Solve your IGNOU Doubts

Question:

State whether the following statement are true or false. Justify your answer with the help of a short proof or a counter-example.

i) The initial value problem

$\frac{dy}{dx}=x^2+y^2,y(0)0$

has a unique solution in some interval of the form - $h<x<h.$

ii) The orthogonal trajectories of all the parabolas with vertices at the origin and foci on the

$x-axis$ is $x^2+2y^2=c^.$

iii) The normal form of the differential equation

$y^{''}-4xy'+(4x^2-1)y=-3ex^2 sin 2x is \frac{d^2v}{dx} +v=-3sin2x,$

where $v=ye^{-x2}.$

iv The solution of the pde $\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}=z^2$ is $z=[y+f(x-y)].$

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Question:

Define a relation R on $\mathbb{Z},$ by R = $=\left \{ \left ( n.n+3k \right )\mid k\in \mathbb{Z} \right \}.$ Check whether R is an equivalence relation or not. If it is, find all the distinct equivalence classes. If R is not an equivalence relation, define an equivalence relation on $\mathbb{Z}.$

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Question:

c) Prove that $\lim_{x\rightarrow 0}xsin\frac{2}{x}=0.$

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Question:

b) Let $f(x,y)=\left\{\begin{matrix} \frac{x^2y}{x^4 +y^2} ,if x^4+y^2\neq 0& \\ 0, & if\,x=y=0\\ & \end{matrix}\right.$

Check whethe $\lim_{(x,y)\rightarrow (0,0}f(x,y)$ exists or not.

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Question:

a) Let the $f(x,y)=\left\{\begin{matrix} \frac{xy(x^2-y^2)}{x^2+y^2}\,x,y)\not\equiv (0,0) & \\ 0, & (x,y)=(0,0)\\ & \end{matrix}\right.$

Show that

i) $f_x(0,y)=y,$ for all y

ii) $f_x(x,0)=x,$ for all $x.$

Hence, verify that $f_{xy}(0,0)\neq f_{xy}(0,0).$

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Question:

c) Find fog and gof, if they exist, for the functions

$f(t)=4t,t\epsilon R,g(x,y)=x+y,x,y\epsilon R.$

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Question:

b) Find the point on the ellipse $\frac{x^2}{4}+y^2=1,$ hat is nearest to the origin.

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Question:

a) Find the mass of an object which is in the form of a cuboid $[0,1]\times [2,4]\times [1,3].$ The density at any poin $(x,y,z)$ on the cuboid is given by $\delta (x,y,z)=x(2+y^2+z^2).$

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Question:

Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so.)

i) $\phi(n)=n-1\forall n\in \mathbb{N},$ where $\phi$ is the Euler-phi function.

ii) If $G_{1}$ and $G_{2}$ are groups, and $f:G_{1}\rightarrow G_2$ is a group homomorphism, then $o(G_1)=o(G_2).$

iii) If G is an abelian group, then G is cyclic.

iv) If G is a group and $H\underline{\Delta}G$ ,then $\mid$ $G:H\mid=2.$

v) Every element of $S_n$ has order at most n .

vi) If R is a ring and I is an ideal of R , then I xr = rx ∀ x ∈ I and r ∈ R .

vii) If $\sigma \in S_n\left ( n\geq 3 \right )$ is a product of an even number of disjoint cycles, then sign $\left ( \sigma \right )=1.$

viii) If a ring has a unit, then it has only one unit.

ix) The characteristic of a finite field is zero.

x) The set of discontinuous functions from $\left [ 0,1 \right ]$ to $\mathbb{R}$ form a ring with respect to pointwise addition and multiplication.

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Question:

b) If $f(x,y)=\left\{\begin{matrix} x \,sin\left ( \frac{1}{y} \right )+y\,sin\left ( \frac{1}{x} \right ),&xy\neq 0 \\ 0, & xy=0 \end{matrix}\right.,$