DCA6107 FUNDAMENTALS OF MATHEMATICS APRIL 2025

Description

SESSION	APRIL 2025
PROGRAM	MCA
SEMESTER	I
COURSE CODE & NAME	DCA6107 FUNDAMENTALS OF MATHEMATICS

 

 

SET-I

 

 

1. If , show that

Ans 1.

Step 1: Understanding the Function

We are given:

This is an exponential function, where the base  is Euler’s number — a fundamental constant in mathematics.

Step 2: Differentiating the Function

To find the derivative , we use the basic rule of differentiation:

Its Half solved only

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2. write

 

Ans 2.

Term 1:

We recall a standard integral identity:

So:

 

 

This integration problem demonstrates the effective use of standard techniques to evaluate seemingly complex expressions. It emphasizes the importance of understanding foundational integrals and their application in advanced mathematics. Mastery of these methods equips learners with tools to solve practical problems in physics, engineering, and economics involving continuous change.

 

 

3. Find all the second order derivatives for .

Ans 3.

Step 1: First Order Partial Derivatives

Partial derivative with respect to x:

• Derivative of r.t x is
• r.t x is
• r.t x is 0

 

 

SET-II

 

 

4. Given , , , find the magnitude of: (i) (ii) (iii)

Ans 4.

We are to find:

(i) Magnitude of

Formula for magnitude:

For

 

(ii) Vector sum

 

5. Find the value of (All angles are in degrees)

Ans 5.

Step 1: Recall Trigonometric Values

Function	Value

 

Step 2: Simplify the Numerator

 

 

 

6. If , prove that

Ans 6.

In this problem, the complex number expression under the square root simplifies into another complex number. By taking the modulus on both sides, we eliminate the imaginary component and work only with real positive values. This allows us to apply the rule that the modulus of a square root of a complex fraction is the square root of the modulus of that fraction.

Prove that

Step 1: Let z = complex expression