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Mathematics · Class 11 Science

Ch 10Complex Numbers — Class 11 Mathematics, concept-first.

Why we need a new kind of number. Consider the equation , i.e. . This has no solution among the real numbers, because the square of every real number — positive, negative, or zero — is never negative.

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Complex Numbers

Complex numbers extend the real number system so that every quadratic equation, even one like x²+1=0 with a negative discriminant, has a solution. Built formally as ordered pairs (a,b) of real numbers and written in…

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1.1

A Complex Number

Why we need a new kind of number. Consider the equation , i.e. . This has no solution among the real numbers, because the square of every real number — positive, negative, or zero — is never negative.…

+Exercise 1.1i59 questions
  1. Q1Simplify : $\sqrt{-16}+3\sqrt{-25}+\sqrt{-36}-\sqrt{-625}$Free
  2. Q2Simplify : $4\sqrt{-4}+5\sqrt{-9}-3\sqrt{-16}$Free
  3. Q3Write the conjugate of the following complex number : $3+i$Free
  4. Q4Write the conjugate of the following complex number : $3-i$Preview
  5. Q5Write the conjugate of the following complex number : $-\sqrt5-\sqrt7\,i$Preview
  6. Q6Write the conjugate of the following complex number : $-\sqrt{-5}$Preview
  7. Q7Write the conjugate of the following complex number : $5i$Preview
  8. Q8Write the conjugate of the following complex number : $\sqrt5-i$Preview
  9. Q9Write the conjugate of the following complex number : $\sqrt2+\sqrt3\,i$Preview
  10. Q10Write the conjugate of the following complex number : $\cos\theta+i\sin\theta$Preview
  11. Q11Find $a$ and $b$ if $a+2b+2ai = 4+6i$Preview
  12. Q12Find $a$ and $b$ if $(a-b)+(a+b)i = a+5i$Preview
  13. Q13Find $a$ and $b$ if $(a+b)(2+i) = b+1+(10+2a)i$Preview
  14. Q14Find $a$ and $b$ if $abi = 3a-b+12i$Preview
  15. Q15Find $a$ and $b$ if $\dfrac{1}{a}+bi = 3-2i$Preview
  16. Q16Find $a$ and $b$ if $(a+ib)(1+i) = 2+i$Preview
  17. Q17Express the following in the form of $a+ib$, $a,b\in\mathbb{R}$, $i=\sqrt{-1}$. State the values of $a$ and $b$ : $(1+2i)(-2+i)$Preview
  18. Q18Express the following in the form of $a+ib$, $a,b\in\mathbb{R}$, $i=\sqrt{-1}$. State the values of $a$ and $b$ : $(1+i)(1-i)^{-1}$Preview
  19. Q19Express the following in the form of $a+ib$, $a,b\in\mathbb{R}$, $i=\sqrt{-1}$. State the values of $a$ and $b$ : $\dfrac{i(4+3i)}{(1-i)}$Preview
  20. Q20Express the following in the form of $a+ib$, $a,b\in\mathbb{R}$, $i=\sqrt{-1}$. State the values of $a$ and $b$ : $\dfrac{(2+i)}{(3-i)(1+2i)…Preview
  21. Q21Express the following in the form of $a+ib$: [the printed source is corrupted at this sub-item's fraction layout — could not reliably recons…Preview
  22. Q22Express the following in the form of $a+ib$, $a,b\in\mathbb{R}$, $i=\sqrt{-1}$. State the values of $a$ and $b$ : $\dfrac{3+2i}{2-5i}+\dfrac…Preview
  23. Q23Express the following in the form of $a+ib$, $a,b\in\mathbb{R}$, $i=\sqrt{-1}$. State the values of $a$ and $b$ : $(1+i)^{-3}$Preview
  24. Q24Express the following in the form of $a+ib$: [the printed source's fraction layout is corrupted at this sub-item — could not reliably recons…Preview
