\(\sin x + i\cos 2x\) and \(\cos x - i\sin 2x\) are conjugate to each other for:
\(x = n\pi\)
\(x = \left(n+\tfrac{1}{2}\right)\tfrac{\pi}{2}\)
\(x = 0\)
No value of \(x\)
The real value of \(\alpha\) for which the expression \(\dfrac{1 - i\sin\alpha}{1 + 2i\sin\alpha}\) is purely real is:
\((n+1)\dfrac{\pi}{2}\)
\((2n+1)\dfrac{\pi}{2}\)
\(n\pi\)
None of these
If \(z = x + iy\) lies in the third quadrant, then \(\overline{z}/z\) also lies in the third quadrant if
\(x > y > 0\)
\(x < y < 0\)
\(y < x < 0\)
\(y > x > 0\)
The value of \((z + 3)(\overline{z} + 3)\) is equivalent to
\(|z + 3|^2\)
\(|z - 3|\)
\(z^2 + 3\)
None of these
If \(\left(\dfrac{1+i}{1-i}\right)^x = 1\), then
\(x = 2n + 1\)
\(x = 4n\)
\(x = 2n\)
\(x = 4n + 1\)
A real value of \(x\) satisfies the equation \(\dfrac{3 - 4ix}{3 + 4ix} = \alpha - i\beta\) (\(\alpha,\beta\in\mathbb{R}\)) if \(\alpha^2 + \beta^2 =\)
1
-1
2
-2
Which of the following is correct for any two complex numbers \(z_1\) and \(z_2\)?
\(|z_1 z_2| = |z_1||z_2|\)
\(\arg(z_1 z_2) = \arg(z_1) \cdot \arg(z_2)\)
\(|z_1 + z_2| = |z_1| + |z_2|\)
\(|z_1 + z_2| \ge |\,|z_1| - |z_2|\,|\)
The point represented by the complex number \(2 - i\) is rotated about origin through an angle \(\dfrac{\pi}{2}\) in the clockwise direction; the new position of point is:
\(1 + 2i\)
\(-1 - 2i\)
\(2 + i\)
\(-1 + 2i\)
Let \(x,y \in \mathbb{R}\). Then \(x + iy\) is a non-real complex number if:
\(x = 0\)
\(y = 0\)
\(x \ne 0\)
\(y \ne 0\)
If \(a + ib = c + id\), then
\(a^2 + c^2 = 0\)
\(b^2 + c^2 = 0\)
\(b^2 + d^2 = 0\)
\(a^2 + b^2 = c^2 + d^2\)
The complex number \(z\) which satisfies the condition \(\left|\dfrac{i + z}{i - z}\right| = 1\) lies on
circle \(x^2 + y^2 = 1\)
the x-axis
the y-axis
the line \(x + y = 1\)
If \(z\) is a complex number, then
\(|z^2| > |z|^2\)
\(|z^2| = |z|^2\)
\(|z^2| < |z|^2\)
\(|z^2| \ge |z|^2\)
\(|z_1 + z_2| = |z_1| + |z_2|\) is possible if
\(z_2 = \overline{z_1}\)
\(z_2 = \dfrac{1}{z_1}\)
\(\arg(z_1) = \arg(z_2)\)
\(|z_1| = |z_2|\)
The real value of \(\theta\) for which the expression \(\dfrac{1 + i\cos\theta}{1 - 2i\cos\theta}\) is a real number is:
\(n\pi + \dfrac{\pi}{4}\)
\(n\pi + (-1)^n\dfrac{\pi}{4}\)
\(2n\pi \pm \dfrac{\pi}{2}\)
none of these
The value of \(\arg(x)\) when \(x < 0\) is:
0
\(\dfrac{\pi}{2}\)
\(\pi\)
none of these
If \(f(z) = \dfrac{7 - z}{1 - \overline{z}}\), where \(z = 1 + 2i\), then \(|f(z)|\) is
\(\dfrac{|z|}{2}\)
\(|z|\)
\(2|z|\)
none of these