Mathematics: Post your doubts here!

[DragonCub]

In i, Why do we need two points that are on both planes?

We need two points because we need to find the line's vector. The fastest way to find the vector is to obtain two points on the line.

If you are interested in another method of determining the vector, you can use the normal vectors of P1 and P2.
If the vector is parallel to both planes, then it must be perpendicular to both planes' normal vectors, n1 and n2
Thus you can use cross product, n1 x n2, the product is the parallel-to-both-planes vector we are looking for.

Also couldn't it be that the planes were skewed?

In a 3-dimension space, only lines can be skewed to each other. Planes can only be parallel or intersecting.

 

[DragonCub]

Can anyone help with this? Can't wrap my head around this problem.

A staff car park at a school has 13 parking spaces in a row. There are 9 cars to be parked.
i. How many Different arrangements are there for parking the 9 cars and leaving 4 empty spaces?
ii.How many different arrangements are there if the 4 empty spaces are next to each other?
iii. if the parking is random, find the probability that there will not be 4 empty spaces next to each other.

Any help would be appreciated.

I'll assume the 9 cars are all different, so that permutation (rather than combination) is used:

(i)
Choose 9 slots from 13, order is considered.
13P9 = (13!)/(4!) = 259459200

(ii)
Now we need to make sure that in every case, the 4 empty slots are bound together.
We can group the 4 empty slots as a single unit, so that:
- there are 9 units to park cars
- there is a single unit of 4 empty slots
A total of 10 units.

Thus the case is to choose 9 units from 10, order is considered.
10P9 = (10!)/(1!) = 3628800

(iii)
The total number of possible outcomes is the answer to (i), which is 259459200.
The number of possible outcomes with 4 empty spaces next to each other, is the answer to (ii), 3628800.

So the number of outcomes where the 4 empty spaces are not next to each other = 259459200 - 3628800.

Probability = (259459200 - 3628800) / 259459200 = 0.986

 

[Reem15]

Hey everyone I'm new on Math AS and would appreciate your help

The thread is so lengthy so if someone can help me and provide me notes for PURE MATH 1 and MECHANICS 1 I've checked some links but always directs me to 404 error .
Thanks in advance .

Btw is it hard to get A in AS Math ? I'm terrified from it

 

[i_try9621]

Hey everyone I'm new on Math AS and would appreciate your help

The thread is so lengthy so if someone can help me and provide me notes for PURE MATH 1 and MECHANICS 1 I've checked some links but always directs me to 404 error .
Thanks in advance .

Btw is it hard to get A in AS Math ? I'm terrified from it

It's definitely not hard to get an A in AS Math although I believe M1 can be a bit tricky and P1 is really easy if you have done Add Maths in O-levels

 

[i_try9621]

Need help with Q24 part 2

 

[DragonCub]

View attachment 61674 Need help with Q24 part 2

Special thanks to i_try9621 for spotting the mistake! (Post #17891)

 

[i_try9621]

jhz

View attachment 61675

why did you consider a to be (1,2,-1) because this is a direction vector, isn't it? I have tried this sum and my ans was 7x + y - 5z = 17 and I took a to be (1,0,-2) from the line l.

 

[DragonCub]

jhz

why did you consider a to be (1,2,-1) because this is a direction vector, isn't it? I have tried this sum and my ans was 7x + y - 5z = 17 and I took a to be (1,0,-2) from the line l.

Sorry, I made a silly mistake... I meant to take the point on line r but misread it. (There is no j in there but I mistook k as it, and s as k, so I came up with the wrong one.)
Yes, you are right, it should be (1, 0, -2) or (6, -5, 4), and both will arrive at d = 17.

The answer should be as you stated.
Good job finding the error

 

[i_try9621]

Sorry, I made a silly mistake... I meant to take the point on line r but misread it. (There is no j in there but I mistook k as it, and s as k, so I came up with the wrong one.)
Yes, you are right, it should be (1, 0, -2) or (6, -5, 4), and both will arrive at d = 17.

