CSIR NET Practice Set 1

Explanation: By Cantor's diagonal argument, the set of real numbers in [0,1] is uncountable. The integers, rationals, and natural numbers are all countably infinite.

Explanation: This is the alternating harmonic series, which converges conditionally to ln 2. This follows from the Taylor series expansion of ln(1 + x) evaluated at x = 1.

Explanation: For p ≠ 1, ∫₁^∞ x^(−p) dx = [x^(1−p)/(1−p)]₁^∞, which converges only when 1 − p < 0, i.e., p > 1. For p = 1, the integral equals ln x evaluated at infinity, which diverges.

Explanation: A 2×2 symmetric matrix has the form [[a, b], [b, c]] with 3 free parameters (a, b, c). A basis consists of [[1,0],[0,0]], [[0,1],[1,0]], [[0,0],[0,1]], so the dimension is 3.

Explanation: Factoring: λ³ − 6λ² + 11λ − 6 = (λ − 1)(λ − 2)(λ − 3). The eigenvalues are 1, 2, and 3. Verification: sum = 6 (trace), product = 6 (determinant).

Explanation: ‖f‖² = ⟨f, f⟩ = ∫₀¹ x² dx = [x³/3]₀¹ = 1/3. Therefore ‖f‖ = 1/√3.

Explanation: eᶻ is an entire function (analytic everywhere in ℂ). The functions |z|², z̄ (complex conjugate), and Re(z) fail to satisfy the Cauchy-Riemann equations except at isolated points.

Explanation: Using partial fractions: 1/(z²(z − 1)) = −1/z − 1/z² + 1/(z − 1). The residue at z = 0 is the coefficient of 1/z in the Laurent expansion, which is −1.

Explanation: The permutation (1 2 3 4) is a single 4-cycle. The order of a permutation equals the LCM of its cycle lengths. A 4-cycle has order 4.

Explanation: By Sylow's third theorem, n₃ divides |G|/3 = 4 and n₃ ≡ 1 (mod 3). The divisors of 4 are 1, 2, 4. Only 1 and 4 satisfy n₃ ≡ 1 (mod 3), so n₃ ∈ {1, 4}.

Explanation: ℤ[x]/⟨x, 2⟩ ≅ ℤ₂, which is a field. An ideal is maximal if and only if the quotient ring is a field. The other quotients (ℤ, ℤ₂[x], ℤ[i]) are integral domains but not fields.

Explanation: A topology must contain ∅ and X, and be closed under arbitrary unions and finite intersections. {∅, {1}, {1,2}, X} satisfies all axioms: {1} ∩ {1,2} = {1} and {1} ∪ {1,2} = {1,2}, all present in the collection.

Explanation: By the Heine-Borel theorem, a subset of ℝ is compact if and only if it is closed and bounded. [0, 1] is both. (0, 1) is bounded but not closed, [0, ∞) is closed but unbounded, and ℝ is unbounded.

Explanation: The standard method for solving first-order linear ODEs is the integrating factor method. The integrating factor is e^(∫P(x) dx), which when multiplied through makes the left side an exact derivative.

Explanation: The characteristic equation r² − 5r + 6 = (r − 2)(r − 3) = 0 has distinct real roots r = 2 and r = 3. The general solution is c₁e²ˣ + c₂e³ˣ.

Explanation: For the general second-order PDE Au_xx + 2Bu_xt + Cu_tt = ..., the discriminant B² − AC determines the type. For the heat equation, the discriminant equals zero, classifying it as parabolic.

Explanation: With boundary conditions X(0) = X(L) = 0, the solutions are Xₙ(x) = sin(nπx/L) with eigenvalues λₙ = (nπ/L)² for n = 1, 2, 3, ... These are the only values satisfying both boundary conditions.

Explanation: Newton-Raphson uses the tangent line at (xₙ, f(xₙ)) to approximate the root. Setting the tangent line to zero gives xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ), which has quadratic convergence near simple roots.

Explanation: Simpson's 1/3 rule passes a quadratic polynomial (parabola) through three equally spaced points and integrates it exactly. The formula is (h/3)[f(x₀) + 4f(x₁) + f(x₂)] where h = (b − a)/2.

Explanation: A polynomial of degree n has n + 1 coefficients (a₀, a₁, ..., aₙ). By the Lagrange interpolation theorem, n + 1 distinct points uniquely determine a polynomial of degree at most n.

Explanation: The Euler-Lagrange equation is the necessary condition for a functional to have an extremum. It is derived by setting the first variation of the functional to zero.

Explanation: A Fredholm integral equation of the second kind has the unknown function both outside and inside the integral with fixed limits: f(x) = g(x) + λ∫ₐᵇ K(x,t)f(t) dt. Option A is the first kind (no g(x)), and option C is Volterra type (variable upper limit).

Explanation: The Lagrangian is defined as L = T − V (kinetic energy minus potential energy). The equations of motion are obtained from the Euler-Lagrange equation d/dt(∂L/∂q̇) − ∂L/∂q = 0.

Explanation: A central force acts along the radial direction, producing zero torque about the center. Therefore angular momentum about the center is always conserved for any central force.

Explanation: For mutually exclusive events, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B) = 0.6 + 0.4 = 1.0.

Explanation: For a Poisson random variable, P(X = k) = e^(−λ)·λᵏ/k!. At k = 0: P(X = 0) = e^(−4)·4⁰/0! = e^(−4) ≈ 0.0183.

Explanation: The Central Limit Theorem states that √n(X̄ₙ − μ)/σ converges in distribution to the standard normal N(0, 1) as n → ∞, regardless of the distribution of the individual observations (given finite variance).

Explanation: An estimator θ̂ is unbiased if its expected value equals the true parameter value E(θ̂) = θ for all θ. This means the estimator is correct on average, though individual estimates may deviate.

Explanation: The Neyman-Pearson lemma gives the most powerful test of a given size for testing a simple null hypothesis against a simple alternative hypothesis. It uses the likelihood ratio as the test statistic.

Explanation: R² represents the proportion of variance in the dependent variable explained by the independent variable. As a proportion, it lies between 0 (no explanatory power) and 1 (perfect fit).