FORMULAS FOR REFERENCE PURE MATHEMATICS A-LEVEL PAPER 1

2000-AL P MATH

PAPER 1

HONG KONG EXAMINATIONS AUTHORITY HONG KONG ADVANCED LEVEL EXAMINATION 2000

PURE MATHEMATICS A-LEVEL PAPER 1

8.30 am ? 11.30 am (3 hours) This paper must be answered in English

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 the AL(E) answer book. 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.

4. The four AL(D) answer 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.

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Hong Kong Examinations Authority All Rights Reserved 2000

2000-AL-P MATH 1?1

FORMULAS FOR REFERENCE

sin( A ? B) = sin A cos B ? cos A sin B cos( A ? B) = cos A cos B P sin A sin B tan( A ? B) = tan A ? tan B

1P tan A tan B

sin A + sin B = 2 sin A + B cos A - B

2

2

sin A - sin B = 2 cos A + B sin A - B

2

2

cos A + cos B = 2 cos A + B cos A - B

2

2

cos A - cos B = -2 sin A + B sin A - B

2

2

2 sin A cos B = sin( A + B) + sin( A - B) 2 cos A cos B = cos( A + B) + cos( A - B) 2 sin A sin B = cos( A - B) - cos( A + B)

2000-AL-P MATH 1-2

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SECTION A (40 marks) Answer ALL questions in this section. Write your answers in the AL(E) answer book.

1.

1 0 Let M = b

0 a

where

b2 + ac = 1 . Show by induction that

c - b

M

2n

=

1

n[ (1 + b)

+

a]

0 1

0 0 for all positive integers n .

n[ c + (1 - b)] 0 1

1 0 0 2000 Hence or otherwise, evaluate - 2 3 2 .

1 - 4 - 3

(5 marks)

2. (a) Let p and q be positive numbers. Using the fact that ln x is increasing on (0, ) , show that ( p - q)(ln p - ln q) 0 .

(b) Let a , b and c be positive numbers. Using (a) or otherwise, show that

a ln a + b ln b + c ln c a + b + c (ln a + ln b + ln c) .

3 (6 marks)

3. Let n be a positive integer.

(a) Expand (1+ x)n -1 in ascending powers of x . x

(b) Using (a) or otherwise, show that

C

n 2

+ 2C3n

+

3C

n 4

+

/

+

(n

-

1)C

n n

= (n - 2)2n-1 + 1

.

(5 marks)

2000-AL-P MATH 1-3

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4. Consider the circle zz = (2 + 3i)z + (2 - 3i)z + 12 ( z C ) ..........(*).

Rewrite (*) in the form of z - a = r where a C and r > 0 .

Hence or otherwise, find the shortest distance between the point -4 - 5i and the circle.

(5 marks)

5. Let f(x) = 2x 4 - x 3 + 3x 2 - 2x +1 and g(x) = x 2 - x + 1 .

(a) Show that f(x) and g(x) have no non-constant common factors.

(b) Find a polynomial p(x) of the lowest degree such that f(x) + p(x) is divisible by g(x) . (5 marks)

6. A transformation T in R2 transforms a vector x to another vector

y = Ax + b

where

A

=

cos sin

3 3

- sin 3

cos 3

and b = -13 .

(a)

Find y when x = 02 .

(b) Describe the geometric meaning of the transformation T .

(c) Find a vector c such that y = A(x + c) .

(7 marks)

2000-AL-P MATH 1-4

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7. Suppose the equation x3 + px2 + qx +1 = 0 has three real roots.

(a) If the roots of the equation can be written as a , a and ar , show that r

p=q .

(b) If p = q , show that -1 is a root of the equation and the three roots of

the equation can form a geometric sequence.

(7 marks)

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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.

8. Consider the system of linear equations

x -

y- z=a

(S) : 2x +

y - 2z = b where R .

x + (2 + 3) y + 2 z = c

(a) Show that (S) has a unique solution if and only if -2 .

Solve (S) for = -1 .

(7 marks)

(b) Let = -2 .

