1.
Given that y = xe−x,
dy
;
(a) find
dx
(2)
n
(b)
+
use mathematical induction to prove that, for n
,
d y
; = (–1)n+1 e–x (n – x).
dx n
(10)
(Total 12 marks)
2.
(a)
Use mathematical induction to prove that
(1)(1!) + (2)(2!) + (3)(3!) + ...+ (n)(n!) = (n +1)!−1 where n
+
.
(8)
(b)
9
Find the minimum number of terms of the series for the sum to exceed 10 .
(3)
(Total 11 marks)
3.
(a)
Use mathematical induction to prove that
n
1
n

, n
(2
r

1)
(2
r

1)
2
n
1
r 1

+
.
(6)
(b)
Hence show that the sum of the first (n + 1) terms of the series
1
1
1
1
(n  1)
+
+
+
+ ... is
.
3
15
(2 n  3)
35
63
(3)
(Total 9 marks)
4.
Use mathematical induction to prove that 5 + 9 + 2 is divisible by 4, for n 
n
n
+
.
(Total 9 marks)
5.
(a)
Use mathematical induction to prove De Moivre’s theorem
(cos + i sin)n = cos (n) + i sin (n), n  +.
(b)
Consider z5 – 32 = 0.
(7)
(i)
(ii)
(iii)
(iv)

 2π 
 2π  
Show that z1 = 2  cos    i sin    is one of the complex roots of this
 5 
 5 

equation.
Find z12, z13, z14, z15, giving your answer in the modulus argument form.
Plot the points that represent z1, z12, z13, z14 and z15, in the complex plane.
The point z1n is mapped to z1n+1 by a composition of two linear transformations,
where n = 1, 2, 3, 4. Give a full geometric description of the two transformations.
(9)
(Total 16 marks)
6.
1 .
(a)
Evaluate (1 + i)2, where i =
(b)
Prove, by mathematical induction, that (1 + i)4n = (–4)n, where n 
(2)
*.
(6)
(c)
32
Hence or otherwise, find (1 + i) .
(2)
(Total 10 marks)
1