1.
Given ABC, with lengths shown in the diagram below, find the length of the
line segment [CD].
diagram not to scale
(Total 5 marks)
2.
The vertices of an equilateral triangle, with perimeter P and area A, lie on a circle with radius r.
P
k
+
Find an exp

The Deadline of HW6 is 17:00, 27th, Mar. You should submit HW6 with the solution of Question 1 in
HW5 again. Please hand in your homework to T.A.
Stochastic Process HW6
Q1. Dr. Wong plays TV game, starting from level 0. This game has countable infinite

11trialsa
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
2
The angle satisfies the equation 2 tan 5 sec 10 = 0, where is in the second quadrant.
Find the value of sec .

DP1 Mathematics HL
Assignment 4 Exponential and logarithmic functions(21 st August2015)
1. Solve the equations
a)
log 6 x+ log 6 ( x 5 )=1
b)
4 log 3 x=log x 3
c)
x+ log 2 27= 3
3 log 2
d)
9 x 6 ( 3 x )16=0
e)
log 4 x +12 log x 47=0
2. Solve the equation

P1
1. Solve the equation log3(x + 17) 2 = log3 2x.
(Total 5 marks)
2.
Solve the equation 2
2x+2
x
10 2 + 4 = 0, x
.
(Total 6 marks)
3.
The diagram below shows two straight lines intersecting at O and two circles, each with
centre O. The outer circle has

1.
Consider the arithmetic sequence 8, 26, 44, .
(a)
th
Find an expression for the n term.
(1)
(b)
Write down the sum of the first n terms using sigma notation.
(1)
(c)
Calculate the sum of the first 15 terms.
(2)
(Total 4 marks)
2.
A geometric sequence h

Grade 11 WS Complex Numbers1
Paper1
1.
z
Given that z 2 = 2 i, z
, find z in the form a + ib.
(Total 4 marks)
2.
The complex numbers z1 = 2 2i and z2 = 1 i 3 are represented by the points A and B
respectively on an Argand diagram. Given that O is the ori

Rev.Ws Vectors.
1.
Line L1 passes through points A(1, 1, 4) and B(2, 2, 5).
(a)
Find AB .
(2)
(b)
Find an equation for L1 in the form r = a + tb.
(2)
Line L2 has equation r =
(c)
2 2
4 s 1
7 3
.
Find the angle between L1 and L2.
(7)
(d)
The lines

Paper1h
1.
(a)
CB = b c, AC = b + c
A1A1
Note: Condone absence of vector notation in (a).
(b)
AC CB = (b + c) (b c)
2
2
= b c
= 0 since b=c
M1
A1
R1
Note: Only award the A1 and R1 if working indicates that they understand
that they are working with vector

Paper1
1.
In the diagram below, [AB] is a diameter of the circle with centre O. Point C is on the
circumference of the circle. Let OB b and OC c .
(a)
Find an expression for CB and for AC in terms of b and c.
(2)
(b)
Hence prove that ACB is a right angle.

Findtheinfinitelimit.
lim
x2
x csc x
x
f(x)
2 0.1
61.9350
2 0.01
627.3289
2 0.001
6282.1864
2 0.0001
62830.8532
Asbasedfromthetable,asxapproaches2fromtheleft,thevalueofthe
limitapproaches
lim
x2
lim
x2
lim
x2
x csc x =
lim
x2
x
=
sin x
x csc x = 62830.853

The Deadline of HW7 is 17:00, 27th, April. Please hand in your homework to T.A.
Stochastic Process HW8
1. Let
event. Find
be a Poisson process with rate
Let
denote the time of the nth
(a)
(b)
(c)
2. The number of hours between successive train arrivals at

Stochastic Process Solution of HW8
5. We want Prcfw_ 30 < X(16) X(0) < 34. In this case mt = (2 cm/sec)(16 sec) = 32 cm and (4
sec1/2) = 2 cm. We know that X(16) X(0) is normal with mean mt and standard deviation s. So
Z = (X(16) X(0) 32)/2 is approximate

