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Hong Kong Advanced Level Examination

Pure Mathematics Past Paper ( 1990 – 2006 ) Sorted by Topic

Mathematical Induction

……………………………………………………

P.2

Polynomials

……………………………………………………

P.10

Binomial Theorem

……………………………………………………

P.28

Inequalities

……………………………………………………

P.33

Complex Number

……………………………………………………

P.55

Determinants and Matrices

……………………………………………………

P.75

Functions

…………………………………………………… P.115

Sequences

…………………………………………………… P.124

Limit

…………………………………………………… P.145

Differentiation

…………………………………………………… P.151

Integration

…………………………………………………… P.191

Coordinate Geometry

…………………………………………………… P.237


HKAL Pure Mathematics Past Paper Topic: Mathematical Induction

Contents: Classification

Chronological Order

Type 1: Verify a Formula 1. 2. 3. 4. 5. 6.

92I05 90I07 93I02 02I01 06I01 97I02 Type 2: Divisibility / Rationality 7. 98I05 8. 04I06 9. 05I01 Type 3: Derive a Formula 10. 96I06 11. 01I02 Type 4: Strong Induction 12. 94I05 13. 95I06 Type 5: Binomial Expansion 14. 91I11 15. 97I12 Type 6: Others 16. 03I04 17. 94I13

P. 2

90I07

Q. 2

91I11 92I05 93I02 94I05 94I13 95I06 96I06 97I02 97I12 98I05 01I02 02I01 03I04 04I06 05I01 06I01

Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q.

14 1 3 12 17 13 10 6 15 7 11 4 16 8 9 5


HKAL Pure Mathematics Past Paper Topic: Mathematical Induction

1.

Type 1: Verify a Formula (92I05) Consider the sequence

un 

in which

u1  0 , un1  2n  un for n = 1, 2, … . Using mathematical induction or otherwise, show that 2un  2n  1  (1)n for n = 1, 2, … .

Hence find lim

n 

un . n (4 marks)

2.

(90I07) A sequence

a0 , a1, a2 ,

of real numbers is defined by

a0  0 , a1  1 and an  an1  an2 for all n  2, 3, . Show that for all non-negative integers n , an 

1 ( n   n ) , 5

where  ,  are roots of x2  x  1  0 with   0 ,   0 . a Also prove that lim n 1   . n  a n (7 marks) 3.

(93I02) Let u1  1 , u2  3 and un  un2  un1 for n  3 . Using mathematical induction, or otherwise, prove that un   n   n for n  1 ,

where  and  are the roots of x2  x  1 = 0 . (5 marks) 4.

(02I01) A sequence {an} is defined by a1 = 1 , a2 = 3 and

an2  2an1  an for n  1, 2, 3,... . Prove by mathematical induction that an 

(1  2)n  (1  2) n for n  1, 2, 3,... . 2

(5 marks) P. 3


HKAL Pure Mathematics Past Paper Topic: Mathematical Induction

5.

(06I01) A sequence {an} is defined by a1 = 1 , a2 = 3 and an + 2 = 3an + 1 + 2an for all n = 1 , 2 , 3 , … . Using mathematical induction,

1 prove that an  17

 3  17 n  3  17 n       for any positive  2   2    

integer n . (6 marks) 6.

(97I02) Let {an} be a sequence of real numbers, where

a0  1 , a1  6 , a2  45 and 1 1 an  an1  an 2  an3  0 for n = 0, 1, 2, … . 3 27 Using mathematical induction, or otherwise, show that an  3n (n2  1) for n = 0, 1, 2, … .

(4 marks) Type 2: Divisibility / Rationality 7.

(98I05) Let  ,  be the roots of x2  14x + 36 = 0 . Show that  n +  n

is divisible by 2n for n = 1, 2, 3, … . (5 marks)

8.

(04I06) Let   R . For each n  N , define xn  sin n   cosn  . (a) Find a function f ( ) , which is independent of n , such that

xn1 x1  xn2  f ( ) xn . Also express f ( ) in terms of x1 . (b) Suppose that x1 is a rational number. Using mathematical induction, prove that xn is a rational number for every n . (7 marks)

P. 4


HKAL Pure Mathematics Past Paper Topic: Mathematical Induction

9.

(05I01) For each positive integer n , define Sn  (1  5)n  (1  5)n . Prove that (a) Sn2  2Sn1  4Sn , (b) S n is divisible by 2n . (6 marks)

Type 3: Derive a Formula 10. (96I06) A sequence

 xn 

is defined by x0  1 , x1  2 and xn 

xn 1  xn 2 2

for n  2 . (a) Write down the values of x2  x1 , x3  x2 and x4  x3 . (b) For n = 1, 2, 3, … , guess an expression for xn  xn1 in terms of n and prove it. Hence find lim xn . n 

(7 marks) 11. (01I02) (a) Show that

n

r

2

r 1

(b) A sequence

an 

1 n(n  1)(2n  1) . 6

is defined as follows:

a1  6 , ak 1  ak  3k 2  9k  6 for k = 1, 2, 3, … .

Find an in terms of n . (5 marks)

P. 5


HKAL Pure Mathematics Past Paper Topic: Mathematical Induction

Type 4: Strong Induction 12. (94I05) Let {an} be a sequence of positive numbers such that  1  an  a1  a2    an     2  for n = 1, 2, 3, … .

2

Prove by induction that an = 2n  1 for n = 1, 2, 3, … . (5 marks) 13. (95I06) Let

an 

be a sequence of non-negative integers such that n

n   ak 2  n  1  (1)n for n = 1, 2, 3, … . k 1

Prove that an  1 for n  1 . (4 marks)

P. 6


HKAL Pure Mathematics Past Paper Topic: Mathematical Induction

Type 5: Binomial Expansion 14. (91I11) (a) For n = 1, 2, … , prove that there exist unique positive integers pn and qn such that

( 3  2)2n  pn  qn 6 ……… (*) and

( 3  2)2n  pn  qn 6 .

(b) Hence deduce that 2 pn  1  ( 3  2)2n  2 pn . [Hint: Use the fact that 0  3  2  1 .] (6 marks) (c) For n = 1, 2, … , show that the following integers are positive multiples of 10 : (i)

25n  2n ,

(ii)

34 n  1 ,

(iii)

2 p2 n  (23n1 )(3n ) where p2n is given by (*) .

(5 marks) (d) By using (a) and (b) or otherwise, find the unit of digit when ( 3  2)100 is expressed in the decimal form.

(4 marks)

P. 7


HKAL Pure Mathematics Past Paper Topic: Mathematical Induction

15. (97I12) (a) Show that for any positive integer n , there exist unique positive integers an and bn such that

(1  3)2n  an  bn 3 and that (i) an 2  3bn 2  22 n , (ii) an and bn are both divisible by 2n . (8 marks) (b) For an and bn as determined in (a), show that

(1  3)2n  an  bn 3 . (2 marks) an  3 . n  b n

(c) Using (b), or otherwise, prove that lim

(3 marks) (d) Using (a) and (b), or otherwise, prove that for any positive integer n , the smallest integer greater than (1  3)2 n is divisible by 2n + 1 . (2 marks) Type 6: Others 16. (03I04) Let {xn} be a sequence of positive real numbers, where x1 = 2 and xn1  xn 2  xn  1 for all n = 1, 2, 3, … . n

Define Sn   i 1

1 for all n = 1, 2, 3, … xi

(a) Using mathematical induction, prove that for any positive integer n , (i)

xi > n ,

(ii)

Sn  1 

1 xn 1  1

.

(b) Using (a), or otherwise, prove that lim Sn exists. n 

(7 marks)

P. 8


HKAL Pure Mathematics Past Paper Topic: Mathematical Induction

17. (94I13) Let Z + be the set of all positive integers and m, n  Z + .

A(m, n)  (1  xm )(1  xm1 )(1  xmn1 ) ,

Let

B(n)  (1  x)(1  x 2 )(1  x n ) . (a) Show that A(m  1, n  1)  A(m, n  1) is divisible by

(1  xn1 )A(m  1, n) . (2 marks) (b) Suppose P(m, n) denote the statement. “ A(m, n) is divisible by B(n) .” (i)

Show that P(1, n) and P(m, 1) are true.

(ii)

Using (a), or otherwise, show that if P(m, n  1) and P(m  1, n) are true, then P(m  1, n  1) is also true.

(iii)

Let k be a fixed positive integer such that P(m, k ) is true for all m  Z . Show by induction that P(m, k + 1) is true for all m  Z (10 marks)

(c) Using (b), or otherwise, show that P(m, n) is true for all

m, n  Z . (3 marks)

P. 9


HKAL Pure Mathematics Past Paper Topic: Polynomials

Contents: 1.

Classification Type 1: Coefficients and Roots 90I03

2. 3. 4. 5. 6. 7. 8. 9.

94I03 95I03 96I07 00I07 03I06 98I12 06I09 94I12 Type 2: Degree 10. 06I03 11. 91I04 12. 01I06 Type 3: H.C.F. 13. 98I07 14. 00I05 15. 02I05 16. 93I12 Type 4: Remainder Theorem / Division Algorithm 17. 99I04 18. 04I04 19. 05I04 20. 95I10 Type 5: Partial Fractions 21. 00I12 Type 6: Repeated Roots 22. 04I09 23. 98I11 Type 7: Solving Polynomial Equations 24. 97I04 25. 26. 27. 28. 29. 30.

96I11 03I11 05I10 02I11 99I11 01I13

P. 10

Chronological Order 90I03 Q. 1 91I04 Q. 11 93I12 Q. 16 94I03 Q. 2 94I12 Q. 9 95I03 Q. 3 95I10 Q. 20 96I07 Q. 4 96I11 Q. 25 97I04 Q. 24 98I07 Q. 13 98I11 Q. 23 98I12 Q. 7 99I04 Q. 17 99I11 Q. 29 00I05 Q. 14 00I07 Q. 5 00I12 Q. 21 01I06 Q. 12 01I13 Q. 30 02I05 02I11 03I06 03I11 04I04 04I09 05I10 06I03 06I09

Q. Q. Q. Q. Q. Q. Q. Q. Q.

15 28 6 26 18 22 19 27 10


HKAL Pure Mathematics Past Paper Topic: Polynomials

1.

Type 1: Coefficients and Roots (90I03) (a) If  ,  and  are the roots of x3  Ax2  Bx  C  0 , express

 2   2   2 and  2  2   2 2   2 2 in terms of A , B and C . (b) Find a cubic equation whose roots are the squares of the roots of x 3  3x  1  0 .

(5 marks) 2.

(94I03) (a) If  ,  and  are the roots of x3 + px + q = 0 , find a cubic equation whose roots are  2 ,  2 and  2 .

x 2 3 (b) Solve the equation 2 x 3  0 . 2 3 x

3.

Hence, or otherwise, solve the equation x3  38x2  361x  900  0 . (6 marks) (95I03) (a) If a1 , a2 , a3 , a4 , p, q,  ,  are real numbers such that

x4  a1 x3  a2 x2  a3 x  a4  ( x 2  px  q)2  ( x   )2 for all x ,

show that

 2 a12  2q  a2   4  1    (a1q  a3 ) . 2  2    q 2  a4  

(b) Find the possible real values of p, q,  ,  such that

x4  4 x3  12 x2  24 x  9  ( x  px  q)2  ( x   )2 for all x . (c) Solve x4  4 x3  12 x2  24 x  9  0 . (7 marks)

P. 11


HKAL Pure Mathematics Past Paper Topic: Polynomials

4.

(96I07) Let the roots of x3  px2  qx  r  0 be  ,  and  . (a) Show that  p  q     2 . (b) If the roots of x3  Px 2  Qx  R  0 are    2 ,    2 and

   2 , express P , Q and R in terms of p , q and r . (7 marks) 5.

(00I07) 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 , r

show that 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) 6.

(03I06) (a) Suppose the cubic equation x3  px2  qx  r  0 , where p , q and r are real numbers, has three real roots. Using relations between coefficients and roots, or otherwise, prove that the three roots form an arithmetic sequence if and only if

p is a root of the equation. 3

(b) Find the two values of p such that the equation

x3  px2  21x  p  0 has three real roots that form an arithmetic sequence. (8 marks)

P. 12


HKAL Pure Mathematics Past Paper Topic: Polynomials

7.

(98I12) (a) Let  ,  and  be positive numbers. Suppose

( x   )( x   )  x2  2 px  q and ( x   )( x   )( x   )  x3  3bx 2  3cx  d for all x . (i)

Show that p2  q .

(ii)

By expressing b , c and d in terms of  , p and q , or otherwise, show that b2  c > 0 and c2  bd . Hence, or otherwise, show that b  c  3 d . (10 marks)

(b) Let A , B and C be the angles of a triangle. Show that

tan

A B B C C A tan  tan tan  tan tan  1 . 2 2 2 2 2 2

Using (a), or otherwise, show that

tan

A B C 3 A B C  tan  tan  3 and tan tan tan  . 2 2 2 9 2 2 2 (5 marks)

8.

(06I09) (a) Let b , c , d  R and  ,  and  be the roots of the equation x3 + bx2 + cx + d = 0 . For every positive integer k , define Sk   k   k   k . (i)

Using relations between coefficients and roots, express S1 , S2 and S3 in terms of b , c and d .

(ii)

Prove that Sk + 3 + bSk + 2 + cSk + 1 + dSk = 0 for any positive integer k .

(iii)

Suppose d = bc . Using the results of (a)(i) and (a)(ii) , prove that S2n + 1 + bS2n = (1)n 2bcn and S2n + bS2n  1 = (1)n 2cn for any positive integer n . (11 marks)

(b) Find three numbers such that their sum is 3 , the sum of their squares is 3203 and the sum of their cubes is 9603 . (4 marks)

P. 13


HKAL Pure Mathematics Past Paper Topic: Polynomials

9.

(94I12) Let p( x)  x4  a1 x3  a2 x2  a3 x  a4 , where a1 , a2 , a3 , a4 R . Suppose z1  cos 1  i sin 1 and z2  cos 2  i sin 2 are two roots of p( x)  0 , where 0  1  2   .

(a) Show that (i)

p( x)  ( x2  2 x cos 1  1)( x 2  2 x cos 2  1) ,

(ii)

  x  cos 1 x  cos  2 p' ( x)  2p( x)  2  2  .  x  2 x cos 1  1 x  2 x cos  2  1 

(5 marks) (b) Suppose p(w) = 0 , by considering p(x)  w(x) , show that

p( x)  x3  ( w  a1 ) x 2  (w2  a1w  a2 ) x  (w3  a1w2  a2 w  a3 ) . xw (3 marks) (c) Let sn  z1n  z1n  z2n  z2n , using (a)(ii) and (b) , show that

p' ( x)  4 x3  (s1  4a1 ) x2  (s2  a1s1  4a2 ) x  (s3  s2a1  s1a2  4a3 ) . [Hint:

2( x  cos  r ) 1 1   , r = 1, 2 . ] x  2 x cos  r  1 x  zr x  zr 2

Hence show that sn  a1sn1    an1s1  nan  0 for n  1, 2, 3, 4 . (7 marks)

P. 14


HKAL Pure Mathematics Past Paper Topic: Polynomials

Type 2: Degree 10. (06I03) Let p(x) be a polynomial of degree 4 with real coefficients satisfying p(0)  0 , p(1) =

1 2 3 4 , p(2) = , p(3) = and p(4) = . 2 3 4 5

(a) Let q(x) = (x + 1)p(x)  x . (i)

Evaluate q(0) , q(1) , q(2) , q(3) and q(4) .

(ii)

Express q(x) as a product of linear polynomials.

(b) Evaluate p(5) . (6 marks) 11. (91I04) Let a1 , a2 ,, an be n distinct non-zero real numbers, where n  2 . (a) Define P( x)  a1

( x  a2 ) ( x  an ) (a1  a2 ) (a1  an )

   ai

( x  a1 ) ( x  ai 1 )( x  ai 1 ) ( x  an ) (ai  a1 ) (ai  ai 1 )(ai  ai 1 ) (ai  an )

   an

( x  a1 ) ( x  an 1 ) . (an  a1 ) (an  an 1 )

(i)

Evaluate P(ai ) for i = 1, 2, … , n .

(ii)

Show that the equation P(x)  x = 0 has n distinct roots.

(iii)

Deduce that P( x)  x  0 for all x  R .

(b) Prove that 1 1   (a1  a2 ) (a1  an ) (ai  a1 ) (ai  ai 1 )(ai  ai 1 ) (ai  an )

 

1 0 . (an  a1 ) (an  an 1 ) (5 marks)

12. (01I06) Let f (x) = ax2 + bx + c where a, b, c are real numbers and a  0 . Show that if f[f ( x)]  [f ( x)]2 for all x , then f (x) = x2 . (4 marks)

P. 15


HKAL Pure Mathematics Past Paper Topic: Polynomials

Type 3: H.C.F 13. (98I07) It is given that f (x) = 2x4 + x3 + 10x2 + 2x + 15 and g(x) = x3 + 2x  3 . Let d(x) be the H.C.F. of f (x) and g(x) . (a) Using Euclidean Algorithm, or otherwise, find d(x) . (b) Find polynomials u(x) and v(x) of degree  1 such that u( x)f ( x)  v( x)g( x)  d( x) for all x . (6 marks) 14. (00I05) Let f (x) = 2x4  x3 + 3x2  2x + 1 and g(x) = x2  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) 15. (02I05) (a) Let f (x) and g(x) be polynomials. Prove that a non-zero polynomial u(x) is a common factor of f (x) and g(x) if and only if u(x) is a common factor of f (x)  g(x) and g(x) . (b) Let f (x) = x4  3x3  6 x2  5x  3 and g( x)  x 4  4 x3  8x 2  7 x  4 . Using (a) or otherwise, find the H.C.F. of f (x) and g(x) . (7 marks)

P. 16


HKAL Pure Mathematics Past Paper Topic: Polynomials

16. (93I12) Let  be the set of all polynomials with real coefficients. Let f , g  \{0} and A = {mf + ng : m , n } . Suppose r  A\{0} has the property that deg r  deg p for all p  A\{0} . (a) Show that r divides every polynomial in A . Deduce that r is a G.C.D. of f and g ( i.e. r divides both f and g , and if h divides both f and g then h divides r ). (6 marks) (b) Let B = {hr : h  } . Show that A = B . (4 marks) (c) If deg r = 0 , i.e. r is a non-zero constant, show that there exist m0 , n0  such that m0f + n0g = 1 , and also A =  . (5 marks) Type 4: Remainder Theorem / Division Algorithm 17. (99I04) Let P(x) be a polynomial. When P(x) is divided by x2  4x  21 , the remainder is 11x  10 . When P(x) is divided by x2  6x  7 . the remainder is 9x + c , where c is a constant. (a) Find a common factor of x2  4x  21 and x2  6x  7 . Hence find c . (b) Find the remainder when P(x) is divided by x2 + 4x + 3 . (6 marks)

P. 17


HKAL Pure Mathematics Past Paper Topic: Polynomials

18. (04I04) Let f ( x)  x3  px 2  qx  r , where p , q and r are non-zero real numbers. (a) If f (x) is divisible by x2 + q , find r in terms of p and q . (b) Suppose that f (x) is divisible by both x  a and x + a , where a is a non-zero real number. (i)

Factorize f (x) as a product of three linear polynomials with real coefficients.

(ii)

If f (x) and f (x + a) have a non-constant common factor, find p in terms of a . (7 marks)

19. (05I04) Let f (x) be a polynomial of degree 4 with real coefficients. When f (x) is divided by x  2 , the remainder is 4 . When f (x) is divided by x + 3 , the remainder is 6 . Let r(x) be the remainder when f (x) is divided by (x  2)(x + 3) . (a) Find r(x) . (b) Let g(x) = f (x)  r(x) . It is known that g(x) is divisible by x2 + 1 and g(1) = 16 . Find g(x) . (7 marks)

P. 18


HKAL Pure Mathematics Past Paper Topic: Polynomials

20. (95I10) Let  ,  and  be real and distinct and

( x   )( x   )( x   )  x3  px 2  qx  r . (a) Show that (i)

1 1 1 3x 2  2 px  q    3 ; x   x   x   x  px 2  qx  r

(ii)

3 2  2 p  q  (   )(   ) . (4 marks)

(b) Let f (x) be a real polynomial. Suppose Ax2 + Bx + C is the remainder when (3x2  2 px  q)f ( x) is divided by x3  px2  qx  r . (i)

f ( ) f (  ) f ( ) Ax 2  Bx  C    Prove that . x   x   x   x3  px 2  qx  r

(ii)

Express A , B and C in terms of  ,  ,  , f ( ) , f (  ) and f ( ) . (11 marks)

P. 19


HKAL Pure Mathematics Past Paper Topic: Polynomials

Type 5: Partial Fractions 21. (00I12) (a) Resolve

x3  x 2  3x  2 into partial fractions. x 2 ( x  1)2 (3 marks)

(b) Let P(x) = m(x  1)(x  2)(x  3)(x  4) where

m, 1 ,  2 , 3 ,   R and m  0 . Prove that (i)

n

i 1

(ii)

1

 x 

 i

P' ( x ) , and P( x)

1 [P' ( x)]2  P( x)P'' ( x)  .  2 [P( x)]2 i 1 ( x   i ) 4

(3 marks) (c) Let f (x) = ax4  bx2 + a where ab > 0 and b2 > 4a2 . (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 β1 , β2 , β3 and β4 . Find

 i 3   i 2  3 i  2 in terms of a and b .   i 2 (  i  1)2 i 1 4

(9 marks) Type 6: Repeated Roots 22. (04I09) (a) Let f (x) be a polynomial with real coefficients. Prove that a real number r is a repeated root of f (x) = 0 if and only if f (r )  f' (r )  0 . (5 marks) (b) Let g( x)  x3  ax 2  bx  c , where a , b and c are real numbers. If a 2  3b , prove that all the roots of g(x) = 0 are distinct. (6 marks) (c) Let k be a real constant. If the equation 12 x3  8x2  x  k  0 has a positive repeated root, find all the roots of the equation. (4 marks)

P. 20


HKAL Pure Mathematics Past Paper Topic: Polynomials

23. (98I11) Consider the equation x3  3px + 2q = 0

……… (*) ,

where p , q are real numbers. (a) (i)

If (*) has a repeated root, show that p3 = q2 .

(ii)

If q 

p3 , show that

p is a repeated root of (*) .

(iii)

If q   p3 , show that (*) has a repeated root. (8 marks)

(b) Consider the equation 2x3 + 3x2 + x + c = 0 where c is a real number.

……… (**) ,

(i)

Transform (**) into the form y3  3py + 2q = 0 by using the substitution x = y  h for some constant h .

(ii)

Find c > 0 such that (**) has a repeated root. Solve (**) for this value of c . (7 marks)

Type 7: Solving Polynomial Equations 24. (97I04) Suppose r  r is a root of the cubic equation x3  ax  b  0 , where a , b , r are rational numbers and r is not a rational number. (a) Show that r 3  3r 2  ar  b  0 and 3r 2  r  a  0 . (b) Using (a), or otherwise, show that (i)

r  r is also a root of the equation, and

(ii)

r

4 8a  9b if a  . 3 2(3a  4) (7 marks)

P. 21


HKAL Pure Mathematics Past Paper Topic: Polynomials

25. (96I11) Suppose the equation x4  3x2  k  0 ……… (*) has two roots  ,  such that  +  = 2 .

(a) Show that    . (5 marks) (b) Show that  2 ,  2 are two distinct roots of the equation

y2  3y  k  0 . Hence find the value of k . (5 marks) (c) Solve (*) and express the roots in the form

a  b where a ,

b are rationals. Hence find the values of  and  . (5 marks)

P. 22


HKAL Pure Mathematics Past Paper Topic: Polynomials

26. (03I11) (a) Consider the equation …………. (*) , x4  ax2  bx  c where a , b and c are real numbers. (i)

Suppose b = 0 . Solve (*) .

(ii)

Suppose b  0 . (1)

Prove that (*) can be written as

( x2  t )2  (a  2t ) x2  bx  (c  t 2 ) , where t is any real number. (2)

Prove that there exists a real number t0 such the equation

(a  2t0 ) x2  bx  (c  t02 )  0 has a repeated root. Hence, deduce that (*) can be written as

( x2  t0 )2  (a  2t0 )( x   )2 for some real number  . (9 marks) (b) Consider the equation …………. (**) . x4  6 x2  12 x  8 Find a real value of t such that the equation

(6  2t ) x2  12 x  (8  t 2 )  0 has a repeated root. Hence solve (**) . (6 marks)

P. 23


HKAL Pure Mathematics Past Paper Topic: Polynomials

27. (05I10) (a) Let f (x) = x4 + 2ax2 + 4bx + c , where a , b , c  R with b  0 . It is known that f ( x)  ( x2  2tx  r )( x 2  2tx  s) , where r , s , t  R . (i)

Prove that t  0 .

(ii)

Express r and s in terms of a , b and t .

(iii)

Prove that 4t 6  4at 4  (a 2  c)t 2  b2  0 . (6 marks)

(b) Consider the equation

y 4  4 y3  2 y 2  52 y  9  0 (i)

Find a constant h such that when y = x + h , (*) can be written as x4  8x2  64 x  48  0

(ii)

………(*) .

………(**) .

Using the results of (a), solve (**) in (b)(i). Hence write down all the roots of (*) . (9 marks)

P. 24


HKAL Pure Mathematics Past Paper Topic: Polynomials

28. (02I11) (a) Let f (x) = x3  3px + 1 , where p  R . (i)

Show that the equation f (x) = 0 has at least one real root.

(ii)

Using differentiation or otherwise, show that if p  0 , then the equation f (x) = 0 has one and only one real root.

(iii)

If p > 0 , find the range of values of p for each of the following cases: (1) (2)

the equation f (x) = 0 has exactly one real root, the equation f (x) = 0 has exactly two distinct real roots, (3) the equation f (x) = 0 has three distinct real roots. (9 marks) (b) Let g(x) = x4 + 4x + a , where a  R . (i)

Prove that the equation g(x) = 0 has at most two real roots.

(ii)

Prove that the equation g(x) = 0 has two distinct real roots if and only if a < 3 . (6 marks)

P. 25


HKAL Pure Mathematics Past Paper Topic: Polynomials

29. (99I11) Let  be a complex cube root of 1 . (a) Prove the identities (i) a2 + b2 + c2  ab  bc  ca = (a + b + c 2)(a + b 2 + c ) , (ii) a3 + b3 + c3  3abc = (a + b + c)(a + b + c 2)(a + b 2 + c) . (4 marks) (b) Consider the equation z3  9z + 12 = 0

……… (*) .

(i)

Find real numbers p and q such that p3 + q3 = 12 and pq = 3 .

(ii)

Using (a) or otherwise, find the roots of (*) in terms of  . (5 marks)

(c) Consider the equation

y3  3 y 2  12 y  10 5  14  0

……… (**) .

Using the substitution y = z  h with a suitable constant h , rewrite (**) as z 3  sz  t  0 where s , t are constants. Hence solve (**) . (6 marks)

P. 26


HKAL Pure Mathematics Past Paper Topic: Polynomials

30. (01I13) (a) Let P(x) = x4 + ax3 + bx2 + cx + d where a, b, c, d  R . (i)

Show that if  is a complex root of P(x) = 0 , then  is also a root of P(x) = 0 .

(ii)

For any   C , show that ( x   )( x   ) is a quadratic polynomial in x with real coefficients. Hence show that P(x) can be factorized as a product of two quadratic polynomials with real coefficients. (6 marks)

(b) Let f (x) = x4 + 8x3 +23x2 + 26x + 7 and g(x) = f (x + k) where k  R and the coefficient of x3 in g(x) is zero. (i)

Find k and the coefficients of g(x) .

(ii)

Suppose g(x) = (x2 + px + q)(x2 + rx + s) where p, q, r , s R . By comparing coefficients or otherwise, show that p6  2 p 4  5 p 2  4  0 . Hence find p , q , r and s .

(iii)

Find all roots of f (x) = 0 . (9 marks)

P. 27


HKAL Pure Mathematics Past Paper Topic: Binomial Theorem

Contents: Classification

Chronological Order

Type 1: Properties of nCr

90I04

Q. 3

1.

03I02

92I04

2.

95I02

94I07

3.

90I04

95I02

4.

94I07

96I02

5.

00I03

99I02

6.

99I02

00I03

7.

06I02

01I05

Type 2: Inequalities

03I02

8.

01I05

05I02

9.

05I02

06I02

Q. Q. Q. Q. Q. Q. Q. Q. Q. Q.

Type 3: Others 10.

92I04

11.

96I02

P. 28

10 4 2 11 6 5 8 1 9 7


HKAL Pure Mathematics Past Paper Topic: Binomial Theorem

Type 1: Properties of nCr (03I02) n For any positive integer n , let Ck be the coefficient of xk in the

expansion of (1  x)

n

. Evaluate

n n n n (a) C1  C2  C3    Cn ,

n n n n (b) C1  2C2  3C3    nCn ,

n 2 n 2 n 2 n (c) C1  2 C2  3 C3    n Cn .

(6 marks) 1.

(95I02) Let n be an integer and n > 1 . By considering the binomial expansion of (1  x)

n

, or otherwise,

n n n n n 1 (a) show that C1  2C2  3C3    nCn  2 n ;

(b) evaluate

1 2 3 (1) n1 n     . (n  1)! 2!(n  2)! 3!(n  3)! n! (5 marks)

2.

(90I04) Let k and n be non-negative integers. Prove that

(a) Ckn  (b)

n 1

 (1) C k

k 0

(c)

k  1 n 1 Ck 1 , where 0  k  n ; n 1 n 1 k

0 ;

(1) k n 1 Ck  .  n 1 k 0 k  1 n

P. 29


HKAL Pure Mathematics Past Paper Topic: Binomial Theorem

3.

