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.
(*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
