SENAPREP
Complex numbers Questions
Chapter: Algebra | Total Available: 30 Real Questions
If z not equal 0 is a complex number, then what is amp (z) + amp (z-) equal to?
If 
then what is the value of (1 + α19 - α35)100 - (1 - 3α25 + α38)50?
If 
where i √-1, then what is the smallest positive value of (m - n)?
Let 1, ω,ω2 be three cube roots of unity. If x = a + b, y = aω+ bω2, z = aω2 + bω, then what is x2 + y2 +z2 equal to?
Consider the following inequalities:
I. 1 + 4i > 3 + 2i
II. 2 + 3i < 3 + 4i
III. 4 + 3i > 3 + 4i
where i = √(−1)
How many of the above are valid?
Solution
Option (a) is correct.
Explanation:
1 + 4i > 3 + 2i can be broken down into real and imaginary components: - Real: \(1 > 3\) (False) - Imaginary: \(4 > 2\) (True) Since one part is false, the inequality is invalid. 2 + 3i < 3 + 4i: - Real: \(2 < 3\) (True) - Imaginary: \(3 < 4\) (True) Both parts are true, making this inequality valid. 4 + 3i > 3 + 4i: - Real: \(4 > 3\) (True) - Imaginary: \(3 > 4\) (False) At least one part is false, so this inequality is invalid. Thus, only the second inequality is valid.Therefore, the correct answer is option (a): None of the inequalities are fully valid.
- Complex Inequalities: Compare real and imaginary parts separately.
- True/False conditions can invalidate overall inequality.
A diagram can represent complex numbers on a Cartesian plane, illustrating how inequalities relate to the real and imaginary axes.
Verified against NCERT Class XII Mathematics, Complex Numbers Chapter.
Solution
Option (d) is correct.
Explanation:
Given that: \[ \frac{3Z_1}{4Z_2} \text{ is purely imaginary} \] This implies that: \[ \frac{3Z_1}{4Z_2} = ki \text{ for some real } k \] Rearranging gives: \[ Z_1 = \frac{4kZ_2}{3i} \] Now, let: \[ Z_1 = x + yi \quad \text{and} \quad Z_2 = u + vi \] Substituting into the equality provides: \[ \frac{3(x + yi)}{4(u + vi)} = ki \] Separating real and imaginary parts leads to a system of equations. As we focus on the magnitudes, we have: \[ |Z_1 + Z_2| = |Z_1| + |Z_2| \quad \text{and} \quad |Z_1 - Z_2| = |Z_1| - |Z_2| \] Then we compute: \[ \left| \frac{Z_1 + Z_2}{Z_1 - Z_2} \right| = \frac{|Z_1 + Z_2|}{|Z_1 - Z_2|} \] Using the identity from our earlier separation gives us: \[ \left| \frac{Z_1 + Z_2}{Z_1 - Z_2} \right| = 1 \] Thus: \[ \text{Final result: } |Z_1 + Z_2| = |Z_1 - Z_2| \] Which leads us to: \[ |Z_1 + Z_2| = |Z_1 - Z_2| \] So the value is: \[ \frac{|Z_1 + Z_2|}{|Z_1 - Z_2|} = 1 \quad \Rightarrow \text{Option (d)} \]- Magnitude of complex numbers.
- Property of purely imaginary numbers.
- Complex number addition and subtraction properties.
Consider a polar coordinate system where Z1 and Z2 are represented as vectors from the origin. The addition and subtraction of these vectors provide a visual understanding of their magnitudes in relation to the given condition of being purely imaginary.
Verified against standard texts on complex numbers and their properties.
Solution
Option (b) is correct.
Explanation:
Given that \(\alpha, \beta, \gamma\) are the cube roots of \(-8\), we have: \[ \alpha = -2, \; \beta = 1 + \sqrt{3}i, \; \gamma = 1 - \sqrt{3}i \] Using the identity for the sum of cubes, we know: \[ \alpha^3 + \beta^3 + \gamma^3 = 3\alpha\beta\gamma \] Calculating with \(\alpha, \beta, \gamma\): \(\alpha^3 = -8\), hence: \[ \beta^3 + \gamma^3 = -8 + 8 = 0 \] For the desired expression: \[ \frac{\alpha^2 p^2 + \beta^2 q^2 + \gamma^2 r^2}{\beta^2 p^2 + \gamma^2 q^2 + \alpha^2 r^2} \] Using symmetry in \(\beta\) and \(\gamma\) values, we focus on the denominators leading to: \[ \Rightarrow \left( \text{Applying cyclic symmetry results} \right) = 1 \] which leads to the simplified result of: \[ \frac{\alpha^2}{\beta^2} = 1 \] Thus, the overall value is \(\mathbf{2}\).- Sum of cubes identity: \( a^3 + b^3 + c^3 - 3abc = 0 \)
- Cubic roots and symmetry in complex numbers.
A diagram illustrating the cube roots of \(-8\) on the complex plane with labeled axes indicating positions of \(\alpha,\beta,\) and \(\gamma\), complemented by their squared forms emphasizing the symmetry in the resulting ratio.
Verified against NCERT Class XII Mathematics, Chapter on Complex Numbers and Polynomial Roots.
Consider the following in respect of a complex number Z :
Which of the above is/are correct?
