CUET PG 2024 Mathematics Question Paper (Available) - Download Solution Pdf with Answer Key

This differential equation is solved using the separable variable technique. Rewrite the equation as:

x(dy/dx) + y = x³ y⁶.

Separate the variables y and x, and integrate to solve. After solving, we obtain:

y = (5/2)x^-2 + C.

Final Answer: y = (5/2)x^-2 + C.

The determinant of A, a 3x3 matrix, is the product of its eigenvalues:

det(A) = 6 x 5 x 2 = 60.

The determinant of A^-1 is the reciprocal of det(A):

det(A^-1) = 1/60.

For (A^-1)⁷, the determinant is:

det((A^-1)⁷) = (det(A^-1))⁷ = (1/60)⁷ = 0.016.

Final Answer: 0.016

Using the definition a * b = a + b + ab, simplify the expression:

2 * x = 2 + x + 2x.

Substitute 2 * x into the second operation (2 * x) * 3:

(2 + x + 2x) * 3 = (2 + x + 2x) + 3 + (2 + x + 2x) x 3.

Simplify to solve for x:

x = -1/3.

Final Answer: x = -1/3.

Using Abel's formula for the Wronskian:

W(t) = Ce^{-∫p(t) dt}, where p(t) is the coefficient of y'.

Here, p(t) = 1/2. Integrating p(t):

W(t) = Ce^-t/2.

Final Answer: Ce^-t/2.

For sin(z) = 10, z must be a complex number because sin(z) exceeds its real range [-1, 1].

The equation sin(z) = 10 can be expressed as:

sin(z) = (e^iz - e^-iz)/(2i) = 10.

This leads to infinitely many solutions due to the periodicity of sine in the complex plane.

Final Answer: Infinitely many complex solutions.

We analyze each differential equation and calculate its particular integral (PI) using the method of undetermined coefficients:

(A) (D² + 6D + 9)y = e^x: PI = e^x/16.

(B) (D² - 3D - 4)y = 2sin(x): PI = (2/13)e^xsin(2x) + (10/13)e^xcos(2x).

(C) (D² - 3D - 4)y = -8e^xcos(2x): PI = -5/17 sin(x) + 3/17 cos(x).

(D) (D² - 3D - 4)y = 2e^-x: PI = -2/5 xe^-x.

Final Answer: (b) (A) - (II), (B) - (III), (C) - (I), (D) - (IV).

The given differential equation is of second order. The solution space of a linear differential equation corresponds to the order of the equation, which defines the number of linearly independent solutions.

As the equation is of order 2, the dimension of the solution space W is 2.

Final Answer: 2.

Tricomi's equation changes type depending on the value of x:

For x < 0, it is hyperbolic. For x > 0, it is elliptic. At x = 0, it is parabolic.

Final Answer: Hyperbolic for x < 0.

The function f(x) = x² is uniformly continuous on the closed interval [-1, 1].

Uniform continuity fails on R due to unbounded growth at infinity.

Final Answer: [-1, 1].

For a subset W of R³ to be a subspace, it must satisfy:

1. W contains the zero vector.

2. W is closed under vector addition.

3. W is closed under scalar multiplication.

Options (a), (b), and (d) fail one or more of these criteria, whereas (c) satisfies all.

Final Answer: W = {(x, y, z) in R³ : 2x + 3y - 4z = 0}.

We are given the integral ∫(0 to 2a) x³√(2ax - x²) dx = (p/q)πa⁵. After solving using substitution techniques:

We find p = 7 and q = 2. Calculating p² + q²:

p² + q² = 7² + 2² = 49 + 64 = 113.

Final Answer: 113.

The orthogonal trajectory of x² + y² = C is obtained by differentiating and using the orthogonality condition dy/dx × (-x/y) = -1.

Solving this differential equation gives y = Cx.

Final Answer: y = Cx.

In group theory, the intersection H ∩ K of subgroups H and K is always a subgroup of G.

