Question 1

If $z = 2 + 3i$, what is the multiplicative inverse of $z$?

Accepted Answer

Correct Option: A. Solution: The multiplicative inverse of a complex number $z = a + ib$ is $\frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i$. For $z = 2 + 3i$, the inverse is $\frac{2}{2^2 + 3^2} - \frac{3}{2^2 + 3^2}i = \frac{2}{13} - \frac{3}{13}i$.

Question 2

What is the value of $i^3$ where $i$ is the imaginary unit?

Accepted Answer

Correct Option: B. Solution: The powers of $i$ cycle every four terms: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$. Therefore, $i^3 = -i$.

Question 3

What is the modulus of the complex number $z = 3 + 4i$?

Accepted Answer

Correct Option: A. Solution: The modulus of a complex number $z = a + ib$ is $|z| = \sqrt{a^2 + b^2}$. For $z = 3 + 4i$, $|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$.

Question 4

Which of the following statements is true about the roots of the equation $x^2 + 1 = 0$?

Accepted Answer

Correct Option: C. Solution: The roots of $x^2 + 1 = 0$ are $i$ and $-i$, which are complex conjugates.

Question 5

What is the value of $i^{23}$?

Accepted Answer

Correct Option: A. Solution: The powers of $i$ cycle every four: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$. Therefore, $i^{23} = i^{20+3} = (i^4)^5 \cdot i^3 = 1^5 \cdot (-i) = -i$.

Question 6

If $z = 1 + i$, what is the value of $z^4$?

Accepted Answer

Correct Option: C. Solution: To find $z^4$, first calculate $z^2 = (1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i$. Then $z^4 = (z^2)^2 = (2i)^2 = 4i^2 = 4(-1) = -4$.

Question 7

Given a complex number $z = 3 + 4i$, what is the modulus of its conjugate?

Accepted Answer

Correct Option: C. Solution: The modulus of a complex number $z = a + ib$ is given by $|z| = \sqrt{a^2 + b^2}$. For the conjugate $\bar{z} = 3 - 4i$, the modulus is $\sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = 5$.

Question 8

Which of the following represents the square root of $-4$ in the form $a + ib$?

Accepted Answer

Correct Option: A. Solution: The square root of a negative real number $-4$ is $\sqrt{4} \cdot \sqrt{-1} = 2i$. Therefore, the correct representation in the form $a + ib$ is $2i$.

Question 9

If $z_1 = 3 + 4i$ and $z_2 = 1 - 2i$, what is $z_1 + z_2$?

Accepted Answer

Correct Option: A. Solution: The sum of two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ is $(a + c) + i(b + d)$. Thus, $z_1 + z_2 = (3 + 1) + i(4 - 2) = 4 + 2i$.

Question 10

Which property of complex numbers states that the sum of two complex numbers is a complex number?

Accepted Answer

Correct Option: B. Solution: The closure law states that the sum of two complex numbers is a complex number.

Question 11

For the complex number $z = a + ib$, what is the conjugate of $z$?

Accepted Answer

Correct Option: B. Solution: The conjugate of a complex number $z = a + ib$ is $a - ib$.

Question 12

Which of the following is the correct expression for the square root of $-9$?

Accepted Answer

Correct Option: C. Solution: The square root of a negative number $-a$ is $\sqrt{a}i$. Therefore, the square roots of $-9$ are $3i$ and $-3i$, hence $\pm 3i$.

Question 13

What is the result of multiplying the complex numbers $z_1 = 3 + i5$ and $z_2 = 2 + i6$?

Accepted Answer

Correct Option: A. Solution: The product of two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ is given by $z_1 z_2 = (ac - bd) + i(ad + bc)$. Applying this to $z_1 = 3 + i5$ and $z_2 = 2 + i6$, we get $(3 	imes 2 - 5 	imes 6) + i(3 	imes 6 + 5 	imes 2) = -24 + i28$.

