Sample QuestionsComplex Numbers and Quadratic Equations questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The real value of $\alpha$ for which the expression $\frac{1-\text{i}\sin\alpha}{1+2\text{i}\sin\alpha}$ is purely real is:
- A
$(\text{n}+1)\frac{\pi}{2}$
- B
$(2\text{n}+1)\frac{\pi}{2}$
- ✓
$\text{n}\pi$
- D
None of these, where $\text{n}\in\text{N}$
Answer: C.
View full solution →The value of $(\text{z}+3)(\bar{\text{z}}+3)$ is equivalent to:
- ✓
$|z + 3|^2$
- B
$|z - 3|$
- C
$z^2 + 3$
- D
Answer: A.
View full solution →$\sin\text{x}+\text{i}\cos2\text{x}$ and $\cos\text{x}-\text{i}\sin2\text{x}$ are conjugate to each other for:
Answer: D.
View full solution →$|z_1 + z_2| = |z_1| + |z_2|$ is possible if:
- A
$\text{z}_2=\bar{\text{z}_1}$
- B
$\text{z}_2=\frac{1}{\text{z}_1}$
- ✓
$\arg(\text{z}_1)=\arg(\text{z}_2)$
- D
$|\text{z}_1|=|\text{z}_2|$
Answer: C.
View full solution →The complex number $z$ which satisfies the condition $\Big|\frac{\text{i}+\text{z}}{\text{i}-\text{z}}\Big|=1$ lies on:
- A
Circle $x^2+ y^2= 1$
- ✓
The $x-$axis
- C
The $y-$axis
- D
The line $x + y = 1$
Answer: B.
View full solution →Let $z_1$ and $z_2$ be two complex numbers such that $|z_1 + z_2| = |z_1| + |z_2|,$ then arg $(z_1 - z_2) = 0.$
View full solution →2 is not a complex number.
The order relation is defined on the set of complex numbers.
If $z$ is a complex number such that $z \neq 0$ and Re $(z) = 0,$ then $Im (z^2) = 0.$
View full solution →The inequality |z - 4| < |z - 2| represents the region given by x > 3.
For a positive integer n, find the value of $(1-\text{i})^{\text{n}}\Big(1-\frac{1}{\text{i}}\Big)^{\text{n}}$
View full solution →Find $\Big|(1+\text{i})\frac{(2+\text{i})}{(3+\text{i})}\Big|$
View full solution →If $(1+\text{i})\text{z}=(1-\text{i})\bar{\text{z}},$ then show that $\text{z}=-\text{i}\bar{\text{z}}$
View full solution →Find principal argument of $(1+\text{i}\sqrt{3})^2$
View full solution →If $\Big(\frac{1+\text{i}}{1-\text{i}}\Big)^3-\Big(\frac{1-\text{i}}{1+\text{i}}\Big)^3=\text{x}+\text{iy},$ then find (x, y).
View full solution →If $|z_1| = |z_2|,$ is it necessary that $z_1 = z_2$?
View full solution →If $\Big(\frac{1-\text{i}}{1+\text{i}}\Big)^{10}=\text{a}+\text{ib},$ then find (a, b).
View full solution →If $\text{z}=\text{x}+\text{iy},$ then show that $\text{z}\bar{\text{z}}+2(\text{z}+\bar{\text{z}})+\text{b}=0,$ where $\text{b}\in\text{R},$ represents a circle.
View full solution →$z_1$ and $z_2$ are two complex numbers such that $|z_1| = |z_2|$ and $\text{arg(z}_1)+\text{arg(z}_2)=\pi,$ then show that $\text{z}_1=-\bar{\text{z}}_2$
View full solution →If the real part of $\frac{\bar{\text{z}}+2}{\bar{\text{z}}-1}$ is 4, then show that the locus of the point representing z in the complex plane is a circle.
View full solution →Show that $\Big|\frac{\text{z}-2}{\text{z}-3}\Big|=2$ represents a circle. Find its center and radius.
View full solution →If $\frac{(\text{a}^2+1)^2}{2\text{a}-\text{i}}=\text{x}+\text{iy},$ what is the value of $x^2 + y^2$ ?
View full solution →Evaluate $\sum\limits^{13}_{\text{n}=1}\big(\text{i}^{\text{n}}+\text{i}^{\text{n}+1}\big),$ where $\text{n}\in\text{N}.$
View full solution →The value of $\sqrt{-25}\times\sqrt{-9}$ is ___________.
View full solution →For any two complex numbers $z_1 , z_2$ and any real numbers $a, b, |az_1 - bz_2|^2 + |bz_1 + az_2|^{2 }=\ ....$
View full solution →The sum of the series i$ + i^2 + i^3 + ...$ upto $1000$ terms is $.......$
View full solution →$\arg(\text{z})+\arg\bar{\text{z}}(\bar{\text{z}}\neq0)\text{ is}$ _________.
View full solution →If $\Big|\frac{\text{z}-2}{\text{z}+2}\Big|=\frac{\pi}{6},$ then the locus of z is __________.
View full solution →If $|z_1| = 1(z_1 \neq -1)$ and $\text{z}_2=\frac{\text{z}_1-1}{\text{z}_1+1},$ then show that the real part of $z_2$ is zero.
View full solution →Find the complex number satisfying the equation $\text{z}+\sqrt{2}|(\text{z}+1)|+\text{i}=0$
View full solution →If $\text{a}=\cos\theta+\text{i}\sin\theta,$ find the value of $\Big(\frac{1+\text{a}}{1-\text{a}}\Big)$
View full solution →If $\frac{\text{z}-1}{\text{z}+1}$ is purely imaginary number (z ≠ -1), then find the value of |z|.
View full solution →Write the complex number $\text{z}=\frac{1-\text{i}}{\cos\frac{\pi}{3}+\text{i}\sin\frac{\pi}{3}}$ in polar form.
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