Solve your IGNOU Doubts

Question:

Let G be a finite group and H be a normal subgroup of G . Prove that $H=\cup C_{X}$ where the C_x are all the distinct conjugacy classes of G such that $H\cap C_{X}\neq \varnothing$

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Question:

Let G be the group of all rigid motions of a plane and S be the set of all rectangles in the plane. Show that G acts on S . Also obtain the orbit and stabiliser of a square under this action.

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Question:

Let G be a group and let H $\leq$ G, K $\leq$ G, o(H) o(K) p, a prime. Show that either H $\cap$ K ={e} = or H =K. = Is this result still true if p is not a prime? Justify your answer.

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Question:

Which of the following statements are true? Give reasons for your answers. Marks will only be given for valid justification of your answers.

(i) If G is a finite abelian group and p is a prime factor of o(G), then the number of Sylow p-subgroups of G is a prime.

ii) The minimum polynomial of 1/3 5 over Q is x^1/3 .

iii) $\mathbb{Z}_{mn}\neq \mathbb{Z}_{m}\mathbb{Z}_{n}m,n\in \mathbb{Z}$

iv) If G is a finite group and m o(G), $m\epsilon n$ then G has a subgroup of order m.

v) If $\beta _{1}$ and $\beta _{2}$ are two 15^th roots of unity, then $\mathbb{Q} (\beta _{1}) =\mathbb{Q} (\beta _{2})$

vi) There exists an extension field of $\mathbb{Z}_{3}$ of order 25.

vii) Every group of order 18 has a normal subgroup of order 2.

viii) If I and J are ideals of a ring R, then IJ = $I\cap J$

ix) If f : R S → is a ring homomorphism and I is an ideal of R, then f (I) is an ideal of S.

x) Every prime ideal of an integral domain is a maximal ideal.

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Question:

Set up geometric data tables for a unit cube using only i) vertex and polygon tables and ii) a single polygon table. Compare the two methods for representing the unit cube with a representation using three data tables, and estimate storage requirements for each.

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Question:

Modify the boundary-fill algorithm for a 4-connected region to avoid excessive stacking by incorporating scan line methods.

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Question:

Suppose that a window has its lower-left corner at (-2, -1) and its upper-right corner at (3, 2) . For each of the following line segments, state whether it will be totally visible, totally hidden, or partially visible; and for the partially visible-segments give the coordinates at which it is clipped.

i) (-1, 0) to (1, 1)
ii)(-3, 1) to (4, 1)
iii) (-2, 3) to (1, 4)
iv) (2, 3) to (4, 1.5)
v) (0, 1.5) to (2.5, 0) .

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Question:

Determine a rotation matrix for rotation by an arbitrary angle $\theta$ about a fixed point (r , r ) x y in homogeneous form. Use this matrix to rotate the square with vertices (0, 0), (1, 0), (1,1) and (0, 1) about the point (1, 1)

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Question:

A Bezier curve of degree 4 has to be expressed as a Bezier curve of degree five. Find out the control points of the curve which make it a curve of degree 5 .

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Question:

Determine the blending functions for uniform, periodic B-spline curves for d = 5 and d = 6.

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Question:

Shear a unit square whose opposite vertices are at (0, 0)
and (2, 2) by

i) 2 units along x-axis and reference line
y =0 .
ii) 3 units along y-axis and reference line
x =0 .
iii) 2 units along x-axis and 3 units along y-axis w.r.t. principal axis.
iv) 2 units along x-axis and reference line
y =-1
v) 3 units along y-axis and reference line
x =-1

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Question:

Show that the order in which transformations are performed is important by applying the transformations to the triangle A(1, 0), B(0, 1), C(1, 1) by i) rotating by  45 about the origin and then translating in the direction of the vector (1, 0) . ii) translating first and then rotating by  45 about the origin

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Question:

Prove that the reflection along the line y =x is equivalent to reflection along the x-axis followed by a counter clockwise rotation by 90 .

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Question:

A figure with vertices A(10, 0), B(0, 10), C(-10, 0) and D(0, -10) is scaled twice uniformly w.r.t. O(0, 0) by the scaling factors 3 and $\frac{2}{5}$ Find the transformed coordinates of the corners of the figure and sketch it.

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Question:

Let W be the window with corners at (5, 5), (10, 5), (10, 8)
and (5, 8). Clip the triangle with vertices (4, 6), (12, 6) and
(6, 10) against W using
i) Cohen-Sutherland Algorithm
ii) Liang-Barsky Line Clipping Algorithm.

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Question:

Find the equation of the Bezier curve that passes through (0, 0) and (4, 2) and controlled through (14, 10) and (4, 0).

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Question:

Show that rotation about an arbitrary pivot point (a, b) through an angle $\Theta$ is equivalent to the rotation about the origin through  followed by a translation through (a, b), where a= a(1-cos $\Theta$ ) bsin $\Theta$ and b=b(1-cos $\Theta$ ) - a sin $\Theta$ .