Question 14

What is the additive identity for complex numbers?

Accepted Answer

Correct Option: B. Solution: The additive identity for complex numbers is $0 + i0$, as adding it to any complex number $z$ results in $z$ itself.

Question 15

If $z = 7 + 24i$, find the conjugate of the multiplicative inverse of $z$.

Accepted Answer

Correct Option: B. Solution: The multiplicative inverse of $z = a + ib$ is $\frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i$. For $z = 7 + 24i$, $|z|^2 = 7^2 + 24^2 = 625$. Thus, the inverse is $\frac{7}{625} - \frac{24}{625}i$. The conjugate of this is $\frac{7}{625} + \frac{24}{625}i$, which simplifies to $\frac{7}{25} + \frac{24}{25}i$.

Question 16

If $z = 5 - 3i$, find the multiplicative inverse of $z$ in the form $a + ib$.

Accepted Answer

Correct Option: B. Solution: The multiplicative inverse of $z = 5 - 3i$ is given by $\frac{\bar{z}}{|z|^2}$, where $\bar{z} = 5 + 3i$ and $|z|^2 = 5^2 + (-3)^2 = 34$. Thus, the inverse is $\frac{5 + 3i}{34} = \frac{5}{34} + \frac{3}{34}i$.

Question 17

If the quadratic equation $x^2 + 4x + 5 = 0$ is solved using complex numbers, what are the roots?

Accepted Answer

Correct Option: B. Solution: The discriminant $D = 4^2 - 4 	imes 1 	imes 5 = -4$. The roots are $\frac{-4 \pm \sqrt{-4}}{2 	imes 1} = -2 \pm 2i$.

Question 18

Consider the quadratic equation $ax^2 + bx + c = 0$ where $D = b^2 - 4ac < 0$. If the roots of this equation are $x_1$ and $x_2$, which of the following represents the correct form of these roots?

Accepted Answer

Correct Option: B. Solution: For a quadratic equation with $D < 0$, the roots are complex and given by $x_1, x_2 = \frac{-b \pm i\sqrt{-D}}{2a}$.

Question 19

Which of the following points represents the conjugate of the complex number $z = 2 + 3i$ on the Argand plane?

Accepted Answer

Correct Option: A. Solution: The conjugate of the complex number $z = 2 + 3i$ is $2 - 3i$, which corresponds to the point (2, -3) on the Argand plane.

Question 20

What is the product of $(3 + i5)$ and $(2 + i6)$?

Accepted Answer

Correct Option: A. Solution: The product of two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ is $(ac - bd) + i(ad + bc)$. Thus, $(3 + i5)(2 + i6) = (3 	imes 2 - 5 	imes 6) + i(3 	imes 6 + 5 	imes 2) = -24 + i28$.

Question 21

What is the value of $i^2$ in complex numbers?

Accepted Answer

Correct Option: A. Solution: In complex numbers, $i^2 = -1$ by definition.

Question 22

Given a complex number $z = 3 + 4i$, what is the representation of $z$ in polar coordinates?

Accepted Answer

Correct Option: A. Solution: The modulus $|z| = \sqrt{3^2 + 4^2} = 5$. The argument $	heta = 	an^{-1}(\frac{4}{3})$. Thus, $z$ in polar form is $5(\cos 	heta + i \sin 	heta)$.

Question 23

What is the conjugate of the complex number $z = 5 - 7i$?

Accepted Answer

Correct Option: A. Solution: The conjugate of a complex number $z = a + ib$ is given by $\overline{z} = a - ib$. For $z = 5 - 7i$, the conjugate is $\overline{z} = 5 + 7i$.

Question 24

If $z = 3 + 4i$, what is the modulus of $z$?

Accepted Answer

Correct Option: C. Solution: The modulus of a complex number $z = a + ib$ is given by $|z| = \sqrt{a^2 + b^2}$. For $z = 3 + 4i$, $|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

Question 25

If $z = 2 - 3i$, what is the multiplicative inverse of $z$?

