Business Mathematics Question Paper 2024 (May/June) [Dibrugarh University BCOM 4th SEM CBCS Pattern]

Business Mathematics Question Paper 2024 (May/June)

[Dibrugarh University BCOM 4th SEM CBCS Pattern]

COMMERCE (Core)

Paper: C-409 (Business Mathematics)

Full Marks: 80

Pass Marks: 32

Time: 3 hours

The figures in the margin indicate full marks for the questions.

1. Answer any five questions: (2 x 5 = 10)

(a) Define a matrix. 

(b) If 

=

determine the values of x and y. 

(c) Find the value of

.

(d) What is the first-order derivative of y = 5e3x?  

(e) If u = f (x, y) is a function, then define the partial derivatives ∂u/∂x and ∂u/∂y. 

(f) What do you mean by present worth of an annuity?  

(g) What do you mean by an LPP?

2. (a) (i) If , then determine A + 2B. (2 marks)  

 (ii) Write any three differences between a determinant and a matrix. (3 marks) 

(iii) If , prove that A² - 5A + 7I = 0. (4 marks) 

(iv) Solve using matrix method: (5 marks) 

x + 2y + z = 4 

3x + y + z = 5 

x + y + 2z = 4

OR

(b) (i) Define determinant. (2 marks) 

(ii) If A = (1 2 3 4) and , then determine AB and BA. (3 marks) 

(iii) If , , and AX = B, then determine the values of x1 and x2. (4 marks)  

(iv) Solve using Cramer's rule: (5 marks) 

x + y + z = 6 

y + 3z = 11

x - 2y + z = 0

3. (a) (i) If f(x) = x² - 3|x|, then determine f (0), f (-1) and f (2). (2 marks) 

(ii) If f(x) = (1 - x) / (1 + x), then show that f ((1 - x) / (1 + x)) = x. (3 marks)  

(iii) If y = √ ((1 + x) / (1 - x)), then determine dy/dx. (4 marks) 

(iv) If y = Aemx + Be-mx, then prove that d²y/dx² - m²y = 0. (5 marks)

OR

(b) (i) Write the first principle of derivatives. (2 marks) 

(ii) Evaluate: 11. Evaluate:  (3 marks)  

(iii) A function f(x) is defined as follows:  

f(x) = x when x < 1; 

f(x) = 1 when x = 1; 

f(x) = 2 - x when x > 1. 

Prove that the function is continuous at x = 1. (4 marks) 

(iv) For what value of x will the function f(x) = 2x³ - 9x² + 12x - 1 have the maximum and the minimum values?  (5 marks)

4. (a) (i) What do you mean by total differential?  (2 marks) 

(ii) Verify Euler's theorem for the function f = xy / (x + y). (3 marks)  

(iii) If u = y/z + z/x + x/y, then prove that x(∂u/∂x) + y(∂u/∂y) + z(∂u/∂z) = 0. (4 marks)  

(iv) If u = log (x² + y²), prove that ∂²u/∂x² + ∂²u/∂y² = 0. (5 marks)

OR

(b) (i) If u = x² + 2y, then determine ∂u/∂x and ∂u/∂y. (2 marks) 

(ii) Verify Euler's theorem for the function u = (x - y) / (x + y). (3 marks) 

(iii) If f(x,y) = 2xy / (x² + y²), prove that f is a homogeneous function. (4 marks) 

(iv) Using definition, find ∂f/∂x and ∂f/∂y from the function f = x² - y. (5 marks)

5. (a) (i) Define perpetual annuity and deferred annuity. (2 marks) 

(ii) The difference in simple interest on a certain sum of money at 14.5% p.a. for 3 years and 4.5 years is 348. Find the sum. (3 marks) 

(iii) A man deposited 5,000 at the end of every year in a bank at 6% p.a. rate of compound interest. What sum will he receive from the bank at the end of 10 years?  (4 marks) 

(iv) Discuss about various types of annuities. (5 marks)

OR

(b) (i) Write the relation between effective rate of interest and nominal rate of interest. (2 marks) 

(ii) In what time a sum of money triples itself at 8% p.a. compound interest, if the interest is compounded annually?  (3 marks) 

(iii) What will be the nominal rate of interest convertible half-yearly when the effective rate is 12% p.a.?  (4 marks) 

(iv) The value of a machine depreciates every year by 10%. The value of a new machine was 1,00,000 and after some year of use, the value of the machine is 15,000. For how many years was the machine in use?  (5 marks)

6. (a) (i) Write the assumptions of an LPP. (2 marks) 

(ii) Write the general mathematical model for LPP. (3 marks)  

(iii) Discuss about the limitations of LPP. (4 marks) 

(iv) Using graphical method, solve the following LPP: (5 marks) 

Maximize Z = 4x1 + 3x2 

Subject to: 

5x1 + 3x2 ≤ 15 

3x1 + 5x2 ≤ 15 

x1, x2 ≥ 0

OR

(b) (i) Write two limitations of LPP. (2 marks) 

(ii) Discuss the scope of LPP in solving business commerce problems. (3 marks) 

(iii) Write short notes on: (4 marks)

a) Unbounded solutions

b) Multiple optimal solutions 

(iv) A company produces two commodities A and B. Profits from each unit of A and B are 3 and 4 respectively. Formulate the LPP: (5 marks)