• Best of luck to the class of 2024 for their HSC exams. You got this!
    Let us know your thoughts on the HSC exams here
  • YOU can help the next generation of students in the community!
    Share your trial papers and notes on our Notes & Resources page
MedVision ad

Worksheets and questions (1 Viewer)

Rax

Custom Me up Scotty
Joined
Jul 30, 2005
Messages
229
Location
In the Bush
Gender
Male
HSC
2006
Re: More Questions

Well I am sure CM_Tutor does

But he doesnt seem to be around much longer

May aswell attempt that question myself
 

haque

Member
Joined
Sep 2, 2006
Messages
426
Gender
Male
HSC
2006
Re: More Questions

i've got the wallis' product one but i think there's a mistake in the last bit-the value of root pi in which rather than root n at the denominator it should be root n+1/2. correct me if i'm wrong-how do i post up my solutions, everytime i try and attach my scans it says error website could not be found?
 

Yip

Member
Joined
Sep 14, 2005
Messages
140
Gender
Male
HSC
2006
Re: More Questions

69th post :p (*snigger*) well since no one has posted their solutions...for 2 years lol, let me be the first to post one, i have solutions for most questions except q3, q16 (how would u prove this formally?...), q18 and q19 (dont get the notation, the double integrals i usually see are in the form integral f(x,y)dxdy or something...). feel free to rip my solutions apart...most probably there are a ton of mistakes in there. i think that there are some mistakes in some of the qs, billyMak's statements about q7 seem to be correct, i think there is a missing negative in q15, and i think there is a typo on q8, isnt it tower AB and building CD?...haque, i think cm_tutor was correct in his wallis product q except for errors in the first part which sedated pointed out earlier
edit: i misnamed one of the files, should read q14,15,17
 
Last edited:

vafa

Member
Joined
Mar 16, 2006
Messages
302
Gender
Undisclosed
HSC
N/A
Re: More Questions

in answer to hoque:

I think the size of your images are too large. You need to compress them. because I had the same problem but when i started to compress them it was ok. i guess there is a limitation like 196kb for your images.
 

brendanm88

New Member
Joined
Jul 29, 2006
Messages
4
Gender
Male
HSC
2006
Re: More Questions

Should there be an n in the final answer for 10)c)? nC0 is 1, not n
 

zeus_three

New Member
Joined
Nov 7, 2004
Messages
2
Gender
Male
HSC
2006
Re: More Questions

I'm sure you know this, but I'll post it anyway.

S={5,9,13...}
With a little calculation, we can determine that the terms of this set can be written as:
T=4n+1

Let a,b be two arbitrary (non-negative) integers...Then we can have two terms in the set S that are:
T(a)=4a+1
T(b)=4b+1

Their multiplication yields:
T(a)T(b)=16ab+4a+4b+1= 4(4ab+a+b)+1

Now, since a,b were defined as arbitrary integers, we know that 4ab+a+b must also be an integer,
i.e. T(a)T(b)=T(4ab+a+b)... which is clearly a member of the set S.
 

zeus_three

New Member
Joined
Nov 7, 2004
Messages
2
Gender
Male
HSC
2006
Re: More Questions

Curious... is the first question do-able by complex numbers?
 
Last edited:

Grey Council

Legend
Joined
Oct 14, 2003
Messages
1,426
Gender
Male
HSC
2004
Re: More Questions

ah, I havent been around since end of 1994, but ive felt guilty for a while that i stopped contributing after my hsc was over. anyway, maybe I can start posting up some stuff now and then. ill start off now.

CM_Tutor was active when i was around, I recieved a fair amount of help from him. If one of the moderators can change this thread's topic to something more appropriate (such as "Random Component B questions", I can post up more component B questions from people like Oldman/buchanan/CM_Tutor, but for now, I'll post up CM_tutor stuff. I'd rather not clutter up forums by creating a totally new thread. anyway, here goes:

Suppose that z is a complex number satisfying |z| > 1 and 0 < arg z < pi / 2.

