Solve your IGNOU Doubts

Question:

Discuss the megascopic and microscopic characters of granite and basalt with the help of neat well labelled diagrams.

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Question:

Describe various mechanisms of magmatic differentiation.

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Question:

Discuss the different types of textures found in igneous rocks with the help of neat well labelled diagrams.

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Question:

Phase Rule

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Question:

Extraterrestrial Rocks

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Question:

State whether the following statements are True or False. Justify your answers.
a) The sequence $\left \{ \left ( \frac{1}{n},\frac{1}{n} \right ) : n \in N \right \}$ is convergent in $R^2$ under the discrete metric on $R^2$ .
b) A subset in a metric space is compact if it is closed.
c) Continuous image of a path connected space is path connected.
d) The second derivative of a linear map from $R^n$ to $R^m$ never vanishes.
e) If $\int_{A}f dm = \int_{A}g dm$ for all $A \in M$ , then $f = g$ .

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Question:

Find the fourier series of the function $f$ defined by

$f(x) = \begin{Bmatrix} -x^2, -\pi <x\leq 0\\ x^2, 0<x < \pi \end{Bmatrix}$

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Question:

Verify Bounded Convergence Theorem for the sequence of functions $\left \{ f_n \right \}$ where

$f_n (x) = \frac{1}{\left ( 1 + \frac{x}{n} \right )^n } , 0\leq x \leq 1, n \in N$

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Question:

Show that if f is measurable, then the function $f^a(x)$ given by

$f^a(x) = \begin{Bmatrix} a &if \: f(x)>a \\ f(x) & if \: f(x) \leq a \end{Bmatrix}$
is also measurable.

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Question:

Find the measure of the following sets.

i) $E = \bigcap_{n=1}^{\infty }\left ( a-\frac{1}{n}, b \right )$
ii) $E = Q \cup \left \{ 1 ,2 ,3,4 \right \}$
iii) $E = ]5,7[ \cup [7,7.5]$ .

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Question:

Let A be any set in R , show that $m^*(A) = m^*(A+x)$ where $m^*$ denotes the outer measure.

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Question:

Consider Z and let $F_1$ denote the class of subsets of Z , given by $F_1 = A \subset Z$ either A is finite or $A^c$ is finite}. Check whether $F_1$ is a $\sigma$ algebra or not.

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Question:

Which of the following sets are connected sets in 2 R with the metric given against it?
Justify your answer.
i) $A ={( x,y) : 0\leq x \leq 1, 0 \leq y \leq 2}$ under the standard metric.
ii) $A = \left \{ ( x,y) : x^2 + y^2 = 1\right \}$ under the discrete metric.

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Question:

Which of the following sets are totally bounded? Give reasons for your answer. Are they compact?
i) 2N in (N,d) where d is the discrete metric.
ii) $[0,2] \cup [5,10]$ in (R,d) where d is the Euclidean metric.

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Question:

Let Q be the set of rationals with the metric defined on Q by $d :Q\times Q \rightarrow R$ , defined by $d(x, y) = | x - y |, \forall x, y \in R$ . Show that $\left \{ \left ( 1 + \frac{1}{n}\right )^n \right \}$ is Cauchy sequence in Q, but does not converge in Q and $\left \{ \frac{1}{3^n} \right \}$ is a Cauchy sequence Q which converges in Q to the limit 0 .

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Question:

Show that the components of a metric space is either identical or pairwise disjoint.

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Question:

Give an example of the following with justification
i) A vector-valued function $f : R^3 \rightarrow R^3$ which is not differentiable at $(0,0,0)$ .
ii) A function which is Legesgue measurable on R.

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Question:

Let A be a compact non-empty subset of a metric space $(X, d)$ and let F be a closed subset of X such that $A \cap F = \phi$ , then show that $d(A, F)> 0$ where $d(A, F) = inf \left \{ d(a,b): a\in A, b \in F \right \}$ .

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Question:

Find the directional derivative of the function $f : R^4 \rightarrow R^3$ defined by $f(x, y, z, w) = (x^2y, xyz, x^2 + y^2 +zw^2)$ at $a = (1,2,-1,-2)$ in the direction $v = (0,1,2,-2)$ .