Description
| SESSION | November 2024 |
| PROGRAM | Bachelor of CoMPUTER APPLICATIONS (BCA) |
| SEMESTER | I |
| course CODE & NAME | DCA1105 – Fundamentals of Mathematics |
Set-I
- 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 , where , let’s determine if it satisfies the properties of being reflexive, symmetric, and transitive.
Reflexive
A relation on a set is reflexive if every element satisfies .
Its Half solved only
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- Write the composite function if
- and
- and .
Ans 2.
To find the composite function , substitute into . Below are the steps for both cases:
Case 1
Given:
The composite function is:
Substitute into :
Expand :
- Evaluate the followings:
(i) (ii)
Ans 3.
Let’s evaluate the given limits step-by-step:
(i)
Evaluate:
Step 1: Identify the highest power of in the numerator and denominator:
- In the numerator, the highest power is .
- In the denominator, the highest power is .
Set-II
- Find the derivative of .
Ans 4.
We need to find the derivative of the given function:
Step 1: Use the Quotient Rule
The quotient rule states:
Here:
- Consider the function . Determine where the function is increasing or decreasing.
Ans 5.
To determine where the function is increasing or decreasing, follow these steps:
Step 1: Compute the derivative of
The derivative of , denoted as , indicates the slope of the tangent line to the curve and helps identify intervals of increase or decrease:
- Evaluate (i)
(ii)
Ans 6.
Let’s evaluate the two given integrals step by step.
(i) Evaluate
To solve , we use integration by parts, where:
Here, let:
- (so that ),
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