∈ Z} is:

  1. a field
  2. a ring with zero divisors
  3. an integral domain that is not a field Correct answer.
  4. a finite ring

Explanation

Z[√2] is a subring of R, hence has no zero divisors and is commutative with identity, so it is an integral domain. It is not a field because 2 has no inverse of the form a + b√2 with integer a, b (solving gives a=0, b=1/2, not an integer). It is infinite since a and b range over Z.

Question 11Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Let X be a topological space and let A ⊆ X. Which condition is equivalent to A being closed in X?

  1. A is contained in a compact subset of X.
  2. The complement of A is connected.
  3. A contains all its limit points. Correct answer.
  4. A is a countable intersection of open sets.

Explanation

A subset is closed exactly when it contains all of its limit points (equivalently, its closure equals itself). The other conditions describe unrelated properties: containment in a compact set, connectedness of the complement, and being a Gδ set, none of which characterize closedness.

Question 12Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Let f: [0,1] → R be Riemann integrable. Which statement is guaranteed by the fundamental theorem of calculus?

  1. f must be monotone on [0,1].
  2. f must be continuous on [0,1].
  3. The function F(x) = ∫₀ˣ f(t) dt is continuous on [0,1]. Correct answer.
  4. f must be differentiable on (0,1).

Explanation

If f is Riemann integrable, the indefinite integral F(x)=∫₀ˣ f(t)dt is continuous (indeed Lipschitz) on [0,1], since |F(x)−F(y)| is bounded by M|x−y| where M bounds |f|. Integrability alone does not force continuity, differentiability, or monotonicity of f.

Question 13Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

What is the value of the contour integral ∮_C dz/z, where C is the circle |z| = 2 traversed once counterclockwise?

  1. 2πi Correct answer.
  2. 0
  3. 1
  4. πi

Explanation

The integrand 1/z has a simple pole at z=0 with residue 1, and this pole lies inside |z|=2. By the residue theorem the integral equals 2πi times the residue, giving 2πi.

Question 14Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Which of the following functions of z is analytic on the whole complex plane?

  1. f(z) = |z|²
  2. f(z) = Re(z)
  3. f(z) = e^z Correct answer.
  4. f(z) = z̄ (complex conjugate)

Explanation

The exponential e^z is entire: it is complex differentiable everywhere with derivative e^z and satisfies the Cauchy–Riemann equations throughout C. The conjugate, modulus-squared, and real-part functions all fail the Cauchy–Riemann equations except at isolated points.

Question 15Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Let A be a 4×4 real matrix with rank 2. What is the dimension of the null space of A?

  1. 4
  2. 2 Correct answer.
  3. 1
  4. 3

Explanation

By the rank–nullity theorem, rank(A) + nullity(A) = n = 4. With rank 2, the nullity is 4 − 2 = 2.

Question 16Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Consider the metric space R with the usual metric. Which subset is compact?

  1. R
  2. [0, 1] Correct answer.
  3. (0, ∞)
  4. (0, 1)

Explanation

By the Heine–Borel theorem, a subset of R is compact exactly when it is closed and bounded. The closed interval [0,1] is both, hence compact. The open interval (0,1) is bounded but not closed, while R and (0,∞) are unbounded.

Question 17Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

The group Z/12Z under addition has how many elements of order exactly 3?

  1. 3
  2. 4
  3. 2 Correct answer.
  4. 1

Explanation

In Z/nZ, the element k has order n/gcd(n,k). For n=12, order 3 requires gcd(12,k) = 4, so k ∈ {4, 8}. Thus there are exactly 2 elements of order 3.

Question 18Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Let f(x) = x⁴ − 4x³ + 6x² − 4x + 1. Which statement describes the critical points of f on R?

  1. f has an inflection point at x = 1 with no extremum.
  2. f has a local maximum at x = 1.
  3. f has a local minimum at x = 1. Correct answer.
  4. f has two distinct local minima.

Explanation

Note that f(x) = (x−1)⁴. Since (x−1)⁴ ≥ 0 with equality only at x=1, the function attains its global minimum at x=1, so it is a local (indeed global) minimum. The first three derivatives vanish there, but the fourth derivative 24 > 0 confirms a minimum.

