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