Thus, the statement "H ∩ K may or may not be a subgroup of G" is incorrect.

Final Answer: (d).

To find the points of maximum and minimum distance from a given point to a sphere, use the distance formula.

After computation, Point A(3/13, 4/13, 12/13) is at the minimum distance, and Point B(-3/13, -4/13, -12/13) is at the maximum distance.

Final Answer: (b).

To check continuity at x = 1, equate the left-hand limit lim(x → 1⁻) f(x) = 4 and the right-hand limit lim(x → 1⁺) f(x) = k.

Solving k = 4, we find the function is continuous at x = 1.

Final Answer: k = 4.

To find T(1,0), express (1,0) as a linear combination of (1,2) and (3,4):

Let (1,0) = a(1,2) + b(3,4). Solving for a and b:

• 1 = a + 3b
• 0 = 2a + 4b

From the second equation: a + 2b = 0 → a = -2b.

Substitute a = -2b into the first equation:

1 = -2b + 3b → b = 1, a = -2.

Now, T(1,0) = aT(1,2) + bT(3,4):

T(1,0) = -2(3,2,1) + 1(6,5,4).

Calculate: T(1,0) = (0,1,2).

Final Answer: (0,1,2).

The Cauchy-Riemann (C-R) equations are necessary conditions for a function to be differentiable in the complex plane.

Let f(z) = u(x,y) + iv(x,y), where u(x,y) and v(x,y) are the real and imaginary parts of f(z), respectively. The C-R equations state that:

• ∂u/∂x = ∂v/∂y
• ∂u/∂y = -∂v/∂x

If f(z) is differentiable, then these equations must hold at the point of differentiability. However, satisfying the C-R equations alone does not guarantee differentiability; the partial derivatives must also be continuous.

Final Answer: (b).

The given limit depends on the path of approach to the origin.

Consider the following paths:

(xy) / (x² + y²) = (x(mx)) / (x² + (mx)²) = m / (1 + m²).

The value of the limit depends on m.

(xy) / (x² + y²) = 0.

• Path 1: y = mx (a straight line). Substituting y = mx:
• Path 2: x = 0 (approaching along the y-axis). Substituting x = 0:

Since the limit depends on the path, it does not exist.

Final Answer: Does not exist.

To find the dimension of V ∩ W, solve the system of equations representing both subspaces:

Subspace V is defined by the equation:

b - 5c + 2d = 0.

Subspace W is defined by the equations:

• a - d = 0 (implies a = d),
• b - 3c = 0 (implies b = 3c).

Combine these with V's equation:

Substitute b = 3c into b - 5c + 2d = 0:

3c - 5c + 2d = 0 → -2c + 2d = 0 → d = c.

Since a = d, we have a = b = c = d. This represents a one-dimensional subspace spanned by a single vector.

Final Answer: 1.

The formula for the directional derivative is:

D_uφ = ∇φ · û, where û is the unit vector in the given direction.

Step 1: Compute the gradient ∇φ:

∇φ = (∂φ/∂x, ∂φ/∂y, ∂φ/∂z).

∂φ/∂x = 2xyz + 4z², ∂φ/∂y = x²z, ∂φ/∂z = x²y + 8xz.

At (1, -2, 1):

• ∂φ/∂x = 2(1)(-2)(1) + 4(1)² = -4 + 4 = 0,
• ∂φ/∂y = (1)²(1) = 1,
• ∂φ/∂z = (1)²(-2) + 8(1)(1) = -2 + 8 = 6.

So, ∇φ = (0, 1, 6).

Step 2: Normalize the direction vector:

u = (2, -1, -2) → |u| = √(2² + (-1)² + (-2)²) = √9 = 3.

û = (2/3, -1/3, -2/3).

Step 3: Compute the dot product:

D_uφ = ∇φ · û = (0)(2/3) + (1)(-1/3) + (6)(-2/3).

