12. Inequalities & Graphs
x2
e.g. i Solve 1 x2
x2 1
x2
x2 x 2
x2 x 2 0
x 2 x 1 0
x 2 or x 1
13. Inequalities & Graphs
x2
e.g. i Solve 1 x2
x2 1
x2
x2 x 2
x2 x 2 0
x 2 x 1 0
x 2 or x 1
x2
1
x2
x 2 or 1 x 2
16. (ii) (1990)
Consider the graph y x
a) Show that the graph is increasing for all x 0
dy
Curve is increasing when 0
dx
17. (ii) (1990)
Consider the graph y x
a) Show that the graph is increasing for all x 0
dy
Curve is increasing when 0
dx
y x
dy 1
dx 2 x
18. (ii) (1990)
Consider the graph y x
a) Show that the graph is increasing for all x 0
dy
Curve is increasing when 0
dx
y x
dy 1
dx 2 x
dy
0 for x 0
dx
19. (ii) (1990)
Consider the graph y x
a) Show that the graph is increasing for all x 0
dy
Curve is increasing when 0
dx
y x at x 0, y 0
dy 1
dx 2 x
dy
0 for x 0
dx
20. (ii) (1990)
Consider the graph y x
a) Show that the graph is increasing for all x 0
dy
Curve is increasing when 0
dx
y x at x 0, y 0
dy 1 when x 0, y 0
dx 2 x
dy
0 for x 0
dx
21. (ii) (1990)
Consider the graph y x
a) Show that the graph is increasing for all x 0
dy
Curve is increasing when 0
dx
y x at x 0, y 0
dy 1 when x 0, y 0
dx 2 x
dy
0 for x 0 curve is increasing for x 0
dx
22. b) Hence show that;
n
2
1 2 n xdx n n
0
3
23. b) Hence show that;
n
2
1 2 n xdx n n
0
3
24. b) Hence show that;
n
2
1 2 n xdx n n
0
3
As x is increasing;
Area outer rectangles Area under curve
25. b) Hence show that;
n
2
1 2 n xdx n n
0
3
As x is increasing;
Area outer rectangles Area under curve
n
1 2 n xdx
0
26. b) Hence show that;
n
2
1 2 n xdx n n
0
3
As x is increasing;
Area outer rectangles Area under curve
n
1 2 n xdx
0
n
2 x x
3 0
2
n n
3
27. b) Hence show that;
n
2
1 2 n xdx n n
0
3
As x is increasing;
Area outer rectangles Area under curve
n
1 2 n xdx
0
n
2 x x
3 0
2
n n
3
n
2
1 2 n xdx n n
0
3
28. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
29. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
Test: n = 1
30. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
Test: n = 1
L.H .S 1
1
31. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
Test: n = 1
41 3
L.H .S 1 R.H .S 1
6
1
7
6
32. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
Test: n = 1
41 3
L.H .S 1 R.H .S 1
6
1
7
6
L.H .S R.H .S
33. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
Test: n = 1
41 3
L.H .S 1 R.H .S 1
6
1
7
6
L.H .S R.H .S
Hence the result is true for n = 1
34. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
Test: n = 1
41 3
L.H .S 1 R.H .S 1
6
1
7
6
L.H .S R.H .S
Hence the result is true for n = 1
Assume the result is true for n k where k is a positive integer
35. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
Test: n = 1
41 3
L.H .S 1 R.H .S 1
6
1
7
6
L.H .S R.H .S
Hence the result is true for n = 1
Assume the result is true for n k where k is a positive integer
4k 3
i.e. 1 2 k k
6
36. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
Test: n = 1
41 3
L.H .S 1 R.H .S 1
6
1
7
6
L.H .S R.H .S
Hence the result is true for n = 1
Assume the result is true for n k where k is a positive integer
4k 3
i.e. 1 2 k k
6
Prove the result is true for n k 1
37. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
Test: n = 1
41 3
L.H .S 1 R.H .S 1
6
1
7
6
L.H .S R.H .S
Hence the result is true for n = 1
Assume the result is true for n k where k is a positive integer
4k 3
i.e. 1 2 k k
6
Prove the result is true for n k 1
4k 7
i.e. Prove 1 2 k 1 k 1
6
40. Proof: 1 2 k 1 1 2 k k 1