(94I07) (a) Let m and n be positive integers. Using the identity

(1  x)n  (1  x)n1    (1  x)n m 

(1  x)n m1  (1  x)n , x

n n 1 nm n  m1 where x  0 , show that Cn  Cn    Cn  Cn1 .

(b) Using (a), or otherwise, show that m 4

 r (r  1)(r  2)(r  3)  24 C

m 5 5

r 5

Hence evaluate

 1 .

k

 r (r  1)(r  2)(r  3)

for k  4 .

r 0

(7 marks) 4.

(00I03) 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

C2n  2C3n  3C4n    (n  1)Cnn  (n  2)2n1  1 . (5 marks) 5.

(99I02) n For any positive integer n , let Ck be the coefficient of xk in the

expansion of (1  x)

n

.

n n n 1 (a) Show that Ck  Ck 1  Ck 1 .

(b) By induction on m or otherwise, show that

Cnn  Cnn1  Cnn2    Cnnm  Cnn1m1 for any m  0 . (5 marks)

P. 30


HKAL Pure Mathematics Past Paper Topic: Binomial Theorem

6.

(06I02) n Let Ck be the coefficient of xk in the expansion of (1 + x)n .

(a) Using the identity (1  x2)n = (1 + x)n(1  x)n , prove that the coefficient of xn in the expansion of (1  x2)n is

n

 (1) (C k

k 0

) .

n 2 k

(b) Evaluate (i)

2005

 (1) (C k

k 0

(ii)

2005 2 k

)

2006

 (1) (C k

k 0

2006 2 k

)

, . (7 marks)

Type 2: Inequalities 7.

(01I05) Let the kth term in the binomial expansion of (1 + x)2n in ascending 2 n k 1 powers of x be denoted by Tk , i.e. Tk  Ck 1 x .

(a) If x 

1 , find the range of values of k such that Tk 1  Tk . 3

(b) Find the greatest term in the expansion if x  8.

1 and n = 15 . 3

(05I02) For any two positive integers k and n , let Tr be the rth term in the kn kn r 1 expansion of (1  x) in ascending powers of x , i.e. Tr  Cr 1 x .

(a) Suppose x 

2 . Find, in terms of k and n , the range of values k

of r such that Tr 1  Tr .

(b) Suppose x 

2 . Using the result of (a), find the greatest term in 3

the expansion of (1  x)

51

. (6 marks)

P. 31


HKAL Pure Mathematics Past Paper Topic: Binomial Theorem

Type 3: Others 9.

(92I04) By considering (1  i)

2n

, or otherwise, evaluate

n

 (1) C r

r 0

n 1

 (1) C r

r 0

2n 2 r 1

2n 2r

and

,where n is a positive integer. (5 marks)

10.

(96I02) (a) Let k and n be positive integers. If k > 1 , show that when (1+ k)n is divided by k , the remainder is 1 . (b) If today is Tuesday, what day of the week is 896 days after? (4 marks)

P. 32


HKAL Pure Mathematics Past Paper Topic: Inequalities

Contents: Classification

Chronological Order

Type 1: Elementary Methods

90I06

Q. 13

1.

91I06

90I12

2.

00I02

90II01

3.

04I05

91I06

4.

02I07

91I08

Type 2: Differentiation

92I07

5.

90II01

93I01

6.

97II02

93I07

7.

01I03

94I14

8.

98I06

95I13

9.

03II02

96I04

10.

00II02

96I13

11.

96I04

97I05

12.

90I12

97I09

Type 3: Absolute Values

97I13

13.

90I06

97II02

14.

93I07

98I06

Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q.

12 5 1 26 17 22 14 33 28 11 29 16 18 32 6 8

15.

03I01

99I12

Type 4: A.M.  G.M.

00I02

16.

97I05

00I11

17.

92I07

00II02

18.

97I09

01I03

19.

01I14

01I14

20.

03I10

02I07

21.

05I11

02I10

Type 5: Cauchy-Schwarz’s

03I01

Inequality

03I10

93I01

03II02

23.

06I06

04I05

24.

99I12

04I10

25.

02I10

05I11

26.

91I08

06I06

Type 6: Method of Difference

06I10

Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q.

24 2 27 10 7 19 4 25 15 20 9 3 31 21 23 30

22.

27.

00I11 Type 7: Others

28.

95I13

P. 33


HKAL Pure Mathematics Past Paper Topic: Inequalities

29.

96I13

30.

06I10

31.

04I10

32.

97I13

33.

94II14

P. 34


HKAL Pure Mathematics Past Paper Topic: Inequalities

Type 1: Elementary Methods 1.

(91I06) (a) Let a , b and c be real numbers. (i)

Show that a2  b2  c2  ab  bc  ca .

(ii)

Hence deduce that if a  b  c  0 then

a3  b3  c3  3abc . (b) Let | x |  ln 2 . (i)

Show that 1

1

1

(e x ) 3  (2  e x ) 3  (e x  e x  1) 3  0 . (ii)

Using (a) or otherwise, show that

e x (2  e x )(e x  e x  1)  1 . (7 marks) 2.

(00I02) (a) Le 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.

(04I05) Let n be a positive integer. (a) Let a > 0 . (i)

If k is a positive integer, prove that a + ak  1 + ak + 1 .

(ii)

Prove that (1 + a)n  2n  1(1 + an) .

(b) Let x and y be positive real numbers. Using (a)(ii), or otherwise,

xn  y n  x y prove that  .   2  2  n

(7 marks)

P. 35


HKAL Pure Mathematics Past Paper Topic: Inequalities

4.

(02I07) (a) Let m and k be positive integers and k  m . Prove that

m(m  1) (m  k  1) (m  1)m (m  k  2) for k > 1 .  mk (m  1)k

Prove that the above inequality does not hold when k = 1 . (b) Let m be a positive integer. Using (a) or otherwise, prove that (1 

1 m 1 m1 )  (1  ) . m m 1 (6 marks)

Type 2: Differentiation 5.

(90II01) By differentiating the function

ln x , or otherwise, prove that if x

e  a  b , then ab  ba . (5 marks) 6.

(97II02) By considering the function f ( x)  xe

x

, or otherwise, show that if

1  a  b , then aeb  bea . (4 marks) 7.

(01I03) (a) Let 0 <  < 1 . Show that x  (1   )   x for all x > 0 . 

(b) Let a , b , p and q be positive numbers with 1 p

1 q

that a b 

1 1   1 . Prove p q

a b .  p q (5 marks)

P. 36


HKAL Pure Mathematics Past Paper Topic: Inequalities

8.

(98I06) Suppose 0 < p < 1 . (a) Let f (x) = x p  px + p  1 for x > 0 . Find the absolute maximum value of f (x) . (b) Show that for all positive numbers a and b ,

a pb1 p  pa  (1  p)b . (6 marks) 9.

(03II02) 1

(a) Let f ( x)  x x for all x  1 . Find the greatest value of f (x) . (b) Using (a) or otherwise, find a positive integer m , such that 1 m

m n

1 n

for all positive integers n . (7 marks)

10.

(00II02) Show that for x > 0 , x  1 + ln x . Find the necessary and sufficient condition for the equality to hold. (5 marks)

11.

(96I04) (a) Show that f ( x) 

x is an increasing function (1, ) . 1 x

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

rs 1 r  s

r 1 r

s 1 s

for any

real numbers r and s . (5 marks)

P. 37


HKAL Pure Mathematics Past Paper Topic: Inequalities

12.

(90I12) (a) Let p > 1 and f ( x)  x  px for all x > 0 . p

(i)

Find the absolute minimum of f (x) on the interval (0, ) .

(ii)

Deduce that x  1  p( x  1) for all x > 0 . p

(4 marks) (b) (i)

Let  ,  ,  and  be positive numbers such that

1

1

 1 and     1 .

By taking x =  and  respectively, prove that, for

p 1 ,

 p1 p   p 1 p  1 , where the equality holds if and only if     1 . (ii)

Deduce that, if a , b , c and d are positive and p > 1 , then

 ab     a 

p 1

 a b  c    b  p

p 1

d p  (c  d ) p . (4 marks)

(c) Suppose

ai i 1, 2,

and

bi i 1, 2,

are two sequences of positive

numbers and p > 1 . 1

1

 n p  n p By considering a    a j p  and   b j p  ,  j 1   j 1  1

1

1

 n p  n p  n p p prove that   ai p     bi p      ai  bi   ,  i 1   i 1   i 1  where the equality holds if and only if

a a1 a2 a    n  . b1 b2 bn b (7 marks)

P. 38


HKAL Pure Mathematics Past Paper Topic: Inequalities

Type 3: Absolute Values 13.

(90I06) Solve the inequality x  1  x  2  2 . (5 marks)

14.

(93I07) Find all ( x, y) in R2 satisfying the following two conditions:

 2 x  1  y  1   y  x  3 . (6 marks) 15.

(03I01) (a) Solve the inequality | x | 6  3 , where x is a real number. (b) Using the result of (a), or otherwise, solve the inequality

|1  2 y | 6  3 , where y is a real number. (6 marks) Type 4: A.M.  G.M 16.

(97I05) Let a , b and c be positive numbers. 3 (a) Using A.M.  G.M. , show that (1  a)(1  b)(1  c)  (1  3 abc ) .

Under what conditions on a , b and c will the equality hold?

 a 0 0   (b) Let P be a 3  3 real matrix such that Q PQ   0 b 0  for 0 0 c   1

some 3  3 real matrix Q . 1 3

1 3

Using (a), show that [det( I  P)]  1  [det P]

. (7 marks)

P. 39


HKAL Pure Mathematics Past Paper Topic: Inequalities

17.

(92I07)

Crn 1 (a) Prove that r  , where n , r are positive integers and n  r . n r! (b) If a1 , a2 , , an are positive real numbers and s  a1  a2    an , using “A.M.  G.M. ” and (a), or otherwise, prove that

(1  a1 )(1  a2 ) (1  an )  1  s  (c) Let cn 

n

1

 (1  2 k 1

sequence

cn 

k

s 2 s3 sn .   2! 3! n!

) . Using (b) or otherwise, show that the

converges. (8 marks)

18.

(97I09) Let x1 , x2 , y1 , y2 , z1 and z2 be positive numbers such that

x1 y1  z12  0 and x2 y2  z22  0 . (a) Let D1 = x1 y1  z12 and D2 = x2 y2  z22 . Using A.M.  G.M. , show that (i)

y2 y D1  1 D2  2 D1D2 , y1 y2

(ii)

y2 y D1  1 D2  x1 y2  x2 y1  2 z1 z2 , y1 y2

(iii)

( x1  x2 )( y1  y2 )  ( z1  z2 )2  4 D1D2 . (9 marks)

(b) Show that

8 1 1   , and if 2 2 ( x1  x2 )( y1  y2 )  ( z1  z2 ) x1 y1  z1 x2 y2  z2 2

the equality holds, then x1 = x2 , y1 = y2 and z1 = z2 . (6 marks)

P. 40


HKAL Pure Mathematics Past Paper Topic: Inequalities

19.

(01I14) (a) If a , b are two real numbers such that a  1  b , show that

a  b  ab  1 and the equality holds if and only if a = 1 and b = 1 . (3 marks) (b) Show by induction that if x1 , x2 , ... , xn are n (n  2) positive real numbers such that x1 x2  xn  1 , then x1  x2    xn  n and the equality holds if and only if x1  x2    xn  1 . (6 marks) (c) Let a1 , a2 , ... , an be n (n  2) positive real numbers. Using (b) or otherwise, show that

a1  a2    an n  a1a2  an and the equality n

holds if and only if a1  a2    an . (3 marks) (d) For u  0 and n = 2, 3, 4, … , using the identity

(1  u)n  1  u[(1  u)n1  (1  u)n2    1] or otherwise, show that (1  u )n  1  nu (1  u )

n 1 2

and the equality holds if and only if u = 0 . (3 marks)

P. 41


HKAL Pure Mathematics Past Paper Topic: Inequalities

20.

(03I10) (a) Let a and b be non-negative real numbers. Prove that

(a  b)n  a n  na n1b

for all n = 2, 3, 4, … .

Write down a necessary and sufficient condition for the equality to hold. (3 marks) (b) Let {a1 , a2 , a3 ,...} be a sequence of positive real numbers satisfying

a1  a2  a3   . For any positive integer n , define

An 

1 a1  a2  a3    an and Gn  (a1a2 a3  an ) n . n

(i)

Prove that Ak 1  Ak for all k = 1, 2, 3, … .

(ii)

k 1 k Using (a), prove that Ak 1  Ak ak 1 for all k = 1, 2, 3, … .

Hence prove that An  Gn and

An  Gn if and only if a1  a2  a3    an for all n = 1, 2, 3, … . (8 marks) (c) Let n be a positive integer. Using (b), prove that n

n  2  n  1  n 1  .  n 1  n  1   Hence deduce that 1    n 1 

n 1

n

 1  1   .  n (4 marks)

P. 42


HKAL Pure Mathematics Past Paper Topic: Inequalities

21.

(05I11) (a) For any positive integer n , prove that

t n  1  n(t  1) for all t > 0 . (3 marks) (b) (i)

Let a , b and c be positive real numbers. 3

abc in (a), prove that ab abc 3 2 a b  abc  (  ab ) . 3 3 2

By putting n = 3 and t 

(ii)

Let y1 , y2 ,..., yk 1 be positive real numbers, where k is a positive integer. Using (a), prove that

yk 1  (k  1)Gk 1  k Gk , where Gk  (iii)

k

y1 y2  yk and Gk 1  k 1 y1 y2  yk 1 .

Using mathematical induction and (b)(ii), prove that

x1  x2    xn n  x1 x2  xn n for any n positive real numbers x1 , x2 ,..., xn . (8 marks) (c) Let n be a positive integer. Using (b)(iii), prove that

nn  (1)(3)(5)(2n  1) . Hence prove that (n  n)  (2n)! 2

n

(4 marks)

P. 43


HKAL Pure Mathematics Past Paper Topic: Inequalities

Type 5: Cauchy-Schwarz’s Inequality 22.

(93I01) Prove the following Schwarz’s inequality: 2

 n   n 2  n 2  a b  i i     ai   bi  ,   i 1   i 1  i 1  where ai , bi  R and n  N . Hence, or otherwise, prove that

1 n 1 n 2 ai    ai . n i 1 n i 1 (5 marks)

23.

(06I06) Let a , b and c be positive real number such that a2 + b2 + c2 = 3 . (a) Using Cauchy-Schwarz’s inequality, prove that (i)

a+b+c3 ,

(ii)

a3 + b3 + c3  3 .

(b) For every n = 2 , 3 , 4 … , let P(n) be the statement an + bn + cn  3 . Prove that for any integer k  2 , (i)

if P(k) is true, then P(2k) is true;

(ii)

if P(k) is true, then P(2k  1) is true. (7 marks)

P. 44


HKAL Pure Mathematics Past Paper Topic: Inequalities

24.

(99I12) (a) Let ai , bi be real numbers where i = 1, 2, … , n . By considering the function f (t ) 

n

 (a t  b ) i 1

i

2

i

, or otherwise, prove Schwarz’s

2

 n   n  n  inequality   ai bi     ai 2   bi 2  .  i 1   i 1  i 1  (4 marks) (b) Let x1 , x2 ,, xn be positive real numbers. Show that 2

 n k 1   n k  2  n k    xi     xi   xi   i 1   i 1  i 1  for any non-negative integer k . (3 marks) (c) Let x1 , x2 ,, xn be positive real numbers such that Prove by induction on p that

n

n

i 1

i 1

n

x i 1

i

1 .

 xi p  n xi p1 for any

non-negative integer p . (5 marks) (d) Let y1 , y2 ,, yn be positive real numbers. Show that n  n  n p  p 1   yi   yi   n yi i 1  i 1  i 1 

for any non-negative integer p . (3 marks)

P. 45


HKAL Pure Mathematics Past Paper Topic: Inequalities

25.

(02I10) (a) Let a1 , a2 ,..., an be real numbers and b1 , b2 ,...., bn be non-zero real numbers. By considering inequality

n i 1

i1 aibi n

holds if and only if

(ai x  bi )2 , or otherwise, prove Schwarz’s

   2

n

a2

i 1 i

  b  , and that the equality n

2

i 1 i

a a1 a2    n . b1 b2 bn (6 marks)

(b) (i)

  n xi Prove that  i 1  n 

2

    

n

x2

i 1 i

, where x1 , x2 ,..., xn are real

n

numbers. (ii)

Prove that

i1 i xi n

       2

n

i 1

i

n i 1

i xi 2

, where

x1 , x2 ,..., xn are real numbers and 1 , 2 ,..., n are positive numbers. Find a necessary and sufficient condition for the equality to hold. (iii)

Using (b)(ii) or otherwise, prove that 2

yn  yn 2 y12 y2 2  y1 y2  t  t 2    t n   t  t 2    t n , where y1 , y2 ,..., yn   are real numbers, not all zero, and t  2 . (9 marks)

P. 46


HKAL Pure Mathematics Past Paper Topic: Inequalities

26.

(91I08) (a) Let ak , bk ( k = 1, 2, … , n ) be non-zero real numbers. (i)

Prove the Schwarz’s inequality 2

 n 2  n 2   n   ak   bk    ak bk  .  k 1   k 1   k 1  (ii)

If p 

bk  q for k = 1, 2, … , n , prove that ak

pqak 2  ( p  q)ak bk  bk 2  0 for k = 1, 2, … , n . Deduce that ( p  q)

n

k 1

(iii)

n

 a b  b k k

k 1

k

n

2

 pq  ak 2 . k 1

If

0  m  ak  M

and

0  m  bk  M for k = 1, 2, … , n ,

prove, by using (ii) or otherwise, that 2

 n 2  n 2  1 M m 2  n   ak   bk   (  )  ak bk  .  k 1   k 1  4 m M  k 1  (10 marks) (b) Using (a) or otherwise, show that

1 1  n 1  169 1 n (n  )2   (1  k )2   (1  k 1 )2   (n  )2 . 9 3   k 1 3 3  k 1  144 (5 marks)

P. 47


HKAL Pure Mathematics Past Paper Topic: Inequalities

Type 6: Method of Difference 27.

(00I11) (a) By considering the derivative of f (x) = (1 + x)α  1  αx , show that (1  x)  1   x for α > 1 , x  1 and x  0 . 

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

 1 1  1    k

(i)

m 1 m

m 1 1   1      1   m k  k

m 1 m

1 ,

m 1 1 m 1 m 1   m  mm1 m  m m m m k  ( k  1)  k  ( k  1)  k     . m 1  m 1   

(ii)

(6 marks) (c) Using (b) or otherwise, show that 1

1

1

1

3

2 12  2 2  3 2    n 2 2  1  2   1   . 3 3 3 n n2 1

Hence or otherwise, find lim

1

1

1

12  2 2  3 2    n 2

n 

n

3 2

. (5 marks)

P. 48


HKAL Pure Mathematics Past Paper Topic: Inequalities

Type 7: Others 28.

(95I13) Let a and b be positive numbers. (a) Prove that

a abb  abba where, if the equality holds, then a = b . (4 marks) (b) Using (a), or otherwise, prove that

 ab     2 

a b

 abba

where, if the equality holds, then a = b . (3 marks) (c) Show that x x (1  x)1 x 

1 for 0 < x < 1 2

where, if the equality holds, then x 

 ab  Deduce that a b     2 

1 . 2

a b

a b

where, if the equality holds, then a = b . (8 marks)

P. 49


HKAL Pure Mathematics Past Paper Topic: Inequalities

29.

(96I13)

a1m  a2 m  a  a2  (a) Prove that  1   2  2  m

where a1 , a2 are positive and m is a positive integer. (4 marks)

a1m  a2 m  a3m  a4 m  a1  a2  a3  a4  (b) Prove that    4 4   m

where a1 , a2 , a3 , a4 are positive and m is a positive integer. (3 marks) (c) For n = 1, 2, 3, … , let P(n) be the statement

a1m  a2 m    an m  a1  a2    an     n n   m

where a1 , a2 , … , an are positive and m is a positive integer. (i)

Prove that for h = 0, 1, 2, … , if P(2h) is true, then P(2h + 1) is true.

(ii)

Prove that for k = 1, 2, 3, … . if P(k + 1) is true, then P(k) is true.

(iii)

Hence prove that P(n) is true for all positive integers n . (8 marks)

P. 50


HKAL Pure Mathematics Past Paper Topic: Inequalities

30.

(06I10) (a) By differentiating f (x) = x ln x  x , prove that x ln x  x + 1  0 for all x  0 . (4 marks) (b) Let a be a positive real number. Define g(x) =

ax 1 for all x > x

0 . Prove that g is increasing. (3 marks) (c) Let p and q be real numbers such that p > q > 0 . (i)

Suppose that a1 , a2 ,…, an are positive real numbers satisfying

n

a k 1

k

q

=n .

Using (b), prove that

n

a k 1

(ii)

k

p

n .

Suppose that b1 , b2 ,…, bn are positive real numbers. 1

1

1 n p  1 n q Using (c)(i), prove that   bk p     bk q  .  n k 1   n k 1  p

q

 1 n 1p   1 n 1q  Hence prove that   bk     bk  .  n k 1   n k 1      (8 marks)

P. 51


HKAL Pure Mathematics Past Paper Topic: Inequalities

31.

(04I10) Let n be a positive integer. (a) Suppose 0 < p < 1 . (i)

By considering the function f (x) = x p  px on (0, ) , or otherwise, prove that x  px  1  p for all x > 0 . p

(ii)

p 1 p

Using (a)(i), or otherwise, prove that a b

 pa  (1  p)b

for all a , b > 0 . (iii)

Let a1 , a2 , a3 ,..., an and b1 , b2 , b3 ,..., bn be positive real numbers. Using (a)(ii), or otherwise, prove that p

n

a i 1

i

p

1 p i

b

1 p

 n   n     ai    bi   i 1   i 1 

. (9 marks)

(b) Suppose 0 < s < 2 . Let x1 , x2 , x3 ,..., xn and y1 , y2 , y3 ,..., yn be positive real numbers. Prove that 1

1

(i)

 n 2  n 2 xi yi    xi s yi 2 s    xi 2 s yi s  ,  i 1  i 1   i 1 

(ii)

 n s 2 s  n 2 s s   n 2  n 2    xi yi   xi yi     xi   yi  .  i 1  i 1   i 1  i 1 

n

(6 marks)

P. 52


HKAL Pure Mathematics Past Paper Topic: Inequalities

32.

(97I13)

1 n

n Let (1  ) 

n

T r 0

, where Tn,r is the (r + 1)-th term in the binomial

n,r

1 n

expansion of (1  ) n in ascending powers of

1 . n

(a) Show that Tn,0 = Tn,1 = 1 and

Tn,r 

1 1 2 r 1 (1  )(1  ) (1  ) for r  2 . r! n n n (2 marks)

(b) For any fixed r , show that the sequences Tr ,r , Tr 1,r , Tr  2,r , … is bounded above and monotonic increasing. Find lim Tn ,r in terms of r . n 

(4 marks) (c) Show that (i)

Tn ,r 1 Tn ,r

 r for r  1 ,

1

(ii)

Tn,k 

(iii)

Tn,r 1  (r  1) Tn,k for r  1 .

r

k  r 1

Tn,r 1 for n  k  r  1 , n

k r

(7 marks) (d) Show that

n

T k r

n,k

1 for r  2 . (r  1)(r  1)! (2 marks)

P. 53


HKAL Pure Mathematics Past Paper Topic: Inequalities

33.

(94II14) (a) f (x) is a continuously differentiable and strictly increasing function on [0, c] such that f (0) = 0 . Let b  [ 0, f (c) ] . Define g(t )  tb  (i)

t 0

f ( x) dx , t [0, c] .

Determine the interval on which g(t) is strictly increasing and the interval on which g(t) is strictly decreasing. Hence 1

show that g(t )  g(f (b)) for t [0, c] . (ii)

Using the substitution y = f (x) and integration by parts, show that

(iii)

b 0

f 1 ( y) dy  g(f 1 (b)) .

If a  [0, c] , prove the inequality

a 0

b

f ( x) dx   f 1 ( x) dx  ab . 0

(10 marks) (b) If a, b, p, q are positive numbers and

1 1   1 , prove that p q

1 p 1 q a  b  ab . p q (5 marks)

P. 54


HKAL Pure Mathematics Past Paper Topic: Complex Number

Contents: Classification

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

21.

Chronological Order

Type 1: Algebraic Problems

90I05

Q.

9

Set 1: Properties 91I05

90I10

Q.

18

91I05

Q.

1

93I04 94I06 95I07 96I03 97I01 92I08 06I11 Set 2: Identities \ 90I05 02I06 98I13 04I12 Set 3: nth of Unity 05I06

91I09 92I08 92I11 93I04 93I11 93I13 94I06 94I11 95I07 95I11 96I03 96I12 97I01 98I04

Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q.

19 7 30 2 22 17 3 20 4 27 5 28 6 23

98I13 99I05 99I13 00I04 00I13 01I08 01I11 02I02 02I06

Q. Q. Q. Q. Q. Q. Q. Q.

11 21 29 24 14 25 16 26

Q.

10

03I12

Q.

15

04I12

Q.

12

05I06 06I11

Q. Q.

13 8

00I13 03I12 01I11 93I13 Set4: Mapping 90I10 91I09 94I11 Type 2: Geometric Problems Set 1: Triangles 99I05

22. 93I11 Set 2: Straight Lines / 23. Circles 98I04 24. 00I04 25. 01I08 26. 02I02 27. 95I11

P. 55


HKAL Pure Mathematics Past Paper Topic: Complex Number

28. 96I12 29. 99I13 Set 3: Others 30. 92I11

P. 56


HKAL Pure Mathematics Past Paper Topic: Complex Number

1.

Type 1: Algebraic Problems Set 1: Properties (91I05) Let u , v be non-zero complex numbers. (a) Show that

uv  uv  0 if and only if

u  ik for some k  R . v

(b) If uv  uv  0 , what is the relationship between arg u and arg v ? (6 marks) 2.

(93I04) (a) If 1  z  2  z , find Re z . (b) Find all z  C such that 1  2 and | z |  z  z  i ( z  z )  2  1 z  2  z . 

(5 marks) 3.

(94I06) Let Arg z denote the principal value of the argument of the complex number z (   Arg z   ) . (a) If z  0 and z  z  0 , show that Arg z   (b) If z1 , z2  0 and z1  z2  z1  z2 , show that hence find all possible values of Arg

 2

.

z1 z1   0 and z2 z2

z1 . z2 (5 marks)

4.

(95I07) Let a  C and a  0 .

z 1 (a) Show that if z  z  a , then Re    . a 2 (b) If z  z  a  a , express z in terms of a . (6 marks)

P. 57


HKAL Pure Mathematics Past Paper Topic: Complex Number

5.

(96I03) Consider the equation z 3  az 2  az  1  0 ……… (*) where a is real.

(a) Find a real root of (*) . (b) Find the range of values of a such that (*) has non-real roots.

6.

(c) Show that all the non-real roots of (*) lies on the unit circle in the complex plane. (5 marks) (97I01) Let z1 , z2 , … , zn be arbitrary complex numbers. (a) Prove that z1 z1  z2 z2  z1 z2  z1 z2 . (b) Using (a), or otherwise, show that z1  z2    zn  Re( z1 z2  z2 z3    zn1 zn  zn z1 ) . 2

2

2

(4 marks) 7.

(92I08) Let u , v  C . (a) Show that |u| + |v|  |u + v| . (3 marks) (b) Suppose uv R . Prove that (i) there exist real numbers  and  , not both zero, such that  u   v  0 . (ii)

 u  v u  v   u  v

if uv  0 , if uv  0 . (6 marks)

(c) Suppose uv R . Given z  C , show that there exist unique  ,   R such that z   u   v . (6 marks)

P. 58


HKAL Pure Mathematics Past Paper Topic: Complex Number

8.

(06I11) Let 0 <  <  . (a) Solve the equation z2 = cos  + i sin  . (2 marks) (b) Let u1 and u2 be the roots of the equation (u + 1)2 = cos

 + i sin  , where Im(u1) < 0 . (i)

Find u1 and u2 .

(ii)

Prove that

u2   i tan . u1 4 n

u  Hence, find all the integers n for which  2  is a real  u1  number.

Prove that u112 = 212cos2 (cos 3 + i sin 3) . 4 Hence, find all the values of  for which u112  u212 is

(iii)

a real number. (13 marks) 9.

Set 2: Identities \ (90I05) (a) Show that cos5  16cos5   20cos3   5cos . (b) Using (a), or otherwise, solve 16cos4   20cos2   5  0 for values of  between 0 and 2 . Hence find the value of

cos 2

 10

cos 2

3 . 10 (7 marks)

P. 59


HKAL Pure Mathematics Past Paper Topic: Complex Number

10. (02I06) For k = 1, 2, 3 , let zk  cos k  i sin k be complex numbers, where 1  2  3  2 . (a) Evaluate z1 z2 z3 .