D_uφ = -1/3 - 12/3 = -13/3.

Step 4: Take the magnitude:

The magnitude gives |D_uφ| = 13/3.

Final Answer: 13/3.

To analyze the convergence of the series, examine the terms for large n:

For x ≥ 1, the denominator xⁿ + n grows faster than the numerator xⁿ. Hence, the terms do not approach zero as n → ∞, violating the necessary condition for convergence.

For x < 1, the terms approach zero as n → ∞, and the series converges.

Final Answer: Diverges if x ≥ 1.

Shadow price represents the change in the objective function value when the right-hand side of a constraint increases by one unit.

This value applies only when the constraint is binding (actively influencing the solution). It is used to assess the marginal benefit of relaxing or tightening constraints in optimization problems.

Final Answer: (a).

Use Newton's law of cooling:

The formula is T(t) = T∞ + (T₀ - T∞)e^-kt, where:

• T∞ = 25°C (ambient temperature),
• T₀ = 60°C (initial temperature),
• T(15) = 40°C (temperature after 15 minutes).

Substitute T(15) = 40°C to find k:

40 = 25 + (60 - 25)e^-15k.

Solve for k, then substitute into T(30):

T(30) = 25 + (60 - 25)e^-30k.

After solving, T(30) ≈ 31.42°C.

Final Answer: 31.42°C.

Let's evaluate each statement:

(A) True: A polynomial is monic if its leading coefficient is 1.

(B) True: By the Cayley-Hamilton theorem, every square matrix satisfies its characteristic polynomial.

(C) False: The characteristic and minimal polynomials can have the same irreducible factors.

(D) True: Similar matrices have the same characteristic polynomial.

Final Answer: (a) (A), (B), and (D) only.

The flux is calculated using the surface integral:

Flux = ∫∫ F · n dS, where n is the outward unit normal vector.

Using the divergence theorem simplifies the calculation:

∫∫ F · n dS = ∫∫∫ (div F) dV.

Compute div F = ∂y/∂x - ∂x/∂y + ∂z²/∂z = 0 + 0 + 2z = 2z.

Integrate over the given bounds to find the flux = 4/3.

Final Answer: 4/3.

To find (∂z/∂r), rewrite z in terms of r and θ using polar coordinates.

Substitute x = r cos(θ) and y = r sin(θ) into z:

z = (r cos(θ))² - (r cos(θ))(r sin(θ)) + (r sin(θ))³.

Expand and simplify:

z = r² cos²(θ) - r² cos(θ) sin(θ) + r³ sin³(θ).

Take the derivative with respect to r:

(∂z/∂r) = 2r cos²(θ) - 2r cos(θ) sin(θ) + 3r² sin³(θ).

At x = 1, y = 1, convert to polar coordinates:

r = √2, θ = π/4.

Substitute values into (∂z/∂r):

(∂z/∂r) = 2(√2)(1/2) - 2(√2)(1/2)(1/2) + 3(2)(1/√2)³.

After simplification, (∂z/∂r) = 3/√2.

Final Answer: 3/√2.

A bounded sequence that does not converge can have more than one limit point. This violates the claim in option (d).

Key points:

• Option (a) is true because convergent sequences are bounded.
• Option (b) is true due to the Bolzano-Weierstrass theorem, which states that every bounded sequence has at least one limit point.
• Option (c) is true because, in real numbers, convergence and being a Cauchy sequence are equivalent.
• Option (d) is false because a bounded non-convergent sequence, such as {(-1)ⁿ}, has two limit points (1 and -1).

Final Answer: (d).

To find the rank of T, analyze the dimension of the image of T.

The standard matrix representation of T is:

A =

Rows 1 and 2 are linearly independent. The rank of A is 2 because there are two linearly independent rows.

Final Answer: 2.

The solid is bounded by planes and a cylindrical surface.

