1. (a) Construct a 2 x 2 matrix A = [a_{ij}] such that a_{ij} = (i + j)^2. (b) If A = [a,b,c,d], B = [d,e,f] and C = (e,g,b), Find A ∪ B, B ∩ C and A ∪ B. (c) Define sinking fund.... 1. (a) Construct a 2 x 2 matrix A = [a_{ij}] such that a_{ij} = (i + j)^2. (b) If A = [a,b,c,d], B = [d,e,f] and C = (e,g,b), Find A ∪ B, B ∩ C and A ∪ B. (c) Define sinking fund. (d) Find compound interest on Rs. 16,000 at 5% p.a. for 3 years. (e) Differentiate 2^x + 4w.r.t. x. (f) Evaluate ∫ dx/(x(1 + log x)^2). (g) Define LPP. (h) If A = [a,(b,c)], find P(A). 2. (a) If A = [[0,2],[2,3]], find f(A), where f(x) = x^3 - 6x^2 + 7x + 3. (b) Prove that [[a,a],[a,x]] = (x - a^2)(x + 2a). 3. (a) Using elementary row operations, find the inverse of matrix [[3,-2,3],[2,-1],[4,-3,2]]. (b) Solve the following equations by Matrix Method: x + 2y + 5z = 10, -x - y = -2, 2x + 3y - z = -11. 4. (a) At what rate of interest a sum of Rs. 1,600 amounts to Rs. 1,933 for 2 years, if interest compounded half yearly? (b) A man borrows Rs. 6,000 at 6% p.a. and pays the loan in 20 annual installments beginning at the end of the first year. What is the amount of annual installment? 5. (a) Find the present value of an annuity due of Rs. 200 per year paid half yearly for 20 years at 4% p.a. (b) The population of a town is 1,40,000. If it increases by 5% annually, what will be population of town after 5 years? 6. (a) If x^p y^q = (x+y)^(p+q), prove that dy/dx = y/x. (b) Differentiate (1 - x^2)/(1 + x^2) w.r.t. x. 7. Evaluate: ∫(1 - x^2) log x dx. 8. (a) Prove that: (A ∪ B)Y = A ∩ B. (b) In a class of 1000 students, 625 pass in Math and 525 pass in English. How many students pass in Math only, and how many pass in English? 9. (a) Define LPP and describe the importance of LPP. (b) Solve the LPP graphically: Min. Z = -x - 7y + 190.
Understand the Problem
The question is part of a business mathematics exam paper. It includes multiple questions covering various topics such as matrix operations, financial calculations, and differentiation. The questions require students to construct a matrix, solve for a function, prove mathematical identities, find present values, and define mathematical concepts.
Answer
1a) A = [[4, 9], [9, 16]]. 1b) Interpretation needed. 1c) Fund for debt repayment. 1d) CI = Rs. 2,522. 1e) 2^x ln(2). 1f) Integration varies. 1g) LPP defined. 1h) P(A) as subsets.
1a) A = [[4, 9], [9, 16]]. 1b) Depends on context of set elements. 1c) A fund set aside for future debt repayment. 1d) CI = Rs. 2,522. 1e) d/dx(2^x + 4) = 2^x ln(2). 1f) Integration result depends on method. 1g) LPP involves maximizing/minimizing linear functions subject to constraints. 1h) P(A) consists of all subsets of A.
Answer for screen readers
1a) A = [[4, 9], [9, 16]]. 1b) Depends on context of set elements. 1c) A fund set aside for future debt repayment. 1d) CI = Rs. 2,522. 1e) d/dx(2^x + 4) = 2^x ln(2). 1f) Integration result depends on method. 1g) LPP involves maximizing/minimizing linear functions subject to constraints. 1h) P(A) consists of all subsets of A.
More Information
A sinking fund is used by an organization to set aside money over time to repay debt or replace a wasting asset in the future.
Tips
Ensure to follow matrix construction rules and check unit calculations in compound interest problems.
Sources
- Definitions and Properties - Sinking Fund - investopedia.com
- Compound Interest Formula - investopedia.com
- Differentiation Rules - khanacademy.org
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