1 1 1 1 (b) Prove that cos  k  ( zk  ) and cos 2 k  ( zk 2  2 ) . 2 zk 2 zk Hence or otherwise, prove that

cos 21  cos 22  cos 23  4cos 1 cos 2 cos 3 1 . (6 marks) 11. (98I13) Let r and  be real numbers. (a) By considering z  r (cos  i sin  ) , or otherwise, simplify r  cos   i sin  . 1  r cos   i r sin  (4 marks) (b) For any positive integer n , show that n

 r  sin   i cos    n   n   n   i sin   n  .    cos   2   2   1  r sin   i r cos   (3 marks) (c) Find r and  , with r  0 , such that 3

 r  sin   i cos   3 i . For such r and  , sketch the    2  1  r sin   i r cos   points representing z  r (cos  i sin  ) on an Argand diagram. (6 marks) (d) Determine with reasons whether there exist r and  , with 3

 r  sin   i cos   r  0 , such that    3 i .  1  r sin   i r cos   (2 marks)

P. 60


HKAL Pure Mathematics Past Paper Topic: Complex Number

12. (04I12) Let n be a positive integer. (a) Assume that  is not an integral multiple of  . (i)

Prove that

n (cos   i sin  )  1 2  cos (n  1)  i sin (n  1)  .     cos   i sin   1 2 2  sin  2 sin

n

(ii)

Using the identity

 zn 1  z  z   for z  1 , or z  1 k 1   n

k

otherwise, prove that

n

 cos k  k 1

n

 sin k  k 1

(iii)

sin

sin

n (n  1) cos 2 2 and  sin 2

n (n  1) sin 2 2 .  sin 2

Using (a)(ii), or otherwise, prove that n  sin n cos(n  1) cos 2 (k )   .  2 2sin  k 1 (9 marks)

(b) For n > 1 , evaluate (i)

n

 sin k 1

(ii)

2

 k    ,  n 

k k    cos   sin  n n  k 1  n

2

. (6 marks)

P. 61


HKAL Pure Mathematics Past Paper Topic: Complex Number

Set 3: nth of Unity 13. (05I06) Let  be a real number. (a) Solve the quadratic equation z2 + 2z cos 6 + 1 = 0 . (b) Using the result of (a), express x6  2 x3 cos 6  1 as a product of quadratic polynomials with real coefficients. (7 marks) 14. (00I13) Let n = 2, 3, 4, … . (a) Evaluate lim x 1

x2n  1 . x2 1

(2 marks) (b) Find all the complex roots of x2n  1 = 0 . Hence or otherwise, show that x2n  1 can be factorized as ( x 2  1)( x 2  2 x cos

 n

 1)( x 2  2 x cos

2 n

 1)  ( x 2  2 x cos

(n  1) n

 1) .

(6 marks) (c) Using (b) or otherwise, show that lim x 1

x2n  1  2 (n  1) .  22 n2 sin 2 sin 2 sin 2 2 x 1 2n 2n 2n

(4 marks) (d) Using (a) and (c), or otherwise, show that  1     2   (n  1) lim  sin   sin  sin  n   2n   2n   2n  n

1

n 1 .   2 

(3 marks)

P. 62


HKAL Pure Mathematics Past Paper Topic: Complex Number

15. (03I12) Let n be a positive integer. (a) (i) (ii)

Find all roots of z 2n  1  0 . By factorizing z 2n  1  0 into a product of quadratic factors with real coefficients, or otherwise, prove that

1 n1  1 (2k  1)     z   2cos  for all z  0 . n z z 2n  k 0  (7 marks) (b) Using (a), or otherwise, prove that zn 

(i)

n 1

  cos  cos k 0

(ii)

n 1

 cos k 1

(2k  1) 2n

 cos n   n1 for any   R , 2 

(2k  1) 2  n . 4n 2 (8 marks)

P. 63


HKAL Pure Mathematics Past Paper Topic: Complex Number

16. (01I11) (a) Show that

1  cos   i sin   .  i cot 1  cos   i sin  2 (3 marks)

(b) Let n be a positive integer. Show that all the roots of equation

( z  1)n  ( z  1)n  0 ……… (*) can be written as i k , where  k  R , k = 0, 1, … , n  1 . (4 marks) (c) If i k ( k = 0, 1, … , n  1 ) are the roots of (*) in (b), using the relations between the roots and coefficients, show that n 1

 k 0

2 k

 n(n  1) . (5 marks)

(d) Let P0 , P1 ,..., Pn1 be the n points in an Argand plane representing the roots of (*) in (b), and O be the origin. Q is the point representing r (cos   i sin  ) where r  0 and   R . If dk is the distance between Pk and Q , show that n 1

d k 0

2 k

is independent of  . (3 marks)

P. 64


HKAL Pure Mathematics Past Paper Topic: Complex Number

17. (93I13) For any n = 1, 2, … , the sets Gn and Hn are defined by

Gn  {z  C : z n  1} , H n  {z  C : z n  1} . Let p , q be any two positive integers. (a) Show that (i) Gp  H p   , (ii) Gp  H p  G2 p . (2 marks) (b) Show that if p is odd and q is even, then H p  H q   . (2 marks) (c) Suppose p = mq where m is an integer. Show that (i) Gq  G p ; (ii) if m is odd, then H q  H p ; (iii) if m is even, then H q  Gp . (5 marks) (d) For any S , T  C , define ST by

ST  z  C : z  st for some s  S and t  T  . Show that (i) GpGp  H p H p  Gp , (ii) Gp H p  H pGp  H p . (6 marks)

P. 65


HKAL Pure Mathematics Past Paper Topic: Complex Number

Set4: Mapping 18. (90I10)

1 . z (a) If z  r (cos  i sin  ) and f ( z )  u  iv where r  0 and  , u, v  R , express u and v in terms of r and  . Let f : C\{0}  C be defined by f ( z )  z 

(2 marks) (b) Find and sketch the image of each of the following circles under f : (i)

z 1 ;

(ii)

z a , 0 < a < 1 . (4 marks)

(c) Show that f is surjective but not injective. (4 marks) (d) Let E   z  C \{0}: z  1 and f E : E  C be defined by f E ( z )  f ( z) for all z  E . Show that f E is injective but not surjective. (5 marks)

P. 66


HKAL Pure Mathematics Past Paper Topic: Complex Number

19. (91I09) A mapping f : C  C is said to be real linear if

f ( z1   z2 )   f ( z1 )   f ( z2 ) for all  ,   R , z1 , z2 C . Let  : C  C be a real linear mapping. (a) Prove that (i)

 ( z)    ( z) for all   R , z C .

(ii)

 (0)  0 . (3 marks)

(b) Let  : C  C be a real linear mapping. Prove that if  (1) =  (1) and  (i ) =  (i ) , then  =  . (2 marks) (c) If, furthermore,  is non-constant such that

 ( z1 z2 )   ( z1 ) ( z2 ) for all z1 , z2  C , show that: (i)

 (1) = 1 and hence  (x) = x for all x  R .

(ii)

Either  (z) = z for all z  C or  ( z )  z for all z C . (10 marks)

P. 67


HKAL Pure Mathematics Past Paper Topic: Complex Number

20. (94I11) A function f : C  C is said to be real linear if

f ( z1   z2 )   f ( z1 )   f ( z2 ) for all  ,   R and z1 , z2 C . (a) Suppose f is a real linear function. Show that (i)

if z = 0 whenever f (z) = 0 , then f is injective;

(ii)

if f (i) = i f (1) and f (i)  0 , then f is bijective. (4 marks)

(b) Suppose  ,  C and g( z )   z   z for all z  C . Show that (i)

g is real linear;

(ii)

g is injective if and only if    .

(8 marks) (c) If f is a real linear function, find a, b C such that

f ( z )  az  bz for all z C . (3 marks) Type 2: Geometric Problems Set 1: Triangles 21. (99I05) A , B and C are three points in the Argand diagram representing the complex numbers z1 , z2 and z3 respectively. If z1 = 0 , z2 = 1 + i and ABC is equilateral , find z3 . (5 marks)

P. 68


HKAL Pure Mathematics Past Paper Topic: Complex Number

22. (93I11) Let Z1 , Z2 and Z3 be 3 distinct points representing the complex numbers z1 , z2 and z3 respectively. (a) Suppose W1 , W2 and W3 are 3 distinct points representing the complex numbers w1 , w2 and w3 respectively. Prove that Z1Z2Z3 is similar to W1W2W3 (vertices anticlockwise) if and only if z3  z1 w3  w1  . z2  z1 w2  w1 (4 marks) (b) Using (a), or otherwise, show that Z1Z2Z3 (vertices anticlockwise) is equilateral if and only if

z1   z2   2 z3  0  2 where   cos   3

  2    i sin   .   3  (5 marks)

(c) A point representing a + i b is said to be an integral point if a and b are integers. Using (b), or otherwise, show that no triangle with distinct integral points as vertices can be equilateral. (6 marks) Set 2: Straight Lines / Circles 23. (98I04) Let z be a complex number satisfying 2 z  2i  z  i . (a) Show that the locus of z on an Argand diagram is a circle. Find its centre and radius. (b) Let S  z  C : 2 z  2i  z  i  . Draw and shade the region which represents S on an Argand diagram. Hence find z0  S such that z0  z for all z  S . (6 marks)

P. 69


HKAL Pure Mathematics Past Paper Topic: Complex Number

24. (00I04) 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) 25. (01I08) Let L be the straight line z  (4  4i)  z and C be the circle

z 1 . (a) Sketch L on an Argand diagram. (b) Let P , Q be points on L and C respectively such that PQ is equal to the shortest distance between L and C . Find PQ and the complex numbers representing P and Q . (6 marks) 26. (02I02) (a) Express

z 1  2 in the form of | z  c | r , where c and r z4

are constants.

  z 1  2 in the (b) Shade the region represented by  z  C : z4   Argand plane. (5 marks)

P. 70


HKAL Pure Mathematics Past Paper Topic: Complex Number

27. (95I11) Let P , Q be two points on a circle with centre C such that P , Q , C are non-collinear and taken anti-clockwise. PCQ   and M is the mid-point of PQ . Let zP , zQ , zC and zM be the complex numbers represented by P , Q , C and M respectively. (a) Show that zC  zM  i( zM  zP ) cot

 2

. (5 marks)

(b) Express zC and the radius r of the circle in terms of zP , zQ and  . (4 marks) (c) (i)

Show that any circle in the complex plane can be represented by an equation of the form zz  az  bz  c  0 where a, b C and c  R .

(ii)

Let C: zz  az  bz  c  0 be a circle passing through the points representing 1  i and i . If the chord joining these two points subtends an angle

 at the centre, 3

find the values of a , b and c . (6 marks)

P. 71


HKAL Pure Mathematics Past Paper Topic: Complex Number

28. (96I12) (a) Let a  C and b  0 . Show that the equation zz  az  az  b

(zC)

can be written in the form of z  a  r where r  0 . (4 marks) (b) Let A and B be two points on the complex plane representing 2  3i and 1  2i respectively. P , representing the complex number z , is a moving point so that PA  2 PB . Show that the equation of the locus of P is a circle with equation

C: zz  iz  iz  3 . Find its radius and centre. (5 marks) (c) Let Q , representing  , be a point on the circle C in (b). (i)

  i  Show that the circle z       1 touches C at Q    i  

externally. (ii)

For any given r > 0 , write down the equations of the two circles with radius r which touch C at Q . (6 marks)

P. 72


HKAL Pure Mathematics Past Paper Topic: Complex Number

29. (99I13) Let  be a complex number and u , v be variable complex numbers satisfying

u   u   and v   v  vv respectively. Let L be the locus of u and C be the locus of v . (a) Show that (i)

the equation of L can be written as u    u ,

(ii)

the equation of C can be written as v     . (4 marks)

(b) For  = 2 + i , sketch L and C on an Argand diagram. (4 marks) (c) (i)

Let z 

1 . Show that z satisfies  z   z  zz for u

some constant  . Hence sketch the locus of z on an Argand diagram for   2  i . (ii)

Let z 

1 . Sketch the locus of z on an Argand v

diagram for   2  i . (7 marks)

P. 73


HKAL Pure Mathematics Past Paper Topic: Complex Number

Set 3: Others 30. (92I11) Let a be a positive real number and n a positive integer. (a) Solve the quadratic equation y 2  2 ya n cos n  a 2n  0 where

 R . Hence show that the polynomial x2n  2 xn a n cos n  a 2n can be n 1  2r  2   factorized as   x 2  2 xa cos    a  . n   r 0   (6 marks) (b) Let P0 , P1 , P2 , … Pn be the n points in the Argand plane representing the nth roots of an , arranged anti-clockwise, with P0 on the positive real axis. Let Q be the point representing x(cos   i sin  ) where x  0 . For r  0, 1, 2, , n 1 , denote the length of the segment QPr by

dr . (i)

Show that

n 1

d r 0

(ii)

 x 2 n  2 x n a n cos n  a 2 n .

If Q lies on the positive real axis, show that n 1

d r 0

(iii)

2 r

r

 xn  an .

If OQ bisects P0OP1 , where O is the origin, show that

n 1

d r 0

r

 xn  an . (9 marks)

P. 74


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

Contents: Classification

Chronological Order

Type 1: General Properties

90I01

Q.

19

1.

91I01

90I08

Q.

42

2.

93I06

91I01

Q.

1

3.

96I01

91I03

Q.

15

4.

97I07

91I10

Q.

12

5.

00I09

92I01

Q.

20

6.

94I08

92I03

Q.

37

7.

02I12

92I09

Q.

9

8.

06I08

92I13

Q.

10

9.

92I09

92II10

Q.

54

10.

92I13

93I03

Q.

21

11.

95I08

93I06

Q.

2

12.

91I10

94I01

Q.

38

Type 2: Systems of Linear Equations

94I02

Q.

18

SECTION A

94I08

Q.

6

13.

97I03

94I09

Q.

34

14.

96I05

95I01

Q.

35

15.

91I03

95I08

Q.

11

16.

98I01

95I09

Q.

22

17.

99I01

96I01

Q.

3

18.

94I02

96I05

Q.

14

19.

90I01

96I08

Q.

39

20.

92I01

96I09

Q.

33

21.

93I03

97I03

Q.

13

SECTION B

97I07

Q.

4

22.

95I09

97I08

Q.

23

23.

97I08

97I10

Q.

46

24.

00I08

98I01

Q.

16

25.

01I09

98I02

Q.

47

26.

03I07

98I08

Q.

30

27.

04I07

98I09

Q.

40

28.

05I07

99I01

Q.

17

29.

06I07

99I06

Q.

53

30.

98I08

99I08

Q.

32

31.

02I08

99I09

Q.

43

32.

99I08

00I01

Q.

36

P. 75


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

33.

96I09

00I06

Q.

48

34.

94I09

00I08

Q.

24

Type 3: Power of a Matrix

00I09

Q.

5

35.

95I01

01I07

Q.

49

36.

00I01

01I09

Q.

25

37.

92I03

02I03

Q.

50

38.

94I01

02I08

Q.

31

39.

96I08

02I12

Q.

7

40.

98I09

03I07

Q.

26

41.

03I08

03I08

Q.

41

42.

90I08

04I03

Q.

51

43.

99I09

04I07

Q.

27

44.

04I08

04I08

Q.

44

45.

05I08

05I05

Q.

52

46.

97I10

05I07

Q.

28

Type 4: Transformation in Cartesian

05I08

Q.

45

06I07

Q.

29

06I08

Q.

8

Plane 47.

98I02

48.

00I06

49.

01I07

50.

02I03

51.

04I03

52.

05I05

53.

99I06

54.

92II10

P. 76


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

Type 1: General Properties 1.

(91I01) Factorize the determinant

a3 a 1

b3 b 1

c3 c . 1 (4 marks)

2.

(93I06) (a) Show that if A is a 3  3 matrix such that At   A , then

det A  0 .  1 2 74    1 67  , use (a), or otherwise, to show (b) Given that B   2  74 67 1    det( I  B)  0 . Hence deduce that det( I  B )  0 . 4

(7 marks) 3.

(96I01)

 0 0 2    Let A   1 2 1  . 1 0 3    (a) Evaluate A3  5 A2  8 A  4I . (b) Hence, or otherwise, find A1 . (6 marks)

P. 77


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

4.

(97I07) (a) Let A be a 3  3 non-singular matrix. Show that

det( A1  xI )  

x3 det( A  x 1I ) . det A

0 1 0   (b) Let A   0 0 1 .  4 17 8    (i)

Show that 4 is a root of det(A  xI ) = 0 and hence find the other roots in surd form.

(ii)

Solve det(A1  xI ) = 0 . (7 marks)

5.

(00I09) n n n 1 (a) Show that Cr  Cr 1  Cr 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 , n

( A  B)n   Crn An r B r

………. (*),

r 0

where A0 and B 0 are by definition the identity matrix I . Would (*) still be valid if AB  BA ? Justify your answer. (6 marks)

 cos  sin 

(c) Let A = 

(i)

 sin    where θ is real. cos   cos n  sin n

Show that An = 

 sin n   for all positive integers cos n 

n . (ii)

Using (*) and the substitution B = A1 , show that n

 Crn cos(n  2r )  2n cosn  and r 0

n

C r 0

n r

sin(n  2r )  0 .

Hence or otherwise, express cos5θ in terms of cos 5θ , cos 3θ and cos θ . (7 marks)

P. 78


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

6.

(94I08)

a b c   Let M   c a b  , where a , b and c are non-negative real b c a   numbers. (a) Show that det( M ) 

1 (a  b  c)[(a  b) 2  (b  c) 2  (c  a) 2 ] and 2

0  det(M )  (a  b  c)3 . (4 marks)

 an  (b) Let M   cn b  n n

bn an cn

cn   bn  for any positive integer n , show that an , an 

bn and cn are non-negative real numbers satisfying

an  bn  cn  (a  b  c)n . (4 marks) (c) If a + b + c = 1 and at least two of a , b and c are non-zero, show that (i)

lim det( M n )  0 ,

(ii)

lim(an  bn )  0 and lim(an  cn )  0 ,

(iii)

lim an 

n 

n 

n 

n 

1 . 3 (7 marks)

P. 79


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

7.

(02I12) (a) Let A be a 3  3 matrix such that

A3  A2  A  I  0 , where I is the 3  3 identity matrix. (i)

Prove that A has an inverse, and find A1 in terms of A .

(ii)

Prove that A4 = I .

(iii)

Prove that ( A )  ( A )  A  I  0 .

(iv)

Find a 3  3 invertible matrix B such that

1 3

1 2

1

B3  B 2  B  I  0 . (6 marks)

1 1 1   (b) Let X   1 1 0  .  1 0 1   (i)

Using (a)(i) or otherwise, find X 1 .

(ii)

Let n be a positive integer. Find X n .

(iii)

Find two 3  3 matrices Y and Z , other than X , such that

Y 3  Y 2  Y  I  0 and Z 3  Z 2  Z  I  0 . (9 marks)

P. 80


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

8.

(06I08)

 m m   , where m > 0 . m m 

Let M = 

(a) Evaluate

M2

. (1 mark)

a b  be a non-zero real matrix such that MX = XM . c d

(b) Let X =  (i)

Prove that c = b and d = a .

(ii)

Prove that X is a non-singular matrix.

(iii)

Suppose X  6 X 1 = 

1 0  . 0 1

(1)

Find X .

(2)

If a > 0 and (M  k X )2 =  M 2 , express k in terms of m . (10 marks)

(c) Using the result of (b)(iii)(2), find two real matrices P and Q , other than M and M , such that P  Q  M 4

4

4

. (4 marks)

P. 81


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

9.

(92I09)

 cos   sin 

(a) Let A  

 sin    . cos  

 cos n  sin n

 sin n   for cos n 

Prove by mathematical induction that An  

n  1, 2,  . (3 marks)

 a b    : a, b  R  and n be a positive integer.  b a  

(b) Let M   (i)

For any X , Y  M , show that (I)

(ii)

XY  M ,

(II)

XY = YX ,

(III)

if X  

0 0 1 1  , then X exists and X  M . 0 0

For any X  M , show that there exist r  0 and   R such

 cos   sin 

that X  r 

 sin    . cos   1 0  . 0 1

Hence find all X  M such that X n   (iii)

If Y , B  M and Y n = Bn , show that there exists X  M

1 0  and Y = BX . 0 1

such that X n  

n

 1 2 Hence find all Y  M such that Y    .  2 1  n

(12 marks)

P. 82


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

10.

(92I13)

 a11  a21

Let M be the set of all 2  2 matrices. For any A  

a12  M , a22 

define tr( A)  a11  a22 . (a) Show that for any A , B , C  M and  ,   R , (i)

tr( A   B)   tr( A)   tr( B) ,

(ii)

tr( AB)  tr( BA) ,

(iii)

the equality “ tr( ABC )  tr( BAC ) ” is not necessarily true. (5 marks)

(b) Let A  M . (i)

Show that A  tr( A) A   det( A) I , where I is the 2  2 2

identity matrix. (ii)

If tr( A )  0 and tr( A)  0 , use (a) and (b)(i) to show that 2

A is singular and A2 = 0 . (5 marks) (c) Let S , T  M such that (ST  TS )S  S (ST  TS ) . Using (a) and (b) or otherwise, show that ( ST  TS )  0 . 2

(5 marks)

P. 83


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

11.

(95I08) Let Mmn be the set of all m  n matrices.

 a1  b1

(a) Let A  

(i)

a2    M 22 . b2   u1    s1  u2 

Show that if A  

s2  , where u1 , u2 , s1 , s2 R , then

det A  0 . (ii)

Conversely, show that if det A = 0 , then A = BC for some

B  M 21 and C  M12 . (5 marks)

 a1 a2  (b) Let D   b1 b2 c c  1 2

(i)

a3   b3   M 33 . c3 

 u1  Show that if D   u2 u  3

v1   s s v2   1 2 t t v3   1 2

s3   , where t3 

ui , vi , si , ti R ( i = 1, 2, 3 ) , then det D = 0 . (ii)

Suppose there are  ,   R such that ci   ai   bi for

i  1, 2, 3 . Find S  M 32 and T  M 23 such that D  ST . (iii)

Show that if det D = 0 , then D = PQ for some P  M 32 and Q  M 23 . (10 marks)

P. 84


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

12.

(91I10) Let M be the set of all 3  1 real matrices.

 u1   x1      For any two 3  1 matrices u   u2  and x   x2  , u  x   3  3  u2 x3  x2u3    we define u  x   u3 x1  x3u1  . u x xu   1 2 1 2 (a) Show that for any u, x, y  M , and any ,   R , (i)

u  ( x   y)   (u  x)   (u  y) ,

(ii)

u  x  ( x  u) . (3 marks)

(b) Show that if u  x  0 for all x  M , then u = 0 . Deduce that if u  x  v  x for all x  M , then u = v . (6 marks) (c) Let A be a 3  3 real matrix and u  M such that

Ae1  u  e1 , Ae2  u  e2 and Ae3  u  e3 1 0 0       where e1   0  , e2   1  , e3   0  . 0 0 1       Show that Ax  u  x for all x  M . (4 marks)

 p   (d) Let u   q  . Find the matrix A such that Ax  u  x for all r  

xM . (2 marks)

P. 85


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

Type 2: Systems of Linear Equations SECTION A 13.

(97I03) Suppose the system of linear equations

 x  ky   y  (*):   x  ky  

 0 z  0 z  0

has nontrivial solution. (a) Show that  satisfies the equation 2 + k  k = 0 . (b) If the quadratic equation in  in (a) has equal roots, find k . Solve (*) for each of these values of k . (6 marks) 14.

(96I05)

Z  (a) Solve  Z  

 Y Y

 

X X

 a  b for X , Y and Z .  c

(b) If a + b  c > 0 , b + c  a > 0 and c + a  b > 0 ,

 xy   solve  xy  

xz  xz 

 a yz  b for x , y and z . yz  c (6 marks)

15.

(91I03) Consider the following system of linear equations:

z  1 x  2 y   y  2z  2 . x    y  q2 z  q  Determine all values of q for each of the following cases: (a) The system has no solution. (b) The system has infinitely many solutions. (4 marks)

P. 86


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

16.

(98I01) Consider the system of linear equations

2 x   (*):  x  kx  

y  2z  0  (k  1) z  0 . y  4z  0

Suppose (*) has infinitely many solutions. (a) Find k . (b) Solve (*) . (6 marks) 17.

(99I01) Suppose the system of linear equations

y  z  0  x   z  0 (*):  x   y   x  y  z  0  has non-trivial solutions. (a) Find all the values of  . (b) Solve (*) for each of the values of  obtained in (a). (6 marks) 18.

(94I02) Consider the following system of linear equations:

z  x 4 x  3 y   (*)  3x  4 y  7 z   y .  x  7 y  6z   z  Suppose  is an integer and (*) has non-trivial solution. Find  and solve (*) . (6 marks)

P. 87


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

19.

(90I01) Consider the following system of linear equations:

y  z  1  3x   (*) 2 x  4 y  5 z  1 , 4 x  2 y  7 z  c  where c  R . Suppose (*) is consistent. Find c and solve (*) . (4 marks) 20.

(92I01) Consider the following system of linear equations:

 x  (t  3) y  5 z  3  9 y  15 z  s (*) 3x   2x  ty  10 z  6  (a) If (*) is consistent, find s and t . (b) Solve (*) when it is consistent. (6 marks) 21.

(93I03) Suppose the following system of linear equations is consistent:

ax bx  (*)   cx  x

 by  cz   cy  ax   ay  bz   y  z 

1 1 , where a , b , c  R . 1 3

(a) Show that a + b + c = 1 . (b) Show that (*) has a unique solution if and only if a , b and c are not all equal. (c) If a = b = c , solve (*) . (6 marks)

P. 88


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

SECTION B 22.

(95I09) Consider the following systems of linear equations

 2x  2 y  z  k  (S):  hx  3 y  z  0 3x  hy  z  0  and

 6x  hx  (T):   3x 5 x

 6 y  3z  3y  z  hy  z  2 y  6z

   

2 0 . 0 h

(a) Show that (S) has a unique solution if and only if h2  9 . Solve (S) in this case. (3 marks) (b) For each of the following cases, find the value(s) of k for which (S) is consistent, and solve (S) : (i)

h=3 ,

(ii)

h = 3 . (7 marks)

(c) Find the values of h for which (T) is consistent. Solve (T) for each of these values of h . (5 marks)

P. 89


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

23.

(97I08) Consider the following two systems of linear equations:

2z  0 (a  1) x  2 y   x  ay  2z  0 (S):   3x  y  (a  7) z  0 

and

2z  6 (a  1) x  2 y   x  ay  2 z  5b  1 . (T):   3x  y  (a  7) z  1  b  (a) If (S) has infinitely many solutions, find all the values of a . Solve (S) for each of these values of a . (7 marks) (b) For the smallest value of a found in (a), find the values of b so that (T) is consistent. Solve (T) for these values of a and b . (4 marks) (c) Solve the system of equations

 x   x   3x   3x

 2y  2 z

6

 2y  2 z

6

 y  9 z  2  4y  z  11

.

(4 marks)

P. 90


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

24.

(00I08) Consider the system of linear equations

y  z  a  x    y  2 z  b where   R . (S): 2 x   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

y  z  3  x   (T): 2 x  2 y  2 z  2   x  y  4z   

 5  0  2  0  1  0

where   R .

Using the results in (b), or otherwise, solve (T) . (4 marks)

P. 91


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

25.

(01I09) Consider the system of linear equations

z  k  x  y   y  z  1 where  , k  R . (S):  x   3x  y  2 z  1  (a) Show that (S) has a unique solution if and only if   0 and

2 . (2 marks) (b) For each of the following cases, determine the value(s) of k for which (S) is consistent. Solve (S) in each case. (i) (ii) (iii)

  0 and   2 ,  0 , 2 . (8 marks)

 x   (c) If some solution of (x , y , z) of  3x  

 z  0 y  z  1 y  2 z  1

satisfies ( x  p)  y  z  1 , find the range of values of p . 2

2

2

(5 marks)

P. 92


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

26.

(03I07) (a) Consider the system of linear equations in x , y , z

ay  z  0  x   y  az  2a , (E):  2 x  2  x  2a y  (a  3) z  2a  where a  R . (i)

Find the range of values of a for which (E) has a unique solution. Solve (E) when (E) has a unique solution.

(ii)

Solve (E) for (1)

a=1 ,

(2)

a = 4 . (10 marks)

(b) Suppose (x , y , z) satisfies

y  z  0  x   y  z  2 .  2x   x  2 y  2 z  2  Find the least value of 24 x  3 y  2 z and the corresponding 2

2

values of x , y , z . (5 marks)

P. 93


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

27.

(04I07) (a) Consider the system of linear equations

 x  (a  2) y  az  1  2 y  4z  1 , (E):  x  ax  y  3z  b  where a , b  R . (i)

Prove that (E) has a unique solution if and only if a  2 and

a  4 . Solve (E) in this case. (ii)

For each of the following cases, determine the value(s) of b for which (E) is consistent, and solve (E) for such value(s) of b . (1)

a=2 ,

(2)

a=4 . (10 marks)

 2z  1  x  (b) If all solutions (x , y , z) of  x  2 y  4 z  1 satisfy 2 x  y  3z  2 

k ( x 2  3)  yz , find the range of values of k . (5 marks)

P. 94


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

28.

(05I07) (a) Consider the system of linear equations in x , y , z

ay  z  b  x   (a  1) z  0 , (E): 2 x  (a  3) y  2  3x  a y  (4a  1) z  b  where a , b  R . (i)

Find the range of values of a for which (E) has a unique solution. Solve (E) when (E) has a unique solution.

(ii)

For each of the following cases, find the value(s) of b for which (E) is consistent, and solve (E) for such value(s) of b . (1)

a=1 ,

(2)

a = 2 . (10 marks)

z  b  x  2y   y  3z  0 (b) Suppose that a real solution of 2 x   3x  4 y  7 z  b  satisfies x  y  z  b  3 , where b  R . Find the range of 2

2

2

values of b . (5 marks)

P. 95


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

29.

