Solve your IGNOU Doubts

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.

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Question:

Find all the 8^th roots of $3i-3$ . Also show any one of them in an Argand diagram.

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Question:

(a) Show that the perpendiculars drawn from the origin to tangent planes to the cone $x^2-y62+5z^2+4xy=0$ lie on the cone $x^2-y^2+z^2+4xy=0.$

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Question:

Find the values of a ∈R for which ia is a solution of $z^{4}-2z^3+7z^{2}-4z+10=0.$ . Also find all the roots of this equation.

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Question:

(d) Find the equation of the cylinder with base $x^2+y^2+z62-3x-6z+9=0,x-2y+2z-6=0.$

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Question:

Consider the linear system

$2x-3y+4z=20\frac{2}{5}$

$x+2y-3z+13.4=0$

$-x-2y+5z=\frac{113}{6}$

Give the two reasons for Cramer’s Rule being applicable for solving this system. Also use the rule to solve the linear system.

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Question:

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Question:

(b) Find the equation of the sphere touching the plane $8x+5y+3z+1=0$ at (3, −1, −1) and cutting the sphere $x^2+y^2+z^2-2x+y-z-6=0$ orthogonally.

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Question:

Give the following:i) a 2× 4 matrix;
ii) the transpose of the matrix in (i) above;
iii) a system of linear equations represented by AX = B, where A is the matrix in (ii) above.

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Question:

In the context of your IGNOU studies, give the following:

i) an example of an implication;
ii) the converse of your statement in (i) above;
iii) the contrapositive of your statement in (i) above;
iv) a statement using ∀ ;
v) a statement using ∃ .

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Question:

Express the following situation in a Venn diagram:
In a survey of 60 women, it is found that 25 have studied upto Class 12 only, 10 have studied till Class 10 only, 26 got scholarships, 9 of those studying till Class 12 got scholarships, 8 of those studying till Class 10 got scholarships, and 11 had completed their BA degree.

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Question:

(a) Show that the plan $2x+y+2z=0$ is a tangent plane to the sphere $x^2+y^2+z^2-2x+2y-2z+2=0.$

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Question:

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Question:

(b) Find the equation of the plane which passes through the line of intersection of the planes $3x+4y-5z=9$ and $2x+6y++z=7$ and which is perpendicular to the plane $3x+2y-5z+6=0.$

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Question:

i) Find A× B, and the number of elements in it, where
$A=\left \{ 3n+2\mid 1\leq n\leq10 \right \}\subseteq \mathbb{Z},$ and
$B=\left \{ n\in \mathbb{Z}\mid 1\leq n\leq 15 \right \}\cap \left \{ m\in \mathbb{Z}\left | 2 \right |m \right \}.$
ii) Given any two sets C and D , under what conditions on them will C× D and D×C have the same number of elements? Give reasons for your answer.