(a) Sketch on the Argand diagram the points A, B, C and D which represent, respectively, z, -1, 1 / z, and 1.
(b) Show that angles CAD and CBD are both arg z - arg (z + 1)
(c) Hence, show that ABCD is a cyclic quadrilateral, and that the centre of the circle through A, B, C and D is [(|z|^2 - 1) / (2 * Im(z))] * i



Suppose that z is a complex number with modulus 1 and argument @ (ie z = cos@ + isin@).
(a) Show that z^n - 1 = 2i * sin(n@ / 2) * [cos(n@ /2) + isin(n@ / 2)], for n a positive integer.
(b) Hence, or otherwise, show that
z + z^2 + z^3 + ... + z^n = {sin(n@ / 2) / sin(@ / 2)] * [cos[(n + 1)@ / 2] + isin[(n + 1)@ / 2]}
(c) Hence, find an expression for cos@ + cos(2@) + cos(3@) + ... cos(n@)
(d) Prove that |cos@ + cos(2@) + cos(3@) + ... cos(n@)| <= |cosec(@ / 2)|
(e) Solve the equation cos@ + cos(2@) + cos(3@) + cos(4@) = 0 over the domain 0 <= @ <= 2 * pi



The polynomial ax^4 + bx^3 + cx + d = 0 has a triple root.
(a) By proving that this root occurs at x = -b / 2a, or otherwise, prove that 4ad = bc.
(b) Is it true to say that any polynomial of this form whose coefficients are related by 4ad = bc must have a triple root? Explain.


1. We know that x^0 = 1, and 0^x = 0, so what is 0^0?
2. Bill and Ted play a game with a fair coin, where each toss the coin n times. Show that the probability that they get the same number of heads is (2n)! / [2^(2n) * (n!)^2]


if w is a complex root of z^3 - 1 =0 where w does not equal to 1
show that 1 + w + w^2 = 0 (easy)
and hence evaluate
(1 - w)(1 - w^2)(1 - w^4)(1 - w^8)
(1 + w)(1 + w2)(1 + w4)(1 + w8)
1. P and Q are points on the Argand Diagram representing the complex numbers z and w, respectively, and z / w is purely imaginary. Draw a diagram to represent this information, and use it to prove that |z + w| = |z - w|.

2. P and Q are points on the Argand Diagram representing the complex numbers z and w, respectively, and
|z| = |w|. Draw a diagram to represent this information, and use it to prove that (z + w) / (z - w) is purely imaginary.

3. z0 is a complex number represented by the point P on the Argand Diagram, and Q represents the point iz0.
(a) Draw a sketch of the locus of z if |z - z0| = |z - iz0|. Include the points P and Q on your sketch.
(b) Draw a sketch of the locus of z if arg(z - z0) = arg(iz0). Include the points P and Q on your sketch.
(c) The loci in (a) and (b) meet at a point R. Find the complex number represented by the point R.

4. If z is any complex number satisfying |z| = 1, show that 1 <= |z + 2| <= 3, and |arg(z + 2)| <= pi / 6.

5. If z0 is a fixed complex number and R is a positive constant, describe the locus of z if
z * z(bar) + z * z0(bar) + z(bar) * z0 + z0 * z0(bar) = R2


There has been a discussion of some of the harder stuff that can turn up at the ends of papers, so I thought I'd post some stuff from later questions of some 4u half-yearlies I have. I've skipped the motion, as people probably haven't done it yet, but please advise if I'm wrong...