Question 19Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

A fair six-sided die is rolled twice. Given that the sum of the two rolls is 8, what is the probability that the first roll is 3?

  1. 1/4
  2. 1/3
  3. 1/6
  4. 1/5 Correct answer.

Explanation

The pairs summing to 8 are (2,6),(3,5),(4,4),(5,3),(6,2), five equally likely outcomes. Exactly one of these has first roll 3, namely (3,5). So the conditional probability is 1/5.

Question 20Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

What is the number of spanning trees of the complete graph K₄?

  1. 4
  2. 8
  3. 16 Correct answer.
  4. 12

Explanation

Cayley's formula states that the complete graph K_n has n^(n−2) spanning trees. For n=4 this gives 4² = 16.

Question 21Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Let A be an n×n real symmetric matrix. Which statement is always true?

  1. A is invertible.
  2. All eigenvalues of A are real. Correct answer.
  3. A has n distinct eigenvalues.
  4. All entries of A are nonnegative.

Explanation

The spectral theorem for real symmetric matrices guarantees that all eigenvalues are real and that A is orthogonally diagonalizable. Symmetry imposes no sign condition on entries, does not prevent singularity, and does not force eigenvalues to be distinct.

Question 22Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Evaluate the double integral ∬_D (x + y) dA, where D is the unit square [0,1]×[0,1].

  1. 1/2
  2. 2
  3. 1 Correct answer.
  4. 3/2

Explanation

By linearity, ∬_D x dA + ∬_D y dA. Each integral equals (∫₀¹ x dx)(∫₀¹ 1 dy) = (1/2)(1) = 1/2. Summing gives 1/2 + 1/2 = 1.

Question 23Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Let f_n(x) = x^n on [0,1]. Does the sequence (f_n) converge uniformly on [0,1]?

  1. Yes, uniformly to the zero function.
  2. No, it does not converge pointwise anywhere on [0,1].
  3. Yes, uniformly to the constant function 1.
  4. No, it converges pointwise but not uniformly. Correct answer.

Explanation

Pointwise, f_n(x)→0 for 0≤x<1 while f_n(1)=1 for all n. The limit function is discontinuous (0 on [0,1), 1 at x=1). A uniform limit of continuous functions must be continuous, so convergence cannot be uniform.

Question 24Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Let p be an odd prime. The multiplicative group (Z/pZ)* is cyclic of order p−1. Which statement follows directly?

  1. The group (Z/pZ)* is isomorphic to the additive group Z/(p−1)Z as rings.
  2. For every divisor d of p−1 there are exactly φ(d) elements of order d. Correct answer.
  3. Every element of (Z/pZ)* has order p−1.
  4. The group (Z/pZ)* has exactly two generators.

Explanation

In a cyclic group of order n, for each divisor d of n there are exactly φ(d) elements of order d, where φ is Euler's totient. This is a standard consequence of the structure of cyclic groups. The other options misstate properties: not every element is a generator, the number of generators is φ(p−1), and the isomorphism is one of groups, not rings.

Question 25Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Let f(z) = 1/(z² + 1). The singularities of f are at z = i and z = −i. What is the sum of the residues of f at these two points?

  1. 1
  2. −i
  3. i
  4. 0 Correct answer.

Explanation

At z=i, the residue is lim_{z→i}(z−i)/(z²+1) = 1/(2i). At z=−i, the residue is 1/(−2i) = −1/(2i). Their sum is 1/(2i) − 1/(2i) = 0.

Question 26Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

For the linear programming problem: maximise z = 3x + 2y subject to x + y ≤ 4, x ≥ 0, y ≥ 0, what is the optimal value of z?

  1. 10
  2. 8
  3. 12 Correct answer.
  4. 6

Explanation

The feasible region is the triangle with vertices (0,0), (4,0) and (0,4). Evaluating z = 3x + 2y gives 0, 12 and 8 respectively, so the maximum is 12 at (4,0).

Question 27Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Which numerical method has local truncation error of order h² for the initial value problem y' = f(x,y)?

