{"id":13526,"date":"2019-12-20T10:53:09","date_gmt":"2019-12-20T05:23:09","guid":{"rendered":"https:\/\/kseebsolutions.guru\/?p=13526"},"modified":"2021-07-02T15:35:46","modified_gmt":"2021-07-02T10:05:46","slug":"1st-puc-maths-question-bank-chapter-5","status":"publish","type":"post","link":"https:\/\/kseebsolutions.guru\/1st-puc-maths-question-bank-chapter-5\/","title":{"rendered":"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations"},"content":{"rendered":"<p>Students can Download Maths Chapter 5 Complex Numbers and Quadratic Equations Questions and Answers, Notes Pdf, <a href=\"https:\/\/kseebsolutions.guru\/1st-puc-maths-question-bank\/\">1st PUC Maths Question Bank with Answers<\/a> helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.<\/p>\n<h2>Karnataka 1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations<\/h2>\n<p>Question 1.<br \/>\nDefine a complex number.<br \/>\nAnswer :<br \/>\nA number of the form a + ib, where a and b are real numbers, is defined to be a complex number, where i<sup>2<\/sup> = -1.<br \/>\nHere \u2018a\u2019 is called the real part, denoted by Re (z) and \u2018b\u2019 is called the imaginary part, denoted by Im (z), of the complex number z = a + ib.<\/p>\n<p>Question 2.<br \/>\nWhen you say that two complex numbers are equal?<br \/>\nAnswer :<br \/>\nTwo complex numbers z<sub>1<\/sub> = a + ib and z<sub>2<\/sub> = c + id are equal if a = c and b &#8211; d.<\/p>\n<p>Question 3.<br \/>\nDefine purely real and purely imaginary numbers.<br \/>\nAnswer :<br \/>\nA complex number z is said to be:<\/p>\n<ul>\n<li>Purely real, if Im(z) = 0,<\/li>\n<li>Purely imaginary, if Re(z) = 0<\/li>\n<\/ul>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 4.<br \/>\nIf 4x + i(3x &#8211; y) = 3 +i (-6), where x and y are real numbers, then find the values of<br \/>\nx and y.<br \/>\nAnswer :<br \/>\nGiven 4x + i(3x &#8211; y) = 3 + i(-6), Equating the real and the imaginary parts,we get<br \/>\n4x = 3 and 3x &#8211; y = -6<br \/>\n\\(\\Rightarrow x=\\frac{3}{4} \\text { and } y=3\\left(\\frac{3}{4}\\right)+6=\\frac{33}{4}\\)<\/p>\n<p>Question 5.<br \/>\nDefine addition of two complex numbers.<br \/>\nAnswer :<br \/>\nLet z<sub>1<\/sub>=a + ib and z<sub>2<\/sub>=c + id be any two complex numbers. Then addition of z<sub>1 <\/sub>and z<sub>2 <\/sub>defined as z<sub>1<\/sub> + z<sub>2<\/sub> = (a + c) + i(b + d) which is again a complex number.<br \/>\nProperties of addition of complex numbers:<\/p>\n<p>(i) Closure property: The sum of two complex numbers is always a complex number, i.e., addition is closed in C set of complex numbers.<\/p>\n<p>(ii) Commutative property: For any two complex numbers z<sub>1<\/sub> and z<sub>2<\/sub><br \/>\nwe have z<sub>1<\/sub> + z<sub>2<\/sub> = z<sub>2<\/sub> + z<sub>1<\/sub><\/p>\n<p>(iii) Associative properly: For any complex numbers z<sub>1<\/sub>,z<sub>2 <\/sub>and Z<sub>3<\/sub>, we have<br \/>\n\\( z_{1}+\\left(z_{2}+z_{3}\\right)=\\left(z_{1}+z_{2}\\right)+z_{3}\\)<\/p>\n<p>(iv) Existence of additive identity: For any complex number z, we have z + 0 = 0 +-z = z<br \/>\nThus, 0 is the additive identity for complex numbers.<\/p>\n<p>(v) Existence of additive inverse: Every complex number z-a + ib has -z = (-a) + i(-b) as its additive inverse, as z + (-z) = (-z) + z = 0.<\/p>\n<p>Question 6.<br \/>\nDefine difference of two complex numbers.