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クイズ #4915

❓ Complex Roots Challenge

10 問
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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10 件中 0 件正解
10 件中 0 件回答済み
質問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.
質問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
質問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
質問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.
質問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
質問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
質問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
質問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
質問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
質問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.
プロンプト: √k +√-k=15 finf k
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