Solve your IGNOU Doubts

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 $S^{\circ}=\bar{S}.$

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

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Question:

Which of the following statements are true or false? Give reasons for your answers in the form of a short proof or counter-example, whichever is appropriate:

i) Every infinite set is an open set.

ii) The negation of p∧ ~ q is p → q.

iii) −1is a limit point of the interval ]−2 ,1],

iv) The necessary condition for a function $f$ to be integrable is that it is continuous.

v) The function $f:\mathbb{R}\rightarrow \mathbb{R}$ defined by $f(x)=\left | x-2 \right |+\left | 3-x \right |$ is differentiable at $x=5.$

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Question:

For the function:

$f(x,y)=x^{3}+xy-2y^{2},$

Find the polynomial given by:

$f_{xx}(1,2)(x-2)^{2}+f_{xy}(1,2)(x-2)(y-1)+f_{yy}(1,2)(y-1)^{2}$

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Question:

Show that the limit of the function ) f (x, y exists at the origin, where:

$f(x,y)=\left\{\begin{matrix} x\, cos\frac{1}{y}+y\, cos\frac{1}{x}, &(x,y)\neq (0,0) \\0 & ,\! (x,y)=(0,0) \end{matrix}\right.$

Do the repeated limits of f (x, y) exist? Justify your answer.

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Question:

Solve the differential equation:

$(2xe^{y}y^{4}+2xy^{3}y)dx+(x^{2}y^{2}e^{y}-x^{2}y^{2}-3x)dy=0$

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Question:

Using the method of variation of parameters, solve the differential equation:

$\frac{\mathrm{d^{2}y} }{\mathrm{d} x^{2}}+y=sec^{3} \, x$

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Question:

Using Lagrange’s method, solve the differential equation:

$(x^{2}-y^{2}-z^{2})p+2xq=2xz$

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Question:

Find the envelope and the characteristic curves of the family of curves:

$x^{2}(y-c)^{2}+z^{2}=c^{2}\, cos^{2}\, \alpha ,$

c and αare constants.

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Question:

Transform the given equation to Clairaut’s form and hence find its general solution:

$xy(y-px)=x-py$

Also find its singular solution, if it exists

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Question:

Find the differential equations of the space curve in which the two families of surfaces:

$u=x^{2}+y^{2}+3z=c_{1}$

and $v= zx+x^{2}+y^{2}=c_{2}$

intersect.

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Question:

The rate of change of the price of a commodity is proportional to the difference between the demand D and the supply S. If D = α − bP and , S = c sin βt where a,b,cand β are constants, determine ). P(t It is given that at $t=0,p=p_{0}$