Download Document

Pui Ying College
F.7 Mock Examination(01-02)
Pure Mathematics 1
Time allowed : 3 hours
Name :
Class : 7 B
No.
Note :
1. This paper consists of Section A and Section B.
2. Answer ALL questions in Section A and any FOUR questions in Section B.
3. You are provided with one AL(E) answer book and four AL(D) answer books.
Section A : Write your answers in AL(E) answer book.
4.
5.
6.
Section B : Use a separate AL(D) answer book for each question and put the
question number on the front cover of each answer book.
The four AL(D) books should be tied together with the green tag provided. The AL(E) answer
book and the four AL(D) answer books must be handed in separately at the end of the
examination.
Unless otherwise specified, all working must be clearly shown.
SECTION A (40 Marks)
Answer ALL questions in this section.
Write your answers in the AL(E) answer book.
1.
Resolve
2x  1
into partial fractions.
x ( x  1) 2
2
Hence or otherwise, show that
n
2r  1
 1.

2
2
r 1 r (r  1)
01-02/F.7 Mock Exam/PMI/LCK/P.1 of 5
(4 marks)
2.
A sequence {an} is defined as follows:
a1 = 3, a2 = 7 and an = 5an –1 – 6a n –2
for n  3.
Prove by induction that an = 2n + 3n –1 for all natural number n.
an
.
3n
(a) Show that x  ln( 1  x) for all x  -1.
Hence evaluate
3.
(5 marks)
lim
n 
(b) Using (a), prove that
1
1 1
1
  ...   ln( n  1).
2 3
n
1 1
1
  ... 
is convergent.
2 3
n
 
  
 

Given two vectors u  i  2 j  k and v  2 i  j  k.
Hence determine whether 1 
4.
(7 marks)


(a) Find a vector which is perpendicular to both u and v.
(b) Find the equation of the plane containing both lines
1 :
5.
x 1 y  2 z  3
x 1 y  2 z  3


and  2 :


1
2
1
2
1
1
Let Cr be the coefficient of xr in the binomial expansion of (1 + x)2n+1.
Prove that Cr = C2n+1-r for all r = 0, 1, 2, …, n.
n
Hence or otherwise, deduce that
C
r 0
6
(4 marks)
r
 4 n.
The equation px 3  qx 2  rx  s  0 has roots ,
(5 marks)
1
and .

Prove that p 2  s 2  pr  qs.
Hence find a root of 6x 3  11x 2  24x  9  0.
7.
Let T : R3→R3 be a transformation defined by T(x, y, z) = (x, y, 0).
(a) Find the matrix representation M of the transformation T.
(b) Determine whether T is an injection or a surjection.
8.
(5 marks)
(4 marks)
(a) If z1 and z2 are complex numbers such that | z1 | = | z2 | = 1, show that
1  z1z 2
is real.
z1  z 2
(b) If w1 , w2 and w3 are complex numbers such that | w1 | = | w2 | = | w3 |, show that
w1  w 2 w 3
is real..
w 1 (w 2  w 3 )
2
01-02/F.7 Mock Exam/PMI/LCK/P.2 of 5
(6 marks)
SECTION B (60 Marks)
Answer any FOUR questions in this section. Each question carries 15 marks.
Use a separate AL(D) answer book for each question.
9.
Consider the system linear equations:

xyza

(E) :  kx  y  kz  b
 k 2 x  y  kz  c

(a) (i)
Find all real values of k such that for any given values of a, b and c, (E) has a unique
solution.
(ii) For k = 0, find the condition that the system (E) is consistent.
(iii) For a = b = c = 0 and k = 1, find two solutions of (E) which are linearly independent
vectors.
(9 marks)
 x  ky  k 2 z  0