4k 3
k k 1
6
41. Proof: 1 2 k 1 1 2 k k 1
4k 3
k k 1
6
4k 3 k 6 k 1
2
6
42. Proof: 1 2 k 1 1 2 k k 1
4k 3
k k 1
6
4k 3 k 6 k 1
2
6
16k 3 24k 2 9k 6 k 1
6
43. Proof: 1 2 k 1 1 2 k k 1
4k 3
k k 1
6
4k 3 k 6 k 1
2
6
16k 3 24k 2 9k 6 k 1
6
k 116k 2 8k 1 1 6 k 1
6
44. Proof: 1 2 k 1 1 2 k k 1
4k 3
k k 1
6
4k 3 k 6 k 1
2
6
16k 3 24k 2 9k 6 k 1
6
k 116k 2 8k 1 1 6 k 1
6
k 14k 12 1 6 k 1
6
45. Proof: 1 2 k 1 1 2 k k 1
4k 3
k k 1
6
4k 3 k 6 k 1
2
6
16k 3 24k 2 9k 6 k 1
6
k 116k 2 8k 1 1 6 k 1
6
k 14k 12 1 6 k 1
6
k 14k 1 6 k 1
2
6
46. Proof: 1 2 k 1 1 2 k k 1
4k 3
k k 1
6
4k 3 k 6 k 1
2
6
16k 3 24k 2 9k 6 k 1
6
k 116k 2 8k 1 1 6 k 1
6
k 14k 12 1 6 k 1
6
k 14k 1 6 k 1
2
6
4k 1 k 1 6 k 1
6
4k 7 k 1
6
47. Hence the result is true for n = k +1 if it is also true for n =k
48. Hence the result is true for n = k +1 if it is also true for n =k
Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2,
and since the result is true for n = 2 then it is also true for n =2+1 i.e.
n=3, and so on for all positive integral values of n
49. Hence the result is true for n = k +1 if it is also true for n =k
Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2,
and since the result is true for n = 2 then it is also true for n =2+1 i.e.
n=3, and so on for all positive integral values of n
d) Use b) and c) to estimate;
1 2 10000 to the nearest hundred
50. Hence the result is true for n = k +1 if it is also true for n =k
Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2,
and since the result is true for n = 2 then it is also true for n =2+1 i.e.
n=3, and so on for all positive integral values of n
d) Use b) and c) to estimate;
1 2 10000 to the nearest hundred
2 4n 3
n n 1 2 n n
3 6
51. Hence the result is true for n = k +1 if it is also true for n =k
Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2,
and since the result is true for n = 2 then it is also true for n =2+1 i.e.
n=3, and so on for all positive integral values of n
d) Use b) and c) to estimate;
1 2 10000 to the nearest hundred
2 4n 3
n n 1 2 n n
3 6
2 410000 3
10000 10000 1 2 10000 10000
3 6
52. Hence the result is true for n = k +1 if it is also true for n =k
Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2,
and since the result is true for n = 2 then it is also true for n =2+1 i.e.
n=3, and so on for all positive integral values of n
d) Use b) and c) to estimate;
1 2 10000 to the nearest hundred
2 4n 3
n n 1 2 n n
3 6
2 410000 3
10000 10000 1 2 10000 10000
3 6
666700 1 2 10000 666700
53. Hence the result is true for n = k +1 if it is also true for n =k
Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2,
and since the result is true for n = 2 then it is also true for n =2+1 i.e.
n=3, and so on for all positive integral values of n
d) Use b) and c) to estimate;
1 2 10000 to the nearest hundred
2 4n 3
n n 1 2 n n
3 6
2 410000 3
10000 10000 1 2 10000 10000
3 6
666700 1 2 10000 666700
1 2 10000 666700 to the nearest hundred
54. Hence the result is true for n = k +1 if it is also true for n =k
Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2,
and since the result is true for n = 2 then it is also true for n =2+1 i.e.
n=3, and so on for all positive integral values of n
d) Use b) and c) to estimate;
1 2 10000 to the nearest hundred
2 4n 3
n n 1 2 n n
3 6
2 410000 3
10000 10000 1 2 10000 10000
3 6
666700 1 2 10000 666700
1 2 10000 666700 to the nearest hundred
Exercise 10F