Solve your IGNOU Doubts

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

is continuous at the origin.

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Question:

a) Examine whether $\lim_{x\rightarrow 0}\frac{e^{1/x}}{e^{1/x}+1}$ exists or not.

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Question:

e) The function $f(x,y)=x^3+y^3$ is integrable on ]2,1[ × ]3,1 $[1,2]\times [1,3].$

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Question:

e) The function $f(x,y)=x^3+y+1x^2+y^2)$ is locally invertible at $(1,2).$

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Question:

d) The function $f (x,y)=x^3+y+1,x^2+y^2)$

is locally invertible at $(1,2).$

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Question:

Domain of $f(x,y)=\frac{xy}{x^4+y^3}$ is $R^2.$

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Question:

b) $f(x,y)=\frac{sin\left ( \frac{x^2y}{x^3+y^3} \right )}{1n\left ( \frac{x+y}{x} \right )}$ is a homogeneous function of degree

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Question:

State whether the following statements are true or false. Justify your answer.

a) $\lim_{x\rightarrow 0}\left ( \frac{1}{x^2} -\frac{1}{sin^2x}\right )$ is in $\left ( \frac{0}{0} \right )$ form.

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Question:

(b) Find the equation of tangent plane to the conicoid $x^2+3y^2=4z at (2,-4,13).$ Represent the tangent plane geometrically.

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Question:

(a) Identify and trace the conicoid $y^2+3z^2=x.$ Describe its sections by the planes $y=0 and z=0$

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Question:

(c) Find the projection of the line segment joining the points (1, −1, 6) and (4, 3, 2) on the line $\frac{x-4}{3}=-y=\frac{z}{5}.$

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Question:

(b) Find the transformation of the equation $12x^2-2y^2+z^2=2xy$ if the origin is kept fixed and the axes are rotated in such a way that the direction ratios of the new axes are 1, −3, 0; 3, 1, 0; 0, 0, 1.

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Question:

(a) Examine which of the following conicoids are central and which are non-central. Also determine which of the central conicoids have centre at the origin.

(i) $x^2+y^2+x^2+4x+3y-z=0$

(ii) $2x^{2}-y^2-z^2+xy+yz-zx=1$

(iii) $x^2+y^2-z^2-2xy-3yz-6zx+x-2y+5z+4=0$

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Question:

Give a real life situation problem, which is mathematically translated into

$2x+y+2z=18,x+3y+3z=24,3y=6.$ Also, explain how this linear system models your problem.

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Question:

Using the method of substitution, obtain the solution set in $\mathbb{R}^{3}$ , of the following:

i) x − π = 5

ii) $2x-y+z=1,x-2y+z=3,y=\sqrt{2}-z$
iii) $x-y=5,x=7,2x-3y=5$

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Question:

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Question:

(b) Transform the equation $x^2+2y^2-6z^2-2x-8y+3=0$ by shifting the origin to (1, 2, 0) without changing the directions of the coordinate axes. What object does this new equation represent? Give a rough sketch of it.