Accepted Answer

Correct Option: A. Solution: The multiplicative inverse of a complex number $z = a + ib$ is given by $\frac{a - ib}{a^2 + b^2}$. For $z = 2 - 3i$, the inverse is $\frac{2 + 3i}{2^2 + (-3)^2} = \frac{2 + 3i}{13}$.

Question 26

If $z = 4 - 3i$, find the multiplicative inverse of $z$.

Accepted Answer

Correct Option: B. Solution: The multiplicative inverse of a complex number $z = a + ib$ is given by $\frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i$. For $z = 4 - 3i$, $a = 4$ and $b = -3$, so $|z|^2 = 4^2 + (-3)^2 = 16 + 9 = 25$. The inverse is $\frac{4}{25} - \frac{3}{25}i$.

Question 27

Which of the following represents the geometric interpretation of the modulus of a complex number $z = x + iy$ in the Argand plane?

Accepted Answer

Correct Option: B. Solution: In the Argand plane, the modulus of a complex number $z = x + iy$ is the distance between the point $(x, y)$ and the origin $(0, 0)$, which is $\sqrt{x^2 + y^2}$.

Question 28

If $z_1 = 3 + 4i$ and $z_2 = 1 - 2i$, what is the product $z_1 z_2$?

Accepted Answer

Correct Option: C. Solution: The product of two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ is given by $z_1 z_2 = (ac - bd) + i(ad + bc)$. For $z_1 = 3 + 4i$ and $z_2 = 1 - 2i$, we have $z_1 z_2 = (3 \cdot 1 - 4 \cdot (-2)) + i(3 \cdot (-2) + 4 \cdot 1) = (3 + 8) + i(-6 + 4) = 10 - 10i$.

Question 29

If $z = 5 - 2i$, what is the conjugate of $z$?

Accepted Answer

Correct Option: A. Solution: The conjugate of a complex number $z = a + ib$ is $\overline{z} = a - ib$. Therefore, the conjugate of $5 - 2i$ is $5 + 2i$.

Question 30

Express the complex number $(3 + 4i)(2 - i)$ in the standard form $a + ib$.

Accepted Answer

Correct Option: B. Solution: The product $(3 + 4i)(2 - i)$ can be expanded as $(3 	imes 2 + 3 	imes (-i) + 4i 	imes 2 + 4i 	imes (-i)) = (6 - 3i + 8i - 4i^2)$. Since $i^2 = -1$, this becomes $6 + 5i + 4 = 10 + 5i$. Therefore, the standard form is $10 + 5i$.

Question 31

What is the imaginary unit $i$ defined as?

Accepted Answer

Correct Option: A. Solution: The imaginary unit $i$ is defined as $i = \sqrt{-1}$, which satisfies $i^2 = -1$.

Question 32

What is the value of the discriminant $D$ for a quadratic equation $ax^2 + bx + c = 0$ if it has complex roots?

Accepted Answer

Correct Option: C. Solution: A quadratic equation has complex roots when the discriminant $D = b^2 - 4ac$ is less than zero.

Question 33

For a complex number $z = 5 - 12i$, what is the argument of its conjugate in degrees?

Accepted Answer

Correct Option: B. Solution: The argument of a complex number $z = a + ib$ is $	an^{-1}(b/a)$. For $\bar{z} = 5 + 12i$, the argument is $	an^{-1}(12/5)$. Converting to degrees, $	an^{-1}(12/5) \approx 67.4^\circ$.

Question 34

If a complex number $z = x + iy$ is represented as a point $P(x, y)$ on the Argand plane, what is the modulus of $z$?

Accepted Answer

Correct Option: A. Solution: The modulus of a complex number $z = x + iy$ is given by $\sqrt{x^2 + y^2}$, which is the distance from the origin to the point $P(x, y)$.

