MATH MJC-3, MJC-4, MIC-3

 

🔴 MATHEMATICS — MJC–3

REAL ANALYSIS

CIA ASSIGNMENT — 

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Q1.

Prove that every bounded monotonic sequence of real numbers is convergent. Explain why the boundedness condition is necessary.

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Q2.

Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function on a closed interval $[a,b]$. Prove that $f$ is uniformly continuous on $[a,b]$. Discuss the importance of this result in real analysis.

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Q3.

If a sequence $\{a_n\}$ converges to a limit $L$, prove that every subsequence of $\{a_n\}$ also converges to $L$. Is the converse true? Justify your answer.

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🔴 MATHEMATICS — MIC–3/MDC-3

REAL ANALYSIS

CIA ASSIGNMENT — 

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Q1.

Let $\{a_n\}$ be a Cauchy sequence in $\mathbb{R}$. Prove that $\{a_n\}$ is convergent. Explain why this result is fundamental to the completeness of real numbers.

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Q2.

Suppose $f : (a,b) \to \mathbb{R}$ is differentiable on $(a,b)$ and continuous on $[a,b]$. Prove that there exists at least one point $c \in (a,b)$ such that

$$f'(c) = \frac{f(b)-f(a)}{b-a}.$$

Also explain the geometric interpretation of this theorem.

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Q3.

Prove that a continuous function on a closed and bounded interval attains its maximum and minimum values. Why is this theorem not necessarily true on an open interval?

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🔴 MATHEMATICS — MJC–4

ORDINARY DIFFERENTIAL EQUATIONS

CIA ASSIGNMENT —

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Q1.

Solve the differential equation

$$(1 + y^2)\frac{dx}{dy} + xy = y^3$$

and discuss the nature of the solution.

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Q2.

Solve the second-order differential equation

$$\frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 4y = e^{2x}$$

and interpret the behavior of the solution.

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Q3.

Using the method of variation of parameters, solve

$$\frac{d^2y}{dx^2} + y = \tan x.$$