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Mathematics · Class 12 Science

Ch 2Complex Numbers — Class 12 Mathematics, concept-first.

The rules for adding, subtracting, multiplying and dividing complex numbers were worked out by the Italian mathematician Rafael Bombelli (1526–1572), generally credited as the first to develop an algebra of complex numbers (a moon crater is named after him).

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Complex Number Arithmetic

Imagine you're trying to solve . You know that no real number squared gives . The square of any real number is either zero or positive. So this equation has no real solution.

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2.1

Introduction to Complex Numbers

The rules for adding, subtracting, multiplying and dividing complex numbers were worked out by the Italian mathematician Rafael Bombelli (1526–1572), generally credited as the first to develop an alge…

2.1.1

Powers of Imaginary Unit i

We now use the defining property to work out every other integer power of : and the negative powers So only ever takes four possible values, and means the pattern repeats every steps.

+Exercise 2.1i6 questions
  1. Q1$i^{1947} + i^{1950}$Free
  2. Q2$i^{1948} - i^{-1869}$Free
  3. Q3$\displaystyle\sum_{n=1}^{12} i^n$Preview
  4. Q4$i^{59} + \dfrac{1}{i^{59}}$Preview
  5. Q5$i \cdot i^2 \cdot i^3 \cdots i^{2000}$Preview
  6. Q6$\displaystyle\sum_{n=1}^{10} i^{n+50}$Preview
2.2

Complex Numbers

We have just seen that has no solution among the real numbers. More generally, there are polynomial equations with real coefficients that have no real solution at all.

2.2.1

Rectangular Form

Definition (rectangular form). A complex number is written (or, equivalently, ), where and are real numbers. Here is called the real part of the complex number and is called the imaginary part.

2.2.2

Argand Plane

A complex number is uniquely determined by the ordered pair of real numbers : for instance and correspond to and respectively.

2.2.3

Algebraic Operations on Complex Numbers

Three operations are defined on complex numbers .

+Exercise 2.2i3 questions
  1. Q1Evaluate the following if $z=5-2i$ and $w=-1+3i$ (i) $z+w$ (ii) $z-iw$ (iii) $2z+3w$ (iv) $zw$ (v) $z^2+2zw+w^2$ (vi) $(z+w)^2$Free
  2. Q2Given the complex number $z=2+3i$, represent the complex numbers in Argand diagram. (i) $z, iz$, and $z+iz$ (ii) $z, -iz$, and $z-iz$Preview
  3. Q3Find the values of the real numbers $x$ and $y$, if the complex numbers $(3-i)x-(2-i)y+2i+5$ and $2x+(-1+2i)y+3+2i$ are equal.Preview
2.3

Basic Algebraic Properties of Complex Numbers

The properties of addition and multiplication of complex numbers turn out to be exactly the same in kind as the familiar properties of addition and multiplication of real numbers — complex numbers for…

2.3.1

Properties of Complex Numbers

Properties under addition (for any complex numbers ): 1. Closure: is again a complex number. 2. Commutative: . 3. Associative: . 4. Additive identity: there is a complex number with for every . 5.

+Exercise 2.3i3 questions
  1. Q1If $z_1=1-3i,\ z_2=-4i$, and $z_3=5$, show that (i) $(z_1+z_2)+z_3=z_1+(z_2+z_3)$ (ii) $(z_1z_2)z_3=z_1(z_2z_3)$.Free
  2. Q2If $z_1=3,\ z_2=-7i$, and $z_3=5+4i$, show that (i) $z_1(z_2+z_3)=z_1z_2+z_1z_3$ (ii) $(z_1+z_2)z_3=z_1z_3+z_2z_3$.Preview
  3. Q3If $z_1=2+5i,\ z_2=-3-4i$, and $z_3=1+i$, find the additive and multiplicative inverse of $z_1, z_2$, and $z_3$.Preview
2.4

Conjugate of a Complex Number

This section studies the conjugate of a complex number: its geometric picture, its algebraic properties, and how it is used (with worked examples) to simplify expressions and divide by complex numbers…

2.4.1

Geometrical Representation of Conjugate of a Complex Number

The complex conjugate of is denoted . To find the conjugate of , simply replace by throughout . For instance, is the conjugate of .