(06I07) Consider the system of linear equations in x , y , z

ay  z  4  x   (E):  x  (2  a) y  (3b  1) z  3 , 2 x  (a  1) y  (b  1) z  7  where a , b  R . (a) Prove that (E) has a unique solution if and only if a  1 and b  0 . Solve (E) in this case. (6 marks) (b) (i)

For a = 1 , find the value(s) of b for which (E) is

consistent, and solve (E) for such value(s) of b . (ii)

Is there a real solution (x , y , z) of

x  y  z  4  2 x  2 y  z  6 4 x  4 y  3z  14  satisfying x2  2y2  z = 14 ? Explain your answer. (7 marks) (c) Is (E) consistent for b = 0 ? Explain your answer. (2 marks)

P. 96


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

30.

(98I08) Consider the system of linear equations in

ax  y  bz  1  bz  1 . (E):  x  ay   x  y  abz  b  (a) Show that (E) has a unique solution if and only if a  2 , a  1 and b  0 . Solve (E) in this case. (7 marks) (b) For each of the following cases, determine the value(s) of b for which (E) is consistent. Solve (E) in each case. (i)

a=2,

(ii)

a=1 . (6 marks)

(c) Determine whether (E) is consistent or not for b = 0 . (2 marks)

P. 97


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

31.

(02I08) (a) Consider the system of linear equations in x , y , z

z  0 ax  2 y   y  2 z  b , where a , b  R . (S):  x   y  az  b  (i)

Show that (S) has a unique solution if and only if a 2  1 . Solve (S) in this case.

(ii)

For each of the following cases, determine the value(s) of b for which (S) is consistent, and solve (S) for such value(s) of b . (1)

a=1 ,

(2)

a = 1 . (9 marks)

(b) Consider the system of linear equations in x , y , z

 ax  2 y  x  y  (T):  y  5 x  2 y

 z  2z  az  z

 0  1 , where a  R .  1  a

Find all the values of a for which (T) is consistent. Solve (T) for each of these values of a . (6 marks)

P. 98


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

32.

(99I08) Consider the system of linear equations

z    x  y   y  (  2) z  7 where   R . (E): 3x   x  y  z  3  (a) Show that (E) has a unique solution if and only if   1 . (3 marks) (b) Solve (E) for (i)

  1 ,

(ii)

  1 ,

(iii)

 1 . (8 marks)

(c) Find the conditions on a , b , c and d so that the system of linear equations

 x 3x    x ax

 y  y  y  by

 z  1  3z  7  z  3  cz  d

is consistent. (4 marks)

P. 99


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

33.

(96I09) Consider the system of linear equations

z  3 x  2 y  . y  2z  4 x 

(*): 

(a) Solve (*) . (3 marks) (b) Find the solutions of (*) that satisfy xy + yz + zx = 2 . (4 marks) (c) Find all possible values of a and  ( a,   R ) so that

z  3  x  2y   y  2z  4  x  ax  y  z    is solvable. (4 marks) (d) Using (b), or otherwise, find all possible values of a and  ( a,   R ) so that

 x  x    xy ax

 2y  z  y  2z  yz  zx  y  z

 3  4  2  

has at least one solution. (4 marks)

P. 100


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

34.

(94I09) (a) Consider

(I)

a11 x  a12 y  a13 z  0  a21 x  a22 y  a23 z  0 a x  a y  a z  0 32 33  31

(II)

a11 x  a12 y  a13  a21 x  a22 y  a23 a x  a y  a 32 33  31

and

(i)

 0  0 .  0

Show that if (I) has a unique solution, then (II) has no solution.

(ii)

Show that (u, v) is a solution of (II) if and only if (ut, vt, t) are solutions of (I) for all t  R .

(iii)

If (II) has no solution and (I) has nontrivial solutions, what can you say about the solutions of (I) ? (5 marks)

(b) Consider

(III)

y  z  0 (3  k ) x   7 x  (5  k ) y  z  0   6 x  6 y  (k  2) z  0 

(IV)

y  1  0 (3  k ) x   7 x  (5  k ) y  1  0 .   6 x  6 y  (k  2)  0 

and

(i)

Find the values of k for which (III) has non-trivial solutions.

(ii)

Find the values of k for which (IV) is consistent. Solve (IV) for each of these values of k .

(iii)

Solve (III) for each k such that (III) has non-trivial solutions. (10 marks)

P. 101


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

Type 3: Power of a Matrix 35.

(95I01)

a 1  where a , b  R and a  b . 0 b  n a n  bn  a  Prove that An   a  b  for all positive integers n .  b n  0

(a) Let A  

1 2 (b) Hence, or otherwise, evaluate   0 3

95

. (6 marks)

36.

(00I01)

1 0 0    Let M =   b a  where b2 + ac = 1 . Show by induction that   c b   

M

2n

1 0 0      n[ (1  b)   a] 1 0  for all positive integers n .  n[ c   (1  b)] 0 1   

0 0 1   Hence or otherwise, evaluate  2 3 2  1 4 3   

2000

.

(5 marks) 37.

(92I03)

1 0   2 0   , B  . 1 3   1 

Let A  

 a 0  , find  , a and b .  0 b

(a) If B1 exists and B 1 AB   (b) Hence find A100 .

(7 marks)

P. 102


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

38.

(94I01)

3 8  2 4   .  and P   1 1  1 5

Let A  

(a) Find P 1 AP . (b) Find An , where n is a positive integer. (6 marks) 39.

(96I08) (a) Solve the equation

 1 3   1 0  det     0 2 2 0 1     

……… (*) . (3 marks)

(b) Let 1 , 2 ( 1  2 ) be the roots of (*) . Find two non-zero vectors (x1 , y1) and (x2, y2) such that

 xi   1 3   xi     y   i  y  ,  2 2  i   i

i = 1, 2.

[Modified]

 x1  y1

Let P  

x2  1  . Show that P is non-singular and find P . y2 

1 3 P .  2 2

Evaluate P 1 

(9 marks) 1996

1 3 (c) Evaluate    2 2

. (3 marks)

P. 103


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

40.

(98I09)

a b  where a, b, c, d  R , a  0 and det A  0 . c d

Let A = 

a b  for some k  R .  ka ka 

(a) Show that A = 

(3 marks)

1 0   such that PA =  r 1 0

(b) Find P in the form of 

,   R .



 for some 0

1 s  such that  0 1

If a + d  0 , find Q in the form of 

 0 PAP 1Q    for some   R .  0 0

(5 marks)

 3 7  1   0  S    for some   R .  6 14   0 0

(c) Find S such that S 

n

3 7  Hence, or otherwise, evaluate   where n is a positive  6 14  integer. (7 marks)

P. 104


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

41.

(03I08)

 2    3 

(a) If det 

3   0 , find the two values of  .   (2 marks)

(b) Let 1 and 2 be the values obtained in (a), where 1   2 . Find 1 and 2 such that

 2  1   3 

3   cos 1   0       , 0  1 <  . 1   sin 1   0 

 2   2   3 

3   cos  2   0       , 0  2 <  .  2   sin  2   0 

 cos 1 cos  2  n  . Evaluate P , where n is a positive sin  sin  1 2  

Let P   integer.

 2  3 

3  d1  P is a matrix of the form  0  0

Prove that P 1 

0  . d2  (8 marks)

 2 (c) Evaluate   3 

n

3  , where n is a positive integer. 0  (5 marks)

P. 105


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

42.

(90I08) (a) Let X and Y be two square matrices such that XY = YX . Prove that (i)

( X  Y )2  X 2  2 XY  Y 2 ,

(ii)

( X  Y )n   Crn X n rY r for n = 3, 4, 5, … .

n

r 0

(Note: For any square matrix A , define A0 = I .) (3 marks)

0 2 4   (b) By using (a)(ii) and considering  0 0 3  , or otherwise, find 0 0 0   100

1 2 4   0 1 3 0 0 1  

.

(4 marks) (c) If X and Y are square matrices, (i)

prove that ( X  Y )  X  2 XY  Y

(ii)

prove that ( X  Y )  X  3 X Y  3 XY  Y

2

3

2

3

2

2

implies XY  YX ;

2

3

does NOT imply

XY  YX . (Hint:

Consider a particular X and Y ,

1 0  b 0  , Y   .) 1 0  0 0

e.g. X  

(8 marks)

P. 106


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

43.

(99I09) (a) Let A and B be two square matrices of the same order. If

AB  BA  0 , show that (A + B)n = A n + B n for any positive integer n . (4 marks)

a b  p q  where a , b are not both zero. If B    ,  0 0  r s show that AB  BA  0 if and only if p = r = 0 and aq + bs = 0 .

(b) Let A  

(4 marks)

x 0

(c) Let C  

y  where x , z are non-zero and distinct. Find z

non-zero matrices D and E such that C = D + E and DE = ED =0 . (3 marks)

 2 5 (d) Evaluate    0 1

99

. (4 marks)

P. 107


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

44.

(04I08)

  k     k  , where  ,  , k  R with    .   k   k

Let A  

Define X 

1 1 ( A   I ) , where I is the 2  2 ( A   I ) and Y     

identity matrix. (a) Evaluate XY , YX , X + Y , X 2 and Y 2 . (4 marks) (b) Prove that A   X   Y for all positive integers n . n

n

n

(4 marks)

 5 4 (c) Evaluate    2 3

2004

. (4 marks)

(d) If  and  are non-zero real numbers, guess an expression for

A1 in terms of  ,  , X and Y , and verify it. (3 marks)

P. 108


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

45.

(05I08)

 p q  , where p , q , r , s  R .  r s

(a) Let M   (i)

Suppose det M = 0 . Prove that M

(ii)

n 1

 ( p  s)n M for any positive integer n .

Suppose qr > 0 . Let  and  be the roots of the quadratic equation x2  (p + s) x + det M = 0 . Denote the 2  2 identity matrix by I . (1)

Prove that  and  are two distinct real numbers.

(2)

Prove that M 2  ( +  ) M +  I = 

(3)

Define A = M   I and B = M   I .

0 0  . 0 0

0 0  and det A = det B = 0 . 0 0 Find real numbers  and  , in terms of  and  , such that M =  A +  B . Prove that AB = BA = 

(11 marks) n

1 2 (b) Evaluate   , where n is a positive integer.  4 3 Candidates may use the fact, without proof, that if X and Y are 2  2 matrices satisfying XY = YX =

0 0 n n n   , then ( X  Y )  X  Y for any positive 0 0 integer n . (4 marks)

P. 109


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

46.

(97I10) (a) (i)

 cos   sin 

Let S  

  sin    :  R . cos   

Show that for any matrices A and B in S , AB is also in S . (ii)

 cos   sin 

 sin    . cos  

Let T( )   Prove that

T( )

n

 T(n ) for any positive integer n . (4 marks)

 a b  2 2  where a, b  R and a + b  0 . b a  

(b) Let M   (i)

Show that M = k T( ) for some real numbers k and  . Express k , cos  and sin  in terms of a and b .

(ii)

If a  0 , prove that there exists a positive integer n such

 p 0  ) if and only  0 q

that M n is diagonal ( i.e. of the form  if (iii)

1

tan 1

b is rational. a

If a = 0 , find all positive integers n such that M n is diagonal. (11 marks)

Type 4: Transformation in Cartesian Plane 47.

(98I02)

 x'   a b  x  a b 2    for any (x, y)  R , then   is said to be  y'   c d  y  c d

If 

the matrix representation of the transformation which transforms (x, y) to ( x' , y' ) . Find the matrix representation of (a) the transformation which transforms any point (x, y) to (x, y) , (b) the transformation which transforms any point (x, y) to (y, x) . (4 marks)

P. 110


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

48.

(00I06) A transformation T in R2 transforms a vector x to another vector

   cos 3 y  Ax  b where A    sin   3 

 sin



 3  3  and b    .  1 cos  3 

 2 0

(a) Find y when x    . (b) Describe the geometric meaning of the transformation T . (c) Find a vector c such that y = A(x + c) . (7 marks) 49.

(01I07) A 2  2 matrix M is the matrix representation of a transformation T in R2 . T transforms (1, 0) and (0, 1) to (1, 1) and (1, 1) respectively. (a) Find M .

  0  a b     0   c d 

(b) Find  > 0 such that M can be decomposed as  where

a b 1 . c d

Hence describe the geometric meaning of T . (5 marks) 50.

(02I03) (a) Write down the matrix A representing the rotation in the Cartesian plane anticlockwise about the origin by 45 . (b) Write down the matrix B representing the enlargement in the Cartesian plane with scale factor

2 .

 x  y

(c) Let X    and V = BAX , where A and B are the matrices

1 0   V  4 , express y in terms of  0 1

defined in (a) and (b). If V t  x .

(5 marks)

P. 111


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

51.

(04I03) (a) Let R be the matrix representing the rotation in the Cartesian plane anticlockwise about the origin by 60 . (i)

Write down R and R 6 .

(ii)

Let A = 

2 0 

1  1  . Verify that A RA is a matrix in which all 3

the elements are integers. (b) Using the results of (a), or otherwise, find a 2  2 matrix M , in which all the elements are integers, such that M 3  I but M  I ,

1 0  . 0 1

where I  

(7 marks) 52.

(05I05) Let T1 be the transformation which transforms a vector x to a vector

  y = Ax , where A =     (a) (i)

3 2 1 2

1   2  . 3  2  0  2

Find y when x =   .

(ii)

Describe the geometric meaning of the transformation T1 .

(iii)

Find A2005 .

(b) For every integer n greater than 1 , let Tn be the transformation which transforms a vector x to a vector y = Anx . Is there a positive integer m such that the transformation Tm

0

 2   ? Explain your answer.   2 

transforms   to    2

(7 marks)

P. 112


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

53.

(99I06) It is given that the matrix representing the reflection in the line

 cos  y  (tan  ) x is   sin 2

sin 2   .  cos 2 

Let T be the reflection in the line y =

1 x . 2

(a) Find the matrix representation of T . (b) The point (4, 7) is transformed by T to another point (x1, y1) . Find x1 and y1 .

(c) The point (4, 10) is reflected in the line y 

1 x  3 to another 2

point ( x2 , y2 ) . Find x2 and y2 . (7 marks)

P. 113


HKAL Pure Mathematics Past Paper Topic: Determinants and Matrices

54.

(92II10) Let  be a Cartesian coordinate system on a plane and  be another Cartesian coordinate system with the same origin, obtained from  by an anti-clockwise rotation through an angle  . Suppose (x, y) and (x, y) are the coordinates of an arbitrary point P with respect to  and  respectively.

 x  y

 x'   .  y' 

(a) Let V    , V'  

 cos   sin 

 sin    . cos  

(i)

Show that V  MV' , where M  

(ii)

If the equation of a conic section in the coordinate system 

a h  , C  (c ) , h b

is given by V t AV  C , where A  

a, b, h, c R , show that this conic section is represented in the coordinate system  by

V' t A' V'  C , where A is a 2  2 matrix such that det A  det A' . Furthermore, show that  can be chosen such that A is a diagonal matrix. (10 marks) (b) The equation of a conic section (H) in  is given by

7 x2  2hxy  13 y 2  16 . Find h if (H) is (i)

an ellipse,

(ii)

a hyperbola,

(iii)

a pair of straight lines,

(iv)

given by x'  4 y'  4 in  . 2

2

(5 marks)

P. 114


HKAL Pure Mathematics Past Paper Topic: Functions

Contents:

1.

Classification

Chronological Order

Type 1: Bisection / Even / Odd / Periodic

92I06

Q.

7

93I09

Q.

10

Functions 95I04

2.

99II06

94II07

Q.

8

3.

02II05

95I04

Q.

1

4.

03II03

97II10

Q.

11

5.

06II02

98II10

Q.

13

6.

04II11

99II06

Q.

2

Type 2: Inverse Function

99II10

Q.

9

7.

92I06

01II13

Q.

12

8.

94II07

02II05

Q.

3

9.

99II10

03II03

Q.

4

Type 3: Others

04II11

Q.

6

10.

93I09

06II02

Q.

5

11.

97II10

12.

01II13

13.

98II10

P. 115


HKAL Pure Mathematics Past Paper Topic: Functions

Type 1: Bisection / Even / Odd / Periodic Functions 1.

(95I04) Let f : [1, 1]  [0,  ] , f (x) = arc cos x and g : R  R , g(x) = f (cos x) . (a) Show that g(x) is even and periodic. (b) Find g(x) for x  [0,  ] . Hence sketch the graph of g(x) for x  [2, 2 ] . (5 marks)

2.

(99II06) (a) Suppose f : R  R is a function satisfying f (a  x)  f (a  x) and

f (b  x)  f (b  x) for all x , where a , b are constants and a > b . Let w = 2(a  b) . Show that w is a period of f , i.e.,

f ( x  w)  f ( x) for all x  R . (b) Suppose g : R  R is a periodic function with period T > 0 satisfying g( x)  g( x) for all x . Show that there exists c with

0  c  T such that g(c + x) = g(c  x) for all x . (6 marks) 3.

(02II05) (a) If the function g: R  R is both even and odd, show that g(x) = 0 for all x  R . (b) For any function f : R  R , define

1 1 F( x)  [f ( x)  f ( x)] and G( x)  [f ( x)  f ( x)] . 2 2 (i)

Show that F is an even function and G is an odd function.

(ii)

If f ( x)  M( x)  N( x) for all x  R , where M is even and N is odd, show that M(x) = F(x) and N(x) = G(x) for all

xR . (6 marks)

P. 116


HKAL Pure Mathematics Past Paper Topic: Functions

4.

(03II03)

1 when x  0 ,  Let f : R  R be defined by f ( x)  0 when x  0 , 1 when x  0 .  (a) Prove that f is an odd function. (b) Is f an injective function? Explain your answer briefly. (c) Sketch the graph of y = f (x + 1) . (d) Let g(x) = f (x + 1) + f (x  1) for all x  R . Sketch that graph of y = g(x) . Write down the value(s) of x at which g(x) is discontinuous. (6 marks) 5.

(06II02) Let f : R  R be defined by

1 when x is an even number, f ( x)   2 when x is not an even number. (a) (i)

Sketch the graph of y = f (x) for 4  x  4 .

(ii)

Is f a periodic function? Explain your answer.

(b) Let g : R  R be defined by g(x) = f (x + 1) + f (x) . (i)

Sketch the graph of y = g(x) for 4  x  4 .

(ii)

Is g an injective function? Explain your answer. (7 marks)

P. 117


HKAL Pure Mathematics Past Paper Topic: Functions

6.

(04II11) For any real number x , let [x] denote the greatest integer not greater than x . Let f : R  R be defined by

1  f ( x)   2  x  [ x]  1  2 (a) (i)

when x is an integer, when x is not an integer.

Prove that f is a periodic function with period 1 .

(ii)

Sketch the graph of f (x) , where 2  x  3 .

(iii)

Write down all the real number(s) x at which f is discontinuous. (6 marks)

(b) Define F( x) 

x 0

f (t ) dt for all real numbers x . x2  x . 2

(i)

If 0  x  1 , prove that F( x) 

(ii)

Is F a periodic function? Explain your answer.

(iii)

Evaluate

 0

F( x) dx . (9 marks)

Type 2: Inverse Function 7.

(92I06) Let f : R  R be bijective and a1  a2    an , where n  2 . (a) Suppose f is strictly increasing. Prove that its inverse f 1 is also

1 n  f (ak )   an .   n k 1 

strictly increasing and deduce that a1  f 1 

(b) Define h( x)  p f ( x)  q , where p , q  R and p  0 . Show that h 1 ( x)  f 1 (

xq ) p

1 n  1  1 n  and deduce that h   h(ak )   f   f ( ak )  .  n k 1   n k 1  1

(5 marks)

P. 118


HKAL Pure Mathematics Past Paper Topic: Functions

8.

(94II07) Let f ( x) 

x 1

sin(cos t ) dt , where x  [0,

 2

) .

(a) Show that f is injective. (b) If g is the inverse function of f , find g(0) . (6 marks) 9.

(99II10) (a) Let f : R  R be a strictly increasing bijective function. (i)

Show that the inverse function f

1

is also strictly

increasing. (ii)

Let a < b and t1 , t2 , , tn [a, b] , n  2 .

1 n  f (ti )   b .   n i 1 

Show that a  f 1 

Find a necessary and sufficient condition on t1 , t2 , , tn such

1 n  f (ti )   b .   n i 1 

that a  f 1 

(8 marks) 1

(b) Let g : R  R be defined by g( x)  x 3 . (i)

Show that g is bijective and strictly increasing. 1 1  1  23    n 3  Hence show that 1   n  

(ii)

3

   n for n  2 .   

Find the area enclosed by graphs of y = g(x) and

y  g 1 ( x) . (7 marks)

P. 119


HKAL Pure Mathematics Past Paper Topic: Functions

Type 3: Others 10.

(93I09) Let f : R  R be a function such that f (x + y) = f (x) + f (y) for all

x, y  R . (a) Show that (i)

f (0) = 0 ,

(ii)

f (x) = f (x) for all x  R ,

(iii)

f (nx) = n f (x) for all n  Z and x  R . (5 marks)

(b) Show that if there exists K > 0 such that f (x) < K for all x  R , then f ( x)  0 for all x  R . (3 marks) (c) Suppose there exists K > 0 such that f (x) < K for all x  [0, 1) . Let g(x) = f (x)  f (1) x for all x  R . Show that, for all x, y  R , (i)

g(x + y) = g(x) + g(y) ,

(ii)

g(x + 1) = g(x) ,

(iii)

g(x) < K + | f (1) | .

Hence, or otherwise, show that f (x) = f (1) x for all x  R . (7 marks)

P. 120


HKAL Pure Mathematics Past Paper Topic: Functions

11.

(97II10) Denote the open interval (0, ) by R+ . Let f : R+  R be a continuous function such that f (xy) = f (x) + f (y) for all x, y  R+ . (a) Show that (i)

f (1) = 0 ,

(ii)

f (x1) = f (x) for all x  R+ ,

(iii)

f (x n) = n f (x) for all n  Z and x  R+ . (6 marks)

(b) Show that f (x r) = r f (x) for all r  Q and x  R+ . (4 marks) (c) Show that f (x  ) =  f (x) for all   R and x  R+ . You may use the fact that for any   R , there exists a sequence rn  in Q such that lim rn   . n 

(3 marks) (d) If f (2) = 1 , show that f (x) = log2 x . (2 marks)

P. 121


HKAL Pure Mathematics Past Paper Topic: Functions

12.

(01II13) Let f : (0, )  (0, ) be a continuous function satisfying f[f ( x)]  x and f (1  x) 

f ( x) for all x . 1  f ( x)

(a) Show that for any n  N and x  R , f (n  x) 

f ( x) . 1  n f ( x) (3 marks)

(b) Define xn  1 

f (1) n 1 for any n  N . Show that f ( xn )  . 1  n f (1) n2

Hence, by considering lim xn , show that f (1) = 1 . n 

Deduce that f (n) 

1 1 and f    n . n n (5 marks)

(c) For any q  N and q  2 , let S(q) be the statement

 p q for all p  N with 0 < p < q .”  q p

“f  (i)

Show that S(2) is true.

(ii)

Assume that S(h) is true for 2  h  q and h  N . Use (a)

 q 1  p for 0 < p < q + 1 .   p  q 1

to show that f 

Hence deduce that S(q + 1) is true. (iii)

Use (a) to show that for any positive rational number x ,

f ( x) 

1 . x (7 marks)

P. 122


HKAL Pure Mathematics Past Paper Topic: Functions

13.

(98II10) (a) Let f : R  R be a continuous function. (i)

Show that

a 0

f (t  b) dt  

a b 0

b

f (t ) dt   f (t ) dt for all 0

a, b  R . (ii)

If f (x + y) = f (x) + f (y) for all x, y  R , show that

x 0

x

f (t  1) dt  f (1) x   f (t ) dt for all x  R . 0

Using (i), or otherwise, show that f (x) = f (1) x for all

xR . (8 marks) (b) Suppose g is a non-constant continuous function defined for all positive real numbers and g(xy) = g(x) + g(y) for all x, y > 0 . By considering the function f (t) = g(et ) for t  R , show that

g( x)  log a x for some a > 0 . (7 marks)

P. 123


HKAL Pure Mathematics Past Paper Topic: Sequences

Contents:

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

18. 19. 20. 21. 22. 23. 24. 25.

Classification

Chronological Order

Type 1: Method of Difference

90I02

Q. 6

90II02 91II02 92I10 94I04 94II05 Type 2: Partial Fractions 90I02 93I05 04I01 99I07 01I01 03I03 06I04 Type 3: Differentiation 95I05 05I03

90I11 90II02 90II11 91I02 91I07 91II02 91II08 92I10 92II13 93I05 93II06 94I04 94II05 95I05 95I12 95II10

Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q. Q.

21 1 27 32 22 2 30 3 33 7 18 4 5 13 29 17

Type 4: Sandwich Theorem 98II06 96II07 95II10 Type 5: Monotone Convergence Theorem SECTION A 93II06 01II03 04I02 SECTION B 90I11 91I07 01I10 02I13 05I09

96II07 96II13 98II06 99I07

Q. Q. Q. Q.

16 28 15 9

01I01

Q. 10

01I10

Q. 23

01II03

Q. 19

02I13 03I03 04I01 04I02 05I03 05I09 05II11 06I04

Q. Q. Q. Q. Q. Q. Q. Q.

06I05

Q. 26

26. 06I05 27. 90II11 28. 96II13

P. 124

24 11 8 20 14 25 31 12


HKAL Pure Mathematics Past Paper Topic: Sequences

29. 95I12 30. 91II08 31. 05II11 Type 6: Others 32. 91I02 33. 92II13

P. 125


HKAL Pure Mathematics Past Paper Topic: Sequences

1.

Type 1: Method of Difference (90II02)

   Let n be a positive integer and x   0,  .  n 1  Show that sin x cot kx  cot(k  1) x  sin kx sin(k  1) x for all k  1, 2, 3, , n . Deduce that 1 1 1 sin nx .     2 sin x sin 2 x sin 2 x sin 3x sin nx sin(n  1) x sin x sin(n  1) x (5 marks) 2.

(91II02) Show that

(1  cos   cos 2    cos n )sin

 2

 sin

(n  1) n cos . 2 2

Hence solve 1  cos  cos 2    cos n  0 , 0    2 . (6 marks)

P. 126


HKAL Pure Mathematics Past Paper Topic: Sequences

3.

(92I10) Let

a1 , a2 , 

,

b1 , b2 , 

be two sequences of real numbers,

and b0  0 . (a) Show that

k

k 1

i 1

i 1

 ai (bi  bi1 )  ak bk   (ai  ai 1 )bi , k  2, 3,  . (4 marks)

(b) Suppose

ai 

is decreasing and bi  K for all i , where K is

a constant. Show that

n

 a (b  b i 1

i

i 1

i

)  K  a1  2 ak  , k = 1, 2, … . (6 marks)

(c) Using (b), or otherwise, show that for any positive integers n and p ,

n p

(1)i 3 .   i 2n i n

(5 marks) 4.

(94I04)

a1 , a2 ,, an 

and

b1 , b2 ,, bn 

are two sequences of real

numbers. Define sk  a1  a2    ak for k = 1, 2, … , n . (a) Prove that

n

a b k 1

k k

 s1 (b1  b2 )  s2 (b2  b3 )    sn 1 (bn 1  bn )  snbn .

(b) If b1  b2    bn  0 and there are constants m and M such that

m  sk  M for k = 1, 2, … , n , n

prove that mb1   ak bk  Mb1 . k 1

(5 marks)

P. 127


HKAL Pure Mathematics Past Paper Topic: Sequences

5.

(94II05) n   For n = 1, 2, 3, … and   R , let sn   3k 1 sin 3  k  . 3  k 1

3 1 Using the identity sin 3   sin   sin 3 , show that 4 4

sn 

3n   1 sin  n   sin  . 4 3  4

Hence, or otherwise, evaluate lim sn . n 

(4 marks) 6.

Type 2: Partial Fractions (90I02) 1 (a) Resolve into partial fractions. x( x  1)( x  2) n

1 . n  k 1 k ( k  1)( k  2)

(b) Evaluate lim 

(6 marks) 7.

(93I05) Express

x4 in partial fractions. x  3x  2 2

Hence evaluate

 1

  k  1  k k 2

2

k 4   .  3k  2  (6 marks)

8.

(04I01) (a) Resolve

1 into partial fractions. (2 x  1)(2 x  1)(2 x  3)

(b) Prove that

n

1

1

1

1

 (2k  1)(2k  1)(2k  3)  12  8(2n  1)  8(2n  3)

for

k 1

all positive integer n . Hence or otherwise, evaluate

1

 (2k 1)(2k  1)(2k  3)

.

k 10

(6 marks)

P. 128


HKAL Pure Mathematics Past Paper Topic: Sequences

9.

(99I07) A sequence

a1 

1 5

an 

is defined as follows:

1 1   2n  5 an 1 an

and

(a) Show that an  (b) Resolve

for n = 1, 2, 3, … .