<br \/>\nAnswer :<br \/>\nLet z<sub>1<\/sub> and z<sub>2<\/sub> be any two complex numbers. The different z<sub>1<\/sub> &#8211; z<sub>2<\/sub> is defined as follows:<br \/>\nz<sub>1 &#8211; <\/sub>z<sub>2 <\/sub>=z<sub>1 <\/sub>+(-z<sub>2<\/sub>)<\/p>\n<p>Question 7.<br \/>\nDefine multiplication numbers.<br \/>\nAnswer :<br \/>\nLet z<sub>1<\/sub>=a + ib and z<sub>2<\/sub> = c + id be any two complex numbers. Then, the product z<sub>1<\/sub>z<sub>2<\/sub> is defined as follows:<br \/>\nz<sub>1<\/sub>z<sub>2<\/sub> = (ac &#8211; bd) + i(ad + bc)<\/p>\n<p>Properties of product of complex numbers:<\/p>\n<ul>\n<li>Closure property: The product of two complex numbers is always a complex number.<\/li>\n<li>Commutative law: For any two complex numbers z<sub>1 <\/sub>and z<sub>2 <\/sub>we have z<sub>1<\/sub> z<sub>2 <\/sub>= z<sub>2<\/sub>z<sub>1<\/sub><\/li>\n<li>Associative law: For any three complex numbers z<sub>1<\/sub> z<sub>2<\/sub> and z<sub>3<\/sub> we have (z<sub>1<\/sub>z<sub>2<\/sub>)z<sub>3<\/sub>=z<sub>1<\/sub>(z<sub>2<\/sub>z<sub>3<\/sub>).<\/li>\n<li>Existence of multiplicative identity: For every complex number z, we have z . 1 = 1. z = z. Thus, 1 is the multiplicative identity.<\/li>\n<li>Existence of multiplicative inverse: Let z = a + ib. Then<br \/>\n\\(z^{-1}=\\frac{1}{z}=\\frac{1}{a+i b}=\\frac{1}{a+i b} \\cdot \\frac{a-i b}{a-i b}=\\frac{a-i b}{a^{2}+b^{2}} \\)<br \/>\nClearly<br \/>\n\\(z \\cdot \\frac{1}{z}=\\frac{1}{z} \\cdot z=1\\)<br \/>\nThus, every z = a + ib has its multiplicative inverse, given<br \/>\n\\( z^{-1}=\\frac{1}{z}=\\frac{a}{a^{2}+b^{2}}-\\frac{a}{a^{2}+b^{2}} i \\quad(z \\neq 0)\\)<\/li>\n<\/ul>\n<p>Question 8.<br \/>\nDefine division of two complex numbers.<br \/>\nAnswer :<br \/>\nLet z<sub>1 <\/sub>and z<sub>2 <\/sub>be two complex numbers and z<sub>2 <\/sub>\u22600.Then \\( \\frac{z_{1}}{z_{2}} \\) is defined by\u00a0 \\( \\frac{z_{1}}{z_{2}}=z_{1}\\left(\\frac{1}{z_{2}}\\right) \\)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-69997\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-1.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 1\" width=\"318\" height=\"239\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-1.png 318w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-1-300x225.png 300w\" sizes=\"(max-width: 318px) 100vw, 318px\" \/><\/p>\n<p>Question 9.<br \/>\nProve the following<br \/>\n(i) \\(\\left(z_{1}+z_{2}\\right)^{2}=z_{1}^{2}+2 z_{1} z_{2}+z_{2}^{2}\\)<br \/>\n(ii) \\(\\left(z_{1}-z_{2}\\right)^{2}=z_{1}^{2}-2 z_{1} z_{2}+z_{2}^{2}\\)<br \/>\n(iii) \\(\\left(z_{1}+z_{2}\\right)^{3}=z_{1}^{3}+3 z_{1}^{2} z_{2}+3 z_{1} z_{2}^{2}+z_{2}^{3}\\)<br \/>\n(iv) \\(\\left(z_{1}-z_{2}\\right)^{3}=z_{1}^{3}-3 z_{1}^{2} z_{2}+3 z_{1} z_{2}^{2}-z_{2}^{3}\\)<br \/>\n(v) \\(z_{1}^{2}-z_{2}^{2}=\\left(z_{1}+z_{2}\\right)\\left(z_{1}-z_{2}\\right) \\)<br \/>\nAnswer:<br \/>\n(i) We have, (z<sub>1<\/sub> + z<sub>2<\/sub>)<sup>2<\/sup> = (z, + z<sub>2<\/sub>)(z<sub>1<\/sub> + z<sub>2<\/sub>)<br \/>\n= (z<sub>1<\/sub> + z<sub>2<\/sub>)z<sub>1<\/sub> + (z<sub>1<\/sub> + z<sub>2<\/sub>)z<sub>2<\/sub> (by distributive law)<br \/>\n= z<sup>2<\/sup><sub>1<\/sub> + z<sub>2<\/sub>z<sub>1<\/sub> +z<sub>1<\/sub>z<sub>2<\/sub> + z<sub>2<\/sub> (by distributive law)<br \/>\n\\( =z_{1}^{2}+2 z_{1} z_{2}+z_{2}^{2} \\quad \\ z_{1} z_{2}=z_{2} z_{1}\\)<br \/>\nRemaining try yourself.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 10.