Question 35

Which of the following represents the additive inverse of the complex number $z = 5 - 3i$?

Accepted Answer

Correct Option: A. Solution: The additive inverse of a complex number $z = a + ib$ is $-a - ib$. Therefore, the additive inverse of $5 - 3i$ is $-5 + 3i$.

Question 36

Which property of complex numbers states that $z_1 + z_2 = z_2 + z_1$?

Accepted Answer

Correct Option: C. Solution: The commutative law states that for any two complex numbers $z_1$ and $z_2$, $z_1 + z_2 = z_2 + z_1$.

Question 37

If $z = 1 + i$, what is the multiplicative inverse of $z$?

Accepted Answer

Correct Option: A. Solution: The multiplicative inverse of $z = a + ib$ is $\frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i$. For $z = 1 + i$, the inverse is $\frac{1}{1^2 + 1^2} - \frac{1}{1^2 + 1^2}i = \frac{1}{2} - \frac{1}{2}i$.

Question 38

Given two complex numbers $z_1 = 2 + 3i$ and $z_2 = 4 - i$, find the modulus of their sum $z_1 + z_2$.

Accepted Answer

Correct Option: A. Solution: First, find the sum: $z_1 + z_2 = (2 + 3i) + (4 - i) = 6 + 2i$. The modulus is $\sqrt{6^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40} = \sqrt{29}$.

Question 39

If a complex number $z = x + iy$ is represented in polar form as $z = r(\cos 	heta + i \sin 	heta)$, which of the following represents the conjugate of $z$?

Accepted Answer

Correct Option: A. Solution: The conjugate of a complex number $z = r(\cos 	heta + i \sin 	heta)$ is $r(\cos 	heta - i \sin 	heta)$, which reflects the imaginary part across the real axis.

Question 40

Consider a complex number $z = x + iy$ represented on the Argand plane. If $|z| = 5$ and the argument of $z$ is $\frac{\pi}{4}$, what are the possible values of $x$ and $y$?

Accepted Answer

Correct Option: C. Solution: The modulus $|z| = \sqrt{x^2 + y^2} = 5$ and the argument $	heta = \frac{\pi}{4}$ implies $x = 5 \cos \frac{\pi}{4} = \frac{5}{\sqrt{2}}$ and $y = 5 \sin \frac{\pi}{4} = \frac{5}{\sqrt{2}}$.

Question 41

What is the result of $(1 + i)^4$ in the form $a + ib$?

Accepted Answer

Correct Option: B. Solution: Using binomial expansion, $(1 + i)^4 = \binom{4}{0}(1)^4 + \binom{4}{1}(1)^3i + \binom{4}{2}(1)^2i^2 + \binom{4}{3}(1)i^3 + \binom{4}{4}i^4 = 1 + 4i - 6 - 4i + 1 = 4 + 0i$.

Question 42

Which of the following represents the square root of $-9$?

Accepted Answer

Correct Option: C. Solution: The square root of a negative number $-a$ is $\sqrt{a}i$. Therefore, the square roots of $-9$ are $3i$ and $-3i$, which is represented as $\pm 3i$.

Question 43

If $z_1 = 3 + 4i$ and $z_2 = 1 + 2i$, what is the result of dividing $z_1$ by $z_2$ in the form $a + ib$?

Accepted Answer

Correct Option: A. Solution: To divide $z_1$ by $z_2$, we multiply by the conjugate of the denominator: $$\frac{z_1}{z_2} = \frac{3 + 4i}{1 + 2i} 	imes \frac{1 - 2i}{1 - 2i} = \frac{(3 + 4i)(1 - 2i)}{1^2 + (2i)^2} = \frac{3 - 6i + 4i + 8}{1 + 4} = \frac{11 - 2i}{5} = \frac{11}{5} + \frac{2}{5}i.$$

Question 44

Which of the following represents the standard form of a complex number?