  25. Q25Express the following in the form of $a+ib$: [the printed source's layout of this product-of-brackets sub-item is corrupted — could not reli…Preview
  26. Q26Express the following in the form of $a+ib$, $a,b\in\mathbb{R}$, $i=\sqrt{-1}$. State the values of $a$ and $b$ : $(2+3i)(2-3i)$Preview
  27. Q27Express the following in the form of $a+ib$: [the printed source's fraction layout is corrupted at this sub-item — could not reliably recons…Preview
  28. Q28Show that $\dfrac{(1-i)^3}{1-i^3}$ is a real number.Preview
  29. Q29Find the value of $\dfrac{3-2i}{(i^6-i^7)(1+i^{11})}$Preview
  30. Q30Evaluate the following : $i^{35}$Preview
  31. Q31Evaluate the following : $i^{888}$Preview
  32. Q32Evaluate the following : $i^{93}$Preview
  33. Q33Evaluate the following : $i^{116}$Preview
  34. Q34Evaluate the following : $i^{403}$Preview
  35. Q35Evaluate the following : $\dfrac{1}{i^{58}}$Preview
  36. Q36Evaluate the following : $i^{-888}$Preview
  37. Q37Evaluate the following : $i^{30}+i^{40}+i^{50}+i^{60}$Preview
  38. Q38Show that $1+i^{10}+i^{20}+i^{30}$ is a real number.Preview
  39. Q39Find the value of $i^{49}+i^{68}+i^{89}+i^{110}$Preview
  40. Q40Find the value of $i+i^2+i^3+i^4$Preview
  41. Q41Simplify : $\dfrac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}$Preview
  42. Q42Find the value of $1+i^2+i^4+i^6+i^8+\ldots+i^{20}$Preview
  43. Q43Show that $1+i^{10}+i^{100}-i^{1000}=0$.Preview
  44. Q44Is $(1+i^{14}+i^{18}+i^{22})$ a real number? Justify your answer.Preview
  45. Q45Evaluate : $i^{37}+\dfrac{1}{i^{67}}$Preview
  46. Q46Prove that $(1+i)^4 \times (1-i)^4 = 16$.Preview
  47. Q47Find the value of $\dfrac{i^6+i^7+i^8+i^9}{i^2+i^3}$Preview
  48. Q48If $a = -\dfrac12+\dfrac{\sqrt3}{2}i$, $b = -\dfrac12-\dfrac{\sqrt3}{2}i$ then show that $a^2=b$ and $b^2=a$.Preview
  49. Q49If $x+iy = (a+ib)^3$, show that $\dfrac{x}{a}+\dfrac{y}{b} = 4(a^2-b^2)$Preview
  50. Q50If $\dfrac{a+3i}{2+ib} = 1-i$, show that $(5a-7b) = 0$.Preview
  51. Q51If $x+iy = \dfrac{a+ib}{c+id}$, prove that $(x^2+y^2)^2 = \dfrac{a^2+b^2}{c^2+d^2}$Preview
  52. Q52If $(a+ib) = \dfrac{1+i}{1-i}$, then prove that $(a^2+b^2) = 1$.Preview
  53. Q53Show that $\dfrac{7+\sqrt3\,i}{7-\sqrt3\,i}+\dfrac{7-\sqrt3\,i}{7+\sqrt3\,i}$ is real.Preview
  54. Q54If $(x+iy)^3 = u+iv$, then show that $\dfrac{u}{x}+\dfrac{v}{y} = 4(x^2-y^2)$Preview
  55. Q55Find the value of $x$ and $y$ which satisfy the following equations ($x,y\in\mathbb{R}$) : $(x+2y)+(2x-3y)i+4i = 5$Preview
  56. Q56Find the value of $x$ and $y$ which satisfy the following equations ($x,y\in\mathbb{R}$) : $\dfrac{x+1}{1+i}+\dfrac{y-1}{1-i} = i$Preview
  57. Q57Find the value of $x$ and $y$ which satisfy the following equations ($x,y\in\mathbb{R}$) : $\dfrac{x+iy}{2+3i}+\dfrac{2+i}{2-3i} = \dfrac{9}…Preview
  58. Q58Find the value of $x$ and $y$ which satisfy the following equations ($x,y\in\mathbb{R}$) : If $x(1+3i)+y(2-i)-5+i^3 = 0$, find $x+y$Preview
  59. Q59Find the value of $x$ and $y$ which satisfy the following equations ($x,y\in\mathbb{R}$) : If $x+2i+15i^6y = 7x+i^3(y+4)$, find $x+y$Preview
1.1(a)

Imaginary Number

Definition. A number of the form , where , , and , is called an imaginary number.