1. (For this question, I'm going to use a in place of alpha, for the sake of clarity.)
(a) Provided sin(a / 2) <> 0, show that cos(a / 2) = sin a / 2sin(a / 2)
Similarly, show that if sin(a / 2) and sin(a / 4) are both <> 0, cos(a / 2) * cos(a / 4) = sin a / 4sin(a / 4)
(b) Prove by mathematical induction that if n is a positive integer, and sin(a / 2n) <> 0, then
cos(a / 2) * cos(a / 4) * ... * cos(a / 2n) = sin a / 2nsin(a / 2n)
(c) Hence, deduce that a = sin a / {[cos(a / 2)] * [cos(a / 4)] * [cos(a / 8)] * ... } and that
pi = 1 / {(1/2) * sqrt(1/2) * sqrt[(1/2) + (1/2)sqrt(1/2)] * sqrt{(1/2) + (1/2)sqrt[(1/2) + (1/2)sqrt(1/2)]} * ...}

2. (a) Let @ = tan-1x + tan-1y. Show that tan@ = (x + y) / (1 - xy)
(b) If tan-1x + tan-1y + tan-1z = pi / 2, show that xy + yz + zx = 1
(c) Let Wn = tan-1x1 + tan-1x2 + ... + tan-1xn, where n is a positive integer.
Show, by mathematical induction or otherwise, that tan Wn = - Im(wn) / Re(wn)
where wn = (1 - ix1)(1 - ix2)...(1 - ixn)

3. The quartic polynomial f(x) = x4 + bx3 + cx2 + dx + e has two double zeroes at alpha and beta, where the coefficients b, c, d, and e are real, but the zeroes alpha and beta may be complex.
(a) Express b, c, d and e in terms of the double zeroes alpha and beta, and hence show that:
(i) d2 = b2e
(ii) b3 + 8d = 4bc

(b) Now, suppose that b = 2 and e = 1, and that the double zeroes alpha and beta are both non-real complex numbers.
(i) Show that alpha and beta are cube roots of 1.
(ii) Write f(x) as the product of polynomials irreducible over the real numbers.

1. Prove by mathematical induction that 13 + 23 + 33 + ... + n3 = (1 + 2 + 3 + ... + n)2, for all positive integers n

2. Prove by mathematical induction on the integers n=>1 that
1 / (1 * 2 * 3) + 1 / (2 * 3 * 4) + 1 / (3 * 4 * 5) + ... + 1 / n(n + 1)(n +2) = n(n + 3) / 4(n + 1)(n + 2)

3. Use induction to prove that cos(x + n * pi) = (-1)ncos x, for all integers n => 1

4. Fred borrows $P from a finance company, which charges interest at a rate of 12 % per annum, compounded monthly, and at the end of each month (after interest has been charged), Fred males a payment of $M. Let $An be the amount owing at the end of n months, after the repayment for that month has been made.

(a) Show that A1 = 1.01P - M, and that A2 = 100M + 1.012(P - 100M)

(b) Prove by mathematical induction that An = 100M + 1.01n(P - 100M)

(c) If the term of the loan was 20 years, and an amount of $50,000 was borrowed, calculate (to the nearest cent) the monthly repayments, $M, that Fred must make.

5. (a) Show that sin x + cos x = sqrt(2) * sin(x + pi / 4)

(b) Show that the derivative of y = exsin x is dy/dx = sqrt(2) * exsin(x + pi / 4)

(c) Given that y = exsin x, use mathematical induction to prove that the nth derivative for positive integral n is given by dny/dxn = [sqrt(2)]nexsin(x + n * pi / 4)

6. The Fibonacci sequence is a sequence of numbers defined by the following rules: F1 = 1, F2 = 1, and
Fn = Fn-1 + Fn-2 for integers n > 2.

(a) Write out the first 12 terms of the Fibonacci sequence

(b) Use mathematical induction to prove that F3k is even, and that F4k is a multiple of 3, for all positive integers k

(c) Hence, or otherwise, prove that F12k is divisible by 6, for all integers k > 0

7. Prove by mathematical induction that 7n > 4n + 5n for all integers n => 2

8. Prove by induction that, for all positive integers n,
2 + (2 + 22) + (2 + 22 + 23) + ... + (2 + 22 + 23 + ... + 2n) = 2n+2 - 2(n + 2)

9. Prove by mathematical indcution, for positive integers n, that
1 / (x - 1) - x-1 - x-2 - x-3 - ... - x-n = 1 / xn(x - 1)

