Some questions ( h2 maths )

lavastar SG

Here are some of the questions for my holiday assignment which im unable to do and i need some help Embarassed

1. A region R is bounded by the curve y=e^x and the line x=ln2, the x axis and the y axis. Find the area of the region R and hence find the exact value of integrate ln y dy ( limit 0 to 1)

The area of region R is 1 which i am able to solve but i need help for the 2nd part (the integration) . (The answer is 2ln2-1)

2. Find the exact area A which is enclosed by the curve y =1/(1+cosx)^0.5 , the y axis and the line y=1

This qn i duno how to integrate y= 1/(1+cosx)^0.5.
(the answer is pi/2 - sq root 2 ln( sq root 2 +1 )

3. Find the number of ways the 5 letters A B C D E and the 3 digits 1,2,3 can be arranged in a row if

ii) the 3 digits must be seperated. ( ans : 14400)
iii) the letters must be in alphabetical order and the digits in ascending numerical order. ( ans : 56 )

Thanx.

eagle SG

I do one question first... later then do the rest

Integrate ln y dy from 0 to 1, you will get the area of the negative portion of the curve (negative x side)

Summarized:
then you get (y ln(y) - y ) from 0 to 1
your main problem is, I presume, you cannot do ln 0
The reason behind this is because the curve tends to infinity at the y=0 line

So instead of integrating with respect to y, what you can do is integrate with respect to x, and subtract the area R off, ie
integrate e^x from negative infinity to ln2
which gives you 2

When you subtract the region R, you get the value of 1

Since it is a negative value (below y-axis), integrating ln y dy ( limit 0 to 1) should give you -1

P.S. I seriously don't think the answer is 2ln2 - 1. My calculator also gives me -1 (it has definite integration function Wink ). Please check the answer again, or check the limit again. Thanks.

eagle SG

change the cos x to 2 [ cos (x/2) ]^0.5 - 1

then you get

y = 1/sqrt(2) sec(x/2)

you should be able to integrate sec(x/2) to solve it

you get sqrt(2) ln | sec(x/2) + tan(x/2) | from x=0 to x = pi/2 (when y=1, x=pi/2)

you get sqrt(2) ln (sqrt(2) + 1)

however, bear in mind that they want the the area between y, y-axis and y=1 line

Hence, you take the square area (pi/2 * 1) - the area calculated

you get pi/2 - sqrt(2) ln (sqrt(2) + 1)

eagle SG

ii)
If not separated:
3 digits can either be together all at once (all 3 considered as 1 unit)
num of ways like that = 6!(5 letters + 1 group) * 3!(within the group) = 4320
or
2 digits together (as 1 unit) and 1 further apart
num of ways = 7! * 3C2 * 2! = 30240
However, this includes 2 instances where all 3 digits are together
eg YYX and XYY (YY represents the 2 digits together, and X is that 1 solo digit)
Hence, total ways without 3 digits together
= 30240 - 4320 - 4320 = 21600

So, for all 3 digits to be separated,
ans = 8! - 4320 - 21600 = 14400

Alternately, after having done iii, there's another more efficient method
3 L 2 L 1 (L denotes a letter)
For these, there are 3! *5C1 * 4C1 ways = 120 ways
(2! not needed between the 2 letters because already included in the choose)

We have then 4 slots to slot in the remaining 3 letters
if 3 letters together, that is 4C3 * 3!
if 2 letters together and 1 letter away, then 4C2 * 3! (exchange letters) * 2! (switch the position between the 2 letter group and the 1 letter)
if all separate, then 4C3 * 3!
total ways = 24 + 72 + 24 = 120

Hence, total number of ways = 120 * 120 = 14400

iii)
A B C D E
we have to insert 321 inside
for 321 to be together, we have 6C1 choices

for 32 to be together, and 1 separate,
we have 6C2 choices = 15

for 3 separate, and 21 together, same as previous, 15 ways

for 3, 2 and 1 to be separate, you can count also
there are 6C3 choices = 20

total = 6 + 15 + 15 + 20 = 56

lavastar SG

wow thanks a lot Smile