1. Obtain the 1st 4 terms in the expansion of (1 + x + x^2)^10
in ascending powers of x. Hence find the approximate value of
1.0101^10, giving your answer to 3 decimal places. (it'll be
ok to just help out with the 1st part)
2. The table shows experimental values of 2 variables x and
y.
x 1.2
1.6
2.5
4.0 6.3
y
1.5 1.8 2.6
3.6 5.0
It is know that x and y are related by the equation y = (Ax)^n
where A and n are constants. Plot a graph of lg y against lg x
for the values given in the table. Estimate the values of A and
n.
Find the value of x when y = 3.
For this part, i subbed in lg 3 = 0.477... rounded off to
0.48
then i looked up 0.48 on the y-axis to find the value of lg x
before doing the necessary to get x = 3.13. Is this method ok? It
doesn't seem right to me, after all we aren't supposed to sub
in values for graph.
3. The equation of a circle C is x^2 + y^2 - 4x - 8y
+ 3 = 0
Find the coordinates of the centre and length of radius
Find the equation of the circle which is a reflection of
C in the line x = -1
Ok,, don't know how to do this since most of the times we get
questions like how to obtain an equation about the x / y axis, in
which case, we have to change the sign of the y / x coeffient
respectively. Am I right? Is the question saying that a
part of the circumference of the circle touches the line x =
-1 and gets reflected there?
And if you are free, I'd like someone to help me verify the
answer for ANOTHER graph. Sorry for the trouble but please do so
only if you've got the time.
The table below show sthe experimental values of 2 variables x
and y.
x
1.0
1.5 2.0
2.5
3.0 3.5
y
3.2 5.2
7.1
9.2 11.2
13.3
It is know that x and y are related by the equation y = hx + k
(x)^0.5, where h and k are constants.
Draw the graph of y/ (x)^0.5 against (x)^0.5 for the given
data.
Find value of h and k. (my ans: 4.49 and -1.25
respectively) ans given - 4.56 and -1.4
***estimate the value of x when y = 5 (x)^0.5
(my ans : 1.18) ans given - 1.96
By drawing a suitable straight line, estimate the values
of x and y which satisfies the simultaneous eqn
y = hx + k(x)^0.5
y = -4(x)^0.5 + 9x
(my ans: x = 0.37515(exact), 0.91875), ans given -
0.3364, y = 0.754
my answers for some of the questions seem so far-fetched..
Hi anpanman,
For the first question, you can take ( 1 + x + x^2 )^10 to be (
1 + ( x + x^2 ) )^10, which you can expand out to get
1 + 10C1( x + x^2 ) + 10C2( x + x^2 )^2 + 10C3( x + x^2 )^3
+...
which is simplified to 1 + 10x + 55x^2 + 210x^3 +...
Notice that if you sub x = 0.01 into ( 1 + x + x^2 )^10, you
will get ( 1.0101 )^10.
Hence to find ( 1.0101 )^10, you can sub x = 0.01 into 1 + 10x +
55x^2 + 210x^3, and you will get 1.10571, which should be cut down
to 1.106
For the second question, if you were to sub in values directly,
you will not get any marks at all. I hope you will not be upset by
me for saying this, but it is likely that your graph is not well
drawn enough. You can sub in values on a piece of rough paper to
check the accuracy of your answer and your graph though.
For the third question, we can first change the equation to a
form which is easier for us, which will be ( x - 2 )^2 + ( y - 4
)^2 = 17
Hence, the midpoint is (2,4) and the radius is sqrt(17)
Note that when any point is reflected across a vertical line,
its y-coordinate will not change, whereas the vertical line will
act as a midpoint for the original x-coordinate and the new
x-coordinate.
In this case, the new midpoint after reflection across x = -1
will be (-4,4)
Hence, the new equation will be ( x + 4 )^2 + ( y - 4 )^2 =
17
As for your graph questions, perhaps you will like to scan and
upload your graph to share with the dear forummers of the homework
forum. Please do not be offended, but I believe that something is
wrong with your graph.
Here are the essential steps for this particular question:
1) Find the values of y / x^(0.5) and x^(0.5) using the x and y
values given
2) Plot the graph y/ x^(0.5) = hx^(0.5) + k using the values you
have found from step 1; this should be a straight line.
3) y = 5x^(0.5) is the same as y/ x^(0.5) = 5, so you will just
need to infer from the line graph which you have drawn
previously.
4) y = -4x^(0.5) + 9x is the same as y/ x^(0.5) = 9x^(0.5) - 4,
so you will just need to draw this as a straight line to find the
required values via the intersection point.
Cheers.