2.4.2

Properties of Complex Conjugates

Ten properties of complex conjugates (for complex numbers and integer ): 1. 2. 3. 4. 5. 6. 7. 8. is real 9. is purely imaginary 10.

+Exercise 2.4i7 questions
  1. Q1Write the following in the rectangular form: (i) $\overline{(5+9i)+(2-4i)}$ (ii) $\dfrac{10-5i}{6+2i}$ (iii) $\overline{3i}+\dfrac{1}{2-i}$Free
  2. Q2If $z=x+iy$, find the following in rectangular form. (i) $\operatorname{Re}\left(\dfrac1z\right)$ (ii) $\operatorname{Re}(i\overline z)$ (ii…Free
  3. Q3If $z_1=2-i$ and $z_2=-4+3i$, find the inverse of $z_1z_2$ and $\dfrac{z_1}{z_2}$.Free
  4. Q4The complex numbers $u,v$, and $w$ are related by $\dfrac1u=\dfrac1v+\dfrac1w$. If $v=3-4i$ and $w=4+3i$, find $u$ in rectangular form.Preview
  5. Q5Prove the following properties: (i) $z$ is real if and only if $z=\overline z$ (ii) $\operatorname{Re}(z)=\dfrac{z+\overline z}{2}$ and $\op…Preview
  6. Q6Find the least value of the positive integer $n$ for which $(\sqrt3+i)^n$ is (i) real (ii) purely imaginary.Preview
  7. Q7Show that (i) $(2+i\sqrt3)^{10}-(2-i\sqrt3)^{10}$ is purely imaginary (ii) $\left(\dfrac{19-7i}{9+i}\right)^{12}+\left(\dfrac{20-5i}{7-6i}\r…Preview
2.5

Modulus of a Complex Number

Just as the absolute value of a real number measures its distance from the origin along the real number line, the modulus of a complex number measures its distance from the origin in the complex plane…

2.5.1

Properties of Modulus of a Complex Number

Eight properties of the modulus (for complex numbers and integer ): 1. 2. (Triangle Inequality) 3. 4. 5. 6. 7. 8.

2.5.2

Square Roots of a Complex Number

To find the square roots of , set to be a square root, so for real . Expanding, Equating real and imaginary parts: and .

+Exercise 2.5i10 questions
  1. Q1Find the modulus of the following complex numbers (i) $\dfrac{2i}{3+4i}$ (ii) $\dfrac{2-i}{1+i}+\dfrac{1-2i}{1-i}$ (iii) $(1-i)^{10}$ (iv) $…Free
  2. Q2For any two complex numbers $z_1$ and $z_2$, such that $|z_1|=|z_2|=1$ and $z_1z_2\ne-1$, then show that $\dfrac{z_1+z_2}{1+z_1z_2}$ is a re…Free
  3. Q3Which one of the points $10-8i,\ 11+6i$ is closest to $1+i$.Free
  4. Q4If $|z|=3$, show that $7\le|z+6-8i|\le13$.Preview
  5. Q5If $|z|=1$, show that $2\le|z^2-3|\le4$.Preview
  6. Q6If $|z|=2$, show that $8\le|z+6+8i|\le12$.Preview
  7. Q7If $z_1, z_2$, and $z_3$ are three complex numbers such that $|z_1|=1, |z_2|=2, |z_3|=3$ and $|z_1+z_2+z_3|=1$, show that $|9z_1z_2+4z_1z_3+…Preview
  8. Q8If the area of the triangle formed by the vertices $z, iz$, and $z+iz$ is $50$ square units, find the value of $|z|$.Preview
  9. Q9Show that the equation $z^3+2\overline z=0$ has five solutions.Preview
  10. Q10Find the square roots of (i) $4+3i$ (ii) $-6+8i$ (iii) $-5-12i$.Preview
2.6

Geometry and Locus of Complex Numbers

This section studies the geometric interpretation of a complex number in the complex plane, and how to convert a condition on (given in terms of or ) into an ordinary Cartesian equation in — the equat…