1 for n = 1, 2, 3, … . n  4n 2

x2 into partial fractions. ( x  4 x) 2 2

n

Hence or otherwise, evaluate lim  (k  2)ak 2 . n 

k 1

(7 marks) 10. (01I01) (a) Resolve

8 into partial fractions. x( x  2)( x  2)

(b) Show that

2001

8

11

 r (r  2)(r  2)  12

.

r 3

(5 marks) 11. (03I03) (a) Resolve (b) (i)

5x  3 into partial fractions. x( x  1)( x  3)

5k  3

n

3

 k (k  1)(k  3)  2

Prove that

for any positive integer

k 1

n . (ii)

Evaluate

5k  3

 k (k  1)(k  3)

.

k 1

(7 marks) 12. (06I04) (a) Resolve

(b) Express

9 x  36 into partial fractions. x( x  2)( x  3) n

9k  36

 k (k  2)(k  3)

in the form A 

k 1

B C D   , n 1 n  2 n  3

where A , B , C and D are constants. (c) Is there a positive integer N such that

n

9k  36

 k (k  2)(k  3)  8

?

k 1

Explain your answer. (7 marks) P. 129


HKAL Pure Mathematics Past Paper Topic: Sequences

Type 3: Differentiation 13. (95I05) (a) For x > 0 , prove that ln x  x 1 where the equality holds if and only if x  1 . (b) Prove that ln

r 1  for r > 1 . r 1 r 1 n 1

1 for n = 2, 3, 4, … . k 1 k

Hence deduce that ln n  

(7 marks) 14. (05I03) (a) By considering the function f (x) = x  ln(x + 1) , or otherwise, prove that x  ln(1  x) for all x > 1 . (b) Using (a), prove that the series

1

n

is divergent.

n 1

(7 marks) Type 4: Sandwich Theorem 15. (98II06) Let a1 = 2 , b1 

3 2n 2n  1 an 1 , bn  bn 1 for n  2 . and an  2n  1 2n 2

(a) Prove that an  bn and anbn  2n  1 for n  1 . (b) Using (a), or otherwise, show that an 2  2n  1 for n  1 .

1 . n  a n

Hence find lim

(7 marks) 16. (96II07) (a) Show that 1  ex  e x  1  x for 0 < x  1 . (b) Using mathematical induction, or otherwise, prove that n 1 r xr ex n1 x x   e    (n  1)! r 0 r ! r 0 r ! n

for n = 0, 1, 2, … and x  (0, 1] .

1 1  Hence show that lim 1  1       e . n  2 n!   (7 marks)

P. 130


HKAL Pure Mathematics Past Paper Topic: Sequences

17. (95II10) For any  > 0 , define a sequence of real numbers as follows:

a1    1 , an  an 1 

 an 1

for n > 1 .

(a) Prove that (i)

an 2  an12  2 for n  2 ;

(ii)

an 2   2  2n  1 for n  1 . (2 marks)

(b) Using (a), show that for n  2 ,

2

n 1

an 2   2  2n  1   k 1

 2  2k   1

. (3 marks)

(c) Prove that for k  1 , k 1 1  dx . 2 2 k  1   2k   1   2 x  1 (2 marks) an 2 exists and find the n  n

(d) Using above results, show that lim limit.

an 2 exists. n  n

State with reasons whether lim

(8 marks)

P. 131


HKAL Pure Mathematics Past Paper Topic: Sequences

Type 5: Monotone Convergence Theorem SECTION A 18. (93II06) (a) Show that if  >  , then n

1 m m 1 2

(b) Let un  

  .   1  1

nm , n = 1, 2, … . n  m 1

Use (a), or otherwise, to show that

un  un1 for n = 1, 2, … . Hence show that lim un exists. n 

(7 marks) 19. (01II03) Let a1 = 1 and an 1 

4  an 2 for n  N . Show that 1  an  2 2

for n  N . Hence show that

an 

is convergent and find its limit. (6 marks)

20. (04I02) Let {an} be a sequence of positive real numbers, where

a1  1 and an 

12an 1  12 , n = 2 , 3, 4 ,… . an 1  13

(a) Prove that an  3 for all positive integers n . (b) Prove that {an} is convergent and find its limit. (6 marks)

P. 132


HKAL Pure Mathematics Past Paper Topic: Sequences

SECTION B 21. (90I11) Let u0 and v0 be real numbers such that 0  v0  u0 . For n = 1, 2, … , define

un  (a) (i) (ii)

2un 1vn 1 un 1  vn 1 , vn  . un 1  vn 1 2 Show that un  vn for n = 0, 1, 2, … . Deduce that

un 

is monotonic decreasing and

vn 

is

monotonic increasing. (iii)

Show that lim un and lim vn exist. n 

n 

(5 marks) (b) (i)

Prove that un  vn 

1 (u0  v0 ) for n = 0, 1, 2, … . 2n

(ii)

Prove that lim un  lim vn .

(iii)

Evaluate lim(un vn ) and lim un .

n 

n 

n 

n 

(10 marks)

P. 133


HKAL Pure Mathematics Past Paper Topic: Sequences

22. (91I07) Given that

an 

is an increasing sequence of positive numbers

and lim an  L . Suppose sequences n 

bn  , cn 

are defined such

that

b1  c1 

1 a1 2

1 and bn  (an 1  cn 1 ) , cn  an1bn1 for n  2 . 2 (a) Show by induction that (i)

bn 

(ii)

bn  an and cn  an for n  1 .

(b) Show that

and

bn 

cn 

and

are strictly increasing,

cn 

are convergent.

Hence evaluate lim bn and lim cn . n 

n 

(7 marks)

P. 134


HKAL Pure Mathematics Past Paper Topic: Sequences

23. (01I10) a  an 1  2bn 1 a  1 Let  1 and  n , n = 2, 3, 4, … . b  a  b b  1 n n  1 n  1  1 

(a) Show that for any positive integer n , (i)

an , bn > 0 and an 2  2bn 2  (1)n ;

(ii)

(1  2)n  an  bn 2 . (4 marks)

(b) For n = 1, 2, 3, … , define un 

an . bn

un  2 . un  1

(i)

Show that un 1 

(ii)

Show that u2 n1  2 and u2 n  2 .

(iii)

Show that un  2 

3un  4 . 2un  3

Hence show that the sequence {u1 , u3 , u5 ,...} is strictly increasing and the sequence {u2 , u4 , u6 ,...} is strictly decreasing. (iv)

Show that the sequences {u1 , u3 , u5 ,...} and {u2 , u4 , u6 ,...} converge to the same limit. Find this limit. (11 marks)

P. 135


HKAL Pure Mathematics Past Paper Topic: Sequences

24. (02I13) Let {xn} be a sequence of real numbers such that x1  x2 and

3xn2  xn1  2 xn  0 for n = 1, 2, 3, … . (a) (i) (ii)

Show that for n  1 , xn 2  xn  (1)n 

2n 1 ( x1  x2 ) . 3n

Show that the sequence {x1 , x3 , x5 ,...} is strictly decreasing and that the sequence {x2 , x4 , x6 ,...} is strictly increasing. (5 marks)

(b) (i)

(ii)

For any positive integer n , show that x2 n  x2 n1 . Show that the sequences {x1 , x3 , x5 ,...} and {x2 , x4 , x6 ,...} converge to the same limit. (6 marks)

(c) By considering

p n 1

( xn 2  xn ) or otherwise, find lim xn in n 

terms of x1 and x2 . [You may use the fact, without proof, that from (b)(ii), lim xn n 

exists.] (4 marks)

P. 136


HKAL Pure Mathematics Past Paper Topic: Sequences

25. (05I09) Let a1 and b1 be real numbers satisfying a1b1 > 0 . For each n = 1 , 2 , 3 , …. , define 2anbn a 2  bn 2 an 1  n and bn 1  . an  bn an  bn (a) Suppose a1  b1 > 0 . (i)

Prove that an  bn for all n = 1 , 2 , 3 , … .

(ii)

Prove that the sequence {an} is monotonic decreasing and that the sequence {bn} is monotonic increasing.

(iii)

Prove that lim an and lim bn both exist.

(iv)

Prove that lim an  lim bn .

(v)

Find lim(an  bn ) and lim an in terms of a1 and b1 .

n 

n 

n 

n 

n 

n 

(12 marks) (b) Suppose a1  b1 < 0 . Do the limits of the sequences {an} and {bn} exist? Explain your answer. (3 marks) 26. (06I05) n

1 and yn = k 1 n  k

For every positive integer n , define xn =  n 1

1

nk

.

k 1

(a) Prove that the sequence {xn} is strictly increasing and that the sequence {yn} is strictly decreasing. (b) Prove that the sequence {xn} and {yn} converge to the same limit. (7 marks)

P. 137


HKAL Pure Mathematics Past Paper Topic: Sequences

27. (90II11) Let

an 

be a sequence of real numbers such that 0 < a1 < 1

and an1  sin(an ) for all n = 1, 2, … . (a) Making use of the fact that sin x < x for 0 < x < 1 , show that lim an exists and find its value. n 

(7 marks) (b) (i) (ii)

Evaluate lim x 0

x 2  sin 2 x . x 2 sin 2 x

 1 1  Hence find lim   2 . 2 n  a  n 1 an 

(5 marks) n

(c) It is known that if lim xn exists and equals L , then lim n 

n 

x i 1

i

n

also exists and equals L . Use this fact, or otherwise, to

show that lim nan 2 n 

exists and find its value. (3 marks)

P. 138


HKAL Pure Mathematics Past Paper Topic: Sequences

28. (96II13) (a) Let x > 1 and define a sequence

an 

an 

by a1  x and

an 12  1 for n  2 . 2an 1

(i)

Show that an > 1 and an > an + 1 for all n .

(ii)

Show that lim an  1 . n 

(8 marks) (b) Let f : [1, )  R be a continuous function satisfying  x2  1  f ( x)  f    2x 

for all x  1 .

Using (a), show that f (x) = f (1) for all x  1 . (7 marks)

P. 139


HKAL Pure Mathematics Past Paper Topic: Sequences

29. (95I12) Let p > 0 and p  1 .

an 

is a sequence of positive numbers

 a0  2  defined by  , n = 1, 2, 3, … . 1 1 an  p n  p an 1  (a) Prove that lim an  0 if the limit exists. n 

(2 marks) (b) (i) (ii)

If 2  a0  a1  a2  , show that lim an does not exist. n 

If ak 1  ak for some k  1 , show that an1  an for

n  k and deduce that lim an  0 . n 

(4 marks) (c) (i) (ii)

If 0  p  1 , show that lim an does not exist. n

If p  2 , show that lim an  0 . n 

(4 marks) (d) Suppose 1 < p < 2 . (i)

Prove by mathematical induction that an 

2 for n  p 1

0 . (ii)

Prove that lim an  0 . n 

(5 marks)

P. 140


HKAL Pure Mathematics Past Paper Topic: Sequences

30. (91II08) (a) Let f (x) = x  ln(1 + x) . Prove that f (x)  0 for all x > 1 . (3 marks)

1 1 1 (b) Let an  1       ln(n  1) 2 3 n 1 1 1 and bn  1       ln n . 2 3 n Show that

an 

is increasing and

Hence prove that

an 

and

bn 

bn 

is decreasing.

are convergent and have

the same limit. (5 marks) (c) (i)

Using (b) or otherwise, show that 1 1   1  k 1  lim      ln    n  kn  1 kn  2 kn  n    k  where k is a positive integer.

(ii)

1 1  1 1 1   . Evaluate lim 1       n  2n  1 2n   2 3 4 (7 marks)

P. 141


HKAL Pure Mathematics Past Paper Topic: Sequences

31. (05II11) Let fn : R  R be defined by f n ( x)  cos x  cos2 x    cosn1 x , where n  1, 2, 3,  . (a) (i)

(ii)

  Prove that fn is strictly decreasing on 0,  .  2 Prove that the equation f n ( x) = 1 has one and only one

root in

   0,  .  2 (4 marks) (b) For each n = 1, 2, 3,… , let n be the root of the equation

  f n ( x) = 1 in  0,  .  2 2 . 3

(i)

Prove that cos 1 

(ii)

Is the sequence {n} monotonic? Explain your answer.

(iii)

Find lim cos n  n

(iv)

Prove that the sequence {n} is convergent and find

n 

its limit. (11 marks)

P. 142


HKAL Pure Mathematics Past Paper Topic: Sequences

Type 6: Others 32. (91I02) 1 Let f ( x)  . ( x  1)(2  x) Express f (x) into partial fractions. Hence, or otherwise, determine ak and bk ( k  0, 1, 2, ) such that 

f ( x)   ak x k when x  1 k 1 

bk when x  2 . k k 0 x

and f ( x)  

(7 marks)

P. 143


HKAL Pure Mathematics Past Paper Topic: Sequences

33. (92II13) Suppose

ak 

is a sequence of positive numbers such that

a0  a1  1 , and ak  ak 1  ak 2 for k = 2, 3, … . n 1 1 Let   x  and Sn ( x)   ak x k . 3 3 k 0

(a) For k = 0, 1, 2, … , prove that

ak 1  2ak , and deduce that ak  2k . Hence prove that Sn ( x)  3 for n = 0, 1, 2, … . (6 marks) (b) Prove that lim Sn ( x) exists and equals n 

1 . 1  x  x2

[Hint: Put y = x for the case when x < 0 .] (5 marks) (c) Evaluate (i)

(ii)

(iii)

1 ak    5 k 0

k

,

1 (1) ak    5 k 0

k

k

 1  a2 k     25  k 0

,

k

. (4 marks)

P. 144


HKAL Pure Mathematics Past Paper Topic: Limit

Contents: Classification

Chronological Order

Type 1: L’Hospital’s Rule

90II04a

(without Diff. of Integrals)

91II05

Q.

1

Q.

14

1.

90II04a

2.

92II01a

92II01a

Q.

2

3.

93II01

92II01b

Q.

21

4.

94II01

93II01

Q.

3

5.

01II01a

94II01

Q.

4

6.

03II01a

95II01a

Q.

13

7.

05II01a

96II06

Q.

20

8.

98II01

97II05a

Q.

10

9.

99II01

97II05b

Q.

15

10.

97II05a

98II01

Q.

8

11.

02II06a

99II01

Q.

9

12.

04II01a

00II05

Q.

16

13.

95II01a

01II01a

Q.

5

14.

91II05

01II01b

Q.

22

Type 2: L’Hospital’s Rule

02II06a

Q.

11

(with Diff. of Integrals)

02II06b

Q.

19

15.

97II05b

16.

00II05

03II01a

Q.

6

17.

05II01b

03II01b

Q.

23

18.

06II01a

04II01a

Q.

12

19.

02II06b

04II01b

Q.

24

20.

96II06

05II01a

Q.

7

Type 3: Sandwich Theorem

05II01b

Q.

17

21.

92II01b

06II01a

Q.

18

22.

01II01b

06II01b

Q.

25

23.

03II01b

24.

04II01b

25.

06II01b

P. 145


HKAL Pure Mathematics Past Paper Topic: Limit

Type 1: L’Hospital’s Rule (without Diff. of Integrals) 1.

(90II04a)

1  1   .  x tan x 

Evaluate lim  x 0

(3 marks) 2.

(92II01a)

tan x  x . x 0 x 2 sin x

Evaluate lim

(3 marks) 3.

(93II01) Evaluate

x

2 , cos x

(a) lim x

2

(1  mx)n  (1  nx)m , where m, n  2 . x 0 x2

(b) lim

(5 marks) 4.

(94II01) Evaluate (a) lim x 1

1 x , 1 5 x

(b) lim (sec x  tan x) . x

2

(4 marks) 5.

(01II01a) Find

e lim x 0

x

 e x 

2

1  cos 2 x

. (4 marks)

6.

(03II01a) Evaluate

1   1   .  sin x tan x 

Evaluate lim  x 0

(3 marks)

P. 146


HKAL Pure Mathematics Past Paper Topic: Limit

7.

(05II01a) Let f (x) = x x for all x > 0 . Prove that f '( x)  x (1  ln x) . x

xx  x . x 1 ln x  x  1

Hence evaluate lim

(5 marks) 8.

(98II01) Evaluate

e x  1  sin x (a) lim , x 0 x 1

 3e x  2  x (b) lim   . x 0  5  (6 marks) 9.

(99II01)

x sin x . x 0 1  cos x

(a) Evaluate lim

(b) Using (a) or otherwise, evaluate lim (1  cos x) x 0

1 ln x

. (6 marks)

10.

(97II05a) Evaluate lim (sin x) x . x 0

(4 marks) 11.

(02II06a)

1 Find lim   x 0  x 

sin x

. (3 marks)

12.

(04II01a) 1

Evaluate lim (tan 3x  cos 4 x) x . x 0

(3 marks)

P. 147


HKAL Pure Mathematics Past Paper Topic: Limit

13.

(95II01a) 1

 ax  bx 1  x (a) Evaluate lim   where a, b > 0 . n  3   (3 marks) 14.

(91II05) 1

 ax 1  x Find lim   for each of the following cases: x   a 1  (a) 0 < a < 1 , (b) a > 1 (6 marks) Type 2: L’Hospital’s Rule (with Diff. of Integrals 15.

(97II05b)

1 3 x

Evaluate lim  x 0

x 0

et dt  2

1   . x2  (3 marks)

16.

(00II05) Let k be a positive integer. Evaluate (a)

d x cos t 2dt , dx  0

(b)

d y2 k cos t 2dt ,  0 dy

(c) lim y 0

1 y 2k

y2 k

cos t 2dt .

0

(6 marks) 17.

(05II01b) Evaluate

 lim

x 0

t sin(sin t ) dt x3

x 0

. (2 marks)

18.

(06II01a) Evaluate

 lim x 0

3 x 1 1

t 5  t 3  1 dt ln( x  1)

. (4 marks)

P. 148


HKAL Pure Mathematics Past Paper Topic: Limit

19.

(02II06b)

 sin t  Let f (t )   t  1 Find

 lim

x 0

if t  0 , if t  0 .

f (t ) dt  x x3

x 0

. (3 marks)

20.

(96II06) Let f (x) be a function with continuous second order derivative.

1 f (t ) dt  ( x  a)(f ( x)  f (a)) f ''(a) a 2  (a) Prove that lim . 3 x a ( x  a) 12

x

(b) Suppose there exists a constant K such that , for all x and a ,

1 f (t ) dt  ( x  a)(f ( x)  f (a))  K ( x  a) 4 . a 2 x

Show that f is a polynomial of degree not greater than 1 . (6 marks) Type 3: Sandwich Theorem 21.

(92II01b) Prove that | x sin

1 |  | x | for all x  0 . x

1 1  sin x . Hence evaluate lim x x 0 1 1  sin x x (3 marks) 22.

(01II01b) Prove that lim x 2 cos x 0

1 0 . x (2 marks)

23.

(03II01b)

x  cos x . x  x  cos x

Evaluate lim

(3 marks)

P. 149


HKAL Pure Mathematics Past Paper Topic: Limit

24.

(04II01b) Evaluate lim (cos 2004  x  cos x ) . x 

(4 marks) 25.

(06II01b) Evaluate lim sin x sin x 0

1 . x (2 marks)

P. 150


HKAL Pure Mathematics Past Paper Topic: Differentiation

Contents: Classification

Chronological Order

Type 1: Differentiability

90II12

Q.

33

1.

99II05

90II13

Q.

14

2.

00II08

91II07

Q.

9

3.

02II02

91II10

Q.

34

4.

04II02

91II13

Q.

16

5.

96II04

92II07

Q.

17

6.

05II02

92II08

Q.

31

7.

97II06

93II07

Q.

13

8.

01II06

93II08

Q.

35

9.

91II07

93II13

Q.

49

Type 2: Higher Derivatives

94II10

Q.

38

10.

95II06

94II12

Q.

22

11.

96II01

95II06

Q.

10

12.

99II04

95II09

Q.

27

13.

93II07

96II01

Q.

11

14.

90II13

96II04

Q.

5

15.

06II09

96II08

Q.

25

16.

91II13

96II10

Q.

41

Type 3: Functional Equations

97II06

Q.

7

17.

92II07

97II08

Q.

39

18.

02II03

98II08

Q.

36

19.

03II08

98II13

Q.

48

20.

05II08

99II04

Q.

12

21.

00II04

99II05

Q.

1

22.

94II12

99II08

Q.

32

Type 4: Curve Sketching

99II13

Q.

42

Set 1

00II04

Q.

21

23.

06II07

00II08

Q.

2

24.

02II08

00II09

Q.

26

25.

96II08

00II12

Q.

46

26.

00II09

00II14

Q.

40

27.

95II09

01II06

Q.

8

28.

03II07

01II08

Q.

37

29.

04II07

01II11

Q.

44

30.

05II07

02II02

Q.

3

Set 2

02II03

Q.

18

P. 151


HKAL Pure Mathematics Past Paper Topic: Differentiation

31. \ 92II08

02II08

Q.

24

32.

99II08

02II12

Q.

47

Set 3

03II07

Q.

28

33.

90II12

03II08

Q.

19

34.

91II10

04II02

Q.

4

35.

93II08

04II07

Q.

29

36.

98II08

04II12

Q.

43

37.

01II08

05II02

Q.

6

38.

94II10

05II07

Q.

30

39.

97II08

05II08

Q.

20

Type 5: Monotoncity

06II07

Q.

23

00II14

06II09

Q.

15

Type 6: Mean Value Theorem

06II11

Q.

45

40.

Set 1 41.

96II10

42.

99II13 Set 2

43.

04II12

44.

01II11

45.

06II11 Set 3

46.

00II12

47.

02II12 Set 4

48.

98II13

49.

93II13

P. 152


HKAL Pure Mathematics Past Paper Topic: Differentiation

Type 1: Differentiability 1.

(99II05)

1  2  x cos Let f ( x)   x  ax  b

if x  0 , if x  0

be differentiable at x = 0 . Find a and b . (6 marks) 2.

(00II08)

 x 2  bx  c if x  0 ,  Let f ( x)   sin x  2 x if x  0 .   x (a) If f is continuous at x = 0 , find c . (b) If f (0) exists, find b . (6 marks) 3.

(02II02)

  if x  , ax 2 Let f ( x)   ebx sin x if x   .  2 b  a , show that e2 . 2 2  Furthermore, if f is differentiable at , find the values of a and b . 2

If f is continuous at

(5 marks) 4.

(04II02)

 x 2  ax  b when x  1 ,  Let f ( x)   sin  x when x  1 .    If f is differentiable at 1 , find a and b . (6 marks)

P. 153


HKAL Pure Mathematics Past Paper Topic: Differentiation

5.

(96II04)

1  3  x sin Let f ( x)   x  0

for x  0

.

for x  0

(a) Evaluate f (x) for x  0 . (b) Prove that f (0) exists. (c) Is f (x) continuous at x = 0 ? Explain. (6 marks) 6.

(05II02) Let a be a constant and f : R  R be defined by

 x2  x  a when x   ,  f ( x)   2 a cos x when x   .  It is known that f is continuous everywhere. (a) Prove that a 

 4

.

(b) Prove that f is differentiable at  . (c) Is f  continuous at  ? Explain your answer. (7 marks) 7.

(97II06) Let f ( x)  x x . (a) Find f (x) for x > 0 and x < 0 respectively. (b) Prove that f (0) exists. (c) Prove that f (x) is continuous at x = 0 . (6 marks)

8.

(01II06)

 x3  12 x when x  2 ,  Let f : R  R be defined by f ( x)   x x  2  x when x  2 . e e (a) Show that lim f' ( x)  lim f' ( x)  0 . x 2

x 2

(b) Is f differentiable at x = 2 ? Explain your answer. (5 marks)

P. 154


HKAL Pure Mathematics Past Paper Topic: Differentiation

9.

(91II07) Let f : R  R . (a) If c  R and f ( x)  f (c)  ( x  c) 2 for all x  R , prove that

f' (c)  0 . (b) If f ( x)  f ( y)  ( x  y) 2 for all x , y  R , prove that f is a constant function. (5 marks) Type 2: Higher Derivatives 10.

(95II06)

 x 1  Let r be a real number. Define y    for x > 1 .  x 1  r

(a) Show that

dy 2ry  . dx x 2  1

(b) For n = 1, 2, 3 … , show that

( x2  1) y ( n1)  2(nx  r ) y ( n)  (n2  n) y ( n1)  0 , (0) where y  y and y ( k ) 

dk y for k  1 . dx k (5 marks)

11.

(96II01) Let f ( x)  x e

n ax

Evaluate f

(2 n )

where a is real and n is a positive integer.

(0) . (4 marks)

12.

(99II04) Let f ( x) 

2x . x 1 2

(a) Resolve f (x) into partial fractions. (b) Find f

(n)

(0) , where n = 1, 2, 3, … . (5 marks)

P. 155


HKAL Pure Mathematics Past Paper Topic: Differentiation

13.

(93II07) Let n be a positive integer and u(x) be a function such that

u' ( x), u'' ( x), , u ( n ) ( x) exist. (a) Given that y( x)  u( x)e

qx

, where q is a real number, express

y( n ) ( x) in terms of u( x), u' ( x), u'' ( x), , u ( n) ( x) . (b) By putting u( x)  e

px

, where p is a real number, use (a) to prove

n the Binomial Theorem, i.e. ( p  q) 

n

C r 0

n r

p r q n r . (6 marks)

P. 156


HKAL Pure Mathematics Past Paper Topic: Differentiation

14.

(90II13) Let f ( x) 

1 1  x2

for all x  R .

Let f ( n ) denote the nth derivative of f for n = 1, 2, … , and f (0)  f . (a) Prove that

(1  x2 )f' ( x)  x f ( x)  0 . Deduce that

(1  x2 )f ( n1) ( x)  (2n  1) x f ( n) ( x)  n2 f ( n1) ( x)  0 for n = 1, 2, … . (2 marks) (b) Define Pn ( x)  (1  x 2 ) (i)

n

1 2

f ( n ) ( x) for n = 0, 1, 2, … .

For n = 0, 1, 2, … , prove that

Pn1 ( x)  (1  x2 )Pn' ( x)  (2n  1) x Pn ( x) . Deduce that Pn(x) is a polynomial of degree n with leading coefficient (1) n ! . n

(ii)

For n = 1, 2, … , show that

Pn1 ( x)  (2n  1) x Pn ( x)  n2 (1  x 2 )Pn1 ( x)  0 and find Pn(0) . (iii)

For n = 1, 2, … , prove that

Pn' ( x)  n2 Pn1 ( x) and deduce that , for r = 1, 2, … , n ,

Pn( r ) ( x)  (1)r (n(n 1)(n  r  1))2 Pnr ( x) . (iv)

Hence show that Pn ( x) is either an odd function or an even function for n  1, 2, ... . (Note:

Pn ( r ) (0) is the coefficient of xr in Pn(x) .) r! (13 marks)

P. 157


HKAL Pure Mathematics Past Paper Topic: Differentiation

15.

(06II09) (a) Prove that lim x 0

xn e

1 x

 0 for any positive integer n . (3 marks)

 0 (b) Let f ( x)   1 x e

when x  0 , when x  0 .

(i)

Find f (x) for all x  0 .

(ii)

Prove that f (0) = 0 . Hence prove that f (x) is continuous at x = 0 . 1

(iii)

1  for any x

For any x > 0 , prove that f (n)(x) = e x p n 

positive integer n , where p n (t ) is a polynomial in t . (iv)

Prove that f (n)(0) = 0 for any positive integer n . (12 marks)

P. 158


HKAL Pure Mathematics Past Paper Topic: Differentiation

16.

(91II13) 2 n (n) For n = 0, 1, 2, … , let R n ( x)  ( x  1) and Pn ( x)  R n ( x) .

(a) (i)

Show that Pn(x) is a polynomial in x of degree n ,

n  0, 1, 2,  . (ii)

Prove by induction that every polynomial of degree n can be expressed as

n

  P ( x) j 0

j

j

where  j  R . (4 marks)

(b) Prove by induction that

(1  x2 )R (nk 2) ( x)  2 x(n  k  1)R (nk 1) ( x)  (k  1)(2n  k )R (nk ) ( x)  0 for k = 0, 1, 2, … . Hence deduce that

[(1  x2 )Pn' ( x)]'  n(n  1)Pn ( x) ……… (*) (5 marks) (c) (i)

Using (*) or otherwise, show that if m  0 , n  0 , then 1

1

1

1

n(n  1)  Pm ( x)Pn ( x) dx   (1  x 2 ) Pn' ( x)Pm' ( x) dx . (ii)

Deduce that if m , n are distinct non-negative integers, then

1

P ( x)Pn ( x) dx  0 .

1 m

(6 marks)

P. 159


HKAL Pure Mathematics Past Paper Topic: Differentiation

Type 3: Functional Equations 17.

(92II07) Let f be a differentiable function such that

f ( x  y)  f ( x)  f ( y)  3xy( x  y) for all x , y  R .

f ( h) . h 0 h

(a) Show that f' (0)  lim

(b) Hence, or otherwise, show that for all x  R , f (x) = f (0) + 3x2 , and deduce that f (x) = f (0) x + x3 . (6 marks) 18.

(02II03) Let f : R  R be a continuous function satisfying the following conditions:

f ( x)  1 1 ; x 0 x

(i)

lim

(ii)

f (x + y) = f (x) f (y)

for all x and y .

(a) Prove that f (x) exists and f (x) = f (x) for every x  R . (b) By considering the derivative of

f ( x) x , show that f ( x)  e . x e (5 marks)

P. 160


HKAL Pure Mathematics Past Paper Topic: Differentiation

19.

(03II08) Let f : R  R be a non-constant function satisfying the following conditions: (1)

f (x + y) = f (x) + f (y) + f (x)f (y) for all x , y  R .

(2)

lim

(a) (i) (ii)

f ( h)  a , where a  R . h 0 h

Prove that f (0) 1  f ( x)   0 for all x  R . Prove that f (0) = 0 and that f (x)  1 for all x  R . (5 marks)

 x x   , or otherwise, prove that 2 2

(b) By considering f ( x)  f 

f ( x)  1 for all x  R . (2 marks) (c) Prove that f is differentiable everywhere and that

f '( x)  a 1  f ( x)  for all x  R . Deduce that a  0 . (4 marks) (d) By considering the derivative of the function ln 1  f ( x)  , prove that f ( x)  e  1 for all x  R . ax

(4 marks)

P. 161


HKAL Pure Mathematics Past Paper Topic: Differentiation

20.

(05II08) Let f : R  R be a function satisfying the following conditions: (1)

f ( x  y)  e x f ( y)  e y f ( x) for all x , y  R ;

(2)

lim

f ( h) = 2005 . h 0 h

(a) Find f (0) . (1 mark) (b) Find lim f (h) . Hence prove that f is a continuous function. h 0

(4 marks) (c) (i)

Prove that f is differentiable everywhere and that

f '( x)  2005e x  f ( x) for all x  R .