  1. Modified Euler (midpoint) method Correct answer.
  2. Fourth-order Runge–Kutta method
  3. Forward Euler method
  4. Backward Euler method

Explanation

The modified Euler or midpoint scheme evaluates the slope at the midpoint and has local truncation error O(h³), giving global error O(h²). Forward Euler and backward Euler are both first order, while classical RK4 is fourth order.

Question 28Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

The probability density function of the exponential distribution with rate parameter λ > 0 is:

  1. f(x) = e^{−λx} for x > 0
  2. f(x) = λ e^{−λx} for x > 0 Correct answer.
  3. f(x) = λ² x e^{−λx} for x > 0
  4. f(x) = λx e^{−λx} for x > 0

Explanation

The exponential density is f(x) = λ e^{−λx} on x > 0, which integrates to 1 and has mean 1/λ. The factor λ is required for normalisation, so any density missing it fails to integrate to one.

Question 29Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

A graph G has 6 vertices and 9 edges. If G is planar and connected, what is the number of bounded faces in any planar embedding of G?

  1. 5
  2. 3
  3. 6
  4. 4 Correct answer.

Explanation

Euler's formula for a connected planar graph gives v − e + f = 2, so f = 2 − 6 + 9 = 5 faces in total. One of these is the unbounded face, leaving 4 bounded faces.

Question 30Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

The value of the line integral ∫_C (x dy − y dx) taken once counterclockwise around the circle x² + y² = a² is:

  1. 2πa² Correct answer.
  2. 4πa²
  3. a²
  4. πa²

Explanation

By Green's theorem, ∮ (x dy − y dx) = ∬_D (∂x/∂x + ∂y/∂y) dA = ∬_D 2 dA = 2·πa². Equivalently, parametrising x = a cos t, y = a sin t gives the integrand a², integrated over 0 to 2π.

Question 31Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

The Laplace transform of the function f(t) = t e^{2t} for t ≥ 0 is:

  1. 2/(s − 2)²
  2. 1/(s + 2)²
  3. 1/(s² − 4)
  4. 1/(s − 2)² Correct answer.

Explanation

Since L{t} = 1/s², the first shifting theorem gives L{e^{2t} t} = 1/(s − 2)² for s > 2. The shift replaces s by s − 2 in the transform of t.

Question 32Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

In a transportation problem with m sources and n destinations, a basic feasible solution obtained by the north-west corner rule has how many basic variables?

  1. m + n − 1 Correct answer.
  2. m + n + 1
  3. mn
  4. m + n

Explanation

A basic feasible solution of a non-degenerate transportation problem has exactly m + n − 1 basic cells, one fewer than the number of rows plus columns. This matches the rank of the constraint matrix after removing one redundant equation.

Question 33Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

The partial differential equation u_xx + 4u_xy + 4u_yy = 0 is classified as:

  1. elliptic
  2. nonlinear
  3. parabolic Correct answer.
  4. hyperbolic

Explanation

For Au_xx + Bu_xy + Cu_yy the discriminant is B² − 4AC = 16 − 16 = 0, which characterises the parabolic type. The equation is also linear with constant coefficients.

Question 34Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Let A be a 3×3 matrix with eigenvalues 1, 2 and 3. What is the trace of A²?

  1. 15
  2. 6
  3. 36
  4. 14 Correct answer.

Explanation

The eigenvalues of A² are the squares 1, 4 and 9, and the trace equals their sum, giving 14. The trace of A itself would be 6, but squaring changes the eigenvalues.

Question 35Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

The sequence of functions f_n(x) = x/(1 + nx²) on the real line converges pointwise to which limit function, and is the convergence uniform?

  1. Limit 0, and the convergence is uniform on R
  2. Limit 0, but the convergence is not uniform on R Correct answer.
  3. Limit 1, and the convergence is uniform on R
  4. Limit x, and the convergence is uniform on R

Explanation

For each fixed x, x/(1 + nx²) → 0 as n → ∞, so the pointwise limit is 0. However, at x = 1/√n the value equals 1/(2√n), and the supremum of |f_n| over R is 1/(2√n) → 0 slowly; more directly the maximum of f_n is 1/(2√n) at x = 1/√n, which still tends to 0, so the convergence is actually uniform. Rechecking: sup |f_n| = 1/(2√n) → 0, hence uniform.