10. Use mathematical induction to prove that, for n an integer, n > 1:
(1 - 1 / 22) * (1 - 1 / 32) * ... * (1 - 1 / n2) = (n + 1) / 2n
and hence evaluate lim(n --> +inf) (1 - 1 / 22) * (1 - 1 / 32) * ... * (1 - 1 / n2)

i cbf editing that post for all the little stuff, but just a little common sense to fix up stuff like "n2" (which obv means n^2)
 

Grey Council

Legend
Joined
Oct 14, 2003
Messages
1,426
Gender
Male
HSC
2004
Re: More Questions

Ah, a few more CM_tutor posted up in 2004. Word for word, the post CM_Tutor made:


Before I post any questions, I would like to make a couple of introductory comments about conics.

Most students don't like conics, saying it is hard. I disagree. This is (conceptual) a relatively straight-forward topic, based mostly in simple co-ordinate geometry and calculus. What makes it difficult is that the algebra is often tangled and lengthy.

So, IMO, the key to conics is to minimise (to the greatest extent possible) the algebra. Easier said than done, you might be thinking. Well, here are some suggestions.

1. Draw a diagram. Make it big. Make sure you have plenty of space to draw onto it, should that be needed.

2. Think about the question. Most questions have an obvious algebra-bash approach. You are trying to find an alternative, because there usually is one.

3. Alternatives:
(a) Look to use properties of the conic sections, like the focus-directrix definition PS = ePM.
(b) Look for recognisable geometry shapes, triangles, etc.
(c) Look for connections between those shapes - similar triangles, congruent triangles, etc
(d) Look to make diagonals into horizontals and verticals - if you have to find a distance, you want it to be horizontal or vertical so that you can avoid the distance formula.
(e) Look for other results, like intercept theorems, or trigonometry.

4. Only after you have exhausted the above may you consider an algebra-bash.


Now, here are some questions to think about - I'm looking for two types of solutions - an elegant solution (if you can find one), or an algebra bash if you can't. Have fun!

1. The point P is any point on the ellipse x2 / a2 + y2 / b2 = 1, which has foci at S and S'. Prove that PS + PS' = 2a. (Note: this is a standard problem. Anyone who does not know a really fast way to do this should learn it as soon as one of your colleagues posts it.)

2. The point P(acos@, bsin@) is a point on the ellipse x2 / a2 + y2 / b2 = 1. The tangent at P meets the x-axis at T, and the normal at P meets the x-axis at N. Draw a diagram to represent this information, and show that
(PT / PN)2 = tan2@ / (1 - e2).

3(a). ABC is a triangle and X is a point on BC. Prove that AX bisects angle BAC if and only if AB:AC = BX:CX, using the sine rule or otherwise.

(b) P(asec@, btan@) is any point on the hyperbola x2 / a2 - y2 / b2 = 1, which is not located on the x-axis. The tangent at P meets the x-axis at T, and S and S' are the foci of the hyperbola.

(i) Show that T has coordinates (acos@, 0)
(ii) Prove that PS and PS' are inclined to the tangent at P at equal angles.

4. P(acos@, bsin@) and Q(acos#, bsin#) are two distinct points on the ellipse x2 / a2 + y2 / b2 = 1, and the chord PQ subtends a right angle at A(a, 0). Find an expression for tan(@ / 2) * tan(# / 2) in terms of a and b only.

5. P(asec@, btan@) is a point on the hyperbola x2 / a2 - y2 / b2 = 1, and tan@ <> 0. The tangent at P meets the x- and y- axes at X and Y, respectively, and meets the asymptotes at K and L.

(a) Find the equation of the tangent at P, and hence find the coordinates of X and Y.

(b) Find the value of
(i) PX / XY, and
(ii) PX / PY

(c) Find the value of PK / PL
 

Grey Council

Legend
Joined
Oct 14, 2003
Messages
1,426
Gender
Male
HSC
2004
Re: More Questions

Actually, ill just post up some more component B questions, title of thread is good enough I guess.

