F.4 MATHS題

1.If (2-i)(a+bi)=5+20i,where a and b are real numbers,find the values of a and b
(2-i)(a+bi)=2a + b +(2b – a)i
2a + b +(2b – a)i = 5 + 20i
2a + b = 5 ------------ (1)
2b – a = 20 ----------- (2)
Solve for a and b
a = -2, b = 9

2.Calculate the value of 0.83(3 repeat)X 2.72(72 repeat) and express the answer as a fraction
Let x = 0.83333
10x = 8.333
100x = 83.3333
100x - 10x = 83.3333 - 8.333 = 75
90x = 75
x = 75/90 = 5/6
Let y =2.727272
100y = 272,7272
100y - y = 272,7272 - 2.7272
99y = 270
y = 270/99 = 30/11

0.83(3 repeat) = 5/6 and 2.72(72 repeat) = 30/11
0.83(3 repeat)X 2.72(72 repeat) = 25/11

3.If 2i is a solution of the equation x^4 + 3x^2 + a =0,find the value of a
Let a = -4
x^4 + 3x^2 - 4 = 0
(x^2)^2 + 3(x^2) - 4 = 0
let k = x^2
k^2 +3 k - 4 = 0
(k + 4)(k - 1) = 0
k + 4 = 0
k = -4
x^2 = - 4
x^2 = (4)(-1)
x = 2i
So a = - 4

4.It is given that (a+bi)+(2-i)=2b-ai,where a and b are real numbers,Find the value of a and b
(a+bi)+(2-i)= (a + 2) + (b – 1)i
b -1 = -a
a + 2 = 2b
Solve for a and b
a = 0, b = 1

5.(a)Let a be a real number.If the result of (2a+7i)-(12-4i) is a pure imaginary number,find the value of a
(2a+7i)-(12-4i) = (2a – 12) + 3i
2a – 12 = 0
a = 6

(b)Let a and b be non-zero real number.If the result of (a+bi)(-10+25i)is a real number,find a:b
(a+bi)(-10+25i) = -10a – 25c+(25a – 10b)i
25a – 10b = 0
25a = 10b
a/b =10/25 =2/5
a:b =2:5