Principle of Management

Grade:11

Section: A/B/C/M/N

Worksheet Number:4

Subject:

Mathematics

Topic:

Name of teacher:

Ms. Sheeba Manoj

Date :26/6/13

REVISION WORKSHEET
Submission date:

3/9/13

Name of student:

PRINCIPLE OF MATHEMATICAL INDUCTION
Prove the following by Mathematical Induction
1.
1
1 + 4 + 7 + …………………….. + (3n – 2 ) = n(3n  1)
2
2.
4 + 8 + …………………………. + 4n = 2n( n + 1 )
3.
1
1.3 + 2.4 + 3.5 + ………………. + n(n +2) = n(n  1)(2n  7)
6
4.
5
5 + 15 + 45 +……………………... +5.3n – 1 = (3 n 1 )
2
5. n a + (a + d) + ( a + 2d) + ……………… + [a +(n – 1)d] = [2a  (n  1)d ]
2
6.
1
12 + 32 + 52 +…………………………..+ (2n – 1)2 = n(4n 2  1)
3
7.
1
1
1
1 n 

 ................................... 

3.7 7.11 11.15
(4n  1)(4n  3) 3(4n  3) n + 15n – 1 is divisible by 9
8.
4
9.
10.
11.

32n + 7 is divisible by 8
2.7n + 3.5n – 5 is divisible by 24.

12.
13.
14.

7 n  3n is divisible by 4 n(n+1)(2n+1) is divisible by 6

15.

2n + 7 < ( n + 3 )2

x 2 n1  y 2 n1 is divisible by x + y.

1 + 2 + 3 + ……………………..+ n <

Sheeba Manoj

1
(2n  1) 2
8

Page 1

TRIGONONOMETRY

Sheeba Manoj

Page 2

COMPLEX NUMBERS
Prove that
1.

1 + i2 + i4 +i6 = 0

2.

6i54 + 5i37 - 2i11 + 6i68 = 7i
1
( 1+ i )4 ( 1 + )4 = 16 i 3.

4.

in. i n+1. i n+2 . i n+3 = -1 , n  N
Solve for x and y.

5.

2x + 3iy = 5 + 6i

6.

( 3 + i ) x + ( 1 – 2i) y + 7i = 0

7.

1  i x  2i
3i

+

2  3i y  i
3i

= i

Express the following numbers in the form x + iy
8.

2  3i 

9.

1  i 

2

1 i
3

10.

11.

1 i3
2
3
2
+
1 i 2  i 1 i

5  12i  5  12i
5  12i  5  12i
Find the multiplicative inverse of the following

12.

(2 – i) (3+i)

13.

2  3i
3  2i

14.

i  1i  2
i  1i  2
Sheeba Manoj

Page 3

Find the modulus and amplitude of the following complex numbers and express in the polar form.
3i

15.

1+

17.