Sure! Let's go through each question step-by-step.
#### Question 17
Given [tex]\(3 + 5i + x - yi = 6 - 2i\)[/tex],
we need to find [tex]\(x^2 - y^2\)[/tex].
First, equate the real and imaginary parts separately:
- Real part: [tex]\(3 + x = 6 \Rightarrow x = 6 - 3 = 3\)[/tex]
- Imaginary part: [tex]\(5 - y = -2 \Rightarrow y = 5 + 2 = 7\)[/tex]
Now, compute [tex]\(x^2 - y^2\)[/tex]:
[tex]\[ x^2 - y^2 = 3^2 - 7^2 = 9 - 49 = -40 \][/tex]
So, the correct answer is:
(E) None of these
#### Question 18
To evaluate [tex]\(i^{216}\)[/tex], we know that:
[tex]\[ i^1 = i \][/tex]
[tex]\[ i^2 = -1 \][/tex]
[tex]\[ i^3 = -i \][/tex]
[tex]\[ i^4 = 1 \][/tex]
The powers of [tex]\(i\)[/tex] cycle every 4 indices. To find [tex]\(i^{216}\)[/tex]:
[tex]\[ 216 \mod 4 = 0 \][/tex]
Thus, [tex]\(i^{216} = (i^4)^{54} = 1^{54} = 1 \)[/tex].
So, the correct answer is:
(E) None of these
#### Question 19
Simplify [tex]\((9 + \sqrt{-2})(6 - \sqrt{-10})\)[/tex].
First, convert the square roots of negative numbers to use the imaginary unit [tex]\(i\)[/tex]:
[tex]\[ \sqrt{-2} = \sqrt{2}i \][/tex]
[tex]\[ \sqrt{-10} = \sqrt{10}i \][/tex]
Now the expression becomes:
[tex]\[ (9 + \sqrt{2}i)(6 - \sqrt{10}i) \][/tex]
Use the distributive property (FOIL method):
[tex]\[
= 9 \cdot 6 + 9 \cdot (-\sqrt{10}i) + \sqrt{2}i \cdot 6 + \sqrt{2}i \cdot (-\sqrt{10}i)
\][/tex]
[tex]\[
= 54 - 9\sqrt{10}i + 6\sqrt{2}i - \sqrt{2} \cdot \sqrt{10} \cdot i^2
\][/tex]
[tex]\[
= 54 - 9\sqrt{10}i + 6\sqrt{2}i - \sqrt{20} \cdot (-1)
\][/tex]
[tex]\[
= 54 - 9\sqrt{10}i + 6\sqrt{2}i + \sqrt{20}
\][/tex]
[tex]\[
= 54 + 2\sqrt{5} + (6\sqrt{2} - 9\sqrt{10})i
\][/tex]
So, the correct answer is:
(C) [tex]\((54 + 2\sqrt{5}) + (6\sqrt{2} - 9\sqrt{10})i\)[/tex]
#### Question 20
Find the roots of the equation [tex]\(x^2 - 2x - 24 = 0\)[/tex].
Use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = -24\)[/tex]:
[tex]\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-24)}}{2 \cdot 1}
\][/tex]
[tex]\[
= \frac{2 \pm \sqrt{4 + 96}}{2}
\][/tex]
[tex]\[
= \frac{2 \pm \sqrt{100}}{2}
\][/tex]
[tex]\[
= \frac{2 \pm 10}{2}
\][/tex]
Two possible solutions:
[tex]\[
x = \frac{2 + 10}{2} = \frac{12}{2} = 6
\][/tex]
[tex]\[
x = \frac{2 - 10}{2} = \frac{-8}{2} = -4
\][/tex]
So, the correct answer is:
(E) None of these
