Solve your IGNOU Doubts

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 $v = (0,1,2,-2)$ .

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Question:

Use the method of Lagrange’s multiplier method to find the shortest possible distance from the ellipse $x^2 + 2y^2 = 2$ to the line $x + y = 2$ .

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Question:

Near what points may the surface $z^2 + xz + y = 0$ be represented uniquely as a graph of a differentiable function $z = k(x,y)$ ? Locate such a point.

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Question:

Let E be an open subset of $R^n$ and $f : E \rightarrow R^m$ be a function such that each of its components function $f_i$ are differentiable, then show that f is differentiable. Is the converse of this result true? Justify your answer.

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Question:

Find the first derivative $f'(a)$ of the function f defined by $f : R^3 \rightarrow R^2$ given by $f(x,y,z) = (xyz, x + y+z^2)$ where $a = (1. -1,2)$ .

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Question:

Show that an infinite discrete metric space X is bounded but not totally bounded.

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Question:

Let $(X_1,d_1)$ and $(X_2,d_2)$ be metric spaces. Show that $f : X \rightarrow Y$ is continuous if and only if $f (\overline{A}) \subseteq \overline{f(A)}$ where A is any subset of X

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Question:

Find the interior, boundary and closure of the following sets A in $\mathbb{R}$ with the usual metric and discrete metric.

i) A = $\mathbb{Q}$ , the set of rationals in $\mathbb{R}$

ii) $A = ]1,2] \cup ]2 ,4[$

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Question:

Let A and B be any two subsets of a metric space $(X, d)$ , then show that
i) int $A = \cup$ {E : is open and $E \subseteq A$ }
ii) int $(A \cap B)$ = int A $\cap$ int B
iii) int(A $\cup$ )B $\supseteq$ int A $\cap$ int B
iv) $\overline{A \cap B} \subseteq \overline{A} \cap \overline{B}$ .

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Question:

Let $(X, d)$ be a metric space and $a \in X$ be a fixed point of $X$ . Show that the function $f_a : X \rightarrow R$ given by $f_a (x) = d(x,a)$ is continuous. Is it uniformly continuous? Justify you answer

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Question:

Let $X = C[0, 1]$ . Define $d : X \times X \rightarrow R$ by $d(f,g) = \int_{0}^{1}\mid f(t) -g(t) \mid dt , f, g \in X$ where the integral is the Riemann integral. Show that d is a metric on X . Find $d(f,g)$ where $f(x) = 4x$ and $g(x) = x^3$ , $x \in [0,1]$

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Question:

What do you understand by file organisation? Explain the methods of file organisation.

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Question:

Explain the meaning of the terms garbage collection, fragmentation, relocation and compaction.

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Question:

Write a function that evaluates an expression in RPN that is given as a string.

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Question:

Define a structure of type hms containing three integer members, called hour, minute and second, respectively. Then define a union containing two members, each a structure of type hms. Call the union members local and home, respectively. Declare a pointer variable called time that points to this union.

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Question:

Write the inorder, preorder and postorder traversals of the following binary search tree

Image ignouassignments-ignouacademy-com--p-your-52872

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