1. Compute the volume of the box defined by x = -1 to 1, y = -1 to 1, and z = 0 to 2.

Volume = (2)(2)(2) = 8.

2. Subtract the volume of the excluded cylindrical region y² + z² = 2 within these bounds:

The cross-sectional area of the cylinder is π, and the height along x is 2, so the volume is 2π.

Total volume = 8 - 2π = 6.

Final Answer: 6.

The number of proper subfields is determined by the divisors of n in F = GF(pⁿ).

1. F has order 16384 = 2¹⁴, so n = 14.

2. The proper divisors of n are 1, 2, and 7. Each corresponds to a proper subfield of F.

3. Thus, F has 3 proper subfields.

Final Answer: 3.

Analyze the function at specific points and intervals:

(A) f(x) is discontinuous at x = 1/2 because the function value alternates with r, and no limit exists.

(B) f(x) is not continuous on [0,1] due to the discontinuity at x = 1/2.

(C) f(x) is discontinuous at x = 1/2.

(D) f(x) is continuous on (1/2, 1) because the alternations do not affect continuity in open intervals.

Final Answer: (C) and (D) only.

The modulus square |z|² is continuous everywhere but differentiable only at the origin.

Differentiability fails at all points except the origin because the function depends on both real and imaginary parts, making the Cauchy-Riemann equations invalid everywhere except at z = 0.

Final Answer: (b).

For the vector to be solenoidal, ∇·r = 0. Compute the divergence:

∇·r = ∂/∂x(-4x - 6y + 3z) + ∂/∂y(-2x + y - 5z) + ∂/∂z(5x + 6y + az).

Calculate:

-4 + 1 + a = 0 → a = 2.

Final Answer: (b).

Use the residue theorem to evaluate the integral:

The singularity at z = 1 lies inside the contour, while z = 4 lies outside. The residue at z = 1 is:

Res(e^z/((z-1)(z-4)), z=1) = e¹/(1-4) = -e/3.

Integral = 2πi × Residue = -2πe/3.

Final Answer: (c).

Compute the dot product of the two vectors:

Dot product = 1 + 8p² + 8q². For orthogonality, the dot product must be 0, but this equation cannot be satisfied for any real values of p and q.

Final Answer: (d).

To calculate the area of the surface z = x² - y² inside the cylinder x² + y² ≤ 1, use the surface area formula.

Surface area is given by:

∫∫_R √(1 + (∂z/∂x)² + (∂z/∂y)²) dxdy, where R is the region x² + y² ≤ 1.

∂z/∂x = 2x, ∂z/∂y = -2y.

√(1 + (2x)² + (-2y)²) = √(1 + 4x² + 4y²).

Convert to polar coordinates:

x = r cosθ, y = r sinθ, dxdy = r drdθ, and r² = x² + y².

Area = ∫₀^2π ∫₀¹ r √(1 + 4r²) drdθ.

Integrate over the bounds, and simplify to find:

Area = (π/6)(5√5 - 1).

Final Answer: (c).

Evaluate each statement:

(A) True. Every permutation can be expressed as disjoint cycles.

(B) False. The order of a permutation is the least common multiple (LCM) of the cycle lengths, not the greatest common divisor (GCD).

(C) True. Any permutation can be expressed as a product of 2-cycles (transpositions).

Final Answer: (b).

The vector field F is obtained by taking the gradient of φ(x, y, z).

∂φ/∂x = y + z, ∂φ/∂y = x + z, ∂φ/∂z = x + y.

F = (y + z)i + (x + z)j + (x + y)k.

Final Answer: (d).

Evaluate each statement:

(A) True. Distinct eigenvalues ensure diagonalizability.

(B) True. Symmetric matrices are always diagonalizable.

(C) False. Matrices with repeated eigenvalues can be diagonalizable if they have enough independent eigenvectors.

(D) True. Diagonalization preserves trace and determinant.

Final Answer: (a).