<br \/>\nExpress the following in the form of a+ib<br \/>\n(i) \\( (-5 i)\\left(\\frac{1}{8} i\\right)\\)<br \/>\n(ii) \\((5 i)\\left(-\\frac{3}{8} i\\right)\\)<br \/>\n(iii) \\((2+i 3)+(-6+i 5)\\)<br \/>\n(iv) \\((6+3 i)-(2-i)\\)<br \/>\n(v) \\((2-i)-(6+3 i)\\)<br \/>\n(vi) \\((3+5 i)(2+6 i)\\)<br \/>\n(vii) \\(3(7+7 i)+i(7+7 i)\\)<br \/>\n(viii) \\((1-i)-(-1+6 i)\\)<br \/>\n(ix)\\(\\left(\\frac{1}{5}+\\frac{2}{5} i\\right)-\\left(4+\\frac{5}{2} i\\right)\\)<br \/>\n(x)\\(\\left[\\left(\\frac{1}{3}+i \\frac{7}{3}\\right)+\\left(4+i \\frac{1}{3}\\right)\\right]-\\left(-\\frac{4}{3}+i\\right) \\)<br \/>\n(xi) \\(i^{9}+i^{19}\\)<br \/>\n(xii)\\( i^{-39}\\)<br \/>\n(xiii)\\((-i)(2 i)\\left(-\\frac{1}{8} i\\right)^{3}\\)<br \/>\n(xiv) \\((5-3 i)^{3}\\)<br \/>\n(xv) \\(\\left(\\frac{1}{3}+3 i\\right)^{3}\\)<br \/>\n(xvi) \\(\\left(-2-\\frac{1}{3} i\\right)^{3}\\)<br \/>\n(xvii) \\((\\mathbf{1}-i)^{4}\\)<br \/>\n(xviii) \\(\\frac{5+\\sqrt{2} i}{1-\\sqrt{2 i}}\\)<br \/>\n(xix) \\( \\frac{(3+i \\sqrt{5})(3-i \\sqrt{5})}{(\\sqrt{3}+\\sqrt{2} i)-(\\sqrt{3}-\\sqrt{2} i)}\\)<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-69998\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-2.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 2\" width=\"305\" height=\"110\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-2.png 305w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-2-300x108.png 300w\" sizes=\"(max-width: 305px) 100vw, 305px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-69999\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-3.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 3\" width=\"373\" height=\"584\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-3.png 373w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-3-192x300.png 192w\" sizes=\"(max-width: 373px) 100vw, 373px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70000\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-4.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 4\" width=\"364\" height=\"465\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-4.png 364w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-4-235x300.png 235w\" sizes=\"(max-width: 364px) 100vw, 364px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70001\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-5.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 5\" width=\"389\" height=\"606\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-5.png 389w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-5-193x300.png 193w\" sizes=\"(max-width: 389px) 100vw, 389px\" \/><\/p>\n<p>Question 11.<br \/>\nDefine modulus of a complex number.<br \/>\nAnswer :<br \/>\nLet. z = a + ib be a complex number. Then, the modulus of z, denoted by |z| to be the non-negative real number \\(\\sqrt{a^{2}+b^{2}}\\)<\/p>\n<p>Question 12.<br \/>\nDefine conjugate of a complex number.<br \/>\nAnswer :<br \/>\nLet z = a + ib be a complex number. Then, the conjugate of complex number z, denoted \\(\\bar{z} \\) as , is the complex number a &#8211; ib, i.e., i= a &#8211; ib<\/p>\n<p>Question 13.<br \/>\nWrite down the modulus of<br \/>\n(i) i<br \/>\n(ii) \\( 3+\\sqrt{-3}\\)<br \/>\n(iii)\\((-1-i)^{3}\\)<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70002\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-6.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 6\" width=\"328\" height=\"199\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-6.png 328w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-6-300x182.png 300w\" sizes=\"(max-width: 328px) 100vw, 328px\" \/><\/p>\n<p>Question 14.