Solved 2025 Pyq - DSC 151 Economics Semester 2

Subject: Economics

Paper Code: ECODSC-151

Semester: 2nd Semester (FYUG)

Time: 3 Hours | Full Marks: 70

UNIT-I

Question 1 (Answer any two) 2 x 2 = 4 Marks

(a) Define Cartesian product.

The Cartesian product of two sets A and B, denoted by A x B, is the set of all ordered pairs (a, b) such that a belongs to A and b belongs to B.

(b) Mention two properties of a function.

Existence: For every element x in the domain, there must exist an image y in the codomain.

Uniqueness: Each element x in the domain is associated with exactly one element y in the codomain.

(c) Find all the subsets of the set S = {1, 2, 3, 4}.

The total number of subsets is 2^4 = 16. The subsets are:

Empty set: ø

Singletons: {1}, {2}, {3}, {4}

Pairs: {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}

Triplets: {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}

Set itself: {1,2,3,4}

Question 2 10 Marks

Option A

(i) Define limit of a function. [2 Marks]

A limit is the value that a function approaches as the input approaches some value. Mathematically, lim x->a f(x) = L means f(x) can be made arbitrarily close to L by taking x sufficiently close to a.

(ii) Find the limit: lim x->∞ [sqrt(x² + 1) - sqrt(x² - 1)]. [3 Marks]

Rationalizing the numerator:

Multiply by [sqrt(x² + 1) + sqrt(x² - 1)] / [sqrt(x² + 1) + sqrt(x² - 1)]

Numerator: (x² + 1) - (x² - 1) = 2

Denominator: sqrt(x² + 1) + sqrt(x² - 1)

As x -> ∞, the denominator approaches ∞.

Limit = 2 / ∞ = 0

(iii) Find the limit: lim x->0 (e^x - 1) / x. [3 Marks]

Using L'Hopital's Rule (since it is 0/0 form):

Derivative of numerator: e^x

Derivative of denominator: 1

Limit x->0 (e^x / 1) = e^0 = 1

(iv) Determine whether f(x) = (x² - 4) / (x - 2) is continuous at x = 2. [2 Marks]

f(2) = (4-4)/(2-2) = 0/0, which is undefined.

Since f(2) is not defined, the function is not continuous at x = 2.

Option B

Given sets U, A, B, C. Solve operations. [10 Marks]

U = {1, 2...10}, A = {3,4,5,6,7}, B = {1,5,7,9}, C = {3,6,9}

(i) A ∩ B = {5, 7}

(ii) A^c ∩ B^c = (A ∪ B)^c = {1,3,4,5,6,7,9}^c = {2, 8, 10}

(iii) (A ∪ B) - C = {1,3,4,5,6,7,9} - {3,6,9} = {1, 4, 5, 7}

(iv) A - C = {4, 5, 7}

(v) (A ∩ C) ∪ (B ∩ C) = {3, 6} ∪ {9} = {3, 6, 9}

UNIT-II

Question 3 (Answer any two) 2 x 2 = 4 Marks

(a) What is transpose of matrix? Give example.

Interchanging rows and columns of matrix A gives A^T. Example: If A = [1, 2; 3, 4], then A^T = [1, 3; 2, 4].

(b) Mention two properties of determinants.

If any two rows or columns are identical, the determinant is zero.

The determinant of a matrix and its transpose are equal.

(c) Define idempotent matrix.

A square matrix A is idempotent if A * A = A.

Question 4 10 Marks

Option A

(i) What is determinant? [2 Marks]

A scalar value that can be computed from the elements of a square matrix.

(ii) Find |B| for symmetric matrix B. [3 Marks]

|B| = a(bc - f²) - h(hc - fg) + g(hf - bg) = abc + 2fgh - af² - bg² - ch²

UNIT-III

Question 5 (Answer any two) 2 x 2 = 4 Marks

(b) Find second-order derivative of y = 5x² + 5/x.

dy/dx = 10x - 5/x²

d²y/dx² = 10 + 10/x³

Question 6 10 Marks

Option B (i) Profit Maximization

C = q³ - 10q² + 17q + 66, Price (P) = 5.

Profit (π) = Total Revenue - Total Cost = 5q - (q³ - 10q² + 17q + 66)

π = -q³ + 10q² - 12q - 66

dπ/dq = -3q² + 20q - 12 = 0

Solving for q: (3q - 2)(q - 6) = 0 => q = 6 (Check 2nd order: -6q + 20 < 0 at q=6).

Profit is maximized at q = 6.

UNIT-IV

Question 8 (a)(i) 10 Marks

Given z = x³e^(2y). Find second-order partial derivatives.

dz/dx = 3x²e^(2y)

dz/dy = 2x³e^(2y)

d²z/dx² = 6xe^(2y)

d²z/dy² = 4x³e^(2y)

d²z/dxdy = 6x²e^(2y)

UNIT-V

Question 10 (b)(ii) Consumer's Surplus 10 Marks

P = 85 - 4x - x², x0 = 5.

P0 = 85 - 4(5) - (5)² = 85 - 20 - 25 = 40

CS = Integral(0 to 5) [85 - 4x - x²] dx - (P0 * x0)

CS = [85x - 2x² - x³/3] (from 0 to 5) - (40 * 5)

CS = [425 - 50 - 41.67] - 200 = 333.33 - 200 = 133.33

Exam Tips

Always check the Second Order Condition (SOC) for Maxima/Minima questions.
In Matrix questions, double-check signs in cofactors.

Common Mistakes

Forgetting the constant 'C' in indefinite integrals.
Incorrectly applying the quotient rule in differentiation.

Important Formulas

Profit Max: MR = MC
Consumer Surplus: Integral(P dx) - P0*Q0

Answer Presentation Strategy: Use bullet points for steps and blockquotes for final laws/formulas to make the paper scannable for examiners.

Solved 2025: FYUG Even Semester Exam, 2025 ECONOMICS: ECODSC-151 (Elementary Mathematics for Economics)