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.
If $z$ is a comp lex num ber, then $|3z – 1|= 3|z – 2|$ represents:
Answer: D.
View full solution →If $z_1 = 2 + 3i$ and $z_2 = 5 + 2i,$ then find sum of two complex numbers:
- A
$4 + 8i$
- B
$3 - i$
- ✓
$7 + 5i$
- D
$7 - 5i$
Answer: C.
View full solution →Choose the correct answer.
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 →What will be the sum of $b + c$ if the equations $x^2 + bx + c = 0$ and $x^2 + 3x + 3 = 0$ have one common root:
Answer: C.
View full solution →If the roots of $x^2− bx + c = 0$ are two consecutive integers, then $b^2 − 4 c$ is:
Answer: B.
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 $x^2 + 3x + 5 = 0$ is $\text{x}=\frac{-3\pm\text{i}\sqrt{11}}{2}.$
Reason: If $x^2 - x + 2 = 0$ is a quadratic equation, then its roots are $\frac{1\pm\text{i}\sqrt{7}}{2}.$
- A
$A$ 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 $A.$
- C
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
Answer: B.
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 $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
Answer: A.
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 $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
Answer: A.
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 $Z_1 = 2 + 3i$ 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 $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
Answer: A.
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 $(1 + i)(x + iy) = 2 - 5i,$ 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 $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
Answer: A.
View full solution →State True or False for the following:
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 →State True or False for the following:
2 is not a complex number.
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 $z \neq 0$ and Re $(z) = 0,$ then Im $(z^2) = 0.$
View full solution →State True or False for the following:
The inequality |z - 4| < |z - 2| represents the region given by x > 3.
Let $z_1 = 2 - i, z_2 = -2 + i.$ 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 $a + ib.$
View full solution →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
$(a^2 + b^2)(c^2 + d^2)(e^2 + f^2)(g^2 + h^2) = A^2 + B^2$
View full solution →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 $z_1 = 2 - i, z_2 = -2 + i.$ Find $\operatorname{Re} \left( {\frac{{{z_1}{z_2}}}{{{{\overline z }_1}}}} \right)$
View full solution →Solve $21x^2- 28x + 10 = 0$
View full solution →Solve: $27x^2 - 10x+ 1 = 0$
View full solution →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 →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 →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 $z_1 , z_2 $ and any real numbers $a, b, |az_1 - bz_2|^2 + |bz_1 + az_2|^2= \_\_\_\_\_\_\_\_\_\_\_.$
View full solution →Fill in the blanks.
The sum of the series $i + i^2 + i^3 + ...$ upto $1000$ terms is __________.
View full solution →Fill in the blanks.
$\arg(\text{z})+\arg\bar{\text{z}}(\bar{\text{z}}\neq0)\text{ is}$ _________.
View full solution →Fill in the blanks.
If $\Big|\frac{\text{z}-2}{\text{z}+2}\Big|=\frac{\pi}{6},$ then the locus of z is __________.
View full solution →