Solve your IGNOU Doubts

Question:

Let F be the ring of all functions from $\mathbb{R}$ to $\mathbb{R}$ w.r.t. pointwise, addition and multiplication. Let S be the set of all differentiable functions in F. Check whether S is

i) a subring of F,

ii) an ideal of F.

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Question:

i ) For the IVP, $\frac{dy}{dx} =f (x,y), y (x^{\circ} ) =y ^{\circ}$ , the continuity of $f ( x,y)$ and $\frac{\partial f}{\partial y}$ guarantees the unique solution of the problem

ii) Equations $\frac{d^{2}y }{dx^{2} } - 2x\frac{dy}{dx} + x^{2}y = e^{x^{2}2} and \frac{d^{2}y}{dx^{2}}+ y =1$ have the same normal form.

iii) Equation $cos (x+y)p+sin (x+y)q =z^{2}+ z$ is a quasi-linear equation.

iv) $\frac{\partial^{2}z }{\partial x^{2} }\: \: \: \frac{\partial^{2}z }{\partial y^{2} } -\left ( \frac{\partial ^{2}z}{\partial x\partial y} \right )^{2}=0$ is a non-linear PDE

v) Every solution of the ordinary differential equation (D² +1)²y=0 is bounded on [ 0, $\infty$ [

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Question:

Let R be a ring, I an ideal of R, J an ideal of I. Show that if J has a unity, then J is an ideal of R. Also give an example to show that if J does not satisfy this condition it need not be an ideal of R.

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Question:

If U (R) denotes the group of units of a ring R, show that U (R₁ × R₂) = U(R₁) × U(R₂) for rings R₁ and R₂.

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Question:

Use FTH to determine all homomorphic images of D₈, upto isomorphism.

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Question:

Define $f:\mathbb{Z}\rightarrow\mathbb{Z}_{m}\times \mathbb{Z}_{n}:$ f(x)=(x mod m,x mod n),m,n∈ $\mathbb{N}$ .

i) If (m, n) = (3, 4), find Ker f.

ii) If (m, n) = (6, 4), find Ker f.

iii) What can you generalize about Ker f from

(i) and (ii)?

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Question:

Can there be a homomorphism from $\mathbb{Z}_{8}\oplus \mathbb{Z}_{2}\; onto\; \mathbb{Z}_{4}\oplus \mathbb{Z}_{4}$ ? Give reasons for your answer

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Question:

Prove, by contradiction, that A₄ has no subgroup of order 6.

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Question:

Check whether the subgroup of reflections and subgroup of rotations in D_2n is normal in D_2n or not. (Note that D_2nis the group of symmetries of an n-gon.)

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Question:

Show that in a group G of odd order, the equation x² = e has a unique solution. Further, show that x² = g has a unique solution ∀g∈G,g ≠e .

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Question:

If G is a group with o(g) < 100 and G has subgroups of order 10 and 25, what is the order of G?

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Question:

Which of the following statements are true or false. Give reasons

a) $Q(x)= X_{1}^{2} - _{2}^{2}$ is the quadratic form of a positive definite matrix

b) In a bivariate normal distribution $N\: \: _{2} ( \mu_{x} ,\mu_{y} ,\sigma_{x} ,\sigma_{y} \int xy ) ,if \int xy =0$ then x and y are independent

c) The relation of accessibility in states is transitive.

d) For any pair of discrete random variables X and $y \sum P [y/y\mid X/x ] < 1.$

e) If {X (t) : t $\geq$ 0 } is a Poisson process, then $N (t)= X (t+a) - X(t)$ , where a is a constant is also a Poisson process.

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Question:

Obtain the left cosets of V₄ = {e, (1 2) (3 4), (1 3) (2 4), (1 4) (2 3)}in A₄.

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Question:

Locate and classify the stationary points of the following:

$(i)f(x,y)=4xy+x^{4}-y^{4}$

$(ii)f(x,y)=xy+\frac{2}{x}+\frac{4}{y},x> 0,y> 0$

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Question:

Suppose the random variables X_{1 ,}X₂ and X₃have the covariance matrix

$\sum =\begin{bmatrix} 1 &-1 &0 \\ -1& 5& 0\\ 0& 0 & 2 \end{bmatrix}$

Find all principal components.

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Question:

Let X have covariance matrix

$\sum =\begin{bmatrix} 16 &-2 &4 \\ -2& 9&1 \\ 4&1 & 25 \end{bmatrix}$

i) Determine p and V ^1/2

ii) Multiply your matrices to check the relation V ^1/2P V ^1/2

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Question:

Does the function

$f(x,y)=\frac{x^{2}-y^{2}}{x^{2}+y^{2}},x\neq 0,y\neq 0$

satisfy the requirement of Schwarz’s theorem at ?)1,1( Justify your answer.

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Question:

Identify the level curves of the following functions:

$(i)\sqrt{x^{2}+y^{2}}$

$(ii)\sqrt{4-x^{2}-y^{2}}$

$(iii)x-y$

$(iv)y/x$

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Question:

A single repairperson looks after the two machines 1 and 2. Each time it is repaired, machine i stays up for an exponential time with rate $\lambda$ , where i = 1, 2 . When machine i fails, it requires an exponentially distributed amount of work with rate $\mu$ to complete its repair. The repairperson will always service machine 1 when it is down. For instance, if machine 1 fails while 2 is being repaired, then the repairperson will immediately stop work on machine 2 and start on

(i) Write down all the states.

(ii) What is the probability that the machine 2 is down.

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Question:

Suppose S and C are subsets of $\mathbb{R}^{3}$ S is the unit open sphere with centre at the origin and C is the open cube = {P(x, y,z ) | −1< x < 1, −1< y < 1, −1< z < 1}.Which of the following is true. Justify your answer.