(ii)

Let n be a positive integer. Using (c)(i), find f

(n)

(0) .

(6 marks)

f ( x) , find f (x) . ex

(d) By considering the derivative of the function

(4 marks) 21.

(00II04) Let f and g be differentiable functions defined on R and satisfying the following conditions: A.

f (x) = g(x) for x  R ;

B.

g(x) = f (x)

C.

f (0) = 0 and g(0) = 1 .

for x  R ;

By differentiating h( x)  [f ( x)  sin x]  [g( x)  cos x] , or otherwise, 2

2

show that f (x) = sin x and g(x) = cos x for x  R . (5 marks)

P. 162


HKAL Pure Mathematics Past Paper Topic: Differentiation

22.

(94II12) Let f : R  R be a continuously differentiable function satisfying the following conditions for all x  R . A.

f (x) > 0 ;

B.

f (x + 1) = f (x) ;

C.

x x 1 f ( )f ( )  f ( x) . 4 4

Define g( x) 

d ln f ( x) for x  R . dx

(a) Show that for all x  R , (i)

f (x + 1) = f (x) ;

(ii)

g(x + 1) = g(x) ;

(iii)

1 x x 1 [g( )  g( )]  g( x) . 4 4 4 (8 marks)

(b) Let M be a constant such that g( x)  M for all x [0,1] . (i)

Using (a), or otherwise, show that g( x) 

M for all 2

xR . Hence deduce that g( x)  0 for all x  R . (ii)

Show that f (x) = 1 for all x  R . (7 marks)

P. 163


HKAL Pure Mathematics Past Paper Topic: Differentiation

Type 4: Curve Sketching Set 1 23.

(06II07)

x2  x  6 Let f ( x)  x6

(x  6) .

(a) Find f (x) and f (x) . (2 marks) (b) Solve each of the following inequalities: (i)

f (x) > 0 ,

(ii)

f (x) < 0 ,

(iii)

f (x) > 0 ,

(iv)

f (x) < 0 . (3 marks)

(c) Find the relative extreme point(s) of the graph of y = f (x) . (2 marks) (d) Find the asymptote(s) of the graph of y = f (x) . (3 marks) (e) Sketch the graph of y = f (x) . (2 marks) (f) Find the area of the region bounded by the graph of y = f (x) and the x-axis. (3 marks)

P. 164


HKAL Pure Mathematics Past Paper Topic: Differentiation

24.

(02II08) Let f ( x)  x 2 

8 x 1

(x  1) .

(a) Find f (x) and f (x) . (2 marks) (b) Determine the range of values of x for each of the following cases: (i)

f (x) > 0 ,

(ii)

f (x) < 0 ,

(iii)

f (x) > 0 ,

(iv)

f (x) < 0 . (3 marks)

(c) Find the relative extreme point(s) and point(s) of inflexion of f (x) . (2 marks) (d) Find the asymptote(s) of the graph of f (x) . (1 marks) (e) Sketch the graph of f (x) . (2 marks) (f) Let g(x) = f (|x|)

(x  1) .

(i)

Is g(x) differentiable at x = 0 ? Why?

(ii)

Sketch the graph of g(x) . (5 marks)

P. 165


HKAL Pure Mathematics Past Paper Topic: Differentiation

25.

(96II08) Let f ( x) 

( x  1)3 . ( x  1) 2

(a) Find f (x) and f (x) for x  1 . (2 marks) (b) Determine the values of x for each of the following cases: (i)

f (x) > 0 ,

(ii)

f (x) < 0 ,

(iii)

f (x) > 0 ,

(iv)

f (x) < 0 . (3 marks)

(c) Find the relative extreme point(s) and point(s) of inflexion of f (x) . (2 marks) (d) Find the asymptote(s) of the graph of f (x) . (2 marks) (e) Sketch the graph of f (x) . (2 marks) (f) Let g( x)  f ( x) . Does g(1) exist? Find the asymptote(s) and sketch the graph of g(x) . (4 marks)

P. 166


HKAL Pure Mathematics Past Paper Topic: Differentiation

26.

(00II09) Let f ( x) 

x . (1  x 2 )2

(a) Find f (x) and f (x) . (2 marks) (b) Determine the values of x for each of the following cases: (i)

f (x) > 0 ,

(ii)

f (x) > 0 . (3 marks)

(c) Find all relative extreme points, points of inflexion and asymptotes of y  f ( x) . (4 marks) (d) Sketch the graph of f (x) . (3 marks) (e) Let g( x)  f ( x) . (i)

Does g(0) exist? Why?

(ii)

Sketch the graph of g(x) . (3 marks)

P. 167


HKAL Pure Mathematics Past Paper Topic: Differentiation

27.

(95II09) Let f ( x) 

| x| , where x  1 . ( x  1) 2

(a) (i)

Find f (x) and f (x) for x > 0 .

(ii)

Find f (x) and f (x) for x < 0 .

(iii)

Show that f (0) does not exist. (4 marks)

(b) Determine the values of x for each of the following cases: (i)

f (x) < 0 ,

(ii)

f (x) > 0 ,

(iii)

f (x) < 0 ,

(iv)

f (x) > 0 . (4 marks)

(c) Find the relative extreme point(s) and point(s) of inflexion of f (x) . (3 marks) (d) Find the asymptote(s) of and sketch the graph of f (x) . (4 marks)

P. 168


HKAL Pure Mathematics Past Paper Topic: Differentiation

28.

(03II07) Let f ( x) 

x | x  1| x2

(a) (i)

Find f (x) for x  1 .

( x  2 ) .

(ii)

Is f differentiable at 1 ? Explain your answer.

(iii)

Find f (x) for x  1 . (4 marks)

(b) Determine the range of values of x for each of the following cases: (i)

f (x) > 0 ,

(ii)

f (x) < 0 ,

(iii)

f (x) > 0 ,

(iv)

f (x) < 0 . (3 marks)

(c) Find the relative extreme point(s) and point(s) of inflexion of f (x) . (3 marks) (d) Find the asymptote(s) of the graph of f (x) . (3 marks) (e) Sketch the graph of f (x) . (2 marks)

P. 169


HKAL Pure Mathematics Past Paper Topic: Differentiation

29.

(04II07)

| x | x3 Let f ( x)  3 x 2 (a) (i)

(x 3 2) .

Find f (x) and f (x) for x > 0 .

(ii)

Write down f (x) and f (x) for x < 0 .

(iii)

Prove that f (0) exists.

(iv)

Does f (0) exist? Explain your answer. (5 marks)

(b) Determine the range of values of x for each of the following cases: (i)

f (x) > 0 ,

(ii)

f (x) < 0 ,

(iii)

f (x) > 0 ,

(iv)

f (x) < 0 . (3 marks)

(c) Find the relative extreme point(s) and point(s) of inflexion f (x) . (2 marks) (d) Find the asymptote(s) of the graph of f (x) . (3 marks) (e) Sketch the graph of f (x) . (2 marks)

P. 170


HKAL Pure Mathematics Past Paper Topic: Differentiation

30.

(05II07) Define f ( x)   x  | x | (a) (i) (ii)

x for x < 2 or x > 0 . x2

Find f (x) and f (x) for x > 0 . Write down f (x) and f (x) for x < 2 . (4 marks)

(b) Solve each of the following inequalities: (i)

f (x) > 0 ,

(ii)

f (x) > 0 . (3 marks)

(c) Find the relative extreme point(s) of the graph of y = f (x) . (2 marks) (d) Find the asymptote(s) of the graph of y = f (x) . (4 marks) (e) Sketch the graph of y = f (x) . (2 marks)

P. 171


HKAL Pure Mathematics Past Paper Topic: Differentiation

Set 2 31. \ (92II08) Let f ( x)  xe x for x  R . 2

(a) Find f (x) and f (x) . (2 marks) (b) Determine the values of x for each of the following cases: (i)

f (x) = 0 ,

(ii)

f (x) > 0 ,

(iii)

f (x) < 0 ,

(iv)

f (x) = 0 ,

(v)

f (x) > 0 ,

(vi)

f (x) < 0 . (3 marks)

(c) Find all relative extrema and points of inflexion of f (x) . (3 marks) (d) Find the asymptote of the graph of f (x) . (1 mark) (e) Sketch the graph of f (x) . (3 marks) (f) Hence sketch the curve x  y  ( x  y )e

1  ( x  y )2 2

. (3 marks)

P. 172


HKAL Pure Mathematics Past Paper Topic: Differentiation

32.

(99II08) 1

Let f ( x)  xe x for x  0 . (a) Find lim f ( x) and show that f (x)   as x  0 + . x 0

(3 marks) (b) Find f (x) and f (x) for x  0 . (2 marks) (c) Determine the values of x for each of the following cases: (i)

f (x) > 0 ,

(ii)

f (x) > 0 . (3 marks)

(d) Find all relative extrema of f (x) . (2 marks) (e) Find all asymptotes of the graph of f (x) . (3 marks) (f) Sketch the graph of f (x) . (2 marks)

P. 173


HKAL Pure Mathematics Past Paper Topic: Differentiation

Set 3 33.

(90II12) 2

Let f ( x)  (2 x  1) x 3 for x  R . (a) Find f (x) and f (x) for x  0 . (2 marks) (b) Show that f (0) does not exist. (1 mark) (c) Determine those values of x such that (i)

f (x) = 0 ,

(ii)

f (x) > 0 ,

(iii)

f (x) < 0 ,

(iv)

f (x) = 0 ,

(v)

f (x) > 0 ,

(vi)

f (x) < 0 . (3 marks)

(d) Find the relative extrema and points of inflexion of the function. (4 marks) (e) Show that the graph of the function has no asymptotes. (2 marks) (f) Using the result of (a), (b), (c), (d) and (e), sketch the graph of the function. (3 marks)

P. 174


HKAL Pure Mathematics Past Paper Topic: Differentiation

34.

(91II10) Let f ( x) 

3

x3  x 2  x  1 .

(a) Find the roots of f (x) = 0 . (1 mark) (b) Find f (x) for x  1, 1 . Show that both f (1) and f (1) do not exist. (3 marks) (c) Determine the values of x for each of the following cases: (i)

f (x) = 0 ,

(ii)

f (x) > 0 ,

(iii)

f (x) < 0 , (3 marks)

(d) Find the relative minimum, the relative maximum and the point of inflexion of f (x) . (4 marks) (e) Find the asymptote of the graph of f (x) . (2 marks) (f) Sketch the graph of f (x) . (2 marks)

P. 175


HKAL Pure Mathematics Past Paper Topic: Differentiation

35.

(93II08) Let f ( x) 

3

x 2  x3 .

(a) Find f (x) and f (x) . (2 marks) (b) Show that both f (0) and f (1) do not exist. (2 marks) (c) Determine the sets of values of x such that: (i)

f (x) = 0 ,

(ii)

f (x) > 0 ,

(iii)

f (x) < 0 ,

(iv)

f (x) = 0 ,

(v)

f (x) > 0 ,

(vi)

f (x) < 0 . (3 marks)

(d) Find the relative extreme point(s) and the point(s) of inflexion on the curve y  f ( x) . (3 marks) (e) Find the asymptote(s) of the curve y = f (x) . (3 marks) (f) Sketch the curve y = f (x) . (2 marks)

P. 176


HKAL Pure Mathematics Past Paper Topic: Differentiation

36.

(98II08) 1

2

Let f ( x)  x 3 ( x  1) 3 . (a) (i) (ii)

Find f (x) for x  1, 0 . Show that f'' ( x) 

2 5 3

9 x ( x  1)

4 3

for x  1, 0 .

(2 marks) (b) Determine with reasons whether f (1) and f (0) exist or not. (2 marks) (c) Determine the values of x for each of the following cases: (i)

f (x) > 0 ,

(ii)

f (x) < 0 ,

(iii)

f (x) > 0 ,

(iv)

f (x) < 0 . (3 marks)

(d) Find all relative extrema and points of inflexion of f (x) . (3 marks) (e) Find all asymptotes of the graph of f (x) . (2 marks) (f) Sketch the graph of f (x) . (3 marks)

P. 177


HKAL Pure Mathematics Past Paper Topic: Differentiation

37.

(01II08) 2

1

Let f ( x)  x 3 (6  x) 3 . (a) (i)

Find f (x) for x  0, 6 .

(ii)

Show that f (0) and f (6) do not exist.

(iii)

Show that f ''( x) 

8 4 3

x (6  x)

5 3

for x  0, 6 .

(4 marks) (b) Determine the values of x for each of the following cases: (i)

f (x) > 0 ,

(ii)

f (x) < 0 ,

(iii)

f (x) > 0 ,

(iv)

f (x) < 0 . (3 marks)

(c) Find all relative extreme points and points of inflexion of f (x) . (3 marks) (d) Find all asymptotes of the graph of f (x) . (2 marks) (e) Sketch the graph of f (x) . (3 marks)

P. 178


HKAL Pure Mathematics Past Paper Topic: Differentiation

38.

(94II10) Let f ( x) 

x2 ,xR . x2  1

(a) (i)

Evaluate f (x) for x  0 .

3

Prove that f (0) does not exist. (ii)

Determine those values of x for which f (x) > 0 and those values of x for which f (x) < 0 .

(iii)

Find the relative extreme points of f (x) . (8 marks)

(b) (i)

Evaluate f (x) for x  0 . Hence determine the points of inflexion of f (x) .

(ii)

Find the asymptote of the graph of f (x) . (4 marks)

(c) Using the above results, sketch the graph of f (x) . (3 marks)

P. 179


HKAL Pure Mathematics Past Paper Topic: Differentiation

39.

(97II08) 2

x3 Let f ( x)  x 1 (a) (i)

(x  1) .

Find f (x) for x  1, 0 . Does f (0) exist? Explain.

(ii)

Show that f ''( x) 

2(2 x 2  8 x  1) 4 3

9 x ( x  1)

for x  1, 0 .

3

(4 marks) (b) Determine the values of x for each of the following cases: (i)

f (x) > 0 ,

(ii)

f (x) < 0 ,

(iii)

f (x) > 0 ,

(iv)

f (x) < 0 . (4 marks)

(c) Find the relative extrema and points of inflexion of f (x) . (3 marks) (d) Find the asymptote(s) of the graph of f (x) . (1 marks) (e) Sketch the graph of f (x) . (3 marks)

P. 180


HKAL Pure Mathematics Past Paper Topic: Differentiation

Type 5: Monotoncity 40.

(00II14) 1  x x x    a  b Let a > b > 0 and define f ( x)   2   ab 

(a) (i)

for x  0 , for x  0 .

Evaluate lim f ( x) . x 0

Hence show that f is continuous at x = 0 . (ii)

Show that lim f ( x)  a . x 

(6 marks) (b) Let h(t )  (1  t )ln(1  t )  (1  t )ln(1  t ) for 0  t < 1 and

g( x)  ln f ( x) for x  0 . (i)

Show that h(t) > h(0) for 0 < t < 1 .

(ii)

For x > 0 , let t 

ax  bx , Show that 0 < t < 1 and ax  bx

 a x ln a x  b x ln b x  2 h(t )  2   ln  x x x x a b  a b  (iii)

  . 

Show that for x > 0 ,

x 2g' ( x) 

a x ln a x  b x ln b x  2   ln  x . x x x  a b  a b 

Hence deduce that f (x) is strictly increasing on [0, ) . (9 marks)

P. 181


HKAL Pure Mathematics Past Paper Topic: Differentiation

Type 6: Mean Value Theorem Set 1 41.

(96II10) (a) Let f (x) be a function such that f (x) is strictly decreasing for

x0 . (i)

Using Mean Value Theorem, or otherwise, show that

f '(k  1)  f (k  1)  f (k )  f '(k ) for k  1 . (ii)

Using (i), show that for any integer n  2,

f '(2)  f '(3)    f '(n)  f (n)  f (1)  f '(1)  f '(2)    f '(n  1) . (5 marks) (b) Define H n  1  (i)

1 1 1     for any positive integer n . 2 3 n

Using (a), or otherwise, show that

H n  1  ln n  H n 

1 n

for n  2 .

 Hn   .  ln n 

Hence, evaluate lim  n 

(ii)

Define  n  H n  ln n . Show that

 n 

is a decreasing sequence and lim  n n 

exists. (10 marks)

P. 182


HKAL Pure Mathematics Past Paper Topic: Differentiation

42.

(99II13) Let f (x) be a differentiable function on R such that f '( x)  f ( x) for all x  R . (a) Suppose a  0 and f (a) = 0 . Let x  (a, a + 1) . (i)

Using Mean Value Theorem or otherwise, show that there exists 1  (a, x) such that f ( x)  f (1 ) ( x  a) .

(ii)

Using (a)(i) or otherwise, show that for each n = 1, 2, 3, … , there exists n  (a, x) such that f ( x)  f (n ) ( x  a)n .

(iii)

Using (a)(ii) or otherwise, show that f (x) = 0 for all

x [a, a  1] . You may use the fact that there is M > 0 such that f ( x)  M for all x [a, a  1] . (9 marks) (b) Suppose f (0) = 0 . (i)

Using (a) or otherwise, show that f (x) = 0 for all x  [0, ) .

(ii)

Show that f (x) = 0 for all x  R . (6 marks)

P. 183


HKAL Pure Mathematics Past Paper Topic: Differentiation

Set 2 43.

(04II12) (a) Let f : R  R be a twice differentiable function. Assume that a and b are two distinct real numbers. (i)

Find a constant k ( independent of x ) such that the function h( x)  f ( x)  f (b)  f '( x)( x  b)  k ( x  b)

2

satisfies

h(a) = 0 . Also find h(b) . (ii)

Let I be the open interval with end points a and b . Using Mean Value Theorem and (a)(i), prove that there exists a real number c  I such that

f (b)  f (a)  f '(a)(b  a) 

f'' (c) (b  a) 2 . 2 (7 marks)

(b) Let g : R  R be a twice differentiable function. Assume that there exists a real number   (0, 1) such that g( x)  g(  )  1 for all x  (0,1) . (i)

Using (a)(ii), prove that there exists a real number   (0,1) such that g(1)  1 

(ii)

g'' ( ) (1   ) 2 . 2

If g'' ( x)  2 for all x  (0,1) , prove that g(0)  g(1)  1 . (8 marks)

P. 184


HKAL Pure Mathematics Past Paper Topic: Differentiation

44.

(01II11) (a) Let f and g be real-valued functions continuous on [a, b] and differentiable in (a, b) . (i)

By considering the function

h( x)  f ( x)[g(b)  g(a)]  g( x)[f (b)  f (a)] on [a, b] , or otherwise, show that there is c  (a, b) such that

f '(c)[g(b)  g(a)]  g' (c)[f (b)  f (a)] . (ii)

Suppose g(x) > 0 for all x  (a, b) . Show that

g( x)  g(a)  0 for any x  (a, b) . If, in addition ,

P( x) 

f '( x ) is increasing on (a, b) , show that g' ( x )

f ( x)  f ( a) is also increasing on (a, b) . g( x)  g(a) (9 marks)

 e x cos x  1   if x   0,  ,  (b) Let Q( x)   sin x  cos x  1  4 1 if x  0 . 

 

Show that Q is continuous at x = 0 and increasing on  0,  .  4

  ,  4 

Hence or otherwise, deduce that for x  0,

x 0

Q(t ) dt  x . (6 marks)

P. 185


HKAL Pure Mathematics Past Paper Topic: Differentiation

45.

(06II11) (a) Let f : R  R and g : R  R be continuous on [a, b] and differentiable in (a, b) , where a < b . Suppose that g(a)  g(b) and g(x)  0 for all x  (a, b) . Define

h( x)  f ( x)  f (a) 

f (b)  f (a)   g( x)  g(a)  for all x  R . g(b)  g(a)

(i)

Find h(a) and h(b) .

(ii)

Using Mean Value Theorem, prove that there exists   (a, b) such that

f '(  ) f (b)  f (a) .  g' (  ) g(b)  g(a) (5 marks)

(b) Let u : R  R be twice differentiable. For each x  R , let F : R  R and G : R  R be defined by

( x  t )2 . For each c  x , 2 F' ( ) F(c) prove that there exists   I such that and  G' ( ) G(c)

F(t )  u( x)  u(t )  u' (t )( x  t ) and G(t ) 

u( x)  u(c)  u' (c)( x  c) 

u'' ( ) ( x  c) 2 , where I is the open 2

interval with end points c and x . (5 marks) (c) Let v : R  R be twice differentiable . It is given that

v( x)  2006 . x 0 x

lim (i)

Prove that v(0) = 0 . Hence find v(0) .

(ii)

Suppose that v(x)  2 for all x  R . Prove that v(x)  2006x + x2 for all x  R . (5 marks)

P. 186


HKAL Pure Mathematics Past Paper Topic: Differentiation

Set 3 46.

(00II12) (a) Let f be a real-valued function defined on an open interval I , and f ''( x)  0 for x  I . (i)

Let a, b, c  I with a < c < b . Using Mean Value Theorem or otherwise, show that Hence show that f (c) 

(ii)

f (c)  f (a) f (b)  f (c)  . ca bc bc ca f ( a)  f (b) . ba ba

Let a, b  I with a < b and   (0, 1) , show that

a   a  (1   )b  b . Hence show that f[ a  (1   )b]  f (a)  (1   )f (b) . (8 marks) (b) Let 0 < a < b . Using (a)(ii) or otherwise, show that (i)

if p > 1 and 0 <  < 1 , then

[ a  (1   )b] p   a p  (1   )b p ;

(ii)

if 0 <  < 1 , then  a  (1   )b  a b

 1

. (7 marks)

P. 187


HKAL Pure Mathematics Past Paper Topic: Differentiation

47.

(02II12) (a) Let g(x) be a function continuous on [a, b] , differentiable in (a, b) , with g(x) decreasing on (a, b) and g(a) = g(b) = 0 . Using Mean Value Theorem, show that there exists c  (a, b) such that g is increasing on (a, c) and decreasing on (c, b) . Hence show that g(x)  0 for all x  [a, b] . (5 marks) (b) Let f be a twice differentiable function and f (x)  0 on an open interval I . Suppose a , b , x  I with a < x < b . By considering the function

g( x)  (b  x)f (a)  ( x  a)f (b)  (b  a)f ( x) or otherwise, show that

f ( x) 

bx xa f ( a)  f (b) . ba ba

Hence, or otherwise, prove that f (1 x1  2 x2 )  1f ( x1 )  2f ( x2 ) for all x1 , x2  I , where 1 , 2  0 with 1 + 2 = 1 . (5 marks) (c) Let x1 and x2 be positive numbers. (i)

If 1 , 2  0 with 1 + 2 = 1 , prove that

1 x1  2 x2  x1 x2 . 1

(ii)

2

If 1 , 2 are positive numbers, prove that

 1 x1   2 x2     1   2 

1   2

 x11 x2 2 . (5 marks)

P. 188


HKAL Pure Mathematics Past Paper Topic: Differentiation

Set 4 48.

(98II13)

 

1

Let I  0,  and g( x)  cos x  cos3 x , where x  I . 3  3 Let x0  I and define xn  g( xn1 ) for n = 1, 2, 3, … . (a) Show that the equation x = g(x) has exactly one root in I . (3 marks) (b) Show that xn  I for all n . (3 marks) (c) Show that g' ( x) 

3 for all x  I . 4 (2 marks)

(d) Let  be the root of x = g(x) mentioned in (a).

3 xn 1   for n = 1, 2, 3, … . 4

(i)

Show that xn   

(ii)

Show that {xn} converges and lim xn   .

(iii)

If x0 

n 

 6

xn   

, find a positive integer n such that

1 . 100 (7 marks)

P. 189


HKAL Pure Mathematics Past Paper Topic: Differentiation

49.

(93II13) Let a , b  R and a < b . Let f (x) be a differentiable function on R such that f (a)  0 , f (b)  0 and f (x) is strictly decreasing. (a) Show that f (a) > 0 . (2 marks) (b) Let x0 = a and x1  x0 

f ( x0 ) . f' ( x0 )

Show that a < x1 < b , f (x1) < 0 and f (x1) > 0 . (6 marks) (c) Let x0 = a and xn  xn 1 

f ( xn 1 ) for n = 1, 2, 3, … . f' ( xn 1 )

Show that a < xn < b , f (xn) < 0 and f (xn) > 0 for n = 1, 2, … . (4 marks) (d) Show that lim xn exists and is a zero of f (x) . n 

(3 marks)

P. 190


HKAL Pure Mathematics Past Paper Topic: Integration

Contents: Classification

Chronological Order

Type 1: Evaluation of Integral

90II03

Q.

17

Set 1: Basic

90II04b

Q.

14

1.

97II04

90II05

Q.

29

2.

94II02

90II07

Q.

22

3.

02II01

90II08

Q.

69

4.

01II12

91II03

Q.

39

Set 2: Integration by Parts

91II04

Q.

19

5.

93II05

91II06

Q.

30

6.

96II05

91II09

Q.

63

7.

97II01

91II12

Q.

46

8.

01II02

92II04

Q.

12

9.

04II03

92II05

Q.

27

10.

05II04

92II09

Q.

50

11.

00II01

92II11

Q.

32

12.

92II04

92II12

Q.

65

13.

02II09

93II05

Q.

5

Set 3: Change of Variables

93II11

Q.

72

14.

90II04b

93II12

Q.

55

15.

95II02

94II02

Q.

2

16.

99II02

94II06

Q.

34

17.

90II03

94II08

Q.

68

18.

96II03

94II11

Q.

44

19.

91II04

94II13

Q.

66

20.

95II11

95II01b

Q.

26

Type 2: Evaluation of Infinite Series

95II02

Q.

15

21.

98II03

95II04

Q.

35

22.

90II07

95II07

Q.

31

23.

03II04

95II08

Q.

64

24.

04II04

95II11

Q.

20

25.

06II04

95II13

Q.

73

26.

95II01b

96II02

Q.

67

27.

92II05

96II03

Q.

18

28.

00II06

96II05

Q.

6

P. 191


HKAL Pure Mathematics Past Paper Topic: Integration

Type 3: Differentiation of Integrals

96II12

Q.

47

Set 1

97II01

Q.

7

29.

90II05

97II04

Q.

1

30.

91II06

97II09

Q.

48

31.

95II07

97II11

Q.

38

32.

92II11

97II13

Q.

61

33.

03II09

98II02

Q.

36

Set 2

98II03

Q.

21

94II06

98II05

Q.

53

Set 3

98II09

Q.

45

35.

95II04

98II11

Q.

74

36.

98II02

99II02

Q.

16

37.

01II05

99II09

Q.

57

38.

97II11

99II11

Q.

70

Type 4: Area / Volume

00II01

Q.

11

39.

91II03

00II06

Q.

28

40.

01II07

00II11

Q.

54

41.

06II05

00II13

Q.

71

Type 5: Reduction Formula

01II02

Q.

8

Set 1

01II05

Q.

37

42.

06II03

01II07

Q.

40

43.

04II08

01II10

Q.

49

Set 2

01II12

Q.

4

44.

94II11

02II01

Q.

3

45.

98II09

02II09

Q.

13

46.

91II12

02II10

Q.

75

47.

96II12

02II13

Q.

56

48.

97II09

03II04

Q.

23

49.

01II10

03II09

Q.

33

Set 3

03II10

Q.

62

50.

92II09

03II12

Q.

51

51.

03II12

04II03

Q.

9

52.

05II10

04II04

Q.

24

Type 6: Convergence of Sequence /

04II08

Q.

43

Series

04II10

Q.

58

34.

Set 1 53.

98II05

05II04

Q.

10

54.

00II11

05II10

Q.

52

P. 192


HKAL Pure Mathematics Past Paper Topic: Integration

Set 2

05II12

Q.

59

55.

93II12

06II03

Q.

42

56.

02II13

06II04

Q.

25

57.

99II09

06II05

Q.

41

58.

04II10

06II08

Q.

60

59.

05II12

60.

06II08 Set 3

61.

97II13

62.

03II10 Set 4

63.

91II09

64.

95II08

65.

92II12

66.

94II13 Type 7: Comparison of Integrals Set 1

67.

96II02

68.

94II08

69.

90II08

70.

99II11

71.

00II13 Set 2

72.

93II11

73.

95II13 Type 8: Others

74.

98II11

75.

02II10

P. 193


HKAL Pure Mathematics Past Paper Topic: Integration

Type 1: Evaluation of Integral Set 1: Basic 1.

(97II04) Show that (sin 2 x  sin 4 x    sin 2nx)sin x  sin nx sin(n  1) x . 

Hence, or otherwise, evaluate



2

4

sin 6 x sin 7 x dx . sin x (6 marks)

2.

(94II02) Evaluate

 tan

(a)

(b)

3

x dx ,

x2  x  2 dx . x( x  2) 2 (6 marks)

3.

(02II01)

3 x

 (1  x)(1  x ) dx

Find the indefinite integral

2

Hence evaluate the improper integral

 0

.

3 x dx . (1  x)(1  x 2 )

Out of Syllabus: Improper Integral You may assume the definition:

 0

u

h( x) dx  lim  h( x) dx . u 

0

(6 marks)

P. 194


HKAL Pure Mathematics Past Paper Topic: Integration

4.

(01II12) (a) Evaluate

1 dx ,  x 1

(i)

x

(ii)

x2  1  ( x2  x  1)( x2  x  1) dx .

