Fsc Part 1 Mathematics (Complete Solution)

Q7
Simplify the following

(i) ( -
1
2
+
 3  ι
2
) 3

SOLUTION:

= (
-1 +  3  ι
 2
) 3

=
(-1+  3  ι)3
(2)3

=
(-1)3 + (√  3 ι)3+3(-1)(  3  ι)(-1+  3 ι)
8

=
-1+3√  3 ι3-3 3  ι(-1+  3 ι)
8

=
-1+3√  3 ι2.ι +3 3  ι-2  (3)(3)
8

=
-1+3√  3 (-1).ι +3 3  ι-3(-1)(3)
8

=
-1-3√  3 ι +3 3  ι+9
8

=
8
8

=1

(ii) ( -
1
 2
-
 3  ι
2
) 3

SOLUTION:

= (
-1-√  3  ι
2
) 3

=
(-1-√  3  ι)3
(2)3

=
(-1)3+(-√  3  ι)3+3(-1)(-√  3  ι)(-1-√  3  ι)
8

=
-1-3√  3 ι3+3 3  ι(-1-  3 ι)
8

=
-1-3√  3 ι2.ι -3 3  ι-2  (3)(3)
8

=
-1-3√  3 (-1).ι -3 3  ι-3(-1)(3)
8

=
-1+3√  3 ι -3 3  ι+9
8

=
8
 8

=1

(iii) ( -
1
 2
-
 3  ι
2
) - 2 + 1

SOLUTION:

= (
-1-√  3  ι
2
) - 1

=
2
-1-√  3  ι

= -
2
1+√  3  ι

= -
2
1+√  3  ι
x
1-√  3  ι
1-√  3  ι

= -
2(1-√  3  ι)
(1+√  3  ι)(1-√  3  ι)

=
-2(1-√  3  ι)
(1)2-(√  3  ι)2

=
-2(1-√  3  ι)
1-3ι2

=
-2(1-√  3  ι)
1-3(-1)

=
-2(1-√  3  ι)
4

=
-1+√  3  ι)
2

(iv) (a+bι)2

Solution:

=(a)2+(bι)2+2(a)(bι)

=a2+b2ι2+2abι

=a2+b2(-1)+2abι

=a2-b2+2abι

(v) (a+bι)-2

SOLUTION:(a+bι)-2

=[(a+bι)2]-1

=[(a)2+(bι)2+2(a)(bι)]-1

=(a2+b2ι2+2abι)-1

=[a2+b2(-1)+2abι]-1

=(a2-b2+2abι)-1

=
1
(a2-b2+2abι)
x
(a2-b2-2abι)
(a2-b2-2abι)

=
1(a2-b2-2abι)
(a2-b2+2abι)(a2-b2-2abι)

=
a2-b2-2abι
(a2-b2)2-(2abι)2

=
a2-b2-2abι
(a4+b4-2a2b2)-4a2b2ι2

=
a2-b2-2abι
a4+b4-2a2b2-4a2b2(-1)

=
a2-b2-2abι
a4+b4-2a2b2+4a2b2

=
a2-b2-2abι
a4+b4+2a2b2

=
(a2-b2)-2abι
(a2+b2)2

=
a2-b2
(a2+b2)2
-
2abι
(a2+b2)2

(vi) (a+bι)3

SOLUTION:(a+bι)3

=(a)3+(bι)3+3(a)(bι)(a+bι)

=a3+b3ι3+3abι(a+bι)

=a3+b3ι3+3abι(a+bι)

=a3+b3ι2.ι+3a2bι+3ab2ι2

=a3+b3ι(-1)+3a2bι+3ab2(-1)

=a3-b3ι+3a2bι-3ab2

=a3-3ab2 +(3a2b-b3

(vii) (a-bι)3

SOLUTION:(a-bι)3

=(a)3+(-bι)3+3(a)(-bι)(a-bι)

=a3-b3ι3-3abι(a-bι)

=a3-b3ι2.ι-3a2bι+3ab2ι2

=a3-b3ι(-1)-3a2bι+3ab2(-1)

=a3+b3ι-3a2bι-3ab2

=a3-3ab2 -(3a2b-b3

(viii) (3 -  -4 )-3

SOLUTION: (3 -  -4 )-3

= [(3 -  -4 )3]-1

= [(3-2ι)3]-1

= [(3)3+(-2ι)3+3(3)(-2ι)(3-2ι)]-1

= [27-8ι3-18ι)(3-2ι)]-1

= [27-8ι2.ι-54ι+36ι2]-1

= [27-8(-1).ι-54ι+36(-1)]-1

= [27+8ι-54ι-36]-1

= [-9-46ι]-1

=
1
-9-46ι
x
-9+46ι
-9+46ι

=
1(-9+46ι)
(-9-46ι)(-9+46ι)

=
-9+46ι
(-9)2-(46ι)2

=
-9+46ι
81-2116ι2

=
-9+46ι
81-2116(-1)

=
-9+46ι
2197

= -
9
2197
+
46ι
2197

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