character sketch of the postmaster in the chapter"the postmaster."

The Postmaster was a
Calcutta boy. He was posted to
a small village Ulapur. He didn't like to live in a
village. He felt ill at ease like fish out of water in the
village. There he didn't find any good company. He missed the
metalled roads, buildings and crowd of people . He was an
educated man and felt odd living in the village
with iiliterate people. He had a helper in his house named
Ratan, who was an orphan girl.
She did domestic work of his house and took
care of him. During the time of Shravan the
postmaster felt ill and missed
his family members- mother  and sister. Ratan
nursed him tenderly and warmly. Ratan was the only
good company for the postmaster. He taught
alphabet to Ratan and shared feeling of memories of his mother and
sister. When the postmaster was ill, ratan put her soft
hands on his head that remembered the
postmaster his mother. In this condition
of sickness he remembered his mother and
sister and wanted to return back so that he resigned from
his job. when he told his decision to Ratan she liked to go with him but according to
postmaster philosophy "separations and deaths
are the part of life "

Solve for x the equation 2^2x=6^(x+1) - 5*3^2x?

We'll move all terms to the left
side:

2^2x - 6^(x+1) + 5*3^2x =
0

We'll write 6^x = 2^x*3^x, also 6^(x+1) =
(6^x)*6

6^(x+1) =
6*2^x*3^x

We'll re-write the
equation:

2^2x - 6*2^x*3^x + 5*3^2x =
0

We'll divide by
3^2x:

(2/3)^2x - 6*(2/3)^x + 5 =
0

We'll substitute (2/3)^x =
t

t^2 - 6t + 5 = 0

We'll apply
quadratic formula:

t1 = [6 +
sqrt(36-20)]/2

t1 = (6+4)/2

t1
= 5

t2 = (6-4)/2

t2 =
1

We'll substitute (2/3)^x = 1, where 1 =
(2/3)^0

(2/3)^x = (2/3)^0 <=> x =
0

(2/3)^x = 5 <=> x*ln (2/3) = ln
5

x = ln
5/ln(2/3)

The values of x that represent the
solutions of the equation are: x = 0 and x = (ln
5)/ln(2/3).

What is the traveller's total displacement if he moves 18.6 km east , 27.6 km south east and 9.1 km south?

We'll assign coordinate axis to all
directions:

- the positive x axis represents east
direction;

- the negative x axis represents west
direction;

- the positive y axis represents north
direction;

- the negative y axis represents south
direction;

South-east is the direction at 45 degrees to he
positive x axis and the negative y axis and it represents the bisectrix of the 4th
quadrant, where x coordinate is positive and y coordinate is
negative.

We'll calculate the x axis
position:

x = 18.6 + 27.6*cos
45

x = 38.11

We'll calculate
the y axis position:

y = -9.1 - 27.6*sin
45

y = -28.61

Now, we'll
calculate the traveler's total displacement using Pythagorean
teorem:

D = sqrt (x^2 + y^2)

D
= sqrt (1452.3721 + 818.5321)

D = sqrt
2270.9042

D = 47.65
Km

The traveler's total displacement is D =
47.65 Km.

Find the inverse function f^-1 (x) if f(x) = (x+1)/(2x-5)?

We'll interchange x and y and we'll
have:

x =  (y+1)/(2y-5)

Now,
we'll multiply both sides by 2y - 5:

x(2y - 5) = y +
1

We'll remove the
brackets:

2xy - 5x = y +
1

We'll keep all the terms in y to the left side, the rest
of term being shifted to the right side.

2xy - y = 5x +
1

We'll factorize by y to the left
side:

y(2x - 1) =5x + 1

We'll
divide by 2x - 1

y = (5x + 1)/(2x -
1)

The inverse function is: f^-1(x) = (5x +
1)/(2x - 1).

Write the fraction as a function of x: sin 2x/(1+cos 2x)

We'll apply double angle identity for sin 2x and cos
2x:

sin 2x = 2sin x*cos
x

1+cos 2x = 1 - 1 + 2(cos
x)^2

1+cos 2x = 2(cos
x)^2

We'll re-write the
fraction:

sin 2x/(1+cos 2x) = 2sin x*cos x/2(cos
x)^2

We'll divide both numerator and denominator by 2 cos
x:

sin 2x/(1+cos 2x) = sin x/cos
x

sin 2x/(1+cos 2x) = tan
x

The fraction written as a function of x is: sin 2x/(1+cos
2x) = tan x.

If (2a+1)u = (b+1)v + (3c-2)w, what constrains a,b and c if w,u and v are not coplanar?

Since the planes u, v and w are not coplanar they do not
all intersect at any one point (or line, or plane).

This
means that when u and v intersect, there is no point on the plane w that lies on this
line.

Suppose that u and v intersect on the plane z = r,
and values of z on w are denoted z(w) then

(3c - 2)z(w) =
r(2a+1-b-1) = r(2a-b)

Since z(w) is never equal to r, this
implies that a,b and c are defined by the fact that

`2a-b
!= 3c-2`   or   `2a-b-3c != -2`

So, if  `b = ma` then `c`
cannot equal  `[(2-m)a + 2]/3`  .

Therefore
a, b, c can lie anywhere in `RR^3` except on the
line

(x,y,z) = (0,0,2/3) +
t(1,m,(2-m)/3)

where m =
b/a

What are some themes in the memoir "The Glass Castle" ?

One of the most prominent discussions presented in the
book is the idea of parenting and how to do it.  Walls and her siblings were not brought
up in a home that displayed "traditional" parent-child relationships and arguably, much
of what they endured as children could be considered abuse or neglect.  However, one
idea explored through this is the question of whether Jeanette turned out "normal"
despite her upbringing.  Many would argue that she did.  It seems, also, that she
defends both her parents love for herself and her brother and
sisters.

Another theme which somewhat results from the
above is the notion of family loyalty.  Consider the humorous and straightforward tone
of the book.  It is clear that though Walls is presenting memories with brutal honesty
and is well aware of how her childhood will appear to the rest of the world, she also
speaks with a defensive tone as if to say that she is the only one who is allowed to
judge her parents based on their parenting.  As a child, she is fiercely loyal
throughout the story to her entire family.

Finally, you
could explore the theme of poverty and survival.  The children in the Walls family are
nothing if not survivors.  Consider the many things they do to "fit in" in their
schools, to find food, and protect their odd parents from the judgemental eye of
others.  They are resilient, both physically and emotionally, and take care of
themselves and each other.