Most of these questions were posted by Oldman, if i remember correctly. If others posted any of these, feel free to PM me and ill give credit etc

The tangent at P(asec@,btan@) on a hyperbola meets the asymptotes at QR. Show that QR is twice the distance of the chord joining point P with the intersection of the asymptotes.

Note: this question is a morph of Geha's question for the special rectangular hyperbola case ie. P(cp,c/p).


The chord AB is normal to the parabola x^2=4ay. Find the point A which minimizes the length of this chord.


Prove that the area of the triangle formed by the tangent to the hyperbola and the asymptotes is a constant.


a)Consider the line y=mx+c and the hyperbola H,
x^2/a^2-y^2/b^2=1.
Show that the conditions for cutting, touching and avoiding are
c^2>(am)^2-b^2, c^2=(am)^2-b^2, and c^2<(am)^2-b^2 respectively.
b)The point M(X_0,Y_0) lies "inside" H when
X_0^2/a^2-Y_0^2/b^2>1.
The line L is given by the equation
xX_0/a^2-yY_0/b^2=1.
(i) Using the result of (a) show that the line L lies entirely "outside" H. That is if (X_1,Y_1) is any pt. on L, then X_1^2/a^2-Y_1^2/b^2<1.
(ii) The chord of contact to the hyperbola from any pt. (X_2,Y_2)
"outside" H has equation
xX_2/a^2-yY_2/b^2=1.
Show that (X_0,Y_0) lies on the chord of contact to H from any point on L. That is if (X_2,Y_2) lies on L, then (X_0,Y_0) will lie on the chord of contact from (X_2,Y_2).


Find the smallest area of the triangle formed by the tangent line to the ellipse (s-major a, s-minor b) with the coordinate axes in the first quadrant.

P is an arbitrary pt. on the ellipse and line L is the tangent to the ellipse at P.
The pts. S' and S are the foci of the ellipse. Let S" be the reflection of S across the L.
i) Prove that the focal chords through P are equally inclined.
ii) Fully describe the path of S" as P moves on the ellipse.


Montana duck hunters area all perfect shots. Ten Montana hunters are
in a duck blind when 10 ducks fly over. All 10 hunters pick a duck at
random to shoot at, and all 10 hunters fire at the same time. How many
ducks could be expected to escape, on average, if this experiment were
repeated a large number of times?


Maximize |z^3-z+2| when |z|=1.


A frictionless* frog jumps from the ground with speed V at an unknown angle to the horizontal. It swallows a fly at a height h. Show that the frog should position itself within a radius of
V/g SQRT(V^2-2gh) of the point below gulp point.
This problem should come with a warning and a promise.
Warning :Quite difficult. It is a dangerous swamp out there, and the problem poses a few traps.
Promise : an earnest attempt yields deeper insights on projectiles and polynomials.


P(x) is a polynomial with integral coefficients. The leading coefficient, the constant term, and P(1) are all odd. Show that P(x) has no rational roots.


Prove that cos(1o), cos(2o),cos(3o),cos(4o),cos(5o) are all irrationals. D’moivres theorem.


Consider the eqn. 2^x=1+x^2
i) find two obvious solutions.
ii) show that there is another solution between 4 and 5.
iii) show that these are the only solutions.

Let P(x) be the quadratic ax^2+bx+c. Suppose that P(x)=x has unequal roots. Show that the roots are also roots of P(P(x)=x. Find a quadratic equation for the other roots of this equation. Hence solve, (x^2-3x+2)^2-3(x^2-3x+2)+2-x=0.


a,b both positive and a+b < ab. Prove a+b > 4.