1.1(b)

Complex Number

Definition. A number of the form , where and (so ), is called a complex number. Here is called the real part of , written or , and is called the imaginary part of , written or .

1.2

Algebra of Complex Numbers

Having pinned down what a complex number is, this section develops the algebra of complex numbers — starting with when two complex numbers count as equal (Section 1.2.1) and what it means to conjugate…

+Exercise 1.2i22 questions
  1. Q60Find the square root of the following complex number : $-8-6i$Free
  2. Q61Find the square root of the following complex number : $7+24i$Free
  3. Q62Find the square root of the following complex number : $1+4\sqrt3\,i$Free
  4. Q63Find the square root of the following complex number : $3+2\sqrt{10}\,i$Preview
  5. Q64Find the square root of the following complex number : $2(1-\sqrt3\,i)$Preview
  6. Q65Solve the following quadratic equation : $8x^2+2x+1=0$Preview
  7. Q66Solve the following quadratic equation : $2x^2-\sqrt3\,x+1=0$Preview
  8. Q67Solve the following quadratic equation : $3x^2-7x+5=0$Preview
  9. Q68Solve the following quadratic equation : $x^2-4x+13=0$Preview
  10. Q69Solve the following quadratic equation : $x^2+3ix+10=0$Preview
  11. Q70Solve the following quadratic equation : $2x^2+3ix+2=0$Preview
  12. Q71Solve the following quadratic equation : $x^2+4ix-4=0$Preview
  13. Q72Solve the following quadratic equation : $ix^2-4x-4i=0$Preview
  14. Q73Solve the following quadratic equation : $x^2-(2+i)x-(1-7i)=0$Preview
  15. Q74Solve the following quadratic equation : $x^2-(3\sqrt2+2i)x+6\sqrt2\,i=0$Preview
  16. Q75Solve the following quadratic equation : $x^2-(5-i)x+(18+i)=0$Preview
  17. Q76Solve the following quadratic equation : $(2+i)x^2-(5-i)x+2(1-i)=0$Preview
  18. Q77Find the value of $x^3-x^2+x+46$, if $x = 2+3i$.Preview
  19. Q78Find the value of $2x^3-11x^2+44x+27$, if $x = \dfrac{25}{3-4i}$.Preview
  20. Q79Find the value of $x^3+x^2-x+22$, if $x = \dfrac{5}{1-2i}$.Preview
  21. Q80Find the value of $x^4+9x^3+35x^2-x+4$, if $x = -5+\sqrt{-4}$.Preview
  22. Q81Find the value of $2x^4+5x^3+7x^2-x+41$, if $x = -2-\sqrt3\,i$.Preview
1.2.1

Equality of Two Complex Numbers

Definition. Two complex numbers and are said to be equal if their corresponding real and imaginary parts are equal:

1.2.2

Conjugate of a Complex Number

Definition. The conjugate of a complex number is defined as , and is denoted by (read "-bar"). In words: keep the real part exactly the same, and flip the sign of the imaginary part.

1.2.3

Addition of Complex Numbers

Let and . Their sum is defined by adding real parts together and imaginary parts together:

1.2.4

Scalar Multiplication

If is any complex number, then for every real number , scalar multiplication is defined by — multiplying a complex number by a real scalar multiplies both its real and imaginary parts by that same .

1.2.5

Subtraction of Complex Numbers

Let and . Subtraction is defined in terms of addition and scalar multiplication (multiplying by the scalar and adding):

1.2.6

Multiplication of Complex Numbers

Let and . Their product, written , is found by expanding like two binomials and then using to simplify:

1.2.7

Powers of i

We already know . Consider for a positive integer . Divide by to get a quotient and remainder : , where .

