What is the conjugate of \( \dfrac{2 - i}{(1 - 2i)^2} \)?
-\(\dfrac{2}{25}\) − i \(\dfrac{11}{25}\)
If \(|z_1| = |z_2|\), is it necessary that \(z_1 = z_2\)?
No
If \( \dfrac{(a^2+1)^2}{2a - i} = x + iy \), what is the value of \(x^2 + y^2\)?
\( \dfrac{(a^2+1)^4}{4a^2 + 1} \)
Find \(z\) if \(|z| = 4\) and \(\arg(z) = \dfrac{5\pi}{6}\).
-2\sqrt{3} + 2i
Find \( \left| \dfrac{(1+i)(2+i)}{3+i} \right| \).
1
Find the principal argument of \((1 + i\sqrt{3})^2\).
\( \dfrac{2\pi}{3} \)
Where does \(z\) lie if \( \left| \dfrac{z - 5i}{z + 5i} \right| = 1 \)?
Real axis
If |z + 1| = z + 2(1 + i), then find z.
\(\dfrac{1}{2} - 2i\)
If arg(z − 1) = arg(z + 3i), then find x − 1 : y, where z = x + iy.
1 : 3
Show that \(\left|\dfrac{z - 2}{z - 3}\right| = 2\) represents a circle. Find its centre and radius.
Centre: \(\left(\dfrac{10}{3}, 0\right)\)
Radius: \(\dfrac{2}{3}\)
If \(\dfrac{z - 1}{z + 1}\) is a purely imaginary number (z ≠ −1), then find |z|.
1
z₁ and z₂ are two complex numbers such that |z₁| = |z₂| and arg(z₁) + arg(z₂) = π, then show that z₁ = − z̄₂.
If |z₁| = 1 (z₁ ≠ −1) and z₂ = \(\dfrac{z₁ − 1}{z₁ + 1}\), show that the real part of z₂ is zero.
If z₁, z₂ and z₃, z₄ are two pairs of conjugate complex numbers, then find \(\arg\left(\dfrac{z₁}{z₄}\right) + \arg\left(\dfrac{z₂}{z₃}\right).\)
0
If |z₁| = |z₂| = … = |zₙ| = 1, then show that
\[|z₁ + z₂ + ⋯ + zₙ| = \left|\dfrac{1}{z₁} + \dfrac{1}{z₂} + ⋯ + \dfrac{1}{zₙ}\right|.\]
For complex numbers z₁ and z₂, if arg(z₁) − arg(z₂) = 0, then show that |z₁ − z₂| = ||z₁| − |z₂||.
Solve the system of equations Re(z²) = 0, |z| = 2.
\(\sqrt{2} + i\sqrt{2},\; \sqrt{2} - i\sqrt{2},\; -\sqrt{2} + i\sqrt{2},\; -\sqrt{2} - i\sqrt{2}\)
Find the complex number satisfying the equation z + √2 |z + 1| + i = 0.
−2 − i
Write the complex number \(z = \dfrac{1 - i}{\cos \dfrac{\pi}{3} + i \sin \dfrac{\pi}{3}}\) in polar form.
\(\sqrt{2}\left(\cos\dfrac{5\pi}{12} + i\sin\dfrac{5\pi}{12}\right)\)
If z and w are two complex numbers such that |zw| = 1 and arg(z) − arg(w) = \(\dfrac{\pi}{2}\), then show that \(\overline{z} w = −i\).
(i) For any two complex numbers \(z_1, z_2\) and any real numbers \(a, b\), \(\; |az_1 - bz_2|^2 + |bz_1 + az_2|^2 = ____\).
\((a^2 + b^2)\bigl(|z_1|^2 + |z_2|^2\bigr)\)
(ii) The value of \(\sqrt{-25} \times \sqrt{-9}\) is ____.
-15
(iii) The number \(\dfrac{(1 - i)^3}{1 - i^3}\) is equal to ____.
-2
(iv) The sum of the series \(i + i^2 + i^3 + \dots\) up to 100 terms is ____.
0
(v) The multiplicative inverse of \(1 + i\) is ____.
\(\dfrac{1}{2} - \dfrac{i}{2}\)
(vi) If \(z_1\) and \(z_2\) are complex numbers such that \(z_1 + z_2\) is a real number, then \(z_2 = ____\).
\(\overline{z_1}\)
(vii) \(\arg(z) + \arg(\overline{z})\) \((\overline{z} \ne 0)\) is ____.
0
(viii) If \(|z + 4| \le 3\), then the greatest and least values of \(|z + 1|\) are ____ and ____ respectively.
6 and 0
(ix) If \(\left|\dfrac{z - 2}{z + 2}\right| = \dfrac{\pi}{6}\), then the locus of \(z\) is ____.
a circle
(x) If \(|z| = 4\) and \(\arg(z) = \dfrac{5\pi}{6}\), then \(z = ____\).
\(-2\sqrt{3} + 2i\)
Match the items in Column A with Column B using the table below.
| Column A | Column B |
|---|---|
(a) The polar form of \(i + \sqrt{3}\) | (i) Perpendicular bisector of segment joining \((-2,0)\) and \((2,0)\) |
(b) The amplitude of \(-1 + \sqrt{3}\) is | (ii) On or outside the circle having centre at \((0,-4)\) and radius 3 |
(c) If \(|z + 2| = |z - 2|\), then locus of \(z\) is | (iii) \(\dfrac{2\pi}{3}\) |
(d) If \(|z + 2i| = |z - 2i|\), then locus of \(z\) is | (iv) Perpendicular bisector of segment joining \((0,-2)\) and \((0,2)\) |
(e) Region represented by \(|z + 4i| \ge 3\) is | (v) \(2\bigl(\cos\tfrac{\pi}{6} + i\sin\tfrac{\pi}{6}\bigr)\) |
(f) Region represented by \(|z + 4| \le 3\) is | (vi) On or inside the circle having centre \((-4,0)\) and radius 3 units |
(g) Conjugate of \(\dfrac{1 + 2i}{1 - i}\) lies in | (vii) First quadrant |
(h) Reciprocal of \(1 - i\) lies in | (viii) Third quadrant |
| Column A | Matched Item from Column B |
|---|---|
(a) The polar form of \(i + \sqrt{3}\) | (v) \(2\bigl(\cos\tfrac{\pi}{6} + i\sin\tfrac{\pi}{6}\bigr)\) |
(b) The amplitude of \(-1 + \sqrt{3}\) is | (iii) \(\dfrac{2\pi}{3}\) |
(c) If \(|z + 2| = |z - 2|\), then locus of \(z\) is | (i) Perpendicular bisector of segment joining \((-2,0)\) and \((2,0)\) |
(d) If \(|z + 2i| = |z - 2i|\), then locus of \(z\) is | (iv) Perpendicular bisector of segment joining \((0,-2)\) and \((0,2)\) |
(e) Region represented by \(|z + 4i| \ge 3\) is | (ii) On or outside the circle having centre at \((0,-4)\) and radius 3 |
(f) Region represented by \(|z + 4| \le 3\) is | (vi) On or inside the circle having centre \((-4,0)\) and radius 3 units |
(g) Conjugate of \(\dfrac{1 + 2i}{1 - i}\) lies in | (viii) Third quadrant |
(h) Reciprocal of \(1 - i\) lies in | (vii) First quadrant |
\(\sin x + i\cos 2x\) 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