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Mathematics: Post your doubts here!

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I NEED HELP IN MAY JUNE 2010 /32 QUESTION NUMBER 7 AND 8

Q7(i).

(e^2t)(dx/dt) = cos^2 x
[1/(cos^2 x)] dx = (e^-2t) dt
(sec^2 x) dx = (e^-2t) dt

Integrate both the sides.

tan x = -(e^-2t)/2 + c

Put 'x=0' and 't=0' in the above equation in order to find the value of the constant 'c'.

tan x = -(e^-2t)/2 + c
0 = -1/2 + c
1/2 = c

Put back the value of 'c' in the integrated equation.

tan x = -(e^-2t)/2 + c
tan x = -(e^-2t)/2 + 1/2
tan x = 1/2 - (e^-2t)/2
x = tan^-1 [ 1/2 - (e^-2t)/2]

Therefore, the expression for 'x' in terms of 't' is 'x = tan^-1 [ 1/2 - (e^-2t)/2]'.

Q7(ii).

In differential equations where they ask us to 'state what happens to the value of [constant] when [constant] becomes very large', all what you need to do is keep the value of the number/expression which has a power with it to '0' and write down the remaining expression. For example in this equations case, x = tan^-1 [ 1/2 - (e^-2t)/2]', keep the value of '(e^-2t)/2' equal to '0' and state the left over expression which is 'tan^-1 (1/2)'.

Q7(iii).

Go back to the main equation and try arranging it in such a way that it becomes an expression for 'dx/dt' in terms of 't' and 'x'.

(e^2t)(dx/dt) = cos^2 x
(dx/dt) = cos^2 x/(e^2t)
(dx/dt) = (cos^2 x)(e^-2t)

The derivative is 'positive' that's why x increases as t increases.

Q8(i).

z = (1 + cos 2x) + i (sin 2x)

Finding the modulus of 'z'.

|z|:

√(1 + cos 2x)^2 + (sin 2x)^2
√(1 + 2 cos 2x + cos^2 2x + sin^2 2x)

Recall the identities 'cos 2x = 2 cos^2 x - 1' and 'sin 2x = 2 sin x cos x'.

cos 2x = 2 cos^2 x - 1
cos^2 2x = (2 cos^2 x - 1)^2
cos^2 2x = 4 cos^4 x - 4 cos^2 x + 1

sin 2x = 2 sin x cos x
sin^2 2x = 4 sin^2 x cos^2 x
sin^2 2x = 4 cos^2 x ( 1 - cos^2 x )
sin^2 2x = 4 cos^2 x - 4 cos^4 x

Replace 'cos^2 2x' with '4 cos^4 x - 4 cos^2 x + 1' and 'sin^2 2x' with '4 cos^2 x - 4 cos^4 x '.

√(1 + 2 cos 2x + cos^2 2x + sin^2 2x)
√(1 + 2 (2 cos^2 x - 1) + 4 cos^4 x - 4 cos^2 x + 1 + 4 cos^2 x - 4 cos^4 x)
√(1 + 4 cos^2 x - 2 + 4 cos^4 x - 4 cos^2 x + 1 + 4 cos^2 x - 4 cos^4 x)
√4 cos^2 x
2 cos x

Therefore, |z| = 2 cos x.

Argument:

tan x = b/a
tan x = (sin 2x)/(1 + cos 2x)
tan x = (2 sin x cos x)/(1 + 2 cos^2 x - 1)
tan x = sin x / cos x
tan x = tan x

Q8(ii).

1/z

[1/((1 + cos 2x) + i (sin 2x)] x [(1 + cos 2x) - i (sin 2x)]/[(1 + cos 2x) - i (sin 2x)]
(1 + cos 2x)/[(1 + cos 2x)^(2) + sin^2 2x]

'(1 + cos 2x)^(2) + sin^2 2x' is equal to '4 cos^2 x' as found in the first part.

(1 + cos 2x)/[(1 + cos 2x)^(2) + sin^2 2x]
(1 + cos 2x)/(4 cos^2 x)
(1 + 2 cos^2 x - 1)/(4 cos^2 x)
1/2
 
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I've got a question..... for Winter 2003, Question paper 7. Question number 2.

2 A certain machine makes matches. One match in 10 000 on average is defective. Using a suitable
approximation, calculate the probability that a random sample of 45 000 matches will include 2, 3 or 4
defective matches.

http://www.xtremepapers.com/CIE/International A And AS Level/9709 - Mathematics/9709_w03_qp_7.pdf

My Doubt: The mark scheme uses Poisson distribution to work the answer out. I used normal distribution as the number (45000) was very large and probability very small. How do we know which one to use?

btw the correct answer is 0.471 and i got 0.421 (using normal approximation)
 
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june 09 paper 3 question 9 part 2 can Any one help me with this??
with explanation :)

r = (4)(2)(-1) + t(2)(-1)(-2) - (0)(2)(4)
r = (4)(0)(-5) + t (2)(-1)(-2)
r = (4+2t) (-t) (-5-2t)

(4+2t) . (2)
(-t) . (-1) = 0
(-5-2t) . (-2)

8 + 4t + t + 10 + 4t = 0
t = -2

r = (4+2t) (-t) (-5-2t)
r = (0)(2)(-1)

Finding the modulus.

Modulus = [ (0)^2 + (2)^2 + (-1)^2 ]^(1/2)
Modulus = (5)^(1/2)
 
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