1.2.8

Division of Complex Number

Let and be complex numbers with (i.e. ). To find , multiply numerator and denominator by the conjugate of the denominator, : so that

1.3

Square Root of a Complex Number

Consider , any complex number, and suppose for some real we want to find. Squaring both sides: Equating real and imaginary parts on the two sides gives two simultaneous real equations: Solving these t…

+Exercise 1.3i34 questions
  1. Q82Find the modulus and amplitude for the following complex number : $7-5i$Free
  2. Q83Find the modulus and amplitude for the following complex number : $\sqrt3+\sqrt2\,i$Free
  3. Q84Find the modulus and amplitude for the following complex number : $-8+15i$Free
  4. Q85Find the modulus and amplitude for the following complex number : $-3(1-i)$Preview
  5. Q86Find the modulus and amplitude for the following complex number : $-4-4i$Preview
  6. Q87Find the modulus and amplitude for the following complex number : $\sqrt3-i$Preview
  7. Q88Find the modulus and amplitude for the following complex number : $3$Preview
  8. Q89Find the modulus and amplitude for the following complex number : $1+i$Preview
  9. Q90Find the modulus and amplitude for the following complex number : $1+i\sqrt3$Preview
  10. Q91Find the modulus and amplitude for the following complex number : $(1+2i)^2(1-i)$Preview
  11. Q92Find real values of $\theta$ for which $\dfrac{4+3i\sin\theta}{1-2i\sin\theta}$ is purely real.Preview
  12. Q93If $z=3+5i$ then represent $z,\ \bar z,\ -z,\ -\bar z$ in Argand's diagram.Preview
  13. Q94Express the following complex number in polar form and exponential form : $-1+\sqrt3\,i$Preview
  14. Q95Express the following complex number in polar form and exponential form : $-i$Preview
  15. Q96Express the following complex number in polar form and exponential form : $-1$Preview
  16. Q97Express the following complex number in polar form and exponential form : $\dfrac{1}{1+i}$Preview
  17. Q98Express the following complex number in polar form and exponential form : $\dfrac{1+2i}{1-3i}$Preview
  18. Q99Express the following complex number in polar form and exponential form : $\dfrac{1+7i}{(2-i)^2}$Preview
  19. Q100Express the following number in the form $x+iy$ : $\sqrt3\left(\cos\dfrac{\pi}{6}+i\sin\dfrac{\pi}{6}\right)$Preview
  20. Q101Express the following number in the form $x+iy$ : $\sqrt2\left(\cos\dfrac{7\pi}{4}+i\sin\dfrac{7\pi}{4}\right)$Preview
  21. Q102Express the following number in the form $x+iy$ : $7\left[\cos\left(-\dfrac{5\pi}{6}\right)+i\sin\left(-\dfrac{5\pi}{6}\right)\right]$Preview
  22. Q103Express the following number in the form $x+iy$ : $e^{i\pi/3}$Preview
  23. Q104Express the following number in the form $x+iy$ : $e^{-4\pi i/3}$Preview
  24. Q105Express the following number in the form $x+iy$ : $e^{5\pi i/6}$Preview
  25. Q106Find the modulus and argument of the complex number $\dfrac{1+2i}{1-3i}$.Preview
  26. Q107Convert the complex number $z = \dfrac{i-1}{\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}}$ in the polar form.Preview
  27. Q108For $z=2+3i$ verify the following : $(\bar{\bar z}) = z$Preview
  28. Q109For $z=2+3i$ verify the following : $z\bar z = |z|^2$Preview
  29. Q110For $z=2+3i$ verify the following : $(z+\bar z)$ is realPreview
  30. Q111For $z=2+3i$ verify the following : $z-\bar z = 6i$Preview
  31. Q112$z_1=1+i,\ z_2=2-3i$. Verify the following : $\overline{z_1+z_2} = \bar z_1+\bar z_2$Preview
  32. Q113$z_1=1+i,\ z_2=2-3i$. Verify the following : $\overline{z_1-z_2} = \bar z_1-\bar z_2$Preview
  33. Q114$z_1=1+i,\ z_2=2-3i$. Verify the following : $\overline{z_1 \cdot z_2} = \bar z_1 \cdot \bar z_2$Preview
  34. Q115$z_1=1+i,\ z_2=2-3i$. Verify the following : $\overline{\left(\dfrac{z_1}{z_2}\right)} = \dfrac{\bar z_1}{\bar z_2}$Preview
1.4