The answer should be as you stated.
Good job finding the error

And thanks for helping me out

 

[i_try9621]

Need help with Q29) part 3

 

[DragonCub]

View attachment 61685 Need help with Q29) part 3

This part does not seem to be solvable.

From the information given we can obtain the following pieces of info:
1. "N is the foot of the perpendicular from B to l" --> N is on line l.
2. "The plane containing A, B and N" --> A and N are both on the plane --> Line l is on the plane.
3. "The line l ... is parallel to OB" --> OB's direction is the same as l --> (and since B is on the plane) Vector OB is also on the plane

--> Point O (the origin) is on the plane.

This is sufficient to mean that, if the plane's equation is in the form of ax + by + cz = d, the right-hand side must be d = 0.

This deduction is, however, contradictary to the requirements of the question.

So unless there is a typo in the question, this part does not have a solution.

------------------------------------------------------------

P.S.
I may have made some mistake during the deduction process. If anyone finds any, do not hesitate to notify me of it.
Also, if you have a marking scheme or solution manual ready at hand, please let me know. I am interested to see the provided solution.

 

[i_try9621]

This part does not seem to be solvable.

From the information given we can obtain the following pieces of info:
1. "N is the foot of the perpendicular from B to l" --> N is on line l.
2. "The plane containing A, B and N" --> A and N are both on the plane --> Line l is on the plane.
3. "The line l ... is parallel to OB" --> OB's direction is the same as l --> (and since B is on the plane) Vector OB is also on the plane

--> Point O (the origin) is on the plane.

This is sufficient to mean that, if the plane's equation is in the form of ax + by + cz = d, the right-hand side must be d = 0.

This deduction is, however, contradictary to the requirements of the question.

So unless there is a typo in the question, this part does not have a solution.

------------------------------------------------------------

P.S.
I may have made some mistake during the deduction process. If anyone finds any, do not hesitate to notify me of it.
Also, if you have a marking scheme or solution manual ready at hand, please let me know. I am interested to see the provided solution.

In 2. How is it that the line l is on the plane?
Also the answer on the markscheme says it's 7x + 8y - 11z = -38

 

[DragonCub]

In 2. How is it that the line l is on the plane?
Also the answer on the markscheme says it's 7x + 8y - 11z = -38

Both A and N are on line l, and both the points are on the plane, so l must be on the plane too.

Also, I tested the answer on the MS using the points and found it to be inconsistent.
Using B's coordinates, (3, -1, -4), we can see that:
7 * 3 + 8 * (-1) + (-11) * (-4)
= 21 - 8 + 44
= 57
The equation shows that B is not on the plane.

 

[Thought blocker]

In 2. How is it that the line l is on the plane?
Also the answer on the markscheme says it's 7x + 8y - 11z = -38

I guess you have a wrong ms. And the question u have posted, check Qten of s06 paper three.

 

[i_try9621]

I guess you have a wrong ms. And the question u have posted, check Qten of s06 paper three.

yup it's wrong, thanks for the help

 

[Shuddho Ukil]

Stuck on this M1 problem from 2009. I solved it but my question is, why aren't we considering the kinetic energy acting on the body?
Its from 9709_s09_qp_04.

 

[DragonCub]

Stuck on this M1 problem from 2009. I solved it but my question is, why aren't we considering the kinetic energy acting on the body?
Its from 9709_s09_qp_04.

The whole process was done at constant speed of C. This means C's kinetic energy does not change at all.

 

[SpawnX-1200]

Hi guys trust you are doing well. I would really need some help for the 2 questions I've attached below.

 

[DragonCub]

Hi guys trust you are doing well. I would really need some help for the 2 questions I've attached below.

 

[SpawnX-1200]

Thanks Sir! It's brilliantly done, but if I may. For the 1st question my only concern is why did you only take the absolute value for a=3 and did not also consider the case where $a$ can be equal to -3 ?