BCA DCA1105 FUNDAMENTALS OF MATHEMATICS

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SESSION November 2024
PROGRAM Bachelor of COMPUTER APPLICATIONS (BCA)
SEMESTER I
course CODE & NAME Fundamentals of Mathematics (DCA1105)
  

 

 

SET-I

 

 

  1. Show that the relation R in the set given by is reflexive but neither symmetric nor transitive.

Ans 1.

To analyze the relation  on the set  defined by , we will examine its properties: reflexivity, symmetry, and transitivity.

Reflexivity

A relation  on a set  is reflexive if every element of  is related to itself, i.e.,  for all .

In this case, , and we observe that:

  • (element  is related to itself),
  • (element  is related

 

 

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  1. Write the composite function if
  2. and  
  3. and   .

 

Ans 2.

To find the composite function , we substitute the function  into . Let’s compute the composite functions for each pair of  and :

 

(a) Given  and :

To compute , substitute  into :

 

 

  1. Evaluate the followings:

(i)                (ii)  

 

Ans 3.

(i)

Step 1: Identify the dominant terms

As , the dominant term in the numerator and denominator is . To simplify, divide every term in the numerator and

 

 

 

 

SET-II

 

 

 

  1. Find the derivative of .

 

Ans 4.

To find the derivative of

we use the quotient rule:

 

  1. Consider the function . Determine where the function is increasing or decreasing.

Ans 5.

Determine where  is increasing or decreasing

To determine where  is increasing or decreasing, follow these steps:

 

Step 1: Find the

 

 

 

 

  1. Evaluate (i)

 

   (ii)

Ans 6.

 

(i)

Use integration by parts: , where:

  • Let , so