Fundamental Theorem of Algebra

This foundational (unproved, stated-as-fact) theorem comes in two equivalent forms:

+Exercise 1.4i34 questions
  1. Q116Find the value of $\omega^{18}$Free
  2. Q117Find the value of $\omega^{21}$Free
  3. Q118Find the value of $\omega^{-30}$Free
  4. Q119Find the value of $\omega^{-105}$Preview
  5. Q120If $\omega$ is a complex cube root of unity, show that $(2-\omega)(2-\omega^2) = 7$Preview
  6. Q121If $\omega$ is a complex cube root of unity, show that $(1+\omega-\omega^2)^6 = 64$Preview
  7. Q122If $\omega$ is a complex cube root of unity, show that $(1+\omega)^3-(1+\omega^2)^3 = 0$Preview
  8. Q123If $\omega$ is a complex cube root of unity, show that $(2+\omega+\omega^2)^3-(1-3\omega+\omega^2)^3 = 65$Preview
  9. Q124If $\omega$ is a complex cube root of unity, show that $(3+3\omega+5\omega^2)^6-(2+6\omega+2\omega^2)^3 = 0$Preview
  10. Q125If $\omega$ is a complex cube root of unity, show that $\dfrac{a+b\omega+c\omega^2}{c+a\omega+b\omega^2} = \omega^2$Preview
  11. Q126If $\omega$ is a complex cube root of unity, show that $(a+b)+(a\omega+b\omega^2)+(a\omega^2+b\omega) = 0$Preview
  12. Q127If $\omega$ is a complex cube root of unity, show that $(a-b)(a-b\omega)(a-b\omega^2) = a^3-b^3$Preview
  13. Q128If $\omega$ is a complex cube root of unity, show that $(a+b)^2+(a\omega+b\omega^2)^2+(a\omega^2+b\omega)^2 = 6ab$Preview
  14. Q129If $\omega$ is a complex cube root of unity, find the value of $\omega+\dfrac{1}{\omega}$Preview
  15. Q130If $\omega$ is a complex cube root of unity, find the value of $\omega^2+\omega^3+\omega^4$Preview
  16. Q131If $\omega$ is a complex cube root of unity, find the value of $(1+\omega^2)^3$Preview
  17. Q132If $\omega$ is a complex cube root of unity, find the value of $(1-\omega-\omega^2)^3+(1-\omega+\omega^2)^3$Preview
  18. Q133If $\omega$ is a complex cube root of unity, find the value of $(1+\omega)(1+\omega^2)(1+\omega^4)(1+\omega^8)$Preview
  19. Q134If $\alpha$ and $\beta$ are the complex cube roots of unity, show that $\alpha^2+\beta^2+\alpha\beta = 0$Preview
  20. Q135If $\alpha$ and $\beta$ are the complex cube roots of unity, show that $\alpha^4+\beta^4+\alpha^{-1}\beta^{-1} = 0$Preview
  21. Q136If $x=a+b$, $y=\alpha a+\beta b$ and $z=a\beta+b\alpha$ where $\alpha$ and $\beta$ are the complex cube roots of unity, show that $xyz = a^3…Preview
  22. Q137Find the equation in cartesian coordinates of the locus of $z$ if $|z| = 10$Preview
  23. Q138Find the equation in cartesian coordinates of the locus of $z$ if $|z-3| = 2$Preview
  24. Q139Find the equation in cartesian coordinates of the locus of $z$ if $|z-5+6i| = 5$Preview
  25. Q140Find the equation in cartesian coordinates of the locus of $z$ if $|z+8| = |z-4|$Preview
  26. Q141Find the equation in cartesian coordinates of the locus of $z$ if $|z-2-2i| = |z+2+2i|$Preview
  27. Q142Find the equation in cartesian coordinates of the locus of $z$: [the printed source's fraction/modulus layout is corrupted at this sub-item…Preview
  28. Q143Use De Moivre's theorem and simplify : $(\cos2\theta+i\sin2\theta)^7(\cos4\theta+i\sin4\theta)^3$Preview
  29. Q144Use De Moivre's theorem and simplify : $(\cos5\theta+i\sin5\theta)(\cos3\theta+i\sin3\theta)^{-2}$Preview
  30. Q145Use De Moivre's theorem and simplify : $\dfrac{\left(\cos\frac{7\pi}{13}+i\sin\frac{7\pi}{13}\right)^4}{\left(\cos\frac{4\pi}{13}+i\sin\frac…Preview
  31. Q146Express the following in the form $a+ib$, $a,b\in\mathbb{R}$, using De Moivre's theorem : $(1-i)^5$Preview
  32. Q147Express the following in the form $a+ib$, $a,b\in\mathbb{R}$, using De Moivre's theorem : $(1+i)^6$Preview
  33. Q148Express the following in the form $a+ib$, $a,b\in\mathbb{R}$, using De Moivre's theorem : $(1-\sqrt3\,i)^4$Preview
  34. Q149Express the following in the form $a+ib$, $a,b\in\mathbb{R}$, using De Moivre's theorem : $(-2\sqrt3-2i)^5$Preview
1.4.1