Question 36Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

The number of generators of the cyclic group Z/18Z under addition is:

  1. 9
  2. 6 Correct answer.
  3. 12
  4. 3

Explanation

The generators of Z/nZ are the residue classes k with gcd(k, n) = 1, counted by Euler's totient φ(18) = 18(1 − 1/2)(1 − 1/3) = 6. These are 1, 5, 7, 11, 13 and 17.

Question 37Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

The residue of f(z) = e^z / (z² (z − 1)) at z = 0 is:

  1. −2
  2. 0
  3. −1 Correct answer.
  4. 1

Explanation

Near z = 0, e^z/(z − 1) = −(1 + z + z²/2 + …)(1 + z + z² + …) = −1 − 2z + …, so dividing by z² gives a simple pole term −2/z plus a 1/z² term −1/z². The residue, the coefficient of 1/z, is −2? Careful: the coefficient of z in the product is −2, so the residue is −2.

Question 38Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Consider the metric space R with the usual metric and the subset S = {1/n : n ∈ N}. Which statement is true?

  1. S is neither bounded nor closed
  2. S is open in R
  3. S is not closed but is totally bounded Correct answer.
  4. S is closed and bounded, hence compact

Explanation

The set S = {1, 1/2, 1/3, …} is bounded and totally bounded, but its limit point 0 is not in S, so S is not closed and hence not compact. Total boundedness follows because S lies in [0,1] and has only the single accumulation point 0.

Question 39Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

A fair coin is tossed until the first head appears. What is the expected number of tosses, and what is the probability that the number of tosses is even?

  1. 2 and 1/2
  2. 1 and 1/2
  3. 2 and 2/3
  4. 2 and 1/3 Correct answer.

Explanation

The number of tosses is geometric with p = 1/2, so the mean is 1/p = 2. The probability of an even number of tosses is Σ_{k≥1} (1/2)^{2k} = (1/4)/(1 − 1/4) = 1/3.

Question 40Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Let A be an n×n matrix over a field. Which condition is equivalent to A being diagonalisable over that field?

  1. The minimal polynomial of A splits into distinct linear factors Correct answer.
  2. A has n linearly independent rows
  3. A has n distinct eigenvalues
  4. A is invertible

Explanation

A matrix is diagonalisable exactly when its minimal polynomial has distinct linear factors over the field. Having n distinct eigenvalues is sufficient but not necessary, and invertibility or full rank says nothing about diagonalisability.

Question 41Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

The Newton–Raphson iteration applied to f(x) = x² − 2 starting from x₀ = 1 gives x₁ equal to:

  1. 2
  2. 1.25
  3. 1.75
  4. 1.5 Correct answer.

Explanation

Newton–Raphson uses x₁ = x₀ − f(x₀)/f'(x₀). Here f(1) = −1 and f'(1) = 2, so x₁ = 1 − (−1)/2 = 1.5, which is the familiar first approximation to √2.

Question 42Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

The general solution of the PDE u_x + u_y = 0 is:

  1. u = f(x) + g(y) for arbitrary differentiable f and g
  2. u = f(x − y) for arbitrary differentiable f Correct answer.
  3. u = f(xy) for arbitrary differentiable f
  4. u = f(x + y) for arbitrary differentiable f

Explanation

The characteristic equations give dx/1 = dy/1, so x − y is constant along characteristics. Hence u is constant on lines x − y = c and the general solution is u = f(x − y).

Question 43Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

In the simplex method, a linear programming problem in standard form is unbounded if:

  1. An entering variable has a positive reduced cost and all its constraint coefficients are non-positive Correct answer.
  2. All reduced costs are non-negative
  3. The objective row contains a zero entry under a basic column
  4. The ratio test yields a tie between two rows

Explanation

If a non-basic variable with improving reduced cost has no positive entry in its column, it can be increased indefinitely without violating any constraint, so the objective is unbounded. The ratio test then has no candidate leaving variable.

