Mathematics: Post your doubts here!

[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 ?

 

[DragonCub]

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 ?