Accepted Answer

Correct Option: A. Solution: The standard form of a complex number is $a + ib$, where $a$ is the real part and $b$ is the imaginary part.

Question 45

What is the modulus of the complex number $z = 5 - 12i$?

Accepted Answer

Correct Option: A. Solution: The modulus of a complex number $z = a + ib$ is given by $\sqrt{a^2 + b^2}$. For $z = 5 - 12i$, the modulus is $\sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.

Question 46

Which of the following represents the multiplicative inverse of a non-zero complex number $z = a + ib$?

Accepted Answer

Correct Option: A. Solution: The multiplicative inverse of $z = a + ib$ is given by $\frac{a}{a^2 + b^2} - i\frac{b}{a^2 + b^2}$.

Question 47

What is the geometric representation of the complex number $z = 3 + 4i$ on the Argand plane?

Accepted Answer

Correct Option: A. Solution: The complex number $z = 3 + 4i$ corresponds to the point (3, 4) on the Argand plane.

Question 48

If $z = 1 + i$ and $w = 2 - 3i$, find the modulus of the product $zw$.

Accepted Answer

Correct Option: D. Solution: The product $zw = (1 + i)(2 - 3i) = 2 + 3 + i(1)(-3) + i(2) = 5 - i$. The modulus is $|5 - i| = \sqrt{5^2 + (-1)^2} = \sqrt{25 + 1} = \sqrt{26}$. However, the modulus of $zw$ is $\sqrt{(1^2 + 1^2)(2^2 + (-3)^2)} = \sqrt{2 	imes 13} = \sqrt{26}$, which is approximately 6.

Question 49

If $z_1 = 1 + 2i$ and $z_2 = 3 - 4i$, what is the modulus of the product $z_1 z_2$?

Accepted Answer

Correct Option: C. Solution: The modulus of the product of two complex numbers $z_1$ and $z_2$ is the product of their moduli. $|z_1| = \sqrt{1^2 + 2^2} = \sqrt{5}$ and $|z_2| = \sqrt{3^2 + (-4)^2} = \sqrt{25} = 5$. Thus, $|z_1 z_2| = |z_1| 	imes |z_2| = \sqrt{5} 	imes 5 = 5\sqrt{5}$, which simplifies to approximately 20.

Question 50

Which of the following points on the Argand plane corresponds to the complex number $3 - 4i$?

Accepted Answer

Correct Option: D. Solution: The complex number $3 - 4i$ has a positive real part and a negative imaginary part, placing it in the fourth quadrant of the Argand plane.

Question 51

If $z = 3 + 4i$, what is the modulus of $z$?

Accepted Answer

Correct Option: C. Solution: The modulus of a complex number $z = a + bi$ is given by $|z| = \sqrt{a^2 + b^2}$. Thus, $|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

Question 52

What is the geometric interpretation of the modulus of a complex number $z = x + iy$ on the Argand plane?

Accepted Answer

Correct Option: B. Solution: The modulus of a complex number $z = x + iy$ is $\sqrt{x^2 + y^2}$, which represents the distance from the origin to the point $(x, y)$ on the Argand plane.

Question 53

Find the modulus of the complex number $z = -3 + 4i$.

Accepted Answer

Correct Option: A. Solution: The modulus of a complex number $z = a + ib$ is given by $|z| = \sqrt{a^2 + b^2}$. For $z = -3 + 4i$, $|z| = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

Question 54

Find the result of $(2 + i)^3$ expressed in the form $a + ib$.

Accepted Answer

Correct Option: B. Solution: Expanding $(2 + i)^3$ using the binomial theorem: $= (2 + i)(2 + i)(2 + i) = (2 + i)(4 + 4i + i^2) = (2 + i)(3 + 4i) = 6 + 8i + 3i + 4i^2 = 6 + 11i - 4 = 2 + 11i$. Thus, the correct form is $-11 + 2i$.

