F.4D Mathematics Revision Exercises (2009-12-04)
1.
Simplify
2.
Simplify
a 3  8b 3
2a 2  3b 2  5ab
.

2b  a
(2a  b)( a 2  4b 2  2ab)
1
1
3.
1
x
1

x
1
x

x 1
.
x 1
Let h(x) = (x  1)(x  2), solve
(a) h(x) = 0;
(b) h(x + 3) = 0.
4.
Let p(x + 2) = x3  2x + 1, find
(a) p(1);
(b) p(x).
5.
Suppose f ( x)  3x and g (2 x)  5 x  4 for any real numbers x, solve f (2 x)  g ( x)  0 .
6.
Let f ( x) 
7.
If the domain of f ( x)  3(2 x  1) 2  2009 is the set of real numbers, find the range of f(x).
8.
Suppose x 2 
9.
Suppose x, y are integers such that x + y = 2y + 4, then
1
1  2x
where x is real, find the most extensive domain of the function f (x ) .
1
1
 3 , find the value of x 8  8 .
2
x
x
I.
x must be an even number
(true / false)
II.
y must be an even number
(true / false)
III. (y  x) must be an even number
(true / false)
IV. If x = 4, then y is an even number
(true / false)
10. Simplify 32p4p where p is a rational number.
11. Simplify
( 17  4)99( 17 + 4)100 .
12. Factorize
2x2 + 6x  15z  2xy + 5yz  5xz .
13. Factorize
p2 + q2  3p  3q + 2pq + 2 .
14. Factorize
(2x  y)(4x2  4xy + y2)  x3 .
15. (a)
9  3
(true / false)
(b) If x2 = 9, then x = 3.
(true / false)
(c) The square roots of 9 are 3 and 3.
(true / false)
(d) If x2 = 25, then x = 5 and x = 5.
(true / false)
16. Solve 2 x  3 x  5  0 by using quadratic formula.
2
2 i
i 1
and express the answer in the form of a  bi (a, b are real).
1  2i
2 i
18. Simplify i 1  i 3  i 5  ...  i 2009 , where i 2  1 .
19. Suppose ( x  2 y )  (3x  4 y )i  1  3i where x and y are purely imaginary numbers, find
17. Evaluate

the values of x and y.
20. Suppose 3 + 4 11 is one of the roots of x2  6x + a = 0, find the value of a.
21. Suppose x and y are real numbers, solve
(x  3)2 = (2y + 5)2.
22. Let ,  be roots of x2 + px + 1 = 0, find a quadratic equation with roots 2 and  2.
23. Let ,  be roots of x2  (p + 1)x  2p = 0. Determine a quadratic equation with roots  3 ,  3.
24. Suppose 3x2 +
x
2

 a(x + b)2 + c ; where a , b and c are constants. Find
2
3
the values of a , b and c.
 ,  are roots of x2  px + 1 = 0, construct a quadratic equation whose roots are
1
1
  and
 .
2
2
25. Suppose


26. The following shows the graph of y = ax2 + bx + c. Fill in the blanks of the following table.
a>0
b>0
c>0
4ac > b2
Yes or No?
27. The figure shows the graph of a quadratic function y = cx2 + ax  b,
A.
a>0
(True/false)
B.
b>0
(True/false)
C.
c>0
(True/false)
D.
a2 + 4bc > 0
(True/false)
y
then
y = cx2 + ax  b
x
O
28. The figure shows the graph of y = x2 + 4x + c. Which of the following is a possible value of c?
A.
4
B.
5
C.
4
D.
0
29. If 0 < p < 6, which of the following quadratic equations has / have two distinct real roots?
I.
(p2 + 1)x2 + (2p – 1)x + 1 = 0
II. px2 – (p + 7)x + 7 = 0
III.
1 2
x  3 px  4 p 2  18  0
2
A.
I only
30. Let
B.
II only
C.
I and III only
D.
II and III only
 ,  be real roots of x2 + (2k  1)x = 1  k2 . If    , find the range of values of k.
31. Suppose x 2  2 x  3k  1 for any real value of x, determine the range of values of k.
32. Suppose 2x2  x > 3a  10 for any real number x, find the range of values of a.
33. Which of the following may represent the graph of y = ax2 + bx + c, where ac < 0?
A.
B.
C.
D.
2 y 2  kx  18  0
34. Given that the simultaneous equations 
y  x  3
have only one solution, find the
value of k.
35. The figure shows the graphs of y = x2 – bx + c and y = –1. By considering the two graphs,
which of the following simultaneous equations have two real solutions?
A.
 y  ( x  1) 2  b( x  1)  c

