2008 HSC questions (1 Viewer)

[steve001]

hey, im just wondering if i did this question right.
its question 6 (b) if anyone has got the paper and ill write the question underneath for anyone that doesn't have it.

here it is:
It can be shown that sin3a=3sina-4sin^3a for all values of a. (Do NOt prove this.)
Use this result to solve sin3a+sin2a=sina for 0<= a => 2pi.

this is what i did:
sin3a+sin2a=sina
3sina-4sin^3a+sin2a=sina
3sina-4sin^3a+2sinacosa=sina
3-4sin^2a+2cosa=1
3-4(cos^2a-1)+2cosa=1
3-4cos^2a-4+2cosa=1
4cos^2a-2cosa-2=0
2(2cos^2a-cosa-1)=0
2(2cosa+1)(cosa-1)=0
cosa=-0.5 or 1

here are some other questions that i had trouble with:
4(b)Barbara and John and six other people go through a doorway one at a time.
(i)In how many ways can the eight people go through the doorway if John goes through the doorway after barbara with no-one in between?
This is simply 7!
(ii)Find the number of ways in which the eight people can go through the doorway if John goes through the doorway after Barbara.
Didnt know this

6(a) From a point A due south of a tower, the angle of elevation of the top of the tower T, is 23 degrees. From another point B, on a bearing of 120 degrees from the tower, the anlge of elevation of T is 32 degrees. the distance AB is 200 metres
(i) Copy or trace the diagram into your writing booklet, adding the given information to your diagram.
Dont worry about this.
(ii)Hence find the height of the tower
Didnt know this

Thanks in advance

 

[anon09]

I tried the first one:

3sina - 4sin^3a + 2sinacosa = sina
4sin^3a - 2sina - 2sinacosa = 0
2sina (2sin^2a - 1 - cosa) = 0

Therefore 2sina = 0, a = 0

Therefore (2sin^2a - 1 - cosa) = 0
2 (1 - cos^2a) - 1 - cosa = 0
2 - 2cos^2a - 1 - cosa = 0
2cos^2a + cosa - 1 = 0
(cosa + 1)(2cosa - 1) = 0

So cosa = -1, a = 180

And cosa = 0.5, a = 60, 300

With yours, I can see that you divided everything by sina but I think you have to factorise and say that sina = 0, a =0.

And I'm not sure about your seventh line of working because I don't know how you added the constants.

 

[steve001]

i fixed mine up but it seems that we have got opposite signs

 

[jet]

http://hsccoaching.com/Resources/2008Ext1.pdf

 

[anon09]

i fixed mine up but it seems that we have got opposite signs

I think that's because you subbed in "cos^2a - 1" for sin^2a and I used "1 - cos^2a" (derived from cos^2a + sin^2a = 1).


I also forgot that from sina = 0, a = 360 as well as 0.

 

[steve001]

whoops....most stupid mistake

 

[steve001]

im just looking at your sign jetblack and there is a total of 17 units there....why sooo many?

 

[jet]

why not?

 

[Timothy.Siu]

i dont think dividing by sin a is a good idea, u can remove solutions.....

(ii)Find the number of ways in which the eight people can go through the doorway if John goes through the doorway after Barbara.

is it 8C2x6!

the 3d trig question u just draw the diagram. and use pythagoras's theorem to find a relationship,

use tan a=o/a for the angles

 

[Templar]

4(b)Barbara and John and six other people go through a doorway one at a time.
(i)In how many ways can the eight people go through the doorway if John goes through the doorway after barbara with no-one in between?
This is simply 7!
(ii)Find the number of ways in which the eight people can go through the doorway if John goes through the doorway after Barbara.
Didnt know this

Consider Barbara and John as a single entity (since John must go after Barbara). There are 7 'people', so 7! possible ways to go through the door (7 choices for first person, 6 for second etc).

For the second one, it's 8!/2, since John either goes before or after Barbara with equal probability.

 

[darkchild69]

hey, im just wondering if i did this question right.
its question 6 (b) if anyone has got the paper and ill write the question underneath for anyone that doesn't have it.

here it is:
It can be shown that sin3a=3sina-4sin^3a for all values of a. (Do NOt prove this.)
Use this result to solve sin3a+sin2a=sina for 0<= a => 2pi.

this is what i did:
sin3a+sin2a=sina
3sina-4sin^3a+sin2a=sina
3sina-4sin^3a+2sinacosa=sina
3-4sin^2a+2cosa=1
3-4(cos^2a-1)+2cosa=1
3-4cos^2a-4+2cosa=1
4cos^2a-2cosa-2=0
2(2cos^2a-cosa-1)=0
2(2cosa+1)(cosa-1)=0
cosa=-0.5 or 1

here are some other questions that i had trouble with:
4(b)Barbara and John and six other people go through a doorway one at a time.
(i)In how many ways can the eight people go through the doorway if John goes through the doorway after barbara with no-one in between?
This is simply 7!
(ii)Find the number of ways in which the eight people can go through the doorway if John goes through the doorway after Barbara.
Didnt know this

6(a) From a point A due south of a tower, the angle of elevation of the top of the tower T, is 23 degrees. From another point B, on a bearing of 120 degrees from the tower, the anlge of elevation of T is 32 degrees. the distance AB is 200 metres
(i) Copy or trace the diagram into your writing booklet, adding the given information to your diagram.
Dont worry about this.
(ii)Hence find the height of the tower
Didnt know this

Thanks in advance

4b) ii)
8!/2! = 20160

6a) ii)

Tan(23) = h/OA

OA = h/Tan(23)

OA = hcot(23)

Tan(32) = h/OB

OB = h/Tan(32)

OB = hcot(32)

AB^2 = OA^2+OB^2-2(OA*OB*Cos(60))

200^2 = h^2(cot^2(23)+cot^2(32)-cot(23)cot(32))

h = 200/((cot^2(23)+cot^2(32)-cot(23)cot(32)^1/2)

h = 96m