Solve your IGNOU Doubts

Question:

Determine the points of discontinuity of the function f and the nature of discontinuity at each of those points: $f(x)=\left\{\begin{matrix} -x^2, &when\,x\leq 0 \\4-5x, &when\,0<x\leq 1 \\ 3x-4x^2, &when\,1<x\leq 2 \\-12x+2x, &when\,x<2 \end{matrix}\right.$

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

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Question:

v) The pde $u_{xx+x^2u_{xy}-\left ( \frac{x^2}{2} +\frac{1}{4}\right )u_{yy}=0}$ is hyperbolic in the entire xy-plane.

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Question:

Are the following statements true or false? Give reasons for your answer.
a) Complement of the open interval ]1,0] is an open set.

b) Every bounded sequences is not convergent.

c) The function [:f − 2,2 ] → R defined by $f(x)=\frac{4x+3}{x^{2}+1}$ is uniformly continuous.

d) If the first derivative of a function at a point vanishes, then it has an extreme value at that point.

e) The function $f:[0,2]\rightarrow\,R$ defined by $f(x)=x+[x]$ is not integrable.

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Question:

Show that there are infinitely many values of α for which $x^{7}+15x^{2}-30x+\alpha$ is irreducible in $\mathbb{Q}[x].$ .

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Question:

Let $R=\mathbb{Z}[\sqrt{2}]\,and\,M={a+b\sqrt{2}\in R \mid 5\mid\,\,and\,\,5\mid b }.$

i) Show that M is an ideal of R .

ii) Show that if a|5 / or b/|5 , then 5| $(a^{2}+b^{2}),$ for ,a . b ∈ $\mathbb{Z}.$

iii) Hence show that if N is an ideal of R properly containing M , then N = R .

iv) Show that $^R/_M$ is a field, and give two distinct non-zero elements of this field.

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Question:

Let $D=\left \{ f\left ( x,y \right ) +g\left ( x,y \right )i\left | f,g\in \mathbb{Z} \right [x,y]\right \}\subseteq \mathbb{C}[x,y].$ Check whether D is a UFD or not.

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Question:

Let R be a commutative ring with unity and r ∈ R . Prove that $\frac{R[x]}{\left \langle x-r \right \rangle}\simeq R$ using the Fundamental Theorem of Homomorphism. Hence show that $\frac{R[x,y]}{\left \langle x-r \right \rangle}\simeq R[x].$

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Question:

Find all the units of $\mathbb{Z}_{12}.$

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Question:

Prove that $\frac{\mathbb{R}[x]}{\left \langle x^{2}+1 \right \rangle}\simeq \mathbb{C}$ as rings.

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Question:

Check whether $\left \{ \begin{bmatrix} m &0 \\ n & 0 \end{bmatrix}\mid m,n\in \mathbb{Z} \right \}$ is a subring of the ring ) ( M2 Z or not. If it is, check whether or not it is an ideal of the ring also. If I is not a subring of the ring, then provide a subring of the ring.

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Question:

Let G be a group such that G Aut is cyclic. Prove that G is abelian.

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Question:

Find a group G , and a homomorphism « of G , so that $\O (G)\simeq S_3$ and Ker $\simeq A_4.$ Is G abelian? Give reasons for your answer.

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Question:

Let G be a group of order 56 . What are all its Sylow p-subgroups? Show that G is not simple, i.e., G must have a proper normal non-trivial subgroup.

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Question:

Explicitly give the elements and structure of the group $S_n/A_n,n\geq 5.$

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Question:

Consider the map $f_{ab}:\mathbb{R}\rightarrow \mathbb{R}:f_{ab}(x)=ax+b.$ Let $b=\left \{ f_{ab}\mid a,b\in \mathbb{R},a\neq 0 \right \}.$ Then B is a group with respect to the composition of functions. Check whether or not $A=\left \{ f_{ab}\mid a\in \mathbb{Q}^{+},b\in \mathbb{R} \right \}$ is a normal subgroup of B .

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Question:

Let G be a group of order n ≥ 2 , with only two subgroups $-\left \{ e \right \}$ and itself. Find a minimal generating set for G . Also, find out whether n is a prime or a composite number, or can be either.