Solve your IGNOU Doubts

Question:

Determine the points of discontinuity of the function f and the nature of discontinuity at each of those points:

$f(x)=left{egin{matrix} -x^{2}, &when: xleq 0 \ 4-x,&when: 0< xleq 1 \ 3x-4x^{2}&when: 1< xleq 2 \ -12x+2x&when: x> 2 end{matrix}ight.$

Also check whether the function f is derivable at x = .1

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Question:

Evaluate

$_{nightarrow infty }^{lim}left [ frac{n}{1+n^{2}}: +: frac{n}{4+n^{2}}+frac{n}{9+n^{^{2}}}+...+frac{n}{2n^{2}} ight ].$

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Question:

Show that $left ( frac{1}{n^{2}+n+1} ight )nepsilon mathbb{N}$ is a Cauchy sequence.

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Question:

Let be any sequence. Show that $_{nightarrow infty }^{lim}: : a_{n}$ = L for every ε > 0, there exists some N ∈ $mathbb{N}$ such that n $geq$ N implies $a_{n}: epsilon : : N_{varepsilon }$ (L).

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Question:

The product of two divergent sequences is divergent. True or false? Justify

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Question:

Give an example of a divergent sequence which has two convergent subsequences. Justify your claim.

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Question:

Give an example for each of the following.

i) A set in $mathbb{R}$ with a unique limit point.

ii) A set in $mathbb{R}$ whose all points except the one are its limit points.

iii) A set having no limit point.

iv) A set S with $S^{o}=ar{S.}$

v) A bijection from $mathbb{N}: _{odd}, to: mathbb{Z}$ to Z.

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Question:

Which of the following statements are true and which are false? Justify your answers with a short proof or a counter-example.

i) If x and y are real numbers such that x < y, then . x² < y²

ii) For every finite set S, sup S ∈ S

iii) There exists an interval with infimum and supremum equal.

iv) The sequence $left ( 1,frac{1}{2},frac{1}{3},frac{1}{4},... ight )$ is unbounded.

v) If a sequence is bounded, then it has at least two convergent subsequences.

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Question:

Differentiate between recharge shafts and recharge trenches.

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Question:

Enumerates major steps for domestic water conservation.

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Question:

What is lining of ponds? Explain its importance in reducing the water losses in the field?

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Question:

Define water use efficiency. Calculate the farm conveyance efficiency, if discharge of 60 litre per second from the source was released and 48 litre per second was delivered to the field.

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Question:

Determine the gross capacity of pond for applying 6 cm depth of irrigation to 10 ha area and meeting water need of 40 buffaloes and 25 cows for 1 month. Assume necessary data.

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Question:

What is irrigation scheduling? Write its role in maximizing irrigation efficiencies.

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Question:

What is artificial groundwater recharge? Write advantages of artificial groundwater recharge?

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Question:

Describe the role of contour vegetative barrier for in-situ water harvesting with the help of neat diagram.

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Question:

Under what conditions embankment type and dug out cum embankment type are constructed.