I{a-->b}f(x)dx integral, upper bound b, lower bound a integrate
f(x) w.r.t. x.
Using sin2x=2sinxcosx or otherwise,
find I{pi/2-->0} Ln(sinx)dx

Prove without using formulas :
nCk = n-1Ck-1 + n-1Ck

w^3 =1. Prove that z_1, z_2,-wz_1-w^2z_2 form the vertices of an equilateral triangle, z_1,z_2 arbitrary complex numbers and w not=1.



there are 2 red, 1black and 1 white marbles in a bag. two marbles are drawn one after the other, and kept hidden in a hand.

a) find the probability that that 2 red marbles are drawn

b) find the probabilty that the second marble is red.

c) one of the drawn marbles slips from the person's hand and it was red, what is the probability that the other one is also red.


What is the reciprocal of i? How is i the reciprocal of i?
Now prove it wrong.


x^3+3x^2+2=0 has roots a, b, c
find the equation with roots : a+1/a, b+1/b, c+1/c





Say you have a pile coins, that all appear identical, and a balance scale. You know that one and only one is counterfeit, and that the counterfeit coin is either heavier or lighter than the others.

If there are a small number of coins, say 5, you can find the counterfeit coin easily in three weighings on the balance scale. But what if there's more? It gets harder and harder to solve. So the question is, what is the maximum number of coins (call it X) which allows you to always be able to find the counterfeit coin in 3 or less weighings, how do you do it with X coins, and prove that you can't always do it with X+1 coins.

If this is too difficult, start with only 2 weighings (it's harder than it sounds!), then work up to 3. Once you've done 3, try 4, and even 5 weighings! Each one has a new trick that you have to use to find the counterfeit, that's one of the reasons I love this problem.

Then, find a formula for the value of X, when you're allowed N weighings. (this is probably the easiest part of the problem)

Then, if you're still after more (by this point I'd been working on the problem for a very long time, but I was enjoying it so much I wanted to make it harder) write an algorithm for finding the counterfeit coin that will work for any number of weighings




Prove |Arg(z)| >= | | |z| -1| - |z-1| | for z a complex number.





i) Find roots of z^5 + 1 = 0
ii) Factor z^5 + 1
iii) Deduce cos(pi/5) + cos(3pi/5) = 1/2
and cos(pi/5)*cos(3pi/5) = -1/4


ii) Find the line that is twice tangent to the curve y=4x^4+14x^3+6x-10.



Di and Dot bet on the total roll of two standard dice. Di bets that a 12 will be rolled first. Dot bets that two consecutive 7's will be rolled first. They keep rolling until one wins. What is the probability that Di will win?


A particle moves in a straight line subject only to a resistive force proportional to its speed. Its speed falls from 1200 m/s to 800 m/s over 1400 m. Find the time taken to the nearest 0.01 sec.


Find the sum :nC1 (1^2) + nC2 (2^2)+...+nCn (n^2)


Explain why the polynomial
b_0+b_1x+...+b_nx^n has at least one real root if,
b_0/1+b_1/2+...+b_n/(n+1)=0.


Bob tosses 11 coins, Penny tosses 10. (i) What is the probability that Bob has more heads than Penny. (ii) Generalize to n+1,n respectively.


I{0--->pi/2}f(x)dx integral upper limit pi/2, lower limit 0 wrt x, and f(x)=1/(1+tan^.25 (x))


Consider the curve y=x^3. The tangent at A meets the curve again at B. Prove that the gradient at B is 4 times the gradient at A.


Show that the coeff. of x^k in the expression,
(1+x+x^2+x^3)^n is SUM(j:0--->int(k/2))[nCj.nC(k-2j)] where int(k/2) is the highest integer <= to k/2.


If 4 distinct points of the curve y=4x^4+14x^3+6x-10 are collinear, then their mean x-coordinates is a constant k. Find k.


Through the magic of compounding, capital C becomes
C(1+r)^n after n years. How much do we need to invest to be able to withdraw $1 at the end of year 1, 4 at the end of year 2, 9 at the end of year 3, 16 at the end of year 4, and so on in perpetuity?


