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.
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 →$\left(\frac{1+i}{\sqrt{2}}\right)^8+\left(\frac{1-i}{\sqrt{2}}\right)^8$ is equal to:
Answer: C.
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 $(3,4),(2,5)$ and $(-9,16)$ are there complex numbers then:
- A
these are the vertices of right angled triangle
- B
these are the vertices of equilateral triangle
- C
these are the vertices of isosceles triangle
- ✓
Answer: D.
View full solution →Let $z$ be a complex number, then $z+\bar{z}$ and $z \bar{z}$ are :
Answer: A.
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 →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 value of $1+i^2+i^4+\ldots\ldots +i^{2 n}.$
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 →Write the expression $\frac{3-\sqrt{-16}}{1-\sqrt{-9}}$ in the form of $a+i b$.
View full solution →If $z=x+i y$ and $\omega=\frac{1-i z}{z-i}$, then what does $|\omega|$ $=1$ show in a complex plane?
View full solution →If two complex numbers $z_1$ and $z_2$ are such that $\left|z_1\right|=\left|z_2\right|$ then is $\left|z_1\right|=\left|z_2\right|$ is necessary?
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 →Write the radius and centre of the circle $z \bar{z}-(2$ $+3 i) z-(2-3 i) \bar{z}+9=0$ where $z=x+ iy$.
View full solution →Find the value of the following :
$(1+i)^8+(1-i)^8$
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 $z_1, z_2, z \in C$, 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 $|z|=1$, 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 →$(1+i)^6+(1-i)^3=$ ________.
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 →| 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) $i$ |