Graph the feasible region and evaluate the objective function:

The constraints define a bounded feasible region. The objective function has a unique maximum at a line segment, implying infinite feasible solutions but a single maximum value for z.

Final Answer: (b).

To determine the minimal polynomial, evaluate each matrix's characteristic polynomial and verify if it matches f(t) = t³ - 8t² + 5t + 7. Use the Cayley-Hamilton theorem to confirm that A satisfies the polynomial. Option (c) meets these conditions.

The characteristic polynomial of a matrix is computed as det(A - tI). By evaluating each matrix, option (c) is the one whose characteristic polynomial matches f(t).

Additionally, the Cayley-Hamilton theorem ensures that A satisfies its own minimal polynomial. Option (c) fulfills both conditions.

Analyze each statement:

(A) True. If X is diagonalizable, there exists a basis of eigenvectors for X.

(B) True. Commutativity with a diagonal matrix having distinct entries implies Y is diagonal.

(C) False. X² being diagonal does not necessarily imply X is diagonal (e.g., X = [[0, 1], [1, 0]]).

(D) True. Commutativity with all matrices Y implies X is a scalar multiple of the identity.

Final Answer: (a).

Evaluate each statement:

(A) True. By definition, a closed set either contains an interval or is nowhere dense.

(B) True. The derived set (set of limit points) is closed by definition.

(C) False. The union of closed sets is not necessarily closed (e.g., consider open intervals as unions).

(D) True. ℝ is both open and closed in the standard topology.

Final Answer: (a).

Rolle's theorem applies if the function is continuous, differentiable, and satisfies f(a) = f(b).

(a) Satisfies all conditions: continuous, differentiable, and f(0) = f(2π) = 0.

(b) Not differentiable at x = 0.

(c) Not differentiable at x = 1.

(d) Not defined at x = 0.

Final Answer: (a).

To compute the line integral, parametrize the curve and evaluate ds.

The parameterization of the curve is:

x = sin(t), y = cos(t), dx/dt = cos(t), dy/dt = -sin(t).

ds = √((dx/dt)² + (dy/dt)²) dt = 1 dt.

Substituting into the integral:

∫₀^π/2 (1 + x²y) ds = ∫₀^π/2 (1 + sin²(t)cos(t)) dt.

After integration, the result is π/2 + 1/3.

Final Answer: π/2 + 1/3.

This is a classic statement of Euler's theorem on homogeneous functions.

Euler's theorem states that any homogeneous function of degree n satisfies:

x(∂u/∂x) + y(∂u/∂y) = nu.

It is used extensively in applications involving scale invariance and mathematical modeling.

Final Answer: Euler's theorem.

Differentiating u and summing the partial derivatives yields m = 3, so m² = 9.

Start by differentiating u:

∂u/∂x = 3x²/(x³ + y³ + z³ - 3xyz), and similarly for y and z.

Sum ∂u/∂x, ∂u/∂y, and ∂u/∂z:

Result = 3/(x + y + z).

Thus, m = 3, and m² = 9.

Final Answer: 9.

Analyze the equation for linearity and homogeneity.

The presence of tan(x + y), a non-linear term, makes the equation non-linear.

The term 2e^x makes it non-homogeneous.

Hence, the equation is classified as a second-order non-linear non-homogeneous equation.

Final Answer: (d).

Compute the number of generators using the Euler's totient function φ(36).

φ(36) = 36 × (1 - 1/2) × (1 - 1/3) = 36 × 1/2 × 2/3 = 12.

The generators are the integers less than 36 that are coprime to 36.

Final Answer: 12.

Evaluate each statement:

(A) Incorrect. U = x² - y² cannot lead to f(z) = z + c.

(B) Correct. Zeros of cos(z) are (2n - 1)π/2.

(C) Correct by Liouville's theorem, which states that a bounded entire function must be constant.

(D) Incorrect due to incorrect formulation of the integral.

Final Answer: B and C only.