+Exercise 2.6i5 questions
  1. Q1If $z=x+iy$ is a complex number such that $\left|\dfrac{z-4i}{z+4i}\right|=1$ show that the locus of $z$ is real axis.Free
  2. Q2If $z=x+iy$ is a complex number such that $\operatorname{Im}\left(\dfrac{2z+1}{iz+1}\right)=0$, show that the locus of $z$ is $2x^2+2y^2+x-2…Free
  3. Q3Obtain the Cartesian form of the locus of $z=x+iy$ in each of the following cases: (i) $[\operatorname{Re}(iz)]^2=3$ (ii) $\operatorname{Im}…Preview
  4. Q4Show that the following equations represent a circle, and, find its centre and radius. (i) $|z-2-i|=3$ (ii) $|2z+2-4i|=2$ (iii) $|3z-6+12i|=…Preview
  5. Q5Obtain the Cartesian equation for the locus of $z=x+iy$ in each of the following cases: (i) $|z-4|=16$ (ii) $|z-4|^2-|z-1|^2=16$.Preview
2.7

Polar and Euler Form of a Complex Number

Rectangular form () is the natural choice for addition and subtraction of complex numbers — real parts add to real parts, imaginary parts add to imaginary parts.

2.7.1

Polar Form of a Complex Number

Polar coordinates form another set of parameters characterising the vector from the origin to a point , using magnitude and direction instead of horizontal/vertical components.

2.7.2

Euler's Form of the Complex Number

Euler's formula identifies the trigonometric bracket in polar form with a complex exponential: Substituting this into the polar form gives the compact Euler (exponential) form

+Exercise 2.7i6 questions
  1. Q1Write in polar form of the following complex numbers (i) $2+i2\sqrt3$ (ii) $3-i\sqrt3$ (iii) $-2-i2$ (iv) $\dfrac{i-1}{\cos\frac\pi3+i\sin\f…Free
  2. Q2Find the rectangular form of the complex numbers (i) $\left(\cos\dfrac\pi6+i\sin\dfrac\pi6\right)\left(\cos\dfrac\pi{12}+i\sin\dfrac\pi{12}\…Free
  3. Q3If $(x_1+iy_1)(x_2+iy_2)(x_3+iy_3)\cdots(x_n+iy_n)=a+ib$, show that (i) $(x_1^2+y_1^2)(x_2^2+y_2^2)(x_3^2+y_3^2)\cdots(x_n^2+y_n^2)=a^2+b^2$…Preview
  4. Q4If $\dfrac{1+z}{1-z}=\cos2\theta+i\sin2\theta$, show that $z=i\tan\theta$.Preview
  5. Q5If $\cos\alpha+\cos\beta+\cos\gamma=\sin\alpha+\sin\beta+\sin\gamma=0$, show that (i) $\cos3\alpha+\cos3\beta+\cos3\gamma=3\cos(\alpha+\beta…Preview
  6. Q6If $z=x+iy$ and $\arg\left(\dfrac{z-i}{z+2}\right)=\dfrac\pi4$, show that $x^2+y^2+3x-3y+2=0$.Preview
2.8

de Moivre's Theorem and its Applications

Abraham de Moivre (1667–1754) was one of the first mathematicians to use complex numbers in trigonometry.

2.8.1

de Moivre's Theorem

de Moivre's Theorem. Given any complex number (which always has modulus ) and any integer ,

2.8.2

Finding nth Roots of a Complex Number

de Moivre's formula can be used to find the roots of a complex number, not just its powers. Suppose is a positive integer and a complex number is an th root of (written ), so that Let , and write in p…

2.8.3

The nth Roots of Unity

34 Q

Definition. For a positive integer , the solutions of are the th roots of unity. In polar form, is written Applying the root formula from §2.8.2 (with ), the th roots of unity are

