Part A:
Given

defined by


but

Since, f(xy) ≠ f(x)f(y)
Therefore, the function is not a homomorphism.
Part B:
Given

defined by

Note that in

, -1 = 1 and f(0) = 0 and f(1) = -1 = 1, so we can also use the formular


and

Therefore, the function is a homomorphism.
Part C:
Given

, defined by


Since, f(x+y) ≠ f(x) + f(y), therefore, the function is not a homomorphism.
Part D:
Given

, defined by


but

Since, h(ab) ≠ h(a)h(b), therefore, the funtion is not a homomorphism.
Part E:
Given

, defined by
![\left([x_{12}]\right)=[x_4]](https://tex.z-dn.net/?f=%5Cleft%28%5Bx_%7B12%7D%5D%5Cright%29%3D%5Bx_4%5D)
, where
![[u_n]](https://tex.z-dn.net/?f=%5Bu_n%5D)
denotes the lass of the integer

in

.
Then, for any
![[a_{12}],[b_{12}]\in Z_{12}](https://tex.z-dn.net/?f=%5Ba_%7B12%7D%5D%2C%5Bb_%7B12%7D%5D%5Cin%20Z_%7B12%7D)
, we have
![f\left([a_{12}]+[b_{12}]\right)=f\left([a+b]_{12}\right) \\ \\ =[a+b]_4=[a]_4+[b]_4=f\left([a]_{12}\right)+f\left([b]_{12}\right)](https://tex.z-dn.net/?f=f%5Cleft%28%5Ba_%7B12%7D%5D%2B%5Bb_%7B12%7D%5D%5Cright%29%3Df%5Cleft%28%5Ba%2Bb%5D_%7B12%7D%5Cright%29%20%5C%5C%20%20%5C%5C%20%3D%5Ba%2Bb%5D_4%3D%5Ba%5D_4%2B%5Bb%5D_4%3Df%5Cleft%28%5Ba%5D_%7B12%7D%5Cright%29%2Bf%5Cleft%28%5Bb%5D_%7B12%7D%5Cright%29)
and
![f\left([a_{12}][b_{12}]\right)=f\left([ab]_{12}\right) \\ \\ =[ab]_4=[a]_4[b]_4=f\left([a]_{12}\right)f\left([b]_{12}\right)](https://tex.z-dn.net/?f=f%5Cleft%28%5Ba_%7B12%7D%5D%5Bb_%7B12%7D%5D%5Cright%29%3Df%5Cleft%28%5Bab%5D_%7B12%7D%5Cright%29%20%5C%5C%20%5C%5C%20%3D%5Bab%5D_4%3D%5Ba%5D_4%5Bb%5D_4%3Df%5Cleft%28%5Ba%5D_%7B12%7D%5Cright%29f%5Cleft%28%5Bb%5D_%7B12%7D%5Cright%29)
Therefore, the function is a homomorphism.
Answer: x = - 2√5, √5/3
Step-by-step explantion
3 √ 5 x 2 + 25 x − 10 √ 5 = 0 35x2+25x−105=0
⇒ 3√5x2 + 30x – 5x – 10√5 = 0
⇒ 3√5x(x + 2√5) – 5(x + 2√5) = 0
⇒ (x + 2√5)(3√5x – 5) = 0
x = - 2√5, √5/3
1. 7x(x + 3)
7x(x) + 7(3)
7x² + 21
2. (7x + 1)(x + 3)
7x² + 21x + x + 3
7x² + 22x + 3
3. (x² + x)²
(x² + x)(x² + x)
x^4 + x³ + x³ + x²
x^4 + 2x³ + x²
The answer is 13.5, hope this helps!
The first term of the geometric sequence is 0.0061
<u>Step-by-step explanation:</u>
The general form of geometric sequence is a, ar, ar²,ar³,.......
where,
- a is the first term of the sequence.
- r is the common ratio. Here, the common ratio r = 4.
- The 8th term of the sequence is 100. Hence n = 8.
<u>The formula to find the nth term of the geometric sequence is given by :</u>
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ a = 0.0061
Therefore, the first term of the geometric sequence is 0.0061