<br \/>\nWrite down the conjugate of<br \/>\n(i) \\((2+\\sqrt{-2})\\)<br \/>\n(ii) \\( i^{3}\\)<br \/>\n(iii) \\((3-4 i)^{2} \\)<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70003\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-7.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 7\" width=\"330\" height=\"178\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-7.png 330w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-7-300x162.png 300w\" sizes=\"(max-width: 330px) 100vw, 330px\" \/><\/p>\n<p>Question 15.<br \/>\nFind the multiplicative inverse of the following complex numbers:<br \/>\n(i) 2-3i<br \/>\n(ii) 4-3i<br \/>\n(iii) \\(\\sqrt{5}+3i\\)<br \/>\n(iv) \\(\\sqrt{5}+3i \\)<br \/>\n(v) \\(-i\\)<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70004\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-8.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 8\" width=\"389\" height=\"500\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-8.png 389w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-8-233x300.png 233w\" sizes=\"(max-width: 389px) 100vw, 389px\" \/><\/p>\n<p>Question 16.<br \/>\nFind the conjugate of\u00a0 \\(\\frac{(3-2 i)(2+3 i)}{(1+2 i)(2-i)}\\)<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70005\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-9.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 9\" width=\"318\" height=\"210\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-9.png 318w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-9-300x198.png 300w\" sizes=\"(max-width: 318px) 100vw, 318px\" \/><\/p>\n<p>Question 17.<br \/>\nEvaluate:<br \/>\n\\( \\left[i^{18}+\\left(\\frac{1}{i}\\right)^{25}\\right]^{3}\\)<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70006\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-10.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 10\" width=\"280\" height=\"103\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 18.<br \/>\nReduce \\( \\left(\\frac{1}{1-4 i}-\\frac{2}{1+i}\\right)\\left(\\frac{3-4 i}{5+i}\\right)\\) to the standard form.<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70007\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-11.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 11\" width=\"327\" height=\"392\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-11.png 327w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-11-250x300.png 250w\" sizes=\"(max-width: 327px) 100vw, 327px\" \/><\/p>\n<p>Question 19.<br \/>\nFind the modulus of<br \/>\n\\(\\frac{1+i}{1-i}-\\frac{1-i}{1+i}\\)<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70008\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-12.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 12\" width=\"390\" height=\"142\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-12.png 390w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-12-300x109.png 300w\" sizes=\"(max-width: 390px) 100vw, 390px\" \/><\/p>\n<p>Question 20.<br \/>\nFind the real numbers x and y if (x &#8211; iy) (3 + 5i) is the conjugate of -6 &#8211; 24i<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70009\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-13.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 13\" width=\"312\" height=\"204\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-13.png 312w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-13-300x196.png 300w\" sizes=\"(max-width: 312px) 100vw, 312px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70010\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-14.