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 argument of $\frac{(1-\text{i}\sqrt{3})}{(1+\text{i}\sqrt{3})}$ is:
View full solution →For any complex number z, the minimum value of |z| + |z – 2i| is equal to:
In z = 4 + i, what is the real part?
If z is a comp lex num ber, then |3z – 1|= 3|z – 2| represents:
- A
- B
- C
- D
a line parallel to y-axis
View full solution →If z1 = 2 + 3i and z2 = 5 + 2i, then find sum of two complex numbers:
State True or False for the following:
Let z1 and z2 be two complex numbers such that |z1 + z2| = |z1| + |z2|, then arg(z1 - z2) = 0.
State True or False for the following:
The order relation is defined on the set of complex numbers.
State True or False for the following:
If z is a complex number such that z ≠ 0 and Re (z) = 0, then Im (z2) = 0.
State True or False for the following:
The inequality |z - 4| < |z - 2| represents the region given by x > 3.
State True or False for the following:
The locus represented by| z - 1| = |z - i| is a line perpendicular to the join of (1, 0) and (0, 1).
Fill in the blanks.
The value of $\sqrt{-25}\times\sqrt{-9}$ is ___________.
View full solution →Fill in the blanks.
For any two complex numbers z1 , z2 and any real numbers a, b, |az1 - bz2|2 + |bz1 + az2|2 = ___________.
Fill in the blanks.
$\arg(\text{z})+\arg\bar{\text{z}}(\bar{\text{z}}\neq0)\text{ is}$ _________.
View full solution →Fill in the blanks.
The number $\frac{(1-\text{i})^3}{1-\text{i}^3}$ is equal to __________.
View full solution →Fill in the blanks.
If $|\text{z}+4|\leq3,$ then the greatest and least values of z + 1 are ..... and ____________.
View full solution →Let z1 = 2 - i, z2 = -2 + i. Find $\operatorname{Im} \left( {\frac{1}{{{z_1}{z_1}}}} \right)$
View full solution →Express the complex numbers (1 - i)4 in the form of a + ib.
Express the complex number$\left[ {\left( {\frac{1}{3} + \frac{7}{3}i} \right) + \left( {4 + \frac{1}{3}i} \right)} \right] - \left[ {\frac{{ - 4}}{3} + i} \right]$ in the form of a + ib.
View full solution →Express the complex number $\left( {\frac{1}{5} + \frac{2}{5}i} \right) - \left( {4 + \frac{5}{2}i} \right)$ in the form of a + ib.
View full solution →Express the complex number (1 + i) - (- 1 + i6) in form of a + ib.
If $\left(\frac{1+i}{1-i}\right)^{m} = 1$ then find the least positive integral value of m.
View full solution →If (a + ib) (c + id) (e+ if) (g + ih) = A + iB then show that
(a2 + b2)(c2 + d2)(e2 + f2)(g2 + h2) = A2 + B2
Find the number of non-zero integral solutions of the equation ${\left| {1 - i} \right|^x} = {2^x}$.
View full solution →Find the modulus of $\frac{{1 + i}}{{1 - i}} - \frac{{1 - i}}{{1 + i}}$.
View full solution →Let z1 = 2 - i, z2 = -2 + i. Find $\operatorname{Re} \left( {\frac{{{z_1}{z_2}}}{{{{\overline z }_1}}}} \right)$
View full solution →Solve 21x2 - 28x + 10 = 0
Solve: ${x^2} - 2x + \frac{3}{2} = 0$
View full solution →Solve $3{x^2} - 4x + \frac{{20}}{3} = 0$
View full solution →Convert in the polar form: $\frac{{1 + 3i}}{{1 - 2i}}$
View full solution →Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: Roots of quadratic equation x2 + 3x + 5 = 0 is $\text{x}=\frac{-3\pm\text{i}\sqrt{11}}{2}.$
Reason: If x2 - x + 2 = 0 is a quadratic equation, then its roots are $\frac{1\pm\text{i}\sqrt{7}}{2}.$
- A is true, R is true; R is a correct explanation of A.
- A is true, R is true; R is not a correct explanation of A.
- A is true; R is false.
- A is false; R is true.
View full solution →Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: If $\text{z}=\frac{1+2\text{i}}{1-3\text{i}},$ then $\mid\text{z}\mid=\frac{1}{\sqrt{2}}.$
Reason: If z = a + ib, then $\mid\text{z}\mid=\sqrt{\text{a}^{2}+\text{b}^{2}}.$
- A is true, R is true; R is a correct explanation of A.
- A is true, R is true; R is not a correct explanation of A.
- A is true; R is false.
- A is false; R is true.
View full solution →Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: If 3x
2 + 4x + 2 = 0, then equation has imaginary roots.
Reason: In a quadratic equation, In a quadratic equation, ax
2 + bx + c = 0, if D = b
2 -4ac is less than zero, then the equation will have imaginary roots.
- A is true, R is true; R is a correct explanation of A.
- A is true, R is true; R is not a correct explanation of A.
- A is true; R is false.
- A is false; R is true.
Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: If Z
1 = 2 + 3i and Z
2 = 3 - 2i, then Z
1 - Z
2 = -1 + 5i.
Reason: If Z, = a + ib and Z
2 = c + id, then Z
1 - Z
2 = (a - c) + i(b - d).
- A is true, R is true; R is a correct explanation of A.