+Exercise 2.8i9 questions
  1. Q1If $\omega\ne1$ is a cube root of unity, show that $\dfrac{a+b\omega+c\omega^2}{b+c\omega+a\omega^2}+\dfrac{a+b\omega+c\omega^2}{c+a\omega+b…Free
  2. Q2Show that $\left(\dfrac{\sqrt3}2+\dfrac i2\right)^5+\left(\dfrac{\sqrt3}2-\dfrac i2\right)^5=-\sqrt3$.Free
  3. Q3Find the value of $\left(\dfrac{1+\sin\frac\pi{10}+i\cos\frac\pi{10}}{1+\sin\frac\pi{10}-i\cos\frac\pi{10}}\right)^{10}$.Free
  4. Q4If $2\cos\alpha=x+\dfrac1x$ and $2\cos\beta=y+\dfrac1y$, show that (i) $\dfrac xy+\dfrac yx=2\cos(\alpha-\beta)$ (ii) $xy-\dfrac1{xy}=2i\sin…Preview
  5. Q5Solve the equation $z^3+27=0$.Preview
  6. Q6If $\omega\ne1$ is a cube root of unity, show that the roots of the equation $(z-1)^3+8=0$ are $-1,\ 1-2\omega,\ 1-2\omega^2$.Preview
  7. Q7Find the value of $\displaystyle\sum_{k=1}^{8}\left(\cos\dfrac{2k\pi}9+i\sin\dfrac{2k\pi}9\right)$.Preview
  8. Q8If $\omega\ne1$ is a cube root of unity, show that (i) $(1-\omega+\omega^2)^6+(1+\omega-\omega^2)^6=128$. (ii) $(1+\omega)(1+\omega^2)(1+\om…Preview
  9. Q9If $z=2-2i$, find the rotation of $z$ by $\theta$ radians in the counter clockwise direction about the origin when (i) $\theta=\dfrac\pi3$ (…Preview
+Exercise 2.9i25 questions
  1. Q1$i^n+i^{n+1}+i^{n+2}+i^{n+3}$ is (1) $0$ (2) $1$ (3) $-1$ (4) $i$Free
  2. Q2The value of $\displaystyle\sum_{n=1}^{13}(i^n+i^{n-1})$ is (1) $1+i$ (2) $i$ (3) $1$ (4) $0$Free
  3. Q3The area of the triangle formed by the complex numbers $z, iz$, and $z+iz$ in the Argand's diagram is (1) $\dfrac12|z|^2$ (2) $|z|^2$ (3) $\…Free
  4. Q4The conjugate of a complex number is $\dfrac1{i-2}$. Then, the complex number is (1) $\dfrac1{i+2}$ (2) $\dfrac{-1}{i+2}$ (3) $\dfrac{-1}{i-…Preview
  5. Q5If $z=\dfrac{(\sqrt3+i)^3(3i+4)^2}{(8+6i)^2}$, then $|z|$ is equal to (1) $0$ (2) $1$ (3) $2$ (4) $3$Preview
  6. Q6If $z$ is a non zero complex number, such that $2iz^2=\overline z$ then $|z|$ is (1) $\dfrac12$ (2) $1$ (3) $2$ (4) $3$Preview
  7. Q7If $|z-2+i|\le2$, then the greatest value of $|z|$ is (1) $\sqrt3-2$ (2) $\sqrt3+2$ (3) $\sqrt5-2$ (4) $\sqrt5+2$Preview
  8. Q8If $\left|z-\dfrac3z\right|=2$, then the least value of $|z|$ is (1) $1$ (2) $2$ (3) $3$ (4) $5$Preview
  9. Q9If $|z|=1$, then the value of $\dfrac{1+z}{1+\overline z}$ is (1) $z$ (2) $\overline z$ (3) $\dfrac1z$ (4) $1$Preview