Question 55

Consider the complex numbers $z_1 = 2 + 3i$ and $z_2 = 4 - 5i$. What is the modulus of the product $z_1 \cdot z_2$?

Accepted Answer

Correct Option: A. Solution: The modulus of a product of two complex numbers is the product of their moduli. So, $$|z_1 \cdot z_2| = |z_1| \cdot |z_2| = \sqrt{(2^2 + 3^2)} \cdot \sqrt{(4^2 + (-5)^2)} = \sqrt{13} \cdot \sqrt{41} = \sqrt{169}$$.

Question 56

The sum of two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ is $z_1 + z_2 = (a + c) + i(b + d)$, which is also a complex number.

Accepted Answer

Answer: True. Solution: According to the closure law of complex numbers, the sum of two complex numbers is also a complex number.

Question 57

The conjugate of a complex number $z = a + ib$ is $z = a + ib$.

Accepted Answer

Answer: False. Solution: The conjugate of a complex number $z = a + ib$ is $\overline{z} = a - ib$. It is the reflection of the point $(a, b)$ across the real axis in the Argand plane.

Question 58

The square roots of a negative real number are always represented by $i$ and $-i$.

Accepted Answer

Answer: True. Solution: The square roots of $-1$ are $i$ and $-i$. Similarly, for any negative real number $-a$, the square roots are $\sqrt{a}i$ and $-\sqrt{a}i$.

Question 59

In the Argand plane, the modulus of a complex number $z = x + iy$ is given by $\sqrt{x^2 + y^2}$, representing the distance from the origin.

Accepted Answer

Answer: True. Solution: The modulus of a complex number $z = x + iy$ is indeed $\sqrt{x^2 + y^2}$, which represents the distance from the origin to the point $(x, y)$ in the Argand plane.

Question 60

The product of two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ is given by $z_1 z_2 = (ac - bd) + i(ad + bc)$.

Accepted Answer

Answer: True. Solution: The multiplication of two complex numbers follows the distributive property of multiplication, resulting in $z_1 z_2 = (ac - bd) + i(ad + bc)$. This is consistent with the definition of complex number multiplication.

Question 61

In the Argand plane, the modulus of a complex number $z = x + iy$ is given by $\sqrt{x^2 + y^2}$, representing the distance from the origin to the point $(x, y)$.

Accepted Answer

Answer: True. Solution: The modulus of a complex number $z = x + iy$ is calculated as $\sqrt{x^2 + y^2}$, which geometrically represents the distance from the origin $(0, 0)$ to the point $(x, y)$ in the Argand plane.

Question 62

The modulus of a complex number $z = a + ib$ is given by $|z| = \sqrt{a^2 + b^2}$.

Accepted Answer

Answer: True. Solution: The modulus of a complex number $z = a + ib$ is defined as the non-negative real number $\sqrt{a^2 + b^2}$, representing the distance from the origin to the point $(a, b)$ in the Argand plane.

Question 63

In the Argand plane, the complex number $z = x + iy$ and its conjugate $\bar{z} = x - iy$ are represented by the same point.

Accepted Answer

Answer: False. Solution: In the Argand plane, the complex number $z = x + iy$ is represented by the point $P(x, y)$, while its conjugate $\bar{z} = x - iy$ is represented by the point $Q(x, -y)$, which is the mirror image of $P(x, y)$ across the real axis.

Question 64

The equation $x^2 + 1 = 0$ has no real solution because the square of every real number is non-negative.

Accepted Answer

Answer: True. Solution: The equation $x^2 + 1 = 0$ implies $x^2 = -1$, which cannot be satisfied by any real number since the square of a real number is always non-negative.

Complex Numbers and Quadr..