Solution of a Quadratic Equation in Complex Number System

Let the given equation be where and . The solution of this quadratic equation is given by the familiar formula so the two roots of are and .

1.5

Argand Diagram or Complex Plane

A complex number , with and , can be represented as a point in a plane whose coordinates are the ordered pair .

1.5.1

Modulus of z

If is a complex number, then the modulus of , denoted or , is defined as From Fig. 1.3, the point represents the complex number , so Hence, the modulus of is the distance of the point from the origin,…

1.5.2

Argument of z

makes an angle with the positive direction of the X-axis. is called the argument or amplitude of the complex number , denoted .

1.5.3

Argument of z in Different Quadrants/Axes

Because alone only ever gives a value in , correctly finding in the standard range requires checking which quadrant (or which axis) the point actually lies in, and adding the matching correction.

1.5.4

Polar Form of a Complex Number

Let the complex number be represented by the point (Fig. 1.4). Let (quadrant-corrected as in Section 1.5.3) and . Then are called the polar coordinates of , and the origin is called the pole.

1.5.5

Exponential Form

It is known (and can be proved using special infinite series — a deeper result not derived here) that This is Euler's identity.

1.6

De Moivre's Theorem

If and , then That is, if two complex numbers are multiplied, their moduli get multiplied and their arguments get added.

1.7

Cube Roots of Unity

Number is often called unity. Let be a cube root of unity, i.e. . Then: From : . From , using the quadratic formula: .

1.8

Set of Points in Complex Plane

If represents the variable point and represents the fixed point , then:

