3 Unit Maths HSC Exam Revision (4 Viewers)

[random-1005]

-20/63

lol, i made a mistake!!!, you we right first time with -33/56

 

[hscishard]

Awsome. Now I can sleep

 

[k02033]

A stone is thrown so that it will hit a bird at the top of a pole. However, the instant the stone is thrown, the bird flies away in a horizontal straight line at a speed of 10 metres per second. The stone reaches a height double that of the pole and, in its descent, (miraculously) touches the bird. Find the horizontal component of the velocity of the stone.

Let be the initial horizontal velocity of the stone and let be the initial vertical velocity of the stone.

Let be the horizontal position of the pole with respect to the point of launch and be the height of the pole. Let be the

time it takes for the stone to reach the top of the pole and be the time it takes to reach its maximum height.

Now because the parabola is symmetrical and so the stone is only at a vertical height of at 2 times, namely and

, so therefore the stone must hit the bird at

using all this...













We get 6 equations with 6 unknowns and we can solve for

 

[k02033]

Prove by induction that 9^(n+2) -4^(n) is divisable by 5 for all positive integers n .

Theorm: for all postive integers .

Proof by induction: Let be the proposition for all postive integers .

Now is clearly true since 5|80

Now suppose is true for some n.

We require to prove that is also true, that is

for all postive integers .

Now





But for some integer k from the inductive assumptions.

so

So is also true and by induction is true for all integer n. QED

 

[bouncing]

inverse ( to minus one), common notation on the net

oh right okay cool thanks haha not used to doing maths on the net

 

[bouncing]

NEW QUESTIONS:

1) Solve the equation4x^3+32x^2+79x+60=0 given that one root is equal to the sum of the other two.

2) Find the cubic equation whose roots are twice those of the equation 3x^2-2x^2+1=0

3) If two of the roots of the equation x^3+qx+r=0 are equal, show that 4q^3+27r^2=0


ps. dont forget to post a question once you've done one?

 

[random-1005]

NEW QUESTIONS:

1) Solve the equation4x^3+32x^2+79x+60=0 given that one root is equal to the sum of the other two.

2) Find the cubic equation whose roots are twice those of the equation 3x^2-2x^2+1=0

3) If two of the roots of the equation x^3+qx+r=0 are equal, show that 4q^3+27r^2=0


ps. dont forget to post a question once you've done one?

1. let roots be a, b and c

a=b+c {given}

we also have sum of roots= -8 { -32/4}

and product of roots= -15 {-60/4}

therefore a-b-c=0 {1}
a+b+c=-8 {2}
abc=-15 {3}

three eqns in three unknowns , i cant be bothered to solve, theres the main stuff

first step you do add {1} and {2} a=-4, etc

2.
for initial eqn

let roots be x, y and z

we have x+y+z= -2/3
xy +yz +xz= 0
xyz= -1/3

for final eqn

2(x+y+z)= -2/3
2(xy +yz +xz)=0
(2x)(2y)(2z)=-1/3

therefore -b/a = -1/3
c/a=0 ---> c=0
-d/a=-1/27

not sure on this one


3.

looks fairly simple and i reckon its similar to number 1, ill let someone else do that

its just going to be simultaneous eqns i reckon

 

[random-1005]

Question: Write down the domain and range of y = pi/2 - sin^-1 (x/2)

 

[random-1005]

Question: Simplify nC(r+1) / (nCr)

 

[random-1005]

QUESTION: Find the term independant of x in the expansion of ( 3x^2 +2/x ) ^12

 

[random-1005]

Also just a question about the program that write maths with (latex or something everyone talks about ), do you have to personally buy it or is it somewhere in the screen when you make a post (if it is can someone direct me to where it is), thanks

 

[undalay]

Let be the initial horizontal velocity of the stone and let be the initial vertical velocity of the stone.

Let be the horizontal position of the pole with respect to the point of launch and be the height of the pole. Let be the

time it takes for the stone to reach the top of the pole and be the time it takes to reach its maximum height.

Now because the parabola is symmetrical and so the stone is only at a vertical height of at 2 times, namely and

, so therefore the stone must hit the bird at

using all this...













We get 6 equations with 6 unknowns and we can solve for

 

[nat_doc]

Prove by induction that 9^(n+2) -4^(n) is divisable by 5 for all positive integers n .














hence by the process of MI... blah blah blah... proven true for all n

NEW QUESTION:

 

[hscishard]

Question: Simplify nC(r+1) / (nCr)

Is that (n-r)/(r+1)?

 

[nat_doc]

Write down the domain and range of y = pi/2 - sin^-1 (x/2)

 

[random-1005]

Prove by induction that 9^(n+2) -4^(n) is divisable by 5 for all positive integers n .














hence by the process of MI... blah blah blah... proven true for all n

NEW QUESTION:

x>= 5 and x<4

 

[k02033]

You are suggesting that their is a unique solution.
What does this mean?
This means if you can solve for a specific solution for U.
It means you can solve for a specific solution for L and W.
But we already know that L and W have no specific solution (in that they can be anything).

So basically your 6 equations are wrong, or not linearly independent.

You can easily check to see that this question has no unique solution.

Take W = 10,000m and and L = 1. Solve for U.

Take W = 1m and L = 1. Solve for U.

I think you may have forgotten to use this part of the question

"A stone is thrown so that it will hit a bird at the top of a pole."

That is, at t=T1 the stone must be at (W,L). This together with the other info, puts restrictions on the whole scenario leading to set of unique solutions, including those for W and L.



So overall, its saying all 6 unknowns including W and L must be of specific values or else the stone cannot possibly satisfy all the required conditions at the same time, and in particular these two: 1.being at (W,L) and 2.hitting the bird. There wont be a set of unique solutions if the condition to be satisfied was just to hit the bird, which is what you are thinking i believe.

 

[random-1005]

A function f is defined as follows:
f(x) = x^2 for 0<=x<=2
f(x) = 4 for 2 < x <=5

Consider a region A bounded by the graph of y=f(x), the x axis and the line x=5

i. Find the volume of the solid formed when the region A is rotated about the x axis.'

ii. If the region A is now rotated about the y axis what is the volume of the solid formed.

This is a past sydney grammar 2 unit question, good question.

 

[k02033]

 

[random-1005]

Evaluate log base 9 (49) -log base 3 (7), make sure you can get full marks on first question, must know two unit inside out, so thought id put up some 2 unit questions