The correct answers are:
(1) 4+5i
(2) 9+58i
(3) 8 - 12i (Option B; The question's options have a typo)
(4) Square root of 208 (Option B).
Explanations:
(1) Given: (6+6i)-(2+i)
We need to simplify the given expression. For that, add real parts with each other, and add imaginary parts of the complex numbers with each other. Remember that the numbers with the symbol "i" are the imaginary parts of the complex number. Therefore,
(6-2) + (6i - i) = 4 + 5i (ans)
(2) Given: (8+i)(2+7i)
Now in this case we will multiply two complex numbers with each other; here in this case, we have to remember that . Now let us find out the multiplication of two complex numbers:
(8+i)(2+7i)
8(2+7i) + i(2+7i)
16 + 58i + 7(-1)
= 9 + 58i
Hence the correct answer is 9+58i.
(3) Given: 8+12i
In simple terms, in order to find the conjugate of the complex number, we take the real number of the complex number as is, but we change the sign of the imaginary part of the complex number. In the given expression, 8 is the real number; hence, we will take it as is, whereas, +12i is the imaginary part of 8+12i. So to find the conjugate, we will change +12i to -12i.
Therefore, the conjugate of the complex number will become 8 - 12i (Option B; The question's options have a typo).
(4) Given: 8+12i
First, we need to find the complex conjugate of the given complex number. Please see the explanation given in Part (3) above to find the complex conjugate. The complex conjugate of 8+12i is 8-12i
Now, to find the absolute value of the complex conjugate 8-12i, follow these steps:
|8-12i|
We will add the square of the real number (8) with the square of the imaginary number (-12) and take the square-root at the end to find the absolute value:
Hence the correct answer is square root of 208 (Option B).