More questions

59 Q
+Show 10 questions10 questions
  1. Q150If $n$ is an odd positive integer then the value of $1+(i)^{2n}+(i)^{4n}+(i)^{6n}$ is : (A) $-4i$ (B) $0$ (C) $4i$ (D) $4$Free
  2. Q151The value of $\dfrac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}$ is equal to : (A) $-2$ (B) $1$ (C) $…Free
  3. Q152$\sqrt{-3}\cdot\sqrt{-6}$ is equal to : (A) $-3\sqrt2$ (B) $3\sqrt2$ (C) $3\sqrt2\,i$ (D) $-3\sqrt2\,i$Free
  4. Q153If $\omega$ is a complex cube root of unity, then the value of $\omega^{99}+\omega^{100}+\omega^{101}$ is : (A) $-1$ (B) $1$ (C) $0$ (D) $3$Preview
  5. Q154If $z=r(\cos\theta+i\sin\theta)$, then the value of $\dfrac{z}{\bar z}+\dfrac{\bar z}{z}$ is : (A) $\cos2\theta$ (B) $2\cos2\theta$ (C) $2\c…Preview
  6. Q155If $\omega(\neq1)$ is a cube root of unity and $(1+\omega)^7 = A+B\omega$, then $A$ and $B$ are respectively the numbers : (A) $0,1$ (B) $1,…Preview
  7. Q156The modulus and argument of $(1+i\sqrt3)^8$ are respectively : (A) $2$ and $\dfrac{2\pi}{3}$ (B) $256$ and $\dfrac{8\pi}{3}$ (C) $256$ and $…Preview
  8. Q157If $\arg(z) = \theta$, then $\arg(\bar z) =$ : (A) $-\theta$ (B) $\theta$ (C) $\pi-\theta$ (D) $\pi+\theta$Preview
  9. Q158If $-1+\sqrt3\,i = re^{i\theta}$, then $\theta =$ ................. . (A) $-\dfrac{2\pi}{3}$ (B) $\dfrac{\pi}{3}$ (C) $-\dfrac{\pi}{3}$ (D)…Preview
  10. Q159If $z=x+iy$ and $|z-zi|=1$ then : (A) $z$ lies on X-axis (B) $z$ lies on Y-axis (C) $z$ lies on a circle (D) $z$ lies on a rectanglePreview
+Show 49 questions49 questions
  1. Q160Simplify the following and express in the form $a+ib$ : $3+\sqrt{-64}$Free
  2. Q161Simplify the following and express in the form $a+ib$ : $(2i^3)^2$Free
  3. Q162Simplify the following and express in the form $a+ib$ : $(2+3i)(1-4i)$Free
  4. Q163Simplify the following and express in the form $a+ib$ : $\dfrac{5}{2}i(-4-3i)$Preview
  5. Q164Simplify the following and express in the form $a+ib$ : $(1+3i)^2(3+i)$Preview
  6. Q165Simplify the following and express in the form $a+ib$ : $\dfrac{4+3i}{1-i}$Preview
  7. Q166Simplify the following and express in the form $a+ib$ : $\left(1+\dfrac2i\right)\left(3+\dfrac4i\right)(5+i)^{-1}$Preview
  8. Q167Simplify the following and express in the form $a+ib$ : $\dfrac{5-3i}{5+3i}$Preview
  9. Q168Simplify the following and express in the form $a+ib$ : $\dfrac{3i^5+2i^7+i^9}{i^6+2i^8+3i^{18}}$Preview
  10. Q169Simplify the following and express in the form $a+ib$ : $\dfrac{5+7i}{4+3i}+\dfrac{5+7i}{4-3i}$Preview
  11. Q170Solve the following equation for $x,y\in\mathbb{R}$ : $(4-5i)x+(2+3i)y = 10-7i$Preview
  12. Q171Solve the following equation for $x,y\in\mathbb{R}$ : $\dfrac{x+iy}{2+3i} = 7-i$Preview
  13. Q172Solve the following equation for $x,y\in\mathbb{R}$ : $(x+iy)(5+6i) = 2+3i$Preview
  14. Q173Solve the following equation for $x,y\in\mathbb{R}$: [the printed source's layout of this equation is corrupted — could not reliably reconst…Preview
  15. Q174Evaluate $(1-i+i^2)^{-15}$Preview
  16. Q175Evaluate $(i^{131}+i^{49})$Preview
  17. Q176Find the value of $x^3+2x^2-3x+21$, if $x = 1+2i$.Preview
  18. Q177Find the value of $x^4+9x^3+35x^2-x+164$, if $x = -5+4i$.Preview
