Mathematics: Post your doubts here!

[areeba240]

in 4(ii) how to find the initial value of x

 

[Aathif Shaffy]

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Thanks a lot ! I got stuck while trying to equate -2to the difference in x's and y's.

 

[areeba240]

how to solve 9(ii) with the method posted above (by ratio)

 

[i_try9621]

Need help with this. It's differential equation related.

 

[DragonCub]

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in 9(ii)(a) how to get the negavtive angle

You should get a single value of θ when you used the calculator, and the value is positive.

Now recall that cosine is an even function, which means cos -θ = cos θ.

So, the negative value is -1 * (whatever value you got for θ).

 

[DragonCub]

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in 4(ii) how to find the initial value of x

You don't find the initial value. You arbitrarily pick one, but within a reasonable domain.
In this case, it is clear that 0 ≤ x ≤ π/2. So randomly pick an initial value within this region, and you will be good to find the final value of x.

 

[DragonCub]

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how to solve 9(ii) with the method posted above (by ratio)

You know the principle where a point and a direction are enough to determine a line.
The method provided uses this principle.

The first step is to find a point that is on both the planes, which sufficiently means the point is also on the line of intersection.
To do this, solve the simultaneous equations { x + 2y - 2z = 7 | 2x + y + 3z = 5 } (which are just the equations of the planes).
You may find infinite solutions to them, but don't worry, any one of the solutions is OK.
This one solution you pick shall represent a point on the line of intersection.
The example in the method is (1, 3, 0).

The next step is to find the direction of the line, which is the "ratio" mentioned in the method, and by you.
a + 2b - 2c = 0 and 2a + b + 3c = 0 can actually be seen as (1 2 -2) . (a b c) = 0 and (2 1 3) . (a b c) = 0 where (1 2 -2) and (2 1 3) are the normal vectors to the planes respectively. (a b c) is the direction vector of the intersection line.
But why do (1 2 -2) . (a b c) and (2 1 3) . (a b c) equal zero?
Recall the dot product: a . b = |a| |b| cos θ
When two vectors are perpendicular to each other, θ = pi/2, and thus the dot product is 0.
It's the same as in the problem: the line of intersection is on both planes, so it is also perpendicular to both normal vectors.
Now we only need to solve a + 2b - 2c = 0 and 2a + b + 3c = 0 together.
Again, we will get infinite solutions, but there is a ratio of a : b : c in them. This ratio thus works as the direction vector of the line of intersection.

Once we find a point on the line, and the direction of the line, we can express the line's equation.

 

[DragonCub]

View attachment 61526 Need help with this. It's differential equation related.

I suppose you know how to do part (i), since it doesn't require differential equations at all.
I'll jump straight to part (ii).

------------------------------------------------------------------------------
One thing worth noticing is that, when we use substitution to shift the integral from the h domain to the x domain, the boundaries need to change too.
In this case, on the h domain, we integrate from 81 to 64. However, when we are on to the x domain, the integral is then from 5 to 4.

 

[i_try9621]

I suppose you know how to do part (i), since it doesn't require differential equations at all.
I'll jump straight to part (ii).
View attachment 61529
------------------------------------------------------------------------------
One thing worth noticing is that, when we use substitution to shift the integral from the h domain to the x domain, the boundaries need to change too.
In this case, on the h domain, we integrate from 81 to 64. However, when we are on to the x domain, the integral is then from 5 to 4.

Thanks for that

Als, a slight confusion in (i), when they say "the rate at which water flows out is constant" do they mean the (Inlet rate) - (Outlet rate) or just the rate of water flowing out ( i.e 900cm^3) ?

 

[DragonCub]

Thanks for that

Als, a slight confusion in (i), when they say "the rate at which water flows out is constant" do they mean the (Inlet rate) - (Outlet rate) or just the rate of water flowing out ( i.e 900cm^3) ?

It only means the outlet rate. Inlet rate always stays constant at 400, in both (i) and (ii).

 

[areeba240]

can someone plzz solve for me 10b(ii)...

 

[areeba240]

how to do 7(iii).can someone plzz solve for me?
i got AP: PB as 3:5 but not getting OA:OB same as that

 

[shahmir khan]

Help with part (ii) ????

 

[shahmir khan]

how to do 7(iii).can someone plzz solve for me?
i got AP: PB as 3:5 but not getting OA:OB same as that

Magnitude of OA is 3 and magnitude of OB is 5 so 3:5.

 

[DragonCub]

can someone plzz solve for me 10b(ii)...

If you did part (b) (i) correctly, you should arrive at the following result:



Here, x = Re z and y = Im z.

The shaded area should be a circle centred at (2, -2), with radius R = 2, but partially cut off by the lines x = 1 and y = - π/4.

Now it is easy to find the largest value of Re z inside the shaded region - simply the point on the right edge of the shape.

This point corresponds to x = 4.


P.S. Plot generated using wolframalpha.com.

 

[DragonCub]

Help with part (ii) ????

This is the same as reply #17847, posted by areeba240 several days ago.

I have posted my explanation to it here, on #17851.

If you want to solve it in another way (different from the method provided), you can use the principle where two points are enough to determine a line.

This method is actually much easier than the one in my previous post. Just find 2 points that are on both planes.
Solve the equations of the planes simultaneously, { x + 2y - 2z = 7 | 2x + y + 3z = 5 }. You'll get infinite solutions (which is supposed to happen). Choose 2 from them.
For example, one solution can be (1, 3, 0) and the other (0, 31/8, 3/8).

Now find the difference between the two points: (1, 3, 0) - (0, 31/8, 3/8) = (1, -7/8, -3/8). This is the direction vector of the line.
To make it pretty, scale it up so that all the elements are integers: (8, -7, -3).

Then we have the line: r = (1, 3, 0) + λ (8, -7, -3).


P.S. to areeba240 : you can take a look at this way of solving your problem too, in case you are curious. And personally, I think this one is much simpler and quicker.

 

[areeba240]

how to do the (iii) part?

 

[DragonCub]

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how to do the (iii) part?

 

[areeba240]

View attachment 61574

from where did u get 5- 2t

 

[DragonCub]

from where did u get 5- 2t

Ouch forgot to tell you this is from the answer to part (ii), sorry about that. The solution to that DE is r = 25 / (5 - 2t).

If you are interested in how this function is obtained, here's my work for part (ii).