  10. Q10The solution of the equation $|z|-z=1+2i$ is (1) $\dfrac32-2i$ (2) $-\dfrac32+2i$ (3) $2-\dfrac32i$ (4) $2+\dfrac32i$Preview
  11. Q11If $|z_1|=1,\ |z_2|=2,\ |z_3|=3$ and $|9z_1z_2+4z_1z_3+z_2z_3|=12$, then the value of $|z_1+z_2+z_3|$ is (1) $1$ (2) $2$ (3) $3$ (4) $4$Preview
  12. Q12If $z$ is a complex number such that $z\in\mathbb C\setminus\mathbb R$ and $z+\dfrac1z\in\mathbb R$, then $|z|$ is (1) $0$ (2) $1$ (3) $2$ (…Preview
  13. Q13$z_1, z_2$, and $z_3$ are complex numbers such that $z_1+z_2+z_3=0$ and $|z_1|=|z_2|=|z_3|=1$ then $z_1^2+z_2^2+z_3^2$ is (1) $3$ (2) $2$ (3…Preview
  14. Q14If $\dfrac{z-1}{z+1}$ is purely imaginary, then $|z|$ is (1) $\dfrac12$ (2) $1$ (3) $2$ (4) $3$Preview
  15. Q15If $z=x+iy$ is a complex number such that $|z+2|=|z-2|$, then the locus of $z$ is (1) real axis (2) imaginary axis (3) ellipse (4) circlePreview
  16. Q16The principal argument of $\dfrac3{-1+i}$ is (1) $-\dfrac{5\pi}6$ (2) $-\dfrac{2\pi}3$ (3) $-\dfrac{3\pi}4$ (4) $-\dfrac\pi2$Preview
  17. Q17The principal argument of $(\sin40^\circ+i\cos40^\circ)^5$ is (1) $-110^\circ$ (2) $-70^\circ$ (3) $70^\circ$ (4) $110^\circ$Preview
  18. Q18If $(1+i)(1+2i)(1+3i)\cdots(1+ni)=x+iy$, then $2\cdot5\cdot10\cdots(1+n^2)$ is (1) $1$\n(2) $i$\n(3) $x^2+y^2$\n(4) $1+n^2$Preview
  19. Q19If $\omega\ne1$ is a cubic root of unity and $(1+\omega)^7=A+B\omega$, then $(A,B)$ equals (1) $(1,0)$ (2) $(-1,1)$ (3) $(0,1)$ (4) $(1,1)$Preview
  20. Q20The principal argument of the complex number $\dfrac{(1+i\sqrt3)^2}{4i(1-i\sqrt3)}$ is (1) $\dfrac{2\pi}3$ (2) $\dfrac\pi6$ (3) $\dfrac{5\pi…Preview
  21. Q21If $\alpha$ and $\beta$ are the roots of $x^2+x+1=0$, then $\alpha^{2020}+\beta^{2020}$ is (1) $-2$\n(2) $-1$\n(3) $1$\n(4) $2$Preview
  22. Q22The product of all four values of $\left(\cos\dfrac\pi3+i\sin\dfrac\pi3\right)^{3/4}$ is (1) $-2$\n(2) $-1$\n(3) $1$\n(4) $2$Preview
  23. Q23If $\omega\ne1$ is a cubic root of unity and $\begin{vmatrix}1&1&1\\1&-\omega^2-1&\omega^2\\1&\omega^2&\omega^7\end{vmatrix}=3k$, then $k$ i…Preview
  24. Q24The value of $\left(\dfrac{1+\sqrt3i}{1-\sqrt3i}\right)^{10}$ is (1) $\operatorname{cis}\dfrac{2\pi}3$ (2) $\operatorname{cis}\dfrac{4\pi}3$…Preview
  25. Q25If $\omega=\operatorname{cis}\dfrac{2\pi}3$, then the number of distinct roots of $\begin{vmatrix}z+1&\omega&\omega^2\\\omega&z+\omega^2&1\\…Preview
2.9