(i) Find the conditions on a , b and c so that (S) has infinitely many solutions.

(ii) Solve (S) when a = -1 , b = -2 and c = 3 .

(4 marks)

(c) Consider the system of linear equations

x - y - z + 3 - 5 = 0 (T) : 2x - 2 y - 2z + 2 - 2 = 0 where R .

x - y + 4z - - 1 = 0

Using the results in (b), or otherwise, solve (T) .

(4 marks)

2000-AL-P MATH 1-6

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9.

(a)

Show that

C rn

+ Crn+1

=

C

n+1 r +1

where n ,

r are positive integers and

n r +1 .

(2 marks)

(b) Let A , B be two square matrices of the same order. If AB = BA , show

by induction that for any positive integer n ,

( A + B) n =

n r=0

C

n r

A

n-r

B

r

,

.....(*)

where A0 and B 0 are by definition the identity matrix I .

Would (*) still be valid if AB BA ? Justify your answer. (6 marks)

(c)

Let

A

=

cos sin

- sin cos

where

is real.

(i)

Show that

An

=

cos sin

n n

- sin n cos n

for all positive integers

n .

(ii) Using (*) and the substitution B = A-1 , show that

n r =0

C

n r

cos(n - 2r)

=

2n

cos n

and

n r =0

C

n r

sin(n - 2r)

=0

.

Hence or otherwise, express cos5 in terms of cos 5 , cos 3 and cos .

(7 marks)

2000-AL-P MATH 1-7

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10. A , B , C are the points (1, 1, 0) , (2, -1, 1) , (-1, -1, 1) respectively and O is the origin. Let a = OA , b = OB and c = OC .

(a) Show that a , b and c are linearly independent.

(3 marks)

(b) Find

(i) the area of OAB , and

(ii) the volume of tetrahedron OABC .

(3 marks)

(c) Find the Cartesian equation of the plane 1 containing A , B and C . (3 marks)

(d) Let 2 be the plane r a = 2 where r is any position vector in R3 . P is a point on 2 such that OP ? b = c .

(i) Find the coordinates of P .

(ii) Find the length of the orthogonal projection of OP on the plane 1 in (c). (6 marks)

2000-AL-P MATH 1-8

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11. (a) By considering the derivative of f(x) = (1 + x) -1- x , show that (1+ x) > 1+ x for > 1 , x -1 and x 0 . (4 marks)

(b) Let k and m be positive integers. Show that

m+1

m+1

(i)

1 - 1 -

1 k

m

<

m +1 m

1 k

<

1 +

1 k

m

-1 ,

(ii)

m m +1

k

m+1 m

m+1

1

- (k -1) m < k m

<

m m+

1

(k

+

1)

m+1 m

m+1 -k m

.

(6 marks)

(c) Using (b) or otherwise, show that

1

1

1

1

3

2 < 12 +22 +32 +/ +n2

3

3

<

2 3

1 +

1 n

2

.

n2

1

1

1

1

12 +22 +32 +/ +n 2

Hence or otherwise, find lim

.

n

3

n2

(5 marks)

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12. (a) Resolve x3 - x2 - 3x + 2 into partial fractions. x 2 (x -1)2

(b) Let P(x) = m(x -1 )(x - 2 )(x - 3 )(x - 4 ) where m, 1, 2, 3, 4 R and m 0 . Prove that

(i)

4 1 i=1 x - i

= P(x) P(x)

, and

(3 marks)

(ii)

4

1 = [P(x)]2 - P(x) P(x) .

i=1 (x - i ) 2

[P(x)]2

(3 marks)

(c) Let f(x) = ax 4 - bx 2 + a where ab > 0 and b 2 > 4a 2 .

(i) Show that the four roots of f(x) = 0 are real and none of them is equal to 0 or 1 .

(ii) Denote the roots of f(x) = 0 by 1 , 2 , 3 and 4 . Find

4 i 3 - i 2 - 3 i + 2 in terms of a and b . i=1 i 2 ( i -1) 2 (9 marks)

2000-AL-P MATH 1-10

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