  19. Q178Find the square roots of $-16+30i$Preview
  20. Q179Find the square roots of $15-8i$Preview
  21. Q180Find the square roots of $2+2\sqrt3\,i$Preview
  22. Q181Find the square roots of $18i$Preview
  23. Q182Find the square roots of $3-4i$Preview
  24. Q183Find the square roots of $6+8i$Preview
  25. Q184Find the modulus and amplitude of the following complex number and express it in the polar form : $8+15i$Preview
  26. Q185Find the modulus and amplitude of the following complex number and express it in the polar form : $6-i$Preview
  27. Q186Find the modulus and amplitude of the following complex number and express it in the polar form : $\dfrac12+\dfrac{\sqrt3}{2}i$Preview
  28. Q187Find the modulus and amplitude of the following complex number and express it in the polar form : $-\dfrac12-\dfrac{\sqrt3}{2}i$Preview
  29. Q188Find the modulus and amplitude of the following complex number and express it in the polar form : $2i$Preview
  30. Q189Find the modulus and amplitude of the following complex number and express it in the polar form : $-3i$Preview
  31. Q190Find the modulus and amplitude of the following complex number and express it in the polar form : $\dfrac{1}{\sqrt2}+\dfrac{1}{\sqrt2}i$Preview
  32. Q191Represent $1+2i,\ 2-i,\ -3-2i,\ -2+3i$ by points in Argand's diagram.Preview
  33. Q192Show that $z = \dfrac{5}{(1-i)(2-i)(3-i)}$ is purely imaginary number.Preview
  34. Q193Find the real numbers $x$ and $y$ such that $\dfrac{x}{1+2i}+\dfrac{y}{3+2i} = \dfrac{5+6i}{-1+8i}$Preview
  35. Q194Show that $\left(\dfrac{1}{\sqrt2}+\dfrac{i}{\sqrt2}\right)^{10}+\left(\dfrac{1}{\sqrt2}-\dfrac{i}{\sqrt2}\right)^{10} = 0$Preview
  36. Q195Show that: [the printed source's nested-radical layout is corrupted at this item — could not reliably reconstruct the verbatim stem]Preview
  37. Q196Convert the complex number in polar form and also in exponential form : $z = \dfrac{2+6\sqrt3\,i}{5+\sqrt3\,i}$Preview
  38. Q197Convert the complex number in polar form and also in exponential form : $z = -6+\sqrt2\,i$Preview
  39. Q198Convert the complex number in polar form and also in exponential form : $z = -\dfrac32+\dfrac{3\sqrt3}{2}i$Preview
  40. Q199If $x+iy = \dfrac{a+ib}{a-ib}$, prove that $x^2+y^2 = 1$.Preview
  41. Q200Show that $z = \left(\dfrac{-1+\sqrt{-3}}{2}\right)^3$ is a rational number.Preview
  42. Q201Show that $\dfrac{1-2i}{3-4i}+\dfrac{1+2i}{3+4i}$ is real.Preview
  43. Q202Simplify : $\dfrac{i^{29}+i^{39}+i^{49}}{i^{30}+i^{40}+i^{50}}$Preview
  44. Q203Simplify : $\dfrac{i^{65}+1}{i^{145}}$Preview
  45. Q204Simplify : $\dfrac{i^{238}+i^{236}+i^{234}+i^{232}+i^{230}}{i^{228}+i^{226}+i^{224}+i^{222}+i^{220}}$Preview
  46. Q205Simplify $\dfrac{1}{1-2i}+\dfrac{3}{1+i}+\dfrac{3+4i}{2-4i}$Preview
  47. Q206If $\alpha$ and $\beta$ are complex cube roots of unity, prove that $(1-\alpha)(1-\beta)(1-\alpha^2)(1-\beta^2) = 9$Preview
  48. Q207If $\omega$ is a complex cube root of unity, prove that $(1-\omega+\omega^2)^6+(1+\omega-\omega^2)^6 = 128$Preview
  49. Q208If $\omega$ is the cube root of unity then find the value of $\left(-\dfrac12+\dfrac{\sqrt3}{2}i\right)^{18}+\left(-\dfrac12-\dfrac{\sqrt3}{…Preview