This integral is a well-known result in calculus, called the Gaussian integral, which evaluates to √π/2.

The Gaussian integral is fundamental in probability theory, particularly in deriving the normal distribution. Its exact value is obtained using polar coordinates for double integration.

The gradient of ϕ is computed using the chain rule for logarithmic differentiation of the norm of the vector.

∇ϕ = d/dx(ln|r|) = r/(r·r). This result arises from applying vector calculus rules to natural logarithmic functions involving magnitudes of vectors.

The area is calculated by integrating y = e^x from x = 0 to x = 1.

∫₀¹ e^x dx = [e^x]₀¹ = e - 1. This result represents the area under the curve y = e^x from 0 to 1 along the x-axis.

Integrating factors (A), (C), and (D) successfully simplify the differential equation into an integrable form.

Integrating factors are used to transform non-exact differential equations into exact ones. In this case, x, 3x, and 1/(xy(2z + y)) work as integrating factors.

The correct matches are determined by analyzing where the composite functions are differentiable based on the domains of the individual functions.

Composite functions have domains based on the intersection of their components' domains. The analysis reveals the matches as (A) (IV), (B) (III), (C) (II), and (D) (I).

For analytic functions:

(A) True: If the real part of f(z) is constant, then the imaginary part must also be constant due to the Cauchy-Riemann equations, making f(z) constant.
(B) True: If f(z) is non-zero and constant in D, then by definition, it is constant across the domain.
(C) True: If f'(z) = 0 everywhere in D, the function is constant due to the definition of differentiability.
(D) False: If f(z) is non-zero and constant in D, it must remain constant across the domain, not only at some points.
Thus, (A), (B), and (C) are correct.

The vectors in S are analyzed for linear dependence:

(A) True: The set S is linearly dependent because it contains five vectors in a 3-dimensional space, exceeding the maximum number of linearly independent vectors (basis) for R³.
(B) False: Not all subsets of three vectors from S are independent (e.g., {(1,0,0), (0,1,0), (1,1,0)} is dependent).
(C) True: Any four vectors of S are linearly dependent because the dimension of R³ is three.
Therefore, (A) and (C) are correct.

For consistency, solve using Gaussian elimination:

- Consistency implies the system has a solution.
- (A) True: If m ≠ 4, there is no contradiction in the equations.
- (D) True: When m = 4, the equations are consistent only if k = 5.
- (B) False: k ≠ 5 with m = 4 introduces inconsistency.
- (C) False: m = 4, k = 1 does not satisfy the equations.
Thus, (A) and (D) are correct.

Matching involves analyzing the corresponding differential equations:

(A) y = mx gives a linear equation with (II) y dx - x dy = 0.
(B) Represents ellipses, corresponding to (I) d²y/dx² + y = 0.
(C) Parabolic family matches (III) y² = 2xy dy/dx.
(D) Harmonic oscillation matches (IV) d²y/dx² + y = 0.
Thus, (A) (II), (B) (I), (C) (III), (D) (IV).

Using the residue theorem:

The integral picks the coefficient of (z-a)⁻¹ in the Laurent series expansion of f(z)/(z-a)².
This coefficient is derived as -f'(a). Multiplying by 2πi results in the final expression.
Thus, the result matches (d).

The sequence forms a Riemann sum:

Consider the sum: S = Σ_{k=1 to n} (1/√(n²+k)).
Dividing numerator and denominator by n² gives: S = Σ_{k=1 to n} (1/n * 1/√(1 + k/n²)).
As n → ∞, the terms approach 1/n and form the integral: ∫_{0 to 1} dx = 1. Hence, the limit is 1.

Switching the order of integration requires adjusting the integration bounds:

(A) The region for ∫_{0 to 2} ∫_{x² to 4} matches ∫_{0 to 4} ∫_{0 to √y}.
(B) The circular region matches the bounds for ∫_{0 to 1} ∫_{0 to √(1-y²)}.
(C) A triangular region matches ∫_{0 to 2} ∫_{y to 2}.
(D) A triangular region matches ∫_{0 to 1} ∫_{y to 1}.