P_1,P_2,P_3,...,P_n represent the complex numbers z1,z2,z3,...,zn (zn=1) and are the vertices of a regular polygon on a unit circle. Prove that
(z1-z2)^2+(z2-z3)^2+(z3-z4)^2+...+(zn-z1)^2=0


A clock's minute and hour hands are lengths 4 and 3 respectively.
At the moment when the distance between the two tips is increasing most rapidly:
i) what is the distance?
ii) what is the speed?
iii) what is the time after 3 o'clock does this first happen?


Two players bet on the outcome of a toss of two coins. Bob bets that double heads will be tossed first. Penny bets that two consecutive single head will be tossed first (that is, exactly one head and one tail, Penny wins if this happens twice, one after the other). They keep tossing until one player wins. What is the probability that Bob wins?


There are five letters, two A's, B, C & D. How many arrangements if two A's cannot be next to each other and B cannot be first.

There are two six letters: two A's, two B's, one C and one D. How many arrangements if two A's and two B's cannot be next to each other?

There are two questions; find the square roots of 3+4i and find sqrt(3+4i). Do u think the solutions are same?

On Argand diagram, points A, B and C represent the complex numbers z1, z2 and z3, respectively. Show that if
(z2-z1) / (z3-z1) =cis (pie/3), then triangle ABC is equilateral triangle?

A biased coin, the probability of the head for a toss is p, where p not equal to 0.5 and 0<P
Q1) Find the probability for exact two heads
Q2)Find the probability for exactly consecutive two heads only

Find x and y is (x+iy)^2=3-4i where x and y are real number?



Let A, B, C, ..., G represent 1, w, w^2, ..., w^6 where w = cis(2pi/7). Let H represent -1

Prove HA*HB*...*HG = 2



1) Let A_1, A_2, ..., A_n represent the nth roots of unity w_1, w_2, ..., w_n. Suppose P represents z such that |z| = 1

(btw, w_1 is omega subscript 1, etc)

i) Prove w_1 + w_2 + ... + w_n = 0
ii) Show that |PA_i|^2 = (z-w_i)(z(bar) - w_i(bar)) (for all i = 1, 2, ..., n)
iii) Hence prove |PA_1|^2 + |PA_2|^2 + ... + |PA_n|^2 = 2n


Prove that x^3 + 3px^2 + 3qx + r has a double root if and only if:

(pq - r)^2 = 4(p^2 - q)(q^2 - pr)


1.solve sin ^nx + cos^nx = 2 ^(2-n)/2

2. solve 3arctg(x) - arctg (3x) = pi/2

3. pi^/sinx^(0.5)/ = /cosx/



An interesting binomial probability question with a twist.

A die is thrown n times. What is the probability of getting an odd number of sixes.

Here's the twist : use (p-q)^n rather than the usual binomial distribution of (p+q)^n.


We had a thread last year on elegance -Spice Girl and ND were the main correspondents.

To do well in Ext. 2, students need to develop their EQ elegance quotient.

Here are three problems to practise on, the first two could easily be done by a Year 11 doing the preliminary, the third - well, lets just say it could happily be embedded in a Question 8 Ext2. But all three share things in common (indeed most maths problems) , multiple approaches -choose the most elegant.


1) P lies on 8y = 15x. Q lies on 10y = 3x and the midpoint of PQ is (8,6). Find distance PQ.

2) A line through the origin divides the parallelogram with vertices (10,45), (10,114), (28,153), (28,84) into two congruent pieces. Find its slope.

3) Let P be the point (a,b) with 0 < b < a. Find Q on the x-axis and R on y=x, so that PQ+QR+RP is minimized.



In a similar vein to the other questions : a car travels at 2/3 km/min due east. A circular storm, w/ radius 51, starts with its center 110 kms due north of the car and travels southeast at 1/sqrt(2) km/min. The car enters the circle of the storm at time t1 and leaves at t2 (mins). Find (t1+t2)/2.
 

jkwii

Ultimate Member
Joined
Jan 4, 2007
Messages
216
Gender
Male
HSC
2008
Re: Some Worksheets

but u cant have 0^0. explain? is it infinity?
 

Users Who Are Viewing This Thread (Users: 0, Guests: 1)

Top