2

(6 marks)

x 6 r 5 (b) For n = 1, 2, 3, … and 0  x < 1 , define g n ( x)   and r 1 6r  5 n

x 6 r 1 h n ( x)   . For any fixed x  [0, 1) , r 1 6r  1 n

(i)

show that the sequences {g n ( x)} and {h n ( x)} are increasing;

(ii)

deduce that lim g n ( x) and lim h n ( x) exist. n 

n 

(3 marks)

(c) For n = 1, 2, 3, … and 0  x < 1 , define f n ( x) 

 x 6 r 5 x 6 r 1      6r  1  r 1  6r  5 n

and let f ( x)  lim f n ( x) . n 

(i)

For any fixed x  (0, 1) . evaluate f n' ( x) and show that

lim f n' ( x) exists. n 

(ii)

Assuming that f '( x)  lim f n' ( x) for any fixed x  (0, 1) and n 

f is continuous on [0, 1) , show that f '( x) 

1  x2 1  x2  x4

for

x  (0,1) . Hence find f (x) . (6 marks)

P. 195


HKAL Pure Mathematics Past Paper Topic: Integration

Set 2: Integration by Parts 5.

(93II05) Evaluate

e

2x

(sin x  cos x)2 dx . (7 marks)

6.

(96II05) (a) Evaluate (i)

 x ln x dx

(ii)

 

;

 ln x   dx . x 

x 2 ln x (b) Consider the curve y  where 1  x  e .  2 4 [Out of Syllabus: Surface Area] Find the area of the surface generated by rotating the curve about the x-axis. (6 marks) 7.

(97II01) (a) Show that

d x 1 tan  . dx 2 1  cos x

(b) Using (a), or otherwise, find

x  sin x

 1  cos x dx

. (5 marks)

8.

(01II02) Evaluate (a)

x3  1  x2 dx ,

(b)

x

2

tan 1 x dx . (5 marks)

9.

(04II03) (a) Evaluate

 sec  d 3

. (3 marks)

(b) [Out of syllabus: Arc Length] Consider the curve C : x2 = 2y , 0  x  1 . Find the length of C . (4 marks)

P. 196


HKAL Pure Mathematics Past Paper Topic: Integration

10.

(05II04) (a) (i) (ii)

Evaluate

Prove that

1  x dx .

1 2 0

arcsin x 2 dx  2 6  4  . 6 1 x (4 marks)

(b) [Out of syllabus: Surface Area] 11.

(00II01) Evaluate

 x cos x dx

Hence evaluate

2 0

.

x cos x dx . (4 marks)

12.

(92II04) Evaluate

2 0

xe

x 1

dx . (4 marks)

P. 197


HKAL Pure Mathematics Past Paper Topic: Integration

13.

(02II09) (a) Find

e

x

sin x dx .

(b) Let f : R  [0, ) be a periodic function with period T . (i)

Prove that

b  kT a  kT

b

e x f ( x) dx  e kT  e x f ( x) dx for any positive a

integer k . (ii)

Let I n 

nT 0

e x f ( x) dx .

Prove that I n  (iii)

1  e nT I1 for any positive integer n . 1  e T

If  is a positive number and n is a positive integer such that nT    (n  1)T , prove that  1  e nT 1  e ( n1)T x I1   e f ( x) dx  I1 . 0 1  e T 1  e T

Hence find the improper integral

 0

e x f ( x) dx in terms of

I1 and T . (9 marks) (c) Using the result of (a) and (b)(iii), evaluate

 0

e x | sin x | dx . (3 marks)

Out of Syllabus: Improper Integral You may assume the definition:

 0

u

h( x) dx  lim  h( x) dx . u 

0

Set 3: Change of Variables 14.

(90II04b) Evaluate

dx x  4x  2 2

. (3 marks)

P. 198


HKAL Pure Mathematics Past Paper Topic: Integration

15.

(95II02) (a) Using the substitution x  sin 2  ( 0   

 2

) , prove that

f ( x) dx  2 f (sin 2  ) d . x(1  x)

(b) Hence, or otherwise, evaluate

dx and x(1  x)

x dx . 1 x (5 marks)

16.

(99II02) (a) Let f be a continuous function. Show that

 0

f ( x) dx   f (  x) dx . 0

(b) Evaluate

 0

x sin x dx . 1  cos 2 x (6 marks)

17.

(90II03) Suppose f (x) and g(x) are real-valued continuous functions on

[0, a] satisfying the conditions that f ( x)  f (a  x) and g( x)  g(a  x)  K where K is a constant. Show that

a 0

f ( x)g( x) dx 

a 1 K  f ( x) dx . 0 2

Hence, or otherwise, evaluate

 0

x sin x cos 4 x dx . (5 marks)

18.

(96II03) (a) Suppose f (x) is continuous on [0, a] . Show that

a 0

a

f ( x) dx   f (a  x) dx . 0

Further, if f ( x)  f (a  x)  K for all x  [0, a] , where K is a constant, prove that (i) (ii)

a K  2 f( ) ; 2

a 0

a f ( x) dx  a f ( ) . 2

(b) Hence, or otherwise, evaluate

2 0

1 e

sin x

1

dx . (6 marks)

P. 199


HKAL Pure Mathematics Past Paper Topic: Integration

19.

(91II04) Evaluate the following definite integral:

 2 0

d . 1  2sin  (6 marks)

20.

(95II11) (a) Evaluate

 2 0

d . 2sin   cos   2 (5 marks)

(b) Let f ( )  a sin   b cos   c and

g( )  A sin   B cos   C where A , B are not both zero. Show that there exist real numbers p , q and r such that

f ( )  p g( )  q g' ( )  r for all real numbers  . (5 marks) (c) Hence, or otherwise, evaluate

 2 0

7sin   4cos   3 d . 2sin   cos   2 (5 marks)

Type 2: Evaluation of Infinite Series 21.

(98II03) Evaluate

 ln(1  x) dx

.

Hence, or otherwise, find lim n 

n

1

k

 n ln(1  n )

.

k 1

(5 marks) 22.

(90II07) (a) Evaluate

(b) Let un 

 ln(1  x ) dx 2

.

1 2n 1 2n 2 1  k2  2 n ( n  k ) . Prove that ln u  ln 1  2  .   n n4 k 1 k 1 n  n 

Hence, or otherwise, find the value of lim un . n 

(8 marks)

P. 200


HKAL Pure Mathematics Past Paper Topic: Integration

23.

(03II04) Using the substitution t  tan

x , evaluate 2

Hence, or otherwise, evaluate lim

n 

dx

 2  cos x

n

 k 1

1 k   n  2  cos( )  2n  

. .

(7 marks) 24.

(04II04) Using the substitution u 

1 , prove that x

2 1 2

ln x dx  0 . 1  x2

 1 k  ln  2(  )   2 2n  . Hence, or otherwise, evaluate lim  n  1 k   k 1 2n  1  (  ) 2  2 2n   3n

(6 marks) 25.

(06II04) (a) Using the substitution t  1  x 2 , find

x3 1  x2

dx .

1 n k3 .  n  n3 k 1 n2  k 2

(b) Evaluate lim

(7 marks) 26.

(95II01b) By considering a suitable definite integral, evaluate

 12 22 n2  lim  3 3  3 3    3  . n  n  1 n 2 n  n3   (3 marks) 27.

(92II05)

2n 2  k 2 .  3 3 n  k 1 n  k

Using a definite integral, or otherwise, evaluate lim

n

(7 marks) 28.

(00II06)

1 1   1     . nn  n 1 n  2

Use a suitable integral to evaluate lim  n 

(4 marks)

P. 201


HKAL Pure Mathematics Past Paper Topic: Integration

Type 3: Differentiation of Integrals Set 1 29.

(90II05)

d xn f (t ) dt , where f is continuous and n is a positive dx  0

(a) Evaluate integer. (b) If F( x) 

x2 x

3

et dt , find F(1) . 2

(6 marks) 30.

(91II06) (a) Evaluate

2 d u ( 2)t dt .  du 0

(b) Define F( x) 

sec x tan x

( 2)t dt for  2

 2

x

 2

.

Solve F' ( x)  0 . (6 marks) 31.

(95II07) Let f : R  (1, ) be a differentiable function. (a) Differentiate ln[1  f ( x)] . (b) If f ( x)  x3 

x 0

3t 2 f (t ) dt for all x  R , by considering f (x) , find

f ( x) . (6 marks)

P. 202


HKAL Pure Mathematics Past Paper Topic: Integration

32.

(92II11) (a) Let f (x) be a polynomial and n a positive integer such that

deg f ( x)  n . Prove that for any a  R , if f (a)  f '(a)  f ''(a)    f

( n 1)

( a)  0 ,

then f (x) is divisible by (x  a)n . (8 marks) (b) Let p(x) , q(x) , r(x) and s(x) be polynomials and

F( x) 

  p(t)r(t) dt   q(t)s(t) dt     p(t)q(t) dt   r(t)s(t) dt  . x

1

x

x

1

1

x

1

Prove that if deg F(x)  4 , then F(x) is divisible by (x  1)4 . (7 marks)

P. 203


HKAL Pure Mathematics Past Paper Topic: Integration

33.

(03II09) (a) Let u : R  R be a twice differentiable function satisfying the following conditions: (1)

u(x) = u(x)

(2)

u(0) = 0 ,

(3)

u(0) = 1 .

for all x  R .

Define v(x) = u(x)  sin x for all x  R . By differentiating w( x)   v( x)    v' ( x)  , prove that u( x)  sin x 2

2

for all x  R . (5 marks) (b) Let f : R  R and g : R  R be continuous functions such that x

x

0

0

f ( x)  e x  et g (t ) dt and g( x)  e x  e x  et f (t ) dt for all x  R . (i)

Prove that f (x) + f (x) = g(x) for all x  R .

(ii)

(1)

Prove that f (x) + 2f (x) + 2f (x) = 0 for all x  R .

(2)

Let h( x)  e f ( x) for all x  R . x

Prove that h(x) = h(x) . Using (a), find f (x) . (iii)

Find g(x) . (10 marks)

Set 2 34.

(94II06) Let f : R  R be a continuous function. Show that

If

1 0

1 0

x

x f (t ) dt   f (t ) dt for all x  R . 0

f ( xt ) dt  0 for all x  R , show that f (x) = 0 for all x  R . (5 marks)

P. 204


HKAL Pure Mathematics Past Paper Topic: Integration

Set 3 35.

(95II04) For x  0 , define F( x) 

x 0

sin t dt . t 1

(a) Find the value of x0 for which F( x)  F( x0 ) for all x [0, 2 ] . (b) By considering F(0) and F(2 ) , show that F( x)  0 for all

x  (0, 2 ) . (7 marks) 36.

(98II02) Let f : R  R be a continuous periodic function with period T . (a) Evaluate

d dx



x T 0

x

f (t ) dt   f (t ) dt 0

.

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

x T x

T

f (t ) dt   f (t ) dt for all x . 0

(4 marks) 37.

(01II05) Let f be a real-valued function continuous on [0, 1] and differentiable in (0, 1) . Suppose f satisfies A.

f (0) = 0 ,

B.

f (1) =

C.

0 < f (t) < 1 for t  (0, 1) .

Define F( x)  2

x 0

1 , 2

f (t ) dt  f ( x) for x  [0, 1] . 2

(a) Show that F(x) > 0 for x  (0, 1) . (b) Show that

1 0

f (t ) dt 

1 . 8 (6 marks)

P. 205


HKAL Pure Mathematics Past Paper Topic: Integration

38.

(97II11) Define F( x)  (a) (i)

x 1

1 1 t3

dt for any x  1 .

Show that F(x) is strictly increasing.

(ii)

Show that 0 < F(x) < 2 for any x > 1 .

(iii)

Find an x0 such that F(x0) > 1 . (6 marks)

(b) Suppose G(u) is a function on (0, 1) such that F(G(u)) = u .

3 [G(u )]2 . 2

(i)

Show that G' (u)  1  [G(u)]3 and G'' (u ) 

(ii)

Show that G'' (u)  G' (u) .

(iii)

Does the graph of G(u) have any points of inflexion? Explain. (9 marks)

Type 4: Area / Volume 39.

(91II03) Consider the curve

 x  sin 3 t  0  t  , .  3 2  y  cos t (a) [Out of Syllabus: Arc Length] Find the length of the curve. (b) Find the area bounded by the curve, the x-axis and the y-axis. (7 marks) 40.

(01II07) The figure shows the curve

 x  cos3 t , 0  t  2 .  : 3 y  sin t ,  (a) [Out of syllabus: Arc Length] (b) Find the area enclosed by  . (4 marks)

P. 206


HKAL Pure Mathematics Past Paper Topic: Integration

41.

(06II05) (a) Find

 ln y dy

.

(b) Find the volume of the solid of revolution generated by revolving the region bounded by the curve y  2 x and the straight line y = 2

2 about the y-axis. (6 marks) Type 5: Reduction Formula Set 1 42.

(06II03) For any positive integers m and n , define Im, n =

(a) Prove that Im + 2, n + 2

1  1  = n  1  2 

mn

 4 0

sin m  d . cos n 

m 1 Im, n . n 1

(b) Using the substitution u = cos  , evaluate I3, 1 . (c) Using the results of (a) and (b), evaluate I7, 5 . (7 marks) 43.

(04II08) (a) For any non-negative integers m and n , define x

I m,n ( x)   cos m  cos n d for all x  R . 0

Prove that I m1,n 1 ( x) 

cos m1 x sin(n  1) x m 1  I m , n ( x) . mn2 mn2 (6 marks)

(b) Evaluate

 2 0

cos 4  cos 3 d . (4 marks)

(c) Evaluate

 2 0

sin 4  cos 3 d . (5 marks)

P. 207


HKAL Pure Mathematics Past Paper Topic: Integration

Set 2 44.

(94II11) For any non-negative integer n , let I n 

(a) (i)

Show that

1     n 1  4 

n 1

 In 

 4 0

tan n x dx .

1     . n 1  4 

You may assume without proof that

x  tan x 

4x

for x  [0,

4

] .

(ii)

Using (i), or otherwise, evaluate lim I n .

(iii)

Show that I n  I n  2 

n 

1 for n = 2, 3, 4, … . n 1 (8 marks)

(b) For n = 1, 2, 3, … , let an 

(1) k 1 .  k 1 2k  1 n

(i)

Using (a)(iii), or otherwise, express an in terms of I2n .

(ii)

Evaluate lim an . n 

(7 marks)

P. 208


HKAL Pure Mathematics Past Paper Topic: Integration

45.

(98II09) Let I m  (a) (i) (ii)

 2 0

cos m t dt where m = 0, 1, 2, … .

Evaluate I0 and I1 . Show that I m 

m 1 I m2 for m  2 . m

Hence, or otherwise, evaluate I 2n and I 2 n 1 for n  1 . (7 marks) (b) Show that I 2 n1  I 2 n  I 2 n1 for n  1 . (2 marks) 2

1  2  4  6 (2n)  (c) Let An  where n = 0, 1, 2, … . 2n  1 1 3  5 (2n  1)  (i)

Using (a) and (b) , show that

(ii)

Show that

(iii)

Evaluate lim

 An 

n 

2n  1  An   An . 2n 2

is monotonic increasing.

1  2  4  6 (2n)    . 2n  1 1 3  5 (2n  1)  (6 marks)

P. 209


HKAL Pure Mathematics Past Paper Topic: Integration

46.

(91II12) For any non-negative integer n , let I n 

(a) (i)

(ii)

 2 0

cos 2 n 1 x dx .

Evaluate I0 and express In in terms of I n 1 for n  1 .

Show by induction that I n 

(n !) 2 22 n for n = 0, 1, 2, … . (2n  1)! (5 marks)

(n !) 2 2n 1 (b) For any non-negative integer m , let Sm   . n  0 (2n  1)! m

(i)

Show that m 1

1  1   cos 2 x   2  Sm   2 2 cos x  0 1 1  cos 2 x 2 (ii)

Deduce that

 (iii)

dx .

 2 0

 2 cos x  2 cos x dx  m  Sm   2 dx . 0 1 1 2 2 2 1  cos x 1  cos x 2 2

Show that

(n !) 2 2n 1  .  n  0 (2n  1)! 

(10 marks)

P. 210


HKAL Pure Mathematics Past Paper Topic: Integration

47.

(96II12) For non-negative integers k and m , define F(k , m) 

1 0

u k (1  u 2 ) m du .

(a) Show that (i)

F(k , 0) 

1 ; k 1

(ii)

F(k , m) 

2m F(k  2, m  1) for m  1 . k 1 (4 marks)

2m (m!) (b) Show that F(k , m)  . (k  1)(k  3) (k  2m  1) (4 marks) (c) Using (b), prove that

 2 0

cos 2 m1  d 

[2m (m!)]2 . (2m  1)! (4 marks)

(d) Show that F(k , m) 

(1)r Crm .  r  0 2r  k  1 m

(3 marks)

P. 211


HKAL Pure Mathematics Past Paper Topic: Integration

48.

(97II09) Let m , n be non-negative integers. (a) Define I m,n 

1 0

x m (1  x)n dx .

Show that (m  1) I m,n  nI m1,n1 for n  1 . Hence, or otherwise, show that I m,n 

m !n ! . (m  n  1)! (7 marks)

(b) Let f be a real-valued function with continuous derivatives on [0, 1] up to order 2n , where n  1 , and f

(k )

(0)  f ( k ) (1)  0 for

k  0,1, ..., 2n  1 . (i)

Show that

1 0

1

f (2 n ) ( x)g( x)dx  (1) n (2n)!  f ( x)dx , where 0

g( x)  x n (1  x)n . (ii)

Suppose there is a constant M such that f (2 n ) ( x)  M for all x [0,1] . Using (i), or otherwise, deduce that

(n !)2 M  0 f ( x) dx  (2n)!(2n  1)! . 1

(8 marks)

P. 212


HKAL Pure Mathematics Past Paper Topic: Integration

49.

(01II10) (a) Prove by induction that lim x(ln x)n  0 for any non-negative x 0

integer n . (3 marks) (b) Let n be a positive integer.

 (ln x) dx  x(ln x)

 n  (ln x)n1 dx .

(i)

Show that

(ii)

Show that the improper integral (i.e. lim h 0

n

1 h

n

1 0

ln x dx is convergent

ln x dx exists) and find its value.

Hence deduce that the improper integral

1 0

(ln x) n dx is

convergent and find its value. (8 marks) (c) Let n be a positive integer and  be a positive real number. For

0  h  1 , show that

1 h

x 1 (ln x)n dx 

1

n 1

Hence show that the improper integral

 1 h

1 h

(ln x)n dx .

x 1 (ln x)n dx is

convergent and find its value. (4 marks) Out of Syllabus: Improper Integral You may assume the definition:

1 0

1

h( x) dx  lim  h( x) dx if h(0) is undefined. u 0

u

P. 213


HKAL Pure Mathematics Past Paper Topic: Integration

Set 3 50.

(92II09) (a) Let g be a continuously differentiable function and p  1 . Prove that

x 0

x

( x  t ) p g' (t ) dt   x p g(0)  p  g(t )( x  t ) p 1 dt for any 0

xR . (2 marks) (b) For any n = 1, 2, … , and x  R , prove that

ex  1 

x x x2 x n1 1    ( x  t )n1 et dt .  1! 2! (n  1)! (n  1)! 0

Hence or otherwise, show that

 1 1 1 1  3 . (e  )  2  1       e (2n)!  (2n)!  2! 4! (7 marks) (c) (i)

Let f0 be a continuous function. For any n = 1, 2, … , and x

x  R , define f n ( x)   f n1 (t ) dt . 0

Prove that f n ( x) 

(ii)

Evaluate

d100 dx100

x 0

x 1 ( x  t ) n1 f 0 (t ) dt  (n  1)! 0

( x  t )99 sin(t 2 ) dt . (6 marks)

P. 214


HKAL Pure Mathematics Past Paper Topic: Integration

51.

(03II12) (a) Let f : (1, 1)  R be a function with derivatives of any order. For each m  1, 2, 3,... and x  (1, 1) , define

Im 

x 1 ( x  t )m1 f ( m ) (t ) dt .  (m  1)! 0

(i)

f ( m ) (0) m Prove that I m1  I m  x . m!

(ii)

Using (a)(i), prove that m 1

f ( k ) (0) k x  Im k! k 0

f ( x)  

where f (0)  f . (6 marks)

1

(b) Define g( x) 

1  x2

for all x  (1, 1) . Let n be a positive

integer. (i)

Prove that (1  x )g' ( x)  x g( x)  0 . 2

Hence deduce that

(1  x2 )g( n1) ( x)  (2n  1) x g( n) ( x)  n2g( n1) ( x)  0 where g

(0)

g . 2

(0)  0 and g

(ii)

Prove that g

(iii)

Using (a), prove that

(2 n1)

(2 n )

 (2n)!  (0)   n  .  (2 )(n !) 

n 1

x Ck2 k 2 k 1 x  ( x  t )2 n1 g (2 n ) (t ) dt . 2k  0 2 (2 n  1)! k 0

g( x)  

(9 marks)

P. 215


HKAL Pure Mathematics Past Paper Topic: Integration

52.

(05II10) 2 n Define g 0 ( x)  1 and g n ( x)  ( x  1) for every positive integer n .

(a) For every non-negative integer n , let I n 

1 1

g n ( x) dx .

Express I n 1 in terms of I n .

Hence prove that I n 1 

(1)n 1 22 n 3  (n  1)!

2

.

(2n  3)!

(5 marks) (k ) (k ) (b) Prove that g n1 (1)  g n1 (1)  0 for all k = 0, 1,…, n ,

where n is a non-negative integer. (3 marks) (c) For every non-negative integer n , let h n ( x)  (i)

g n ( n ) ( x) . 2n n !

Using (b), prove that

1 1

p( x)h n1 ( x) dx 

1 (1)n1 p( n1) ( x)g n1 ( x) dx n 1  2 (n  1)! 1

for any polynomial p(x) . (ii)

Using (c)(i), or otherwise, evaluate

1 1

x h n ( x)h n1 ( x) dx (7 marks)

Note: For any function f , f

(0)

=f .

P. 216


HKAL Pure Mathematics Past Paper Topic: Integration

Type 6: Convergence of Sequence / Series Set 1 53.

(98II05)

(a) In the figure, using the fact that the shaded area is less than the area of the trapezium ACDB , or otherwise, show that

1 1 1 ln b  ln a  (b  a)(  ) . 2 a b (b) Using (a), or otherwise, show that ln n  1 

1 1 1 n 1    for 2 3 n 2n

n  2, 3, 4,... . State with reasons whether lim(1  n 

1 1 1     ) exists. 2 3 n (6 marks)

P. 217


HKAL Pure Mathematics Past Paper Topic: Integration

54.

(00II11) (a) In the figure, SR is tangent to the curve y = ln x at x = r , where

r  2 . By considering the area of PQRS , show that

1 2 1 r 2 r

ln x dx  ln r .

Hence show that

n 3 2

1 ln x dx  ln(n !)  ln n for any integer n  2 . 2

(5 marks) (b) By considering the graph of y = ln x and a suitable trapezium, show that for r  2 , Hence show that

n 1

r r 1

ln x dx 

1 ln(r  1)  ln r  . 2

1 ln x dx  ln(n !)  ln n for any integer n  2 . 2 (4 marks)

(c) Using integration by parts, find

 ln x dx

. n

1 2 n

1 n e Using the results of (a) and (b), deduce that  e n!

3

 3 2    2e 

for any integer n  2 . (6 marks)

P. 218


HKAL Pure Mathematics Past Paper Topic: Integration

Set 2 55.

(93II12) (a) Show that

1 (1)n t 2 n 2 4 n 1 2 n  2  1  t  t    (1) t  1 t2 1 t2 for all t  R and n = 1, 2, 3, … . Deduce that

tan 1 x  x 

n 2n x (1) t x3 x5 (1)n1 2 n 1    x  dt 0 3 5 2n  1 1 t2

for all x  R and n = 1, 2, 3, … . (4 marks) (b) Using (a), or otherwise, show that

 x3 x5 (1)n 1 2 n 1  x 2 n 1 tan x   x      x  3 5 2n  1   2n  1 1

for all x  0 and n = 1, 2, 3, … .

(1) k Hence find  . k  0 2k  1 

(6 marks) (c) Show that tan 1

1 1   tan 1  . 2 3 4

Deduce that

 1 1  1  1 1  1  1 1  ( 1) n 1  1 1        3  3    5  5      2 n 1  2 n 1  4  2 3  3  2 3  5  2 3  2n  1  2 3  

1 n  22 n 1

for n = 1, 2, 3, … . (5 marks)

P. 219


HKAL Pure Mathematics Past Paper Topic: Integration

56.

(02II13) (a) (i)

Let I n ( )  and 

 2

 0

 

tan n u du , where n is a non-negative integer

 2

.

tan 2 n 1  Show that I 2 n ( )   I 2 n 1 ( ) for all n  1 . 2n  1 (ii)

Using the substitution t = tan u , or otherwise, show that

t 2n x 2 n 1 x 2 n 3 x d t      (1)n1  (1)n tan 1 x for any  0 1 t2 2n  1 2 n  3 1 x

positive integer n . (5 marks) (b) (i)

Let x  0 and n be a positive integer. Prove that 2n x t x 2 n1 x 2 n 1  dt  . (2n  1)(1  x 2 )  0 1  t 2 2n  1

(ii)

Using (a) or otherwise, show that

1  n (1) p 1 1 .    2(2n  1) 4 p 1 2 p  1 2n  1 (iii)

Suppose that tan   show that

1 . Evaluate tan 2 and tan 4 , and 5

1  1   4 tan 1    tan 1   . 4 5  239 

Hence prove that

 4

(1) p 1  4 1  1  4 1    2 p 1   2 n 1   . 2 p 1  239 2392 n 1   (2n  1)  5 p 1 2 p  1  5 n



(10 marks)

P. 220


HKAL Pure Mathematics Past Paper Topic: Integration

57.

(99II09) Let n be a positive integer. (a) Show that

1 t 2n 2 2 n 2 for t 2  1 .  1  t    t    2 2 1 t 1 t (2 marks)

(b) For 1 < x < 1 , show that

t 1 dt  ln , 2 0 1 t 1  x2 x

(i)

(ii)

 x2 x4 t 2 n 1 1 x2n  d t  ln        .  0 1 t2 2n  1  x2  2 4 x

(7 marks)

(c) Show that 0  ln 3 

k

1 8 9 8       2n  2  9  k 1 2k  9  n

1 8 Hence evaluate    k 1 2k  9 

n 1

.

k

. (6 marks)

P. 221


HKAL Pure Mathematics Past Paper Topic: Integration

58.

(04II10) (a) (i) (ii)

dx . 2 x  x2

Prove that

 (iii)

 1

Evaluate

1 2 0

2 2 dx   . 1  2 x  x2

Using (a)(ii), deduce that

1 2 0

2 2  4 x  2 2 x2 dx   . 1  x4 (6 marks)

(b) (i)

Let k be a non-negative integer. Prove that k

 x 4 k  4   (1)n x 4 n  n 0

1  x4k 4 1  x4

for all real numbers x . (ii)

Using (b)(i) and (a)(iii), or otherwise, prove that 

1 n  2 2 1       .  4n  1 4 n  2 4 n  3 

( 4 ) n 1

Candidates may use the fact, without proof, that for any given polynomial p(x) ,

lim  k 

1 2 0

x k p( x)dx  0 . (9 marks)

P. 222


HKAL Pure Mathematics Past Paper Topic: Integration

59.

(05II12) (a) Let f : R  R be a function with derivatives of any order. For each m = 0, 1, 2, … and x  R , define

f ( k ) (0) k E m ( x )  f ( x)   x , where f (0)  f . k! k 0 m

(i)

Using integration by parts, prove that

1 x ( x  t ) m f ( m1) (t ) dt .  0 m!

E m ( x)  (ii)

Suppose that there is a constant C such that f ( k ) ( x)  C for all k = 0, 1, 2,… and x  R . Using (a)(i), prove that

E m ( x) 

C m 1 . x (m  1)! (6 marks)

(b) Let n be a non-negative integer. (i)

Using (a)(i), prove that

(1)k x 2 k 1 (1)n 1 x  ( x  t )2 n1 sin t dt  0 (2n  1)! k  0 (2k  1)! n

sin x  

for all x  R . (ii)

Using (b)(i) and the identity sin 3 x 

3 1 sin x  sin 3x , 4 4

prove that

1   (1) k 1 1  k  1 1 1 1    9   sin 3   1  2 n 1  . 3 4 k 0 (2k  1)! 4(2n  2)!  3  n

(9 marks)

P. 223


HKAL Pure Mathematics Past Paper Topic: Integration

60.

(06II08) (a) (i)

For any two positive integers m and n , evaluate

 (ii)

 0

cos mx cos nx dx .

For any positive integer n , evaluate

 0

 0

cos nx dx and

cos 2 nx dx . (5 marks)

(b) Using integration by parts, prove that

 0

x 2 cos nx dx 

(1)2 2 for n2

any positive integer n . (3 marks) (c) Let N be a positive integer and f ( x)  a0 

a0 , a1 ,, aN are constants. It is given that

  f ( x)  x  cos nx dx = 0 

2

0

N

a m 1

m

cos mx , where

  f ( x)  x  dx = 0 

2

0

for all n =1, 2,…, N .

(i)

Find a0 .

(ii)

(1) n 4 Prove that an  for all n =1, 2,…, N . n2

(iii)

For any positive integer k , let

Ik  

 0

and

 f ( x)  x  cos( N  k ) x dx 2

. Evaluate lim I k . k 

(7 marks)

P. 224


HKAL Pure Mathematics Past Paper Topic: Integration

Set 3 61.

(97II13) Let f (x) be a decreasing continuous function on [1, ) , and f (x) > 0 for all x . For any positive integer n , define

an  f (n)   (a) (i)

n 1 n

f ( x) dx and cn  a1  a2    an .