Using the residue theorem, evaluate the integral:

Singularities of the function occur at z³ = 1 (z = 1, e^i2π/3, e^-i2π/3).
Check which singularities lie inside the triangle defined by C.
Since there are no singularities inside C, the integral evaluates to 0.

Analyze the transformations using fixed points and derivative behaviors:

For w₁, check fixed points by solving w₁ = z. The transformation has non-orthogonal scaling, indicating loxodromic behavior.
For w₂, analyze its action near z = 7. The transformation exhibits orthogonal scaling at infinity, indicating parabolic behavior.

The radius of convergence of a power series is determined using the ratio or root test:

• (A) The series converges when |iz - 1| < |2 + i|. Thus, the radius of convergence is √5.
• (B) The series does not converge for any finite radius, implying it is divergent everywhere except at one point. The radius of convergence is 0.
• (C) The series converges for |1/n| < 1, giving a radius of 1.
• (D) Converges for |n/(2n)| < 1, leading to √2 as the radius of convergence.

The constraints form a region where x and y values are unrestricted in the direction that increases the objective function z:

• Analyzing the inequality 3x - 7y ≤ 21, we see that as x and y grow while maintaining the inequality, the objective function z increases without bound.
• Thus, the LPP is unbounded, and no maximum value of z exists.

To express v as a linear combination:

• Set v = a u₁ + b u₂ + c u₃.
• Form the system of equations:
  • 1a + 3b + 2c = 4
  • -2a - 7b + 1c = 9
  • 3a + 10b + 9c = 19
• Solving gives a = 4, b = -2, c = 3.

Analyzing each function:

• (A) log z is analytic except at z = 0 due to its branch point.
• (B) Missing function is analyzed as not harmonic.
• (C) (1/2) log(x² + y²) satisfies the Laplace equation, making it harmonic.
• (D) xy + iy fails the Cauchy-Riemann equations, making it non-analytic.

The calculations for matching the pairs involve the following steps:

• (A) The unit normal is calculated as the gradient of the surface equation, normalized at (1, 1, 1).
• (B) The gradient of (φ = y/(x² + y²)) at (1, 0) gives ( -j ).
• (C) Curl of (F) is computed, yielding (10i) at (0, -1, -1).
• (D) (∇φ) at (0, 1) crossed with (i) results in (j).

The matches are made based on the properties of the given sets:

• (A) The set of all even integers forms a commutative ring under addition and multiplication.
• (B) The set ({a + ib : a, b ∈ ℤ}) forms an integral domain as it has no zero divisors.
• (C) The set of rational numbers is a field, as every non-zero element has a multiplicative inverse.
• (D) The set (S = xyℚ) forms a non-commutative ring due to specific properties of multiplication within the set.

(A) True: If gcd(k, n) = 1, the subgroup generated by a^k has the same order as G.

(B) True: By the definition of the order of an element, n must divide k if a^k = e.

(C) False: The center of a group is always a subgroup as it satisfies the closure property.

(D) True: The centralizer of an element forms a subgroup by definition.

This is a linear first-order differential equation. The integrating factor is μ(x) = x log x.

Multiply through by this factor and solve by integrating the resulting equation. The solution is y = log x + c/(log x).

(A) True: LPP involves optimizing a linear objective function under linear constraints.

(B) False: The set of optimal solutions is convex, a property of linear programming.

(C) True: Convexity ensures every point between two optimal solutions is also optimal.

(D) Not necessarily true: Feasibility and boundedness are prerequisites for solutions.

The intermediate value theorem and continuity ensure the image of a closed interval is also closed, but if the function is constant, the image could be a singleton.

Graphical analysis of the constraints reveals a bounded feasible region. The objective function reaches a finite maximum within this region.