/أسبوع مجانًااختبار #4915
❓ Complex Roots Challenge
10 سؤال
Test your skills with complex numbers by solving an intriguing equation involving square roots. This quiz is perfect for students and enthusiasts of advanced algebra.
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السؤال 1
What is a common first step when solving the equation √k + √-k = 15 in the complex plane?
A.
Assume k is a real number.
B.
Let √k = z.
C.
Square both sides immediately.
D.
Substitute k = x + yi.
السؤال 2
If √k = z, what is √-k in terms of z, assuming principal roots and that √(-1) = i?
A.
z/i
B.
-z
C.
iz
D.
-iz
السؤال 3
After substituting √k = z and √-k = iz, what is the resulting linear equation for z?
A.
z^2 - z = 15i
B.
z + iz = 15
C.
z - iz = 15
D.
z^2 + (iz)^2 = 15
السؤال 4
From the equation z(1 + i) = 15, what is the first step to isolate z?
A.
Subtract i from both sides.
B.
Divide both sides by (1 + i).
C.
Multiply by (1 - i).
D.
Divide both sides by 15.
السؤال 5
To simplify z = 15 / (1 + i), which expression should you multiply the numerator and denominator by?
A.
(1 + i)
B.
i
C.
(1 - i)
D.
-i
السؤال 6
What is the value of z in the equation z = 15 / (1 + i)?
A.
-7.5 - 7.5i
B.
7.5 + 7.5i
C.
15 - 15i
D.
7.5 - 7.5i
السؤال 7
If z = √k, and you've found z, how do you calculate k?
A.
k = z
B.
k = √z
C.
k = z^2
D.
k = 1/z
السؤال 8
If z = a + bi, what is z^2?
A.
a^2 + b^2
B.
a^2 - b^2 + 2abi
C.
a^2 - b^2
D.
a^2 + b^2 - 2abi
السؤال 9
Given z = 7.5 - 7.5i, what is the value of k = z^2?
A.
-112.5i
B.
112.5
C.
112.5i
D.
-112.5
السؤال 10
If the problem specified that 'k' must be a real number, how many solutions would there be for √k + √-k = 15?
A.
Exactly one solution.
B.
Infinitely many solutions.
C.
Exactly two solutions.
D.
No real solutions.
الموجّه: √k +√-k=15 finf k