Summary

In this chapter we studied:

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 48 questions48 questions
  1. Q1The value of $\left[\dfrac{-1+i\sqrt3}{2}\right]^{100} + \left[\dfrac{-1-i\sqrt3}{2}\right]^{100}$ is : (a) $2$ (b) $0$ (c) $-1$ (d) $1$Preview
  2. Q2If $-\bar z$ lies in the third quadrant then $z$ lies in the : (a) first quadrant (b) second quadrant (c) third quadrant (d) fourth quadrantPreview
  3. Q3The points $z_1, z_2, z_3, z_4$ in the complex plane are the vertices of a parallelogram taken in order if and only if : (a) $z_1+z_4=z_2+z_…Preview
  4. Q4If $|z-z_1|=|z-z_2|$ then the locus of $z$ is (a) a circle with centre at the origin (b) a circle with centre at $z_1$ (c) a straight line p…Preview
  5. Q5'P' represents the variable complex number $z$. Find the locus of P if $\text{Re}\left(\dfrac{z+1}{z+i}\right)=1$.Preview
  6. Q6If $(m-5)+i(n+4)$ is the complex conjugate of $(2m+3)+i(3n-2)$ then $(n, m)$ are : (a) $\left(\dfrac{-1}{2}, -8\right)$ (b) $\left(\dfrac{-1…Preview
  7. Q7The principal value of arg $(z)$ lies in the interval : (a) $\left[0, \dfrac{\pi}{2}\right]$ (b) $(-\pi, \pi]$ (c) $[0, \pi]$ (d) $(-\pi, 0]…Preview
  8. Q8If $\omega$ is the cube root of unity then the value of $(1-\omega)(1-\omega^2)(1-\omega^4)(1-\omega^8)$ is : (a) 9 (b) $-9$ (c) 16 (d) 32Preview
  9. Q9If P represents the variable complex number $z$ and if $|2z-1| = 2|z|$ then the locus of P is : (a) the straight line $x = \dfrac{1}{4}$ (b)…Preview
  10. Q10If $\alpha$ and $\beta$ are complex conjugates to each other and $\alpha = -\sqrt2 + i$ then find $\alpha^2 + \beta^2 - \alpha\beta$.Preview
  11. Q11Show that the points representing the complex numbers $7+9i,\ -3+7i,\ 3+3i$ form a right angled triangle on the Argand diagram.Preview
  12. Q12If $|z - z_1| = |z - z_2|$ then the locus of $z$ is : (a) a straight line passing through the origin (b) a circle with centre at the origin…Preview
  13. Q13If $\omega$ is a cube root of unity then the value of $(1 - \omega + \omega^2)^4 + (1 + \omega - \omega^2)^4$ is : (a) $-16$ (b) $0$ (c) $-3…Preview
  14. Q14The modulus and amplitude of the complex number $\left[e^{3 - i\frac{\pi}{4}}\right]^3$ are respectively : (a) $e^6, \dfrac{-3\pi}{4}$ (b) $…Preview
  15. Q15If the point represented by the complex number $iz$ is rotated about the origin through an angle $\dfrac{\pi}{2}$ in the counter clockwise d…Preview
  16. Q16'P' represents the variable complex number $z$. Find the locus of P if $\text{Re}\left[\dfrac{z - 1}{z + i}\right] = 1$.Preview
  17. Q17If $-x - iy$ lies in the first quadrant, then $-ix + y$ lies in the : (a) third quadrant (b) fourth quadrant (c) first quadrant (d) second q…Preview
  18. Q18If $z_1 = 1 + 2i$, $z_2 = 1 - 3i$ and $z_3 = 2 + 4i$ then, the points on the Argand diagram representing $z_1 z_2 z_3$, $2z_1 z_2 z_3$, $-7z…Preview
  19. Q19Find the least positive integer $n$ such that $\left(\dfrac{1+i}{1-i}\right)^n = 1$.Preview
  20. Q20If $n$ is a positive integer, prove that $\left(\dfrac{1+\sin\theta - i\cos\theta}{1+\sin\theta+i\cos\theta}\right)^n = \cos n\left(\dfrac{\…Preview