- A is true, R is true; R is not a correct explanation of A.
- A is true; R is false.
- A is false; R is true.
Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: If (1 + i)(x + iy) = 2 - 5i, then $\text{x}=\frac{-3}{2}$ and $\text{y}=\frac{-7}{2}.$
Reason: If a + ib = c + id, then a = c and b = d.
- A is true, R is true; R is a correct explanation of A.
- A is true, R is true; R is not a correct explanation of A.
- A is true; R is false.
- A is false; R is true.
View full solution →We have, $i=\sqrt{-1}$. So, we can write the higher powers of $i$ as follows
(i) $i^2=-1$
(ii) $i^3=i^2 \cdot i=(-1) \cdot i=-i$
(iii) $i^4=\left(i^2\right)^2=(-1)^2=1$
(iv) $i^5=i^{4+1}=i^4 \cdot i=1 \cdot i=i$
(v) $i^6=i^{4+2}=i^4 \cdot i^2=1 \cdot i^2=-1$
In order to compute $i^n$ for $n>4$, write $i^n=i^{4 q+r}$ for some $q, r \in N$ and $0 \leq r \leq 3$. Then, $i^n=$ $i^{4 q} \cdot i^r=\left(i^4\right)^q \cdot i^r=(1)^q \cdot i^r=i^r$.
In general, for any integer $k, i^{4 k}=1, i^{4 k+1}=i, i^{4 k+2}=-1$ and $i^{4 k+3}=-i$.
On the basis of above information, answer the following questions.
(i) The value of $i^{37}$ is equal to
(a) $i$ (b) $-i$ (c) 1 (d) -1
(ii) The value of $i^{-30}$ is equal to
(a) $i$ (b) 1 (c) -1 (d) $-i$
(iii) If $z=i^9+i^{19}$, then $z$ is equal to
(a) $0+0 i$ (b) $1+0 i$ (c) $0+i$ (d) $1+2 i$
(iv) The value of $\left[i^{19}+\left(\frac{1}{i}\right)^{25}\right]^2$ is equal to
(a) -4 (b) 4 (c) $\mathrm{i}$ (d) 1
(v) If $z=i^{-39}$, then simplest form of $z$ is equal to
(a) $1+0 i$ (b) $0+i$ (c) $0+0 i$ (d) $1+i$
View full solution →A complex number $z$ is pure real if and only if $\bar{z}=z$ and is pure imaginary if and only if $\bar{z}=-z$.
Based on the above information, answer the following questions.
(i) If $(1+i) z=(1-i) \bar{z}$, then $-i \bar{z}$ is
(a) $-\bar{z}$ (b) $z$ (c) $\bar{z}$ (d) $z^{-1}$
(ii) $\overline{Z_1 Z_2}$ is
(a) $\bar{z}_1 \bar{z}_2$ (b) $\bar{z}_1+\bar{z}_2$ (c) $\frac{z_1}{z_2}$ (d) $\frac{1}{z_1 z_2}$
(iii) If $x$ and $y$ are real numbers and the complex number $\frac{(2+i) x-i}{4+i}+\frac{(1-i) y+2 i}{4 i}$ is pure real, the relation between $x$ and $y$ is
(a) $8 x-17 y=16$ (b) $8 x+17 y=16$
(c) $17 x-8 y=16$ (d) $17 x-8 y=-16$
(iv) If $z=\frac{3+2 i \sin \theta}{1-2 i \sin \theta}\left(0<\theta \leq \frac{\pi}{2}\right)$ is pure imaginary, then $\theta$ is equal to
(a) $\frac{\pi}{4}$ (b) $\frac{4}{6}$ (c) $\frac{6}{3}$ (d) $\frac{\pi}{12}$
(v) If $z_1$ and $z_2$ are complex numbers such that $\left|\frac{z_1-z_2}{z_1+z_2}\right|=1$
(a) $\frac{z_1}{z_2}$ is pure real (b) $\frac{z_1}{z_2}$ is pure imaginary
(c) $z_1$ is pure real (d) $z_1$ and $z_2$ are pure imaginary
View full solution →Two complex numbers $Z_1=a+i b$ and $Z_2=c+i d$ are said to be equal, if $a=c$ and $b=d$.
On the basis of above information, answer the following questions.
(i) If $(3 a-6)+2 i b=-6 b+(6+a) i$, then the real values of $a$ and $b$ are respectively
(a) $-2,2$ (b) $2,-2$ (c) $3,-3$ (d) 4,2
(ii) If $(2 a+2 b)+i(b-a)=-4 i$, then the real values of $a$ and $b$ are respectively.
(a) 2,3 (b) $2,-2$ (c) 3,1 (d) $-2,2$
(iii) If $\left(\frac{1-i}{1+i}\right)^{100}=a+i b$, then the values of $a$ and $b$ are respectively
(a) 1,0 (b) 0,1 (c) 1,2 (d) 2,1
(iv) If $\frac{(1+i)^2}{2-i}=x+i y$, then the value of $x+y$ is
(a) $\frac{1}{5}$ (b) $\frac{3}{5}$ (c) $\frac{4}{5}$ (d) $\frac{2}{5}$
(v) If $(x+y)+i(x-y)=4+6 i$, then $x y$ is equal to
(a) 5 (b) -5 (c) 4 (d) -4
View full solution →