xyz 0
(b) Let (x0, y0, z0) be a solution of (F) 
 x  ky  kz  0

Show that if the scalar product (a, b, c)  ( x 0 , y 0 , z 0 )  0, then (E) is inconsistent.
(6 marks)
10. Let
S : zz  cz  cz    0, where ,   R and c  C.
(a) (i) If α = 0, show that S is a straight line in the Argand Diagram.
(ii) If α≠0, find a condition that S can be written in the form
(z  z 0 )( z  z 0 )  r , 2 , where r  0.
In such case, what does S represent in the Argand Diagram?
(7 marks)
1
for z  S.
z
Show that ww  cw  cw    0.
(b) (i) Let w 
What does it possibly represent?
(ii) Let u = z + a for z  C, where a  C.
Show that the image u[S] of S under u is possibly a straight line or a circle.
(iii) Let v = bz for z  C, where b  C.
Show that the image v[S] of S under v is possibly a straight line or a circle.
(6 marks)
(c) Define a mapping f : C → C by
f (z)  bz  a for z  C.
Describe briefly the image f[S] of S under f.
(2 marks)
01-02/F.7 Mock Exam/PMI/LCK/P.3 of 5
11.
Let c be a real number greater than 1. {an} is a sequence satisfying the following conditions:
c2  a n
a1 > c and a n 1 
for n  1, 2, 3,...
1 an
(a) Express a2n+1 in terms of a2n-1.
(b) Prove by induction that a2n-1 > c for n = 1, 2, 3, ….
Hence show that a2n+1 < a2n-1 for n = 1, 2, 3, …..
(c) The sequence {bn} is defined by bn = a2n-1 for n = 1, 2, 3, …
Show that {bn} converges and find lim b n .
n 
(3 marks)
(7 marks)
(5 marks)
12. The position vector of a point R(x, y, z) is given by
 


r  x i  y j  zk.
(a) Consider the vector equations of
  
 
a plane  : r  n   and a line  : r  a  tb, t  R .
(i)
 
If b  n  0, prove that  and  intersect at a point with position vector
 
 an  
a      b.
 bn 
(ii) Find the position vector of the foot of perpendicular from a point R0(x0, y0, z0) to the
plane π.
(8 marks)
(b) The image by reflection of a point P with respect to a plane π is defined as the point P
such that π bisects the line segment P P  perpendicular.
Using (a), or otherwise, find the coordinates of the image P  by reflection of the point
P(α, β, γ) with respect to the plane π : ax + by + cz + d = 0.
(7 marks)
13. A polynomial P(x, y) in x, y with degree n is said to be homogeneous iff P(ta, tb) = tn P(a, b)
for any real numbers t, a, b.
(a) Give an example of homogeneous polynomial in x, y with degree 3.
(1 marks)
(b) P(x, y) is a homogeneous polynomial with degree n > 1 satisfying the two conditions:
(*) P(a+b, c) + P(b+c, a) + P(c+a, b) = 0 for any real numbers a, b, c.
(**) P(1, 0) = 1
(i) Show that (x – 2y) is a factor of P(x, y)
(Hint: Consider a = b = c = y)
(ii) Show that (x + y ) is also a factor of P(x, y).
P ( x , y)
(iii) Let P1 ( x, y) 
. Show that P1(x, y) is also a homogeneous polynomial
xy
satisfying the two conditions (*) and (**).
(iv) Using (b)(i),(iii) and induction, show that P(x, y) = (x – 2y)(x + y)n-1 for n  2.
(14 marks)
01-02/F.7 Mock Exam/PMI/LCK/P.4 of 5
14. (a) Let p, q be two positive numbers such that p + q = 1.
Show that ln(px + qy) ≥ pln x + qln y for any x, y > 0.
(4 marks)
(b) Let p1, p2, …, pn be any n positive numbers satisfying p1+ p2+…+ pn = 1.
 n
 n
Using (a) and induction, or otherwise, show that ln   p i x i    p i ln x i
 i 1
 i 1
for any positive numbers x1, x2,…, xn .
(6 marks)
(c) Using (b), deduce that for any positive numbers x1, x2,…, xn,
x 1  x 2  ...  x n n
 x 1 x 2 ...x n .
n
 n  1
Hence show that n!  
 for any n  N.
 2 
n
END OF PAPER
01-02/F.7 Mock Exam/PMI/LCK/P.5 of 5
(5 marks)