  21. Q21The value of $\displaystyle\sum_{i=1}^{13}\left(i^{n} + i^{n-1}\right)$ is : (a) $0$ (b) $1+i$ (c) $i$ (d) $1$Preview
  22. Q22$\arg(0)$ is : (a) $\infty$ (b) $0$ (c) $\pi$ (d) undefinedPreview
  23. Q23Prove that $\left(\dfrac{1+i}{1-i}\right)^3 - \left(\dfrac{1-i}{1+i}\right)^3 = -2i$.Preview
  24. Q24If $(1+i)(1+2i)\ldots(1+ni) = x+iy$, then prove that $2\cdot5\cdot10\cdots(1+n^2) = x^2+y^2$.Preview
  25. Q25If $(1+i)(1+2i)(1+3i)\ldots(1+ni)=x+iy$ then the value $2\cdot5\cdot10\ldots(1+n^2)$ is : (a) $x^2+y^2$ (b) $1$ (c) $1+n^2$ (d) $i$Preview
  26. Q26The value of $\displaystyle\sum_{n=1}^{12}i^{n}$ is : (a) $0$ (b) $1$ (c) $-1$ (d) $i$Preview
  27. Q27Prove the following properties : $\operatorname{Re}(z)=\dfrac{z+\bar{z}}{2}$ and $\operatorname{Im}(z)=\dfrac{z-\bar{z}}{2i}$Preview
  28. Q28Which one of the points $10-8i$, $11+6i$ is closest to $1+i$.Preview
  29. Q29(a) Show that the locus of $z=x+iy$ if $|z+i|=|z-1|$, is $x+y=0$. **OR** (b) Show that $\displaystyle\int_{0}^{a}\dfrac{f(x)}{f(x)+f(a-x)}\,…Preview
  30. Q30The value of $\left(\dfrac{1+i}{\sqrt2}\right)^8+\left(\dfrac{1-i}{\sqrt2}\right)^8$ is : (a) $8$ (b) $4$ (c) $2$ (d) $6$Preview
  31. Q31If $|z|=1$, then the value of $\dfrac{1+z}{1+\bar{z}}$ is : (a) $\dfrac1z$ (b) $z$ (c) $1$ (d) $\bar{z}$Preview
  32. Q32If $|z|=2$, show that $3\le|z+3+4i|\le7$Preview
  33. Q33Express $e^{\cos\theta+i\sin\theta}$ in $a+ib$ form.Preview
  34. Q34If $z=(2+3i)(1-i)$, then find $z^{-1}$.Preview
  35. Q35(a) Solve the equation $z^3+8i=0$, where $z\in\mathbb{C}$. **OR** (b) Solve : $\left(1+x+xy^2\right)\dfrac{dy}{dx}+\left(y+y^3\right)=0$.Preview
  36. Q36If $|z_1|=1$, $|z_2|=2$, $|z_3|=3$ and $|9z_1z_2+4z_1z_3+z_2z_3|=12$ then the value of $|z_1+z_2+z_3|$ is : (a) $3$ (b) $1$ (c) $4$ (d) $2$Preview
  37. Q37If $(1+i)(1+2i)(1+3i)\ldots(1+ni)=x+iy$ then $2\cdot5\cdot10\ldots(1+n^2)$ is : (a) $x^2+y^2$ (b) $1$ (c) $1+n^2$ (d) $i$Preview
  38. Q38Simplify : $\displaystyle\sum_{n=1}^{12}i^n$Preview
  39. Q39Simplify $\left(\dfrac{1+i}{1-i}\right)^3-\left(\dfrac{1-i}{1+i}\right)^3$ into rectangular form.Preview
  40. Q40The square root of i are : (a) $\pm\dfrac12(1+i)$ (b) $\pm\dfrac{1}{\sqrt2}(1+i)$ (c) $\pm\dfrac12(1-i)$ (d) $\pm\dfrac{1}{\sqrt2}(1-i)$Preview
  41. Q41The value of $\displaystyle\sum_{n=1}^{13}\left(i^n+i^{n-1}\right)$ is : (a) $1$ (b) $1+i$ (c) $0$ (d) $i$Preview
  42. Q42If $z=x+iy$, then find $\text{Re}\left(\dfrac1z\right)$ in rectangular form.Preview
  43. Q43If $|z|=2$ show that $8\le|z+6+8i|\le12$Preview
  44. Q44(a) If $\omega\ne1$ is a cube root of unity, show that the roots of the equation $(z-1)^3+8=0$ are $-1$, $1-2\omega$, $1-2\omega^2$ **OR** (…Preview
  45. Q45If $z$ is a complex number such that $z\in C\setminus R$ and $z+\dfrac1z\in R$, then $|z|$ is : (a) $2$ (b) $0$ (c) $3$ (d) $1$Preview
  46. Q46The product of all four values of $\left(\cos\dfrac{\pi}{3}+i\sin\dfrac{\pi}{3}\right)^{3/4}$ is : (a) $1$ (b) $-2$ (c) $2$ (d) $-1$Preview
  47. Q47Simplify : $\displaystyle\sum_{n=1}^{12}i^n$Preview
  48. Q48If $z_1=\overline{1+i}$ and $\overline{z_2}=1-i$, find the inverse of $\left(\dfrac{z_1}{z_2}\right)^{2026}$Preview