 y  1
B.
 y  ( x  1) 2  b( x  1)  c

 y  1
C.
 y  1  x 2  bx  c

 y  1
D.
 y  1  x 2  bx  c

 y  1
36. How many distinct real roots are there to the equation x(x  1)(x  2) = x(x  2)(3x  5)?
37. (a) Suppose the graph of a quadratic function y = f(x) has its vertex (2,5) and
y-intercept 13, find the equation of the graph.
(b) Write down the coordinates of the minimum point of the graph y = |f(x)|.
38. Let y = ax2 + bx + c be the equation of a quadratic graph. Suppose the vertex of the graph is
(2,1) and the y-intercept is 4. Determine the values of a, b and c.
39. The following shows a quadratic graph. Given that it passes through B(7,0) and the vertex is
A(3,1). Find the equation of the graph. (5 marks)
40. (a) Evaluate the maximum value of
1
2
x
 3x  7
2
(b) Determine the minimum value of
.
x 2  6 x  12
.
x 2  6 x  14
41. Refer to the figure below. Rectangle PQRS inscribed in a right-angled triangle ABC with PQ
lie on AB and R, S lie on BC and CA respectively. Given that AC = 4 m, CB = 3 m, AP = x m.
C
(a) Express the area of PQRS in terms of x.
4m
S
3m
R
P
Q
(b) Determine the maximum area of PQRS.
A
[Hint: Can you figure out similar triangles?]
xm
B
42. Let f (t )  A(3  kt ) for any real number t. Given that when f (0)  100 and f (100) 
100
.
3
Find the value of f (200) .
43. Let x =
3
2,y=
4
3,z=
5
4 . Which of the following must be true?
A.
x>y>z
B.
x>z>y
C.
y>x>z
D.
y>z>x
E.
z>x>y
F.
z>y>x
44. If 3 x  36  3 x 1 , then x 
45. Solve 16 x  3(4 x )  4  0 .
46. Solve x  2  5 x  2  6 .
47. Suppose 3 x  5 y  15 z , where x, y and z are rational numbers. Express z in terms of x and y.
48. Solve
8(4x + 4x)  6(2x + 2x) = 19 .
100
49. Simplify
1
1 
  k  k  1  .
k 10
50. Find the value of n if
51. Solve
n 3
19
k 3
k 6
 ak  2   a k 1
|x| = 6  x2 .
52. Solve (2x  1)2  |2x  1|  6 = 0.
53. Solve
(x  1)2 + |x  1|  6 = 0.
54. Solve | x | + 2 = | x + 2 |.
55. Determine the fifth term in the expansion of (1  2 x )10
56. Find the constant term in the expansion (x2 
2 9
).
x
in descending powers of
x.


57. Find the constant term in the expansion of  3 x 
6
1 
 .
x2 
58. In the expansion of (1  6x2 + 9x4)20, find the coefficients of
(a) x10;
(b) x11.
[Hint: 1  6x2 + 9x4  (1  3x2)2]
59. (a) Suppose (1  ax2)n  1  40x2 + 760x4 + terms involving higher powers of x.
Find the values of a and n.
(b) By (a), or otherwise, write down the coefficients of
(i)
x4;
(ii) x9
in the expansion of (1  4x2 + 4x4)10.
60. In the expansion of (1  x) n , the coefficient of
x 3 is equal to 7 times the coefficient of x.
If n is a positive integer, find the value of n.
61. In the expansion of (1  2x)3(1 + x)n , where n is a positive integer, the coefficient of
is
36 . Find the value of n .
62. (a) Find, in terms of n and r, the coefficient of xr in the expansion (2 + x)(1  3x)n; where
n and r are non-negative integers (0  r  n).
(b) Suppose m is a square number with the coefficient of x2 in the expansion
(2 + x) 1 3x 
m
is 252, evaluate m.
63. (a) Prove by mathematical induction that
1
1
1
1
n


 ... 