Learning Objectives

Learning Objectives

Revision Notes & Summary

Chapter Notes

Complex Numbers

Algebra of Complex Numbers

Modulus and Conjugate of Complex Numbers

Argand Plane and Polar Representation

Powers and Roots of Complex Numbers

Identities and Equations Involving Complex Numbers

Standard Form and Simplification of Complex Expressions

Quadratic Equations with Complex Roots

Exam Tips & Common Mistakes

Common Mistakes & Exam Tips

Complex Numbers

Algebra of Complex Numbers

Modulus and Conjugate

Argand Plane and Polar Representation

Powers and Roots of Complex Numbers

Identities and Equations

Standard Form and Simplification

Quadratic Equations with Complex Roots

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Multiple Choice Questions

Q1.If z=2+3iz = 2 + 3iz=2+3i, what is the multiplicative inverse of zzz?

Correct Answer: A

Q2.What is the value of i3i^3i3 where iii is the imaginary unit?

Correct Answer: B

Q3.What is the modulus of the complex number z=3+4iz = 3 + 4iz=3+4i?

Correct Answer: A

Q4.Which of the following statements is true about the roots of the equation x2+1=0x^2 + 1 = 0x2+1=0?

Correct Answer: C

Q5.What is the value of i23i^{23}i23?

Correct Answer: A

Q6.If z=1+iz = 1 + iz=1+i, what is the value of z4z^4z4?

Correct Answer: C

Q7.Given a complex number z=3+4iz = 3 + 4iz=3+4i, what is the modulus of its conjugate?

Correct Answer: C

Q8.Which of the following represents the square root of −4-4−4 in the form a+iba + iba+ib?

Correct Answer: A

Q9.If z1=3+4iz_1 = 3 + 4iz1​=3+4i and z2=1−2iz_2 = 1 - 2iz2​=1−2i, what is z1+z2z_1 + z_2z1​+z2​?

Correct Answer: A

Q10.Which property of complex numbers states that the sum of two complex numbers is a complex number?

Correct Answer: B

Q11.For the complex number z=a+ibz = a + ibz=a+ib, what is the conjugate of zzz?

Correct Answer: B

Q12.Which of the following is the correct expression for the square root of −9-9−9?

Correct Answer: C

Q13.What is the result of multiplying the complex numbers z1=3+i5z_1 = 3 + i5z1​=3+i5 and z2=2+i6z_2 = 2 + i6z2​=2+i6?

Correct Answer: A

Q14.What is the additive identity for complex numbers?

Correct Answer: B

Q15.If z=7+24iz = 7 + 24iz=7+24i, find the conjugate of the multiplicative inverse of zzz.

Correct Answer: B

Q16.If z=5−3iz = 5 - 3iz=5−3i, find the multiplicative inverse of zzz in the form a+iba + iba+ib.

Correct Answer: B

Q17.If the quadratic equation x2+4x+5=0x^2 + 4x + 5 = 0x2+4x+5=0 is solved using complex numbers, what are the roots?

Correct Answer: B

Q18.Consider the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 where D=b2−4ac<0D = b^2 - 4ac < 0D=b2−4ac<0. If the roots of this equation are x1x_1x1​ and x2x_2x2​, which of the following represents the correct form of these roots?

Correct Answer: B

Q19.Which of the following points represents the conjugate of the complex number z=2+3iz = 2 + 3iz=2+3i on the Argand plane?

Correct Answer: A

Q20.What is the product of (3+i5)(3 + i5)(3+i5) and (2+i6)(2 + i6)(2+i6)?

Correct Answer: A

Q21.What is the value of i2i^2i2 in complex numbers?

Correct Answer: A

Q22.Given a complex number z=3+4iz = 3 + 4iz=3+4i, what is the representation of zzz in polar coordinates?

Correct Answer: A

Q23.What is the conjugate of the complex number z=5−7iz = 5 - 7iz=5−7i?

Correct Answer: A

Q24.If z=3+4iz = 3 + 4iz=3+4i, what is the modulus of zzz?

Correct Answer: C

Q25.If z=2−3iz = 2 - 3iz=2−3i, what is the multiplicative inverse of zzz?