Sample QuestionsComplex Numbers questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
Let $x=\alpha +\beta ,\,y=\alpha \omega +\beta {{\omega }^{2}},\,z=\alpha {{\omega }^{2}}+\beta \omega ,\,\omega $ is an imaginary cube root of unity. Product of xyz is [Orissa JEE 2005]
- A
${{\alpha }^{2}}+{{\beta }^{2}}$
- B
${{\alpha }^{2}}-{{\beta }^{2}}$
- C
${{\alpha }^{3}}+{{\beta }^{3}}$
- ✓
${{\alpha }^{3}}-{{\beta }^{3}}$
Answer: D.
View full solution →If 1, $\omega ,\,{{\omega }^{2}}$ are the cube roots of unity then ${{\omega }^{2}}{{(1+\omega )}^{3}}-(1+{{\omega }^{2}})\omega =$ [Orissa JEE 2005]
Answer: D.
View full solution →If $\omega $ is a cube root of unity but not equal to 1 then minimum value of $|a+b\omega +c{{\omega }^{2}}|$ (where a, b, c are integers but not all equal) is [IIT Screening 2005]
- A
- B
$\frac{\sqrt{3}}{2}$
- ✓
- D
Answer: C.
View full solution →If ${{\tan }^{-1}}(\alpha +i\beta )=x+iy,$ then x = [RPET 2002]
- ✓
$\frac{1}{2}{{\tan }^{-1}}\left( \frac{2\alpha }{1-{{\alpha }^{2}}-{{\beta }^{2}}} \right)$
- B
$\frac{1}{2}{{\tan }^{-1}}\left( \frac{2\alpha }{1+{{\alpha }^{2}}+{{\beta }^{2}}} \right)$
- C
${{\tan }^{-1}}\left( \frac{2\alpha }{1-{{\alpha }^{2}}-{{\beta }^{2}}} \right)$
- D
Answer: A.
View full solution →If $\tan (u+iv)=i$, then the value of v is [RPET 2001]
Answer: B.
View full solution →Write the values of the square root of -i.
If $|\text{z}-5\text{i}|=|\text{z}+5\text{i}|,$ then find the locus of z.
View full solution →Write $-1 + \text{i}\sqrt{3}$ in polar form.
View full solution →Write the least positive integral value of n for which $\Big(\frac{1+\text{i}}{1-\text{i}}\Big)^{\text{n}}$ is real.
View full solution →Write the sum of the series $\text{i}+\text{i}^2+\text{i}^3+...$ upto 1000 terms.
View full solution →Find the number of solutions of $\text{z}^2+|\text{z}|^2=0.$
View full solution →Find the conjugates of the following complex numbers:
$\frac{(3-2\text{i})(2+3\text{i})}{(1+2\text{i})(2-\text{i})}$
View full solution →Express in $\sin\frac{\pi}{5}+\text{i}\Big(1-\cos\frac{\pi}{5}\Big)$ polar form.
View full solution →Express the following complex numbers in the standard form a + ib:
$\frac{(1-\text{i})^3}{1-\text{i}^3}$
View full solution →Express the following complex numbers in the standard form a + ib:
$\frac{(1-\text{i})(1+\sqrt{3}\text{i})}{1-\text{i}}$
View full solution →Write in $(\text{i}^{25})^3$ polar form.
View full solution →Find the conjugates of the following complex numbers:
$\frac{1}{3+5\text{i}}$
View full solution →Find the values of the following expressions:
$\text{i}^{30}+\text{i}^{80}+\text{i}^{120}$
View full solution →Find the conjugates of the following complex numbers:
$\frac{(1+\text{i})(2+\text{i})}{3+\text{i}}$
View full solution →$(1+\text{i})^6+(1-\text{i})^3$
View full solution →Evaluate the following:
$\text{x}^4-4\text{x}^3+4\text{x}^2+8\text{x}+44,$ when $\text{x}=3+2\text{i}$
View full solution →Express the following complex numbers in the form $\text{r}(\cos\theta+\text{i}\sin\theta):$
$1+\text{i}\tan\alpha$
View full solution →If $|\text{z}+1|=\text{z}+2(1+\text{i}),$ find z.
View full solution →Find the smallest positive integer value of n for which $\frac{(1+\text{i})^\text{n}}{(1-\text{i})^{\text{n}-2}}$ is a real number.
View full solution →If $\text{x}+\text{iy}=\frac{\text{a}+\text{ib}}{\text{a}-\text{ib}},$ Prove that $\text{x}^2+\text{y}^2=1$
View full solution →