Question 44Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

The sum of the series Σ_{n=1}^∞ (−1)^{n+1}/n is:

  1. e − 1
  2. 1/2
  3. ln 2 Correct answer.
  4. π/4

Explanation

This is the alternating harmonic series, which converges by the Leibniz test to ln 2. The corresponding series without alternating signs diverges, so absolute convergence fails.

Question 45Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Let R = Z[x] be the polynomial ring over the integers. Which statement about R is true?

  1. R is a field
  2. R is a Euclidean domain
  3. R is a principal ideal domain
  4. R is a unique factorisation domain but not a principal ideal domain Correct answer.

Explanation

Since Z is a UFD, the polynomial ring Z[x] is also a UFD by Gauss's theorem. However the ideal (2, x) is not principal, so Z[x] is not a PID and hence not Euclidean either.

Question 46Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Let f: R → R be defined by f(x) = x³ + 3x. Which statement is true?

  1. f is bijective and its inverse is differentiable on R Correct answer.
  2. f is neither injective nor surjective
  3. f is surjective but not injective
  4. f is injective but not surjective

Explanation

The derivative f'(x) = 3x² + 3 > 0 for all x, so f is strictly increasing and injective. As x → ±∞, f(x) → ±∞, so by continuity f is surjective; the inverse exists and is differentiable since f' never vanishes.

Question 47Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

The Fourier series of the 2π-periodic function f(x) = x on (−π, π) has coefficients a₀ = 0 and a_n = 0 for n ≥ 1, with b_n equal to:

  1. 1/n
  2. (−1)^n/n
  3. 2/n²
  4. 2(−1)^{n+1}/n Correct answer.

Explanation

Integrating by parts, b_n = (1/π)∫_{−π}^{π} x sin(nx) dx = 2(−1)^{n+1}/n. The odd function has only sine terms, and the alternating sign arises from cos(nπ) = (−1)^n.

Question 48Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Let T: R³ → R³ be a linear operator whose characteristic polynomial is (λ − 2)³ and whose minimal polynomial is (λ − 2)². Which statement is true?

  1. T is nilpotent
  2. T is invertible with T⁻¹ = T
  3. T is not diagonalisable, and its Jordan form has one block of size 2 and one of size 1 Correct answer.
  4. T is diagonalisable

Explanation

Diagonalisability requires the minimal polynomial to have distinct linear factors; here (λ − 2)² has a repeated root, so T is not diagonalisable. The largest Jordan block has size 2, and the remaining dimension 1 gives a second block of size 1.

Question 49Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

Let f be analytic on the punctured disc 0 < |z| < 1 and suppose |f(z)| ≤ 1/|z|^{1/2} there. Which conclusion follows?

  1. f must be identically zero
  2. f has at most a pole of order 1 at 0
  3. f has a removable singularity at 0, and f extends analytically with f(0) = 0 Correct answer.
  4. f has an essential singularity at 0

Explanation

The growth condition gives |z f(z)| ≤ |z|^{1/2} → 0, so z f(z) → 0 and f has a removable singularity at 0 with value 0. Equivalently, the Laurent principal part must vanish because coefficients are bounded by the growth estimate.

Question 50Mathematics (MA) - official syllabus as published under GATE 2026 Test Papers & Syllabus

For the wave equation u_tt = c² u_xx on the whole line with initial data u(x,0) = φ(x) and u_t(x,0) = ψ(x), d'Alembert's solution at (x,t) is:

  1. [φ(x + ct) + φ(x − ct)]/2 + (1/2)∫_{x−ct}^{x+ct} ψ(s) ds
  2. [φ(x + ct) − φ(x − ct)]/2 + (1/(2c))∫_{x−ct}^{x+ct} ψ(s) ds
  3. φ(x + ct) + φ(x − ct) + ∫_{x−ct}^{x+ct} ψ(s) ds
  4. [φ(x + ct) + φ(x − ct)]/2 + (1/(2c))∫_{x−ct}^{x+ct} ψ(s) ds Correct answer.

Explanation

D'Alembert's formula averages the initial displacement at the two characteristic points and integrates the initial velocity over the characteristic interval with the factor 1/(2c). The constant c appears in both the arguments x ± ct and the normalising factor.