Show that 0  an  f (n)  f (n + 1) and cn  f (1) . Hence deduce that lim cn exists. n 

(ii)

Prove that 0  ck  cn  f (n + 1) for k > n . (5 marks)

(b) Let c  lim cn . Show that n 

(i)

0  c  cn  f (n + 1) , and

(ii)

f (1)  f (2)    f (n)   f ( x) dx  c  n f (n)

n

1

for some n  [0,1] . (5 marks) (c) Let

1 1 Sn  1      ln n and 2 n 1 1 1 1 1 Tn  1        . 2 3 4 2n  1 2 n

(i)

Using (b)(ii), or otherwise, show that lim Sn exists.

(ii)

Express Tn in terms of S2n and Sn .

n 

Hence find lim Tn . n 

(5 marks)

P. 225


HKAL Pure Mathematics Past Paper Topic: Integration

62.

(03II10) (a) Let f : [1, )  [0, ) be a continuous and decreasing function. For every n = 1, 2, 3, … . define Fn 

(i)

Prove that f ( j  1) 

j 1 j

n 1

n

f ( x) dx and Sn   f ( j ) . j 1

f ( x) dx  f ( j ) for all j = 1, 2, 3, … .

Hence deduce that Sn  f (1)  Fn  Sn1 for all n = 2, 3, 4, … .

(ii)

Using (a)(i), prove that the series

 f ( j)

is convergent if

j 1

and only if there is a constant K ( independent of n ) such that Fn  K for all n = 1, 2, 3,… . (7 marks) (b) Using (a), prove that (i)

1

n3

n 1

(ii)

1

n

is convergent,

is divergent.

n 1

(4 marks) (c) Using (a) to determine whether

1

 n(ln n) n2

2

is convergent. Explain

your answer. (4 marks)

P. 226


HKAL Pure Mathematics Past Paper Topic: Integration

Set 4 63.

(91II09) (a) Prove that lim n n  1 . n 

(3 marks) (b) Show that n

ln1  ln 2    ln(n  1)   ln x dx  ln 2  ln 3    ln n 1

where n  2 . n  n 1

Deduce that (n  1)!  n e

 n! . (7 marks) 1

(n !) n (c) Using (a) and (b), or otherwise, evaluate lim . n  n (5 marks) 64.

(95II08) Let I k 

(1)k (1  x) x3k  0 1  x3 dx , k = 0, 1, 2, … . 1

(a) Evaluate I0 . (4 marks) (b) Prove that 

1 1  Ik  . 3k  1 3k  1 (3 marks)

(c) Express I k 1  I k in terms of k . (3 marks) (d) For n = 0, 1, 2, … , let bn 

(1)k .  k  0 (3k  1)(3k  2) n

Using (a) and (c), express I n 1 in terms of bn . Hence use (b) to evaluate lim bn . n 

(5 marks)

P. 227


HKAL Pure Mathematics Past Paper Topic: Integration

65.

(92II12) (a) For any x > 0 , by considering the integral

1 x 1

1 dt or otherwise, t

prove that

x  ln(1  x)  x , 1 x and deduce that

1  1 1  ln 1    . 1 x  x x (3 marks) x

 1 (b) For any x > 0 , define f ( x)  1   . Using (a) or otherwise,  x prove that f is strictly increasing and 1  f ( x)  e . (4 marks) (c) For x > 0 and n = 2, 3, … , define

Fn ( x)  f ( x)  f (n  1)  

n x

1 dt , t f (t ) 2

x

 1 where f ( x)  1   .  x (i)

For each fixed n , prove that there exists a unique n  R such that Fn ( n )  0 . Does lim  n exist? Explain. n 

(ii)

For each fixed x , prove that lim Fn ( x) exists. n 

(8 marks)

P. 228


HKAL Pure Mathematics Past Paper Topic: Integration

66.

(94II13) 1

  n Let Ln    2 cos n x dx  for any positive integer n .  2  1

(a) Show that Ln   n . (3 marks) (b) For n = 1, 2, 3, … , let rn  cos Find the values of x in [

1 . 2n

 

, ] such that cos x  rn . 2 2 1

 1 n Hence show that Ln  rn   . n (5 marks) (c) Show that 1

(i)

lim n n  1 , n 

(ii) lim Ln  1 . n 

(7 marks) Type 7: Comparison of Integrals Set 1 67.

(96II02)

nx n Let an   dx for n = 1, 2, 3, … . 0 1  x2 1

(a) Show that

n 1 1 nx nx n d x  a  n 0 2  0 2 dx for n = 1, 2, 3, … . 1

(b) Evaluate lim an . n 

(5 marks)

P. 229


HKAL Pure Mathematics Past Paper Topic: Integration

68.

(94II08) (a) Show that for any a, y  R , e  e  e ( y  a) . y

a

a

(b) By taking y = x2 in the inequality in (a), prove that

1 0

1 3

e dx  e . x2

(6 marks) 69.

(90II08) (a) Let I n 

x n 1  0 (1  x)2 dx for n = 0, 1, 2, … . 1

(i)

Find I0 .

(ii)

Prove that lim I n  0 . n 

(5 marks) (b) (i)

Prove that n n  1  ( x) m  1  ( x) n  (1)i  j x d x     0  1  x  i 1 j 1 i  j  1  x  1

for any positive integers m and n . (ii)

(1)i  j Hence evaluate lim  . n  i 1 j 1 i  j n

n

(10 marks)

P. 230


HKAL Pure Mathematics Past Paper Topic: Integration

70.

(99II11) (a) For n = 0, 1, 2, … and y  0 , define In ( y ) 

y 0

t n et dt .

n y Prove that In ( y)   y e  n In1 ( y) for n  1 and y  0 .

Hence deduce that In ( y)  n! for n  0 and y  0 . (5 marks) (b) Let n be a positive integer. (i)

By considering g( x)  n ln(n  x)  n ln(n  x)  2 x for

0  x  n , show that (n  x)n e( n x )  (n  x)n e( n x ) for

0 xn . (ii)

Use (b)(i) to show that Hence deduce that

2n 0

2n n

n

u n eu du   u n eu du . 0

n

t n et dt  2e n  (n  t )n et dt . 0

(8 marks) (c) Using the above results or otherwise, show that

1 n en n t for all positive integers n . ( n  t ) e d t  n!  0 2 (2 marks)

P. 231


HKAL Pure Mathematics Past Paper Topic: Integration

71.

(00II13)

Let n be a positive integer. Define f n

(a) (i)

 ( x)  

x 0 1 0

(1  t 4 ) n dt (1  t 4 ) n dt

.

Show that fn(x) is an odd function.

(ii)

Find f n' ( x) and f n'' ( x) .

(iii)

Sketch the graph of fn(x) for 1  x  1 . (7 marks)

(b) Using the facts A.

t 3 (1  t 4 )n  (1  t 4 )n for 0  t  1 and

B.

(1  t 4 )n 

t3 (1  t 4 )n for 0 < x  t  1 , 3 x

or otherwise, show that 0  1  fn(x) 

(1  x 4 ) n 1 for 0  x  1 . x3 (5 marks)

(c) For each x  [1, 1] , let g( x)  lim f n ( x) . Evaluate g(x) when n 

0  x  1 and when x = 0 respectively. Sketch the graph of g(x) for 1  x  1 . (3 marks)

P. 232


HKAL Pure Mathematics Past Paper Topic: Integration

Set 2 72.

(93II11) (a) Let f and g be real-valued functions defined on (a, ) where

a  0 , and f be twice differentiable satisfying the following conditions: A.

g is decreasing,

B.

g(t)  0 and f (t)  0 for all t  (a, ) ,

C.

lim g(t )f' (t )  0 .

(i)

t 

Using Mean Value Theorem to show that

f (n)  f' (n)(t  n)  f (t )  f (n)  f' (n  1)(t  n) for all t [n, n  1] , where n is a positive integer greater than a . (ii)

Hence, or otherwise, show that

n 1 n

 f (n)  f (n  1)  f' (n  1)  f' (n) f (t ) dt   ,  2 2  

where n is a positive integer greater than a . n 1  n2  f ( j )  f ( j  1)   2 Show that lim   f (t ) dt    g(n )  0 . n n  2 j n     2

(iii)

(9 marks)

ln n n2 1 1 dt  , or otherwise,  n ln t n  n 2

(b) Using (a) and the fact that lim

1 1 1  ln n    .  ln(n2 )  n2  ln(n  1) ln(n  2)

evaluate lim  n 

(6 marks)

P. 233


HKAL Pure Mathematics Past Paper Topic: Integration

73.

(95II13) (a) Suppose f (x) , g(x) are continuously differentiable functions such that f '( x)  0 for a  x  b . (i)

Let w( x) 

 (ii)

b a

x a

g(t ) dt . Show that b

b

a

a

f ( x)g( x) dx  f (b)  g( x) dx   f '( x)w( x) dx .

Using the Theorem (*) below, show that

b a

b

c

c

a

f ( x)g( x) dx  f (b)  g( x) dx  f (a)  g( x) dx for some

c [a, b] . Theorem (*) : If w(x) , u(x) are continuous functions and u( x)  0 for a  x  b , then

b a

b

w( x)u( x) dx  w(c)  u( x) dx for some c [a, b] . a

(5 marks) (b) Let F(x) be a function with a continuous second derivative such that F'' ( x)  0 and F' ( x)  m  0 for a  x  b . Using (a) with

f ( x)  

b a

1 and g( x)  F' ( x) cos F( x) , show that F' ( x)

cos F( x) dx 

4 . m (5 marks)

(c) (i)

Show that

1 0

1

cos( x n ) dx   cos( x n1 ) dx . 0

Hence show that lim n 

(ii)

1 0

cos( x n ) dx exists.

Using (b), or otherwise, show that lim n 

2 0

cos( x n ) dx exists. (5 marks)

P. 234


HKAL Pure Mathematics Past Paper Topic: Integration

Type 8: Others 74.

(98II11) (a) Let f be a non-negative continuous function on [a, b] . Define x

F( x)   f (t ) dt for x  [a, b] . 0

Show that F is an increasing function on [a, b] . Hence deduce that if

b a

f (t ) dt  0 , then f (x) = 0 for all

x [a, b] . (5 marks) (b) Let g be a continuous function on [a, b] . If

b a

g( x)u( x) dx  0 for

any continuous function u on [a, b] , show that g(x) = 0 for all

x [a, b] . (3 marks) (c) Let h be a continuous function on [a, b] . Define

A (i)

b 1 h(t ) dt .  ba a

If v(x) = h(x)  A for all x  [a, b] , show that

 (ii)

b a

If

v( x) dx  0 .

b a

h( x)w( x) dt  0 for any continuous function w on

[a, b] satisfying

b a

w( x) dt  0 , show that h(x) = A for all

x [a, b] . (7 marks)

P. 235


HKAL Pure Mathematics Past Paper Topic: Integration

75.

(02II10) Let f and g be continuous functions defined on [0, 1] such that f is decreasing and 0  g( x)  1 for all x  [0, 1] . For x  [0, 1] , define G( x) 

 ( x)  

G( x ) 0

(a) (i)

x 0

g(t ) dt and

x

f (t ) dt   f (t )g(t ) dt . 0

Prove that G( x)  x . Hence prove that ' ( x)  0 for all

x  (0,1) . (ii)

Evaluate  (0) and hence prove that

1 0

f (t )g(t ) dt  

G(1) 0

f (t ) dt . (7 marks)

(b) Let H( x) 

x 0

[1  g(t )]dt for all x  [0, 1] .

(i)

Prove that G(1) + H(1) = 1 .

(ii)

Using (a)(ii), prove that

1 1G (1)

1

f (t ) dt   f (t )g(t ) dt . 0

(5 marks) (c) Using the results of (a)(ii) and (b)(ii), prove that

1 n n 1

1

f (t ) dt   f (t )  t n dt   0

Hence show that lim n 

1 0

1 n 1 0

f (t ) dt , where n is a positive integer.

f (t )  t n dt  0 . (3 marks)

P. 236


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

Contents: Classification

Chronological Order

Type 1: Tangents

90II09

Q.

15

SECTION A

91II01

Q.

14

1.

96II09

92II06

Q.

11

2.

00II07

93II09

Q.

18

3.

03II05

93II10

Q.

25

4.

05II05

94II09

Q.

16

5.

06II06

95II03

Q.

8

SECTION B

96II09

Q.

1

6.

04II09

96II11

Q.

23

7.

05II09

97II03

Q.

12

Type 2: Relations of Roots

97II12

Q.

21

8.

95II03

98II12

Q.

24

9.

00II10

99II12

Q.

20

10.

06II10

00II07

Q.

2

Type 3: Locus

00II10

Q.

9

SECTION A

01II04

Q.

13

11.

92II06

01II09

Q.

22

12.

97II03

02II11

Q.

17

13.

01II04

03II05

Q.

3

14.

91II01

03II11

Q.

19

SECTION B

04II09

Q.

6

Set 1

05II05

Q.

4

15.

90II09

05II09

Q.

7

16.

94II09

06II06

Q.

5

17.

02II11

06II10

Q.

10

Set 2 18.

93II09

19.

03II11 Type 4: Others

20.

99II12

21.

97II12

22.

01II09

23.

96II11

24.

98II12

25.

93II10

P. 237


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

Type 1: Tangents SECTION A 1.

(96II09) Consider the curves

x2 y 2  1 8 2

E:

P: y  kx  3 2

and (k > 0) .

A common tangent L of E and P touches E at (2, 1) . (a) (i)

Find the equation of L and the value of k .

(ii)

Determine the coordinates of the point at which L touches

P . (4 marks) (b) Find the area enclosed by L , P and the y-axis. (4 marks) (c) Find the equation of the remaining three common tangents of E and P . (7 marks) 2.

(00II07) The curve in the figure has parametric equations

 x  2(t  sin t ) , 0  t  2 .   y  2(1  cos t )

(a) Find the equation of the tangent to the curve at the point where

t

 2

.

(b) [Out of Syllabus] Find the arc length of the curve. (6 marks)

P. 238


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

3.

(03II05)

x2 y 2 Consider the hyperbola H : 2  2  1 , where a and b are positive a b constants, with its asymptotes L1 : y  the point (a sec , b tan  ) , where

b b x and L2 : y  x . Let P be a a

    . 2 2

(a) Prove that P lies on H . (b) The tangents to H at P cuts L1 and L2 at Q1 and Q2 respectively. (i)

Find the coordinates of Q1 and Q2 .

(ii)

Find the areas of OPQ1 and OQ2P , where O is the origin. Hence find Q1P : PQ2 . (7 marks)

4.

(05II05) Let P: y2 = 80x be a parabola. (a) Prove that the straight line y = mx + c is a tangent to P if and only if mc  20 .

 x  3cos  ,  3   . 2 2  y  sin  ,

(b) Consider the curve  : 

Find the coordinates of the two points on  at which the normals to  are tangents to P . (7 marks)

P. 239


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

5.

(06II06)

x2 y 2 Let the equation of the ellipse E be 2  2  1 , where a and b are a b two distinct positive constants. The coordinates of the points P and Q are

(a cos  , b sin  ) and  (a  b) cos  ,(a  b)sin   respectively, where

0  

 2

.

(a) Prove that (i)

P lies on E ,

(ii)

the straight line passing through P and Q is the normal to E at P .

(b) Let c be a constant such that the straight line x sin   y cos  = c is a tangent to E . Express the distance between P and Q in terms of c . (7 marks)

P. 240


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

SECTION B 6.

(04II09)

x2 y 2   1 , where a and b are two positive a 2 b2 constants with a > b . Let P be the point (a cos  , b sin  ) , where Consider the ellipse E :

0  

 2

.

(a) Prove that P lies on E . (1 mark) (b) Let L be the tangent to E at P . L cuts the x-axis and the y-axis at P1 and P2 respectively. Find (i)

the equation of L ,

(ii)

the coordinates of P1 and P2 . (4 marks)

2 2 2 2 2 2 (c) Consider the two circles C1 : x  y  a and C2 : x  y  b .

Also consider the two points P1 and P2 described in (b). For

k  1, 2 , let Lk be the tangents to Ck from Pk , with the point of contact Qk lying in the first quadrant. (i)

Prove that L1 is parallel to L2 .

(ii)

Find the coordinates of Q1 and Q2 .

(iii)

Let l be the straight line passing through Q1 and Q2 . Is l a common normal to C1 and C2 ? Explain your answer. (10 marks)

P. 241


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

7.

(05II09) Let H1 : xy  4 and H 2 : xy  1 be two hyperbolas, and L be the

2 t

tangent to H1 at P(2t , ) . (a) Find the equation of L . (2 marks) (b) Let A( ,

1

1 ) and B(  , ) be the two distinct points where L  

intersects H2 . (i)

Prove that  +  = 4t and  = t 2 .

(ii)

Prove that the length of chord AB is

1  12  t 2  2  . t   (6 marks)

(c) It is given that the tangents to H2 at A and B intersect at Q , where A and B are the points described in (b). (i)

Find the coordinates of Q in terms of t .

(ii)

Using (b)(ii), or otherwise, prove that the area of QAB is independent of t . (7 marks)

P. 242


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

Type 2: Relations of Roots 8.

(95II03) Consider the parabola y  4ax . 2

(a) Prove that the equation of the normal at P(at , 2at ) is 2

y  tx  2at  at 3 ……… (*) . 2 (b) Pi (ati , 2ati ) , i = 1, 2, 3, are three distinct points on the parabola.

Suppose the normals at these points are concurrent. By considering (*) as a cubic equation in t , or otherwise, show that

t1  t2  t3  0 . (5 marks)

P. 243


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

9.

(00II10) The equation of the parabola  is y2 = 4ax . (a) Find the equation of the normal to  at the point (at 2 , 2at) . Show that if this normal passes through the point (h, k) , then

at 3  (2a  h)t  k  0 . (4 marks) 2 (b) Suppose the normals to  at three distinct points (at1 , 2at1 ) ,

(at2 2 , 2at2 ) and (at32 , 2at3 ) are concurrent. Using the result of (a), show that t1  t2  t3  0 . (2 marks) (c) If the circle x2 + y2 + 2gx + 2fy + c = 0 intersects  at (as12 , 2as1 ) , (as2 2 , 2as2 ) , (as32 , 2as3 ) and (as4 2 , 2as4 ) , show that

s1  s2  s3  s4  0 . (4 marks) (d) A circle intersects  at points A , B , C and D . Suppose A , B and C are distinct and the normals to  at these three points are concurrent. (i)

Show that D is the origin.

(ii)

If A , B are symmetric about the x-axis, show that the circle touches  at the origin. (5 marks)

P. 244


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

10.

(06II10) Let the equation of the parabola P be y2 = 4ax , where a is a non-zero constant. (a) Find the equation of the normal to P at the point (at2 , 2at) . (3 marks) (b) The normals to P at two distinct points (at12 , 2at1) and (at22 , 2at2) intersect at the point (h, k) . Let t3 = (t1 + t2) . (i)

Prove that the roots of the equation at 3 + (2a  h)t  k = 0 are t1 , t2 and t3 .

(ii)

Does the normal to P at the point (at32 , 2at3) pass through the point (h, k) ? Explain your answer.

(iii)

Express t1t2  t2t3  t3t1 and t1t2t3 in terms of a , h and k . (8 marks)

(c) Let A and B be two points on P at which the normals to P are perpendicular to each other. Using the results of (b)(iii), or otherwise, find the equation of the locus of the point of intersection of the two normals as A and B vary. (4 marks) Type 3: Locus SECTION A 11.

(92II06) Consider the line (L) : y  2a and the circle (C) : x  y  a 2

2

2

, where

a  0 . Let P be a variable point on (L) . If the tangents from P to (C) touch the circle (C) at points Q and R respectively, show that the mid-point of QR lies on a fixed circle, and find the centre and radius of the circle. (6 marks)

P. 245


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

12.

(97II03) Let P be the parabola y 2 = kx , where k is a non-zero constant. A(ks2 , ks) and B(kt 2 , kt) are two distinct points on P moving in such a way that the tangents drawn to P at A and B are perpendicular to each other. (a) Show that st  

1 . 4

(b) If M is the mid-point of AB , show that M lies on a parabola and find the equation of this parabola. (5 marks) 13.

(01II04) Let P be the parabola y2 = 4ax where a is a non-zero constant, and

A(at12 , 2at1 ) , B(at22 , 2at2 ) be two distinct points on P . (a) Find the equation of chord AB . (b) If A and B move in such a way that chord AB always passes through (a, 0) , find the equation of the locus of the mid-point of AB . (5 marks) 14.

(91II01) Chords with slope equal to 1 are drawn in the ellipse

2 x 2  2 xy  y 2  1 . Prove that the mid-points of these chords lie on a straight line, and find the equation of the line. (4 marks)

P. 246


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

SECTION B Set 1 15.

(90II09) Consider the hyperbola (H) : xy = c2 , c > 0 .

c t1

Let P(ct1 , ) and Q(ct2 ,

c ) be points on (H) where t12  t2 2 , t1  0 t2

and t2  0 . (a) Find the equation of the straight line joining the points P and Q , and hence, or otherwise, obtain the equations of the tangents to (H) at P and Q respectively. (3 marks) (b) Suppose R is the point of intersection of the tangents at P and Q . (i)

Find the coordinates of R .

(ii)

Show that if P and Q are moving in such a way that t1t2 is constant, then R lies on a straight line passing through the mid-point of PQ .

(iii)

If P and Q are moving in such a way that PQ always touches the ellipse 4x2 + y2 = c2 , show that R lies on an ellipse with centre at the origin. Also find the equation of this ellipse. (12 marks)

P. 247


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

16.

(94II09) Given an ellipse (E):

x2 y 2  1 a 2 b2

and a point P(h, k) outside (E) . (a) If y = mx + c is a tangent from P to (E) , show that

(h2  a 2 )m2  2km  k 2  b2  0 . (4 marks) (b) Suppose the two tangents from P to (E) touch (E) at A and B . (i)

Find the equation of the line passing through A and B .

(ii)

Find the coordinates of the mid-point of AB . (6 marks)

(c) Show that the two tangents from P to (E) are perpendicular if and only if P lies on the circle x  y  a  b 2

2

2

2

. (5 marks)

17.

(02II11) 2 2 Consider the parabolas C1 : y  4( x  1) and C2 : y  4 x .

2 Let P( p  1, 2 p) be a point on C1 . The two tangents drawn from P to 2 2 C2 touch C2 at the points S ( s , 2s) and T (t , 2t ) .

(a) Find the equations of PS and PT and hence show that s  t  2 p ,

st  p 2  1 . (4 marks) (b) Q(q , 2q) is a point on the arc ST of C2 . Prove that the area of 2

SQT is a maximum if and only if q = p . (6 marks) (c) Let Q be the point in (b) where the area of SQT is a maximum. If the straight line PQ cuts the chord ST at M , find the equation of the locus of M as P moves along C1 . (5 marks)

P. 248


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

Set 2 18.

(93II09) The equation of the hyperbola H is

x2 y 2   1 , where a , b > 0 . a 2 b2 1 1 1  1 Let P   a(t  ), b(t  )  , where t  0 . t 2 t  2 (a) (i) (ii)

Show that P lies on H . Find the equation of the tangent to H at P . (4 marks)

(b) Let the tangents to H at P meet the asymptotes of H at the points S and T . Let O be the origin. (i)

Show that as t varies the locus of the centre of the circle passing through O , S and T is a hyperbola.

(ii)

Prove that OS  OT  a 2  b2 . Hence show that S , T and the two foci of H are concyclic. (11 marks)

P. 249


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

19.

(03II11) Consider the parabola P : y  4 x . 2

Let A(a , 2a) be a point on P , L1 be the tangent to P at A , and F 2

be the point (1, 0) . (a) (i) (ii)

Find the equation of L1 . Let F' be a point such that L1 is the perpendicular bisector of FF' . Prove that the x-coordinate of F' is 1 . Also find the y-coordinate of F' . (7 marks)

(b) Suppose a = 2 . The straight line x = 1 intersects L1 at B . Let L2 be the straight line passing through B and perpendicular to L1 . (i)

Prove that L2 is tangent to P and find the point of contact.

(ii)

Let L3 be the tangent to P at the point (1, 2) . L3 intersects L1 and L2 at C and D respectively. Prove that B , C , F and D are concyclic. (8 marks)

P. 250


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

Type 4: Others 20.

(99II12) (a) The equation of the ellipse E is

( x  h) 2 y 2  2  1 , where a2 b

a, b, h  R and a, b > 0 . (i)

By integration, find the area enclosed by E .

(ii)

If the straight line y = mx is tangent to E , show that

b2 . m  2 h  a2 2

(6 marks) (b) For n = 1, 2, 3, … , let En be the ellipse given by

( x  hn )2 y2  2 2 1 , an 2 p an where p > 0 , hn > hn + 1 and hn > an > 0 . Suppose for all n ,

En and En 1 touch each other externally and the straight line y = mx is a common tangent to all En as shown in the figure.

(i)

Express hn  hn + 1 in terms of an and an + 1 .

(ii)

Using (a)(ii) and the result of (b)(i), or otherwise, show that

an 1 h1  a1  . an h1  a1 (iii)

Let Sn be the area enclosed by the ellipse En . Evaluate 

S n 1

n

in terms of a1 , h1 and p . (9 marks)

P. 251


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

21.

(97II12) Consider the curves :

xy = 1 (x > 0) ,

:

xy = 1

:

xy = 1 (x < 0) ,

:

xy = 1

(x < 0) , (x > 0) ,

And the region determined by xy  1 in the rectangular coordinate plane as shown in the figure. For 0 < a  1 , let P , Q , R and S be points on , , and respectively , where

1 1 P  ( a, ) , Q  (  , a ) , a a 1 1 R  (a,  ) and S  ( ,  a) . a a (a) Show that PQRS is a square. (3 marks) (b) If the straight line PQ intersects

at another point Q , find the

coordinates of Q in terms of a . Hence find the range of values of a such that the line segment PQ lies inside the region

. (8 marks)

(c) If PQRS lies within the region

, determine a such that its area

is maximized. You may use the fact that PQRS lies within the region and only if PQ lies inside

if

. (4 marks)

P. 252


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

22.

(01II09) In the figure, C is the circle

x2  ( y  1)2  42 and F is the point (0, 1) . For any point P on C , let Lp be the perpendicular bisector of the line segment FP . It appears that as P moves on C , all the Lp ’s are tangents to an ellipse inside C . (a) Suppose for every P on C , the line Lp is tangent to the same ellipse. Write down the equations of the two horizontal tangents and the two vertical tangents to the ellipse. Hence guess the equation of the ellipse. (Note that C is symmetric about the y-axis and F lies on this line of symmetry.) (5 marks) (b) Let E be the ellipse found in (a). (i)

For any point P(p, q) on C , let M be the point (m, n) where m 

3p 4(q  1) and n  . Show that 7q 7q

(I)

M lies on E ,

(II)

the tangent at M to E is the perpendicular bisector of FP .

(ii)

For any point M on E , show that there is a point P on C such that the perpendicular bisector of FP is the tangent to E at M . (10 marks)

P. 253


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

23.

(96II11) Figure A shows a circle with radius b rolling externally without slipping on a fixed circle with radius a and centred at the origin. Let P be a point fixed on the rolling circle with initial position at A(a, 0) . (a) Referring to Figure B, show that the parametric equations of the locus of P are given by

  ab  x  (a  b) cos   b cos  b      .   y  (a  b) sin   b sin  a  b       b  (6 marks) (b) Suppose a = 2b . (i)

Write down the parametric equations of the locus of P . 2

 dx   dy       d   d 

2

in terms of  .

(ii)

Express

(iii)

[Out of Syllabus: Arc Length] Find the distance travelled by P before it meets the fixed circle again. (9 marks)

Figure A

Figure B P. 254


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

24.

(98II12) Figure A shows a circle with radius 1 and centre C touching a line L with slope m > 0 at Q(a, b) . R(x0, y0) is another point on the circle and QCR   .

2     sin (cos  m sin )  x0  a  2 2 2 2 1 m  (a) Show that  . 2    y  b  sin (m cos  sin )  0 2 2 2 1  m2 (7 marks)

(i)

3 2 ( x  1) 2 where x  1 . 3 Find the slope of tangent of  at x = a .

(ii)

[Out of Syllabus: Arc Length] Find the length of arc 

(b) Consider the curve  : y 

from x  1 to x  a .

2 32 Ans: (a  1) 3 (iii)

Figure B shows a circle with radius 1 rolling tangentially below the curve  without slipping. Let P be a point fixed on the circle with initial position at (1, 0) . Find the x-coordinate of P when the circle touches  at x = 4 . (8 marks)

Figure A

Figure B P. 255


HKAL Pure Mathematics Past Paper Topic: Coordinate Geometry

25.

(93II10)

In the figure, O is the origin and  is the curve whose equation is

x3  y3  3axy ( a > 0 ) . L is the asymptote of  . (a) Evaluate lim

x 

y , where (x, y)   . x

(You may assume that lim

x 

y exists.) x

Hence, or otherwise, show that the equation of L is x  y  a  0 . (3 marks) (b) [Out of Syllabus: Polar Coordinate] Find the polar equations of  and L . (3 marks) (c) [Out of Syllabus: Area of region characterized by Polar Equation] Find the area of the region enclosed by  (i.e. 1). (3 marks) (d) Suppose a straight line through O cuts  at A and L at B in the second quadrant. Let  be the angle between OB and the positive x-axis. Let A be the area of the region bounded by the x-axis ,  , L and AB (i.e. 2) . [Out of Syllabus: Area of region characterized by Polar Equation] Show that

 a2  1 3 A     2 . 3 2 1  tan  1  tan  

[Evaluation of limit is included in the syllabus] Hence evaluate lim A . 

3 4

P. 256

(6 marks)


Pure Maths