Stochastic Process Solution of HW3
1.
p ( z ) = 0.5 z + 0.5 0.2 z 2 + 0.5 0.8 (0.5 0.8 z 2 + 0.1z ) n 0.5 0.2 z 4
n =0
+ 0.5 0.8 (0.5 0.8 z 2 + 0.1z ) n 0.4 z 3
n =0
= 0.5 z + 0.1z 2 + (0.16 z 3 + 0.04 z 4 ) (0.5 0.8 z 2 + 0.1z ) n
n =0
= 0.5 z + 0.1z 2 +

Stochastic Process Solution of HW1
!
The rest are 0.
!
!
4a) Because whether or not it rains today depends on previous weather conditions through
1
the last two days but not yesterday only.
!
!
5a) 8
!
! 0.5 0.4 0.1 "
#
$
P = # 0.3 0.4 0.3 $
# 0.2 0.3 0

Stochastic Process Solution of HW2
st
1(a) !
(b) !
(c) !
(d) !
(e) !
P (process enters S 2 for the 1 time at k
P (process never enters S 4 ) =
th
1"1#
trial) = $ %
3& 4'
k 1
7
9
P (process enters S 2 and leaves S 2 on the next trial) =
P (process enters S

The Deadline of HW3 is 17:00, 6th, Mar. Only first 5 questions are required. Please hand in
your homework to T.A.
Stochastic Process HW3
1. Find !
3,3
of the following Markov chain:
!
2. Consider the Markov chain in the below figure
!
where ! , 0 and !

The Deadline of HW1 is 17:00, 20th Feb. Please hand in your homework to T.A.
Stochastic Process HW1
1. Three white and three black balls are distributed in two urns in such a way that each
contains three balls. We say that the system is in state i, i = 0,

Stochastic Process Solution of HW6
1. !
0 = i qi i = i 1 pi 1 , for i 1
i =0
n
Let ! u1 = 1 and !
un = pi .
i =0
So ! i = ui 1 0 . So ! qi i = (1 pi )ui 1 0 = (ui 1 ui ) 0 .
n
n
0 = i qi = lim i qi = 0 lim (ui 1 ui ) = 0 lim(1 un ).
!
i =0
n
n
i =0
n

The Deadline of HW5 is 17:00, 20th, Mar. Please hand in your homework to T.A.
Stochastic Process Solution of HW5
!
1
!
!
2
!
!
5 We have the balance equation
(0) = q (1)
(1) = (0) + p (2) (1) = (2)
(2) = p (1) + q (3) (2) = (3)
(3) = q (2) + p (4)

The Deadline of HW1 is 17:00, 27th Feb. Please hand in your homework to T.A.
Stochastic Process HW2
!
Find
st
th
(a) ! P(process enters S 2 for the 1 time at k trial)
(b) ! P(process never enters S 4 )
(c) ! P(process enters S 2 and leaves S 2 on the next

The Deadline of HW7 is 17:00, 10th, April. Please hand in your homework to T.A.
Stochastic Process HW7
Q1. Consider the three-period model.
!
Let ! r =
(i)
1
4
! ! 1
be the interest rate so that the risk-neutral probabilities are ! p = q = 2 .
P
Determine

The Deadline of HW5 is 17:00, 20th, Mar. Please hand in your homework to T.A.
Stochastic Process HW5
Q1. A particle moves among n + 1 vertices that are situated on a circle in the following manner.
At each step it moves one step either in the clockwise di

a.)Determinewhatiswrongintheequation
x
2
+ x 6
= x + 3
x 2
Thefunctionintheleftsideisdefinedforallvaluesofxexceptforx = 2 .
However,thefunctionontherightsideisoneveryvaluesofx
b.)Provethattheequations lim
x
lim
2
+ x 6
x 2
x2
x
2
+ x 6
lim
x2
x 2
+ x 6
x