Sample QuestionsPART - 1 CH - 4 Complex 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.
If $z=x+i y$ and $\left|\frac{z-5 i}{z+5 i}\right|=1$ then $z$ lies :
- ✓
at $x-$ axis
- B
at $y -$ axis
- C
at line $y=5$
- D
Answer: A.
View full solution →If $z_1, z_2 \in C$ then which is true statement:
- A
$\left|z_1-z_2\right|=\left|z_1\right|+\left|z_2\right|$
- B
$\left|z_1-z_2\right|>\left|z_1\right|+\left|z_2\right|$
- ✓
$\left|z_1+z_2\right| \leq\left|z_1\right|+\left|z_2\right|$
- D
$\left|z_1+z_2\right|=\left|z_1-z_2\right|$
Answer: C.
View full solution →$\left(\frac{1+i}{\sqrt{2}}\right)^8+\left(\frac{1-i}{\sqrt{2}}\right)^8$ is equal to:
Answer: C.
View full solution →Statement $(a+i b)<(c+i d)$ is true for which of the followings :
- A
$a^2+b^2=0$
- B
$b^2+c^2=0$
- C
$a^2+c^2=0$
- ✓
$b^2+d^2=0$
Answer: D.
View full solution →A complex number is pure imaginary if:
- A
its imaginary part is zero
- B
its both real and imaginary parts are zero
- ✓
- D
Answer: C.
View full solution →If $\alpha$ and $\beta$ are the roots of the equation $4 x^2+3 x+$ $7=0$, then $\frac{1}{\alpha}+\frac{1}{\beta}$ is equal to $\frac{3}{7}$.
View full solution →If $a$ and $b$ are the roots of equation $x^2+x+1=0$, then $a^2+b^2=-1$.
View full solution →If $z=\frac{1+i}{1-i}$, then the value of $z^4$ is -2 .
View full solution →The square root of $i$ is $\pm \frac{1}{\sqrt{2}}(1+i)$.
View full solution →$\left|z_1+z_2\right|^2=\left|z_1\right|^2+\left|z_2\right|^2+2 \operatorname{Re}\left(z_1 \bar{z}_2\right)$
View full solution →$1+i^{10}+i^{20}+i^{30}$ is a number equal to _________.
View full solution →$\sqrt{-4} \times \sqrt{\frac{-9}{4}}=$ __________.
View full solution →Additive identity of complex number is ___________.
Conjugate of complex number $z=-i$ is $\bar{z}=$ __________.
View full solution →For any integer $n,(i)^{4 n+2}=$ __________.
View full solution →If $z_1, z_2$ and $z_3 \in C$ then write the value of $\overline{z_1+z_2}$.
View full solution →Write the expression $\frac{3-\sqrt{-16}}{1-\sqrt{-9}}$ in the form of $a+i b$.
View full solution →If $z_1=2+3 i$ and $z_2=1+2 i$ then, write the value of $\frac{z_1}{z_2}$.
View full solution →Write the product of the complex number $3-2 i$ and its conjugate.
View full solution →Find the value of $x$ and $y$ in equation $(3 x-7)+2 i y$ $=-5 y+(5+x) i$.
View full solution →Find the value of $x$ and $y$ for the equation
$\frac{x-1}{3+i}+\frac{y-1}{3-i}=i$
View full solution →Write the conjugate of the following complex numbers :
$2 i, 3 i-5,7+11 i, 12 i+9 .$
View full solution →Find the value of the following :
$(1+i)^8+(1-i)^8$
View full solution →Write the real values of $x$ and $y$ for the equation $(1+i) y^2+(6+i)=(2+i) x$.
View full solution →If $z$ is a complex number and $\bar{z}$ is its conjugate then prove that :
$z^{-1}=\frac{\bar{z}}{|z|^2} \text {, where } z \neq 0$
View full solution →If $z_1, z_2, z \in C$, then prove that:
(i) $\left|z_1-z_2\right| \leq\left|z_1\right|+\left|z_2\right|$
(ii) $\left|z_1+z_2\right| \geq\left|z_1\right|-\left|z_2\right|$
View full solution →If $x+i y=\frac{ C +i}{ C -i}$, where C is a real number, then prove that :
View full solution →If $|z|=1$, then prove that $\frac{z-1}{z+1},(z \neq-1)$ is a pure imaginary number. If $z=1$, then what conclusion do you draw from this?
View full solution →| Part (a) | Part (b) |
| 1. $\begin{array}{l}\text { Value of }(1+i)\left(1+i^2\right) \left(1+i^3\right)\left(1+i^4\right)\end{array}$ | (a) 0 |
| 2. $a=1+i$ then value of $a^2$ | (b) $-i$ |
| 3. Square root of $-i$ | (c) $2 i$ |
| 4. $i^{135}$ | (d) $\pm \frac{1}{\sqrt{2}}(1-i)$ |
| 5. $i^{-999}$ | (e) $i$ |