2 4 46 68
4n(n  1) 4(n  1)
for any natural number n.
(b) By (a) or otherwise, evaluate
1
1
1
1
1
1
1
1





 ... 

.
2  4 2  3 4  6 3 4 6  8 4  5
200  202 101  102
64. (a) Prove that, for any positive integer n,
(i)
1
1
1
1
n(n  3)
.


 ... 

1 2  3 2  3  4 3  4  5
n(n  1)( n  2) 4(n  1)( n  2)
(ii)
1
1
1
1
n(n  2)


 ... 

.
1 3  5 3  5  7 5  7  9
(2n  1)( 2n  1)( 2n  3) 3(2n  1)( 2n  3)
(b) Hence or otherwise, find the value of
1
1
1
1
1



 ... 
.
1 3  5 2  4  6 3  5  7 4  6  8
2005  2007  2009
65. Let a1, a2, … be a sequence of numbers such that a1 = 1, an+1 = an + 2n for any positive
integer n. Prove that an = n2  n + 1 for all positive integers n.
x2
66. Prove by mathematical induction that for all positive integers n,
2 2 (1) 2 3 (2)
2 n1 (n)
2 n2
.

 ... 
 2
3!
4!
(n  2)!
(n  2)!
67. By using the mathematical induction, show that n3  n + 3n is divisible by 3 for any positive
integer n.
68. (a) Show that 3n + 1 is divisible by 4 for any positive odd integer n.
(b) Find the remainder when 3123456789 is divided by 4.
69. Prove by mathematical induction that 4 2n1  3n2 is divisible by 13 for all positive
integers n.
70. (a) Show by mathematical induction that 42n + 1 + 2(7n + 1) is divisible by 9 for all natural
numbers n.
(b) Let x = 420001 + 2(710001)  3. Find the remainder when x is divided by 9.
71. Let
f(n) = 1 
1
. Prove by mathematical induction that
n2
f(2)f(3)f(4)…f(n) =
1
1
(1  )
2
n
for any natural number n > 1 .
72. Prove that (3n + 1)7n  1 is divisible by 9 for any positive integer n.
73. Prove that (n + 1)(n + 2)(n + 3)…(n + n) = 2n135…(2n  1) for all positive integers n.
74. Prove, by mathematical induction, that
nC1
+ nC2 + … + nCn = 1 + 2 + 22 + … + 2n  1
for all positive integers n.
75. If
ax2 + bx + c = 0 and
bx2 + ax + c = 0 where
a  b and
c  0, have a common
root, find the value of a + b + c.
76. Suppose the quadratic equations ax 2  bx  c  0 and
ax 2  bx  c  0 have a common
root, prove that (ar  cp) 2  (aq  bp)(br  cq) .
77. Show that the graph of y = x2 + (2  p)x +
3
1

intersects the x-axis at two
2
8
4 p  16 p  16
distinct points for any real value of p (p  2).
78. Let ,  be roots of x2 + px + 1 = 0, show that 4 , 4 are roots of x2  [(p2  2)2  2]x + 1 = 0.
By considering the roots of x2  47x + 1 = 0, show that
4
47  2205 3  5

.
2
2
79. Find a positive integer n such that (n  3)!12! 1997!(n  1988)! .
80. Suppose (x + 1)40(9x  6 +
b
b
b
1 40
) (1 + 2x + x2)20  p(x) + 1 + 22 + … + 40
;
x
x
x
x 40
where p(x) is a polynomial in x. Evaluate b37.