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 14\" width=\"336\" height=\"300\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-14.png 336w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-14-300x268.png 300w\" sizes=\"(max-width: 336px) 100vw, 336px\" \/><\/p>\n<p>Question 21.<br \/>\nFor any two complex numbers Z<sub>1<\/sub> and Z<sub>2<\/sub>, prove that<br \/>\nRe(z<sub>1<\/sub>z<sub>2<\/sub>) = Re(z<sub>2<\/sub>) . Re(z<sub>2<\/sub>) &#8211; Im(z<sub>1<\/sub>) . Im(z<sub>2<\/sub>)<br \/>\nAnswer :<br \/>\nz<sub>1<\/sub>=a + ib and z<sub>2<\/sub>=c + id<br \/>\nz<sub>1<\/sub>z<sub>2<\/sub> = {a + ib)(c + id) = (ac-bd) + i(ad + bc)<br \/>\n\u2234 Re(z<sub>1<\/sub>z<sub>2<\/sub>) = ac-bd<br \/>\n= Re(z<sub>1<\/sub>).Re(z<sub>2<\/sub>)-1m(z<sub>1<\/sub>)\\m{z<sub>2<\/sub>)<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 22.<br \/>\n\\(\\text { If } z_{1}=2-i, z_{2}=1+i, \\text { find }\\left|\\frac{z_{1}+z_{2}+1}{z_{1}-z_{2}+1}\\right| \\)<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70011\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-15.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 15\" width=\"320\" height=\"120\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-15.png 320w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-15-300x113.png 300w\" sizes=\"(max-width: 320px) 100vw, 320px\" \/><\/p>\n<p>Question 23.<br \/>\nLet z<sub>1<\/sub> = 2 &#8211; i, z<sub>2<\/sub> = -2 + i .Find<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70012\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-16.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 16\" width=\"248\" height=\"63\" \/><br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70013\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-17.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 17\" width=\"287\" height=\"350\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-17.png 287w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-17-246x300.png 246w\" sizes=\"(max-width: 287px) 100vw, 287px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70014\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-18.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 18\" width=\"280\" height=\"122\" \/><\/p>\n<p>Question 24.<br \/>\nIf \\(x+i y=\\frac{a+i b}{a-i b}, \\text { prove that } x^{2}+y^{2}=1\\)<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70015\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-19.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 19\" width=\"264\" height=\"181\" \/><\/p>\n<p>Question 25.<br \/>\nIf \\(x-i y=\\sqrt{\\frac{a-i b}{c-i d}}\\) Prove that<br \/>\n\\(\\left(x^{2}+y^{2}\\right)^{2}=\\frac{a^{2}+b^{2}}{c^{2}+d^{2}}\\)<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70016\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-20.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 20\" width=\"314\" height=\"285\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-20.png 314w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-20-300x272.png 300w\" sizes=\"(max-width: 314px) 100vw, 314px\" \/><\/p>\n<p>Question 26.<br \/>\nIf \\(a+i b=\\frac{(x+i)^{2}}{2 x^{2}+1}\\), Prove that \\(a^{2}+b^{2}=\\frac{\\left(x^{2}+1\\right)^{2}}{\\left(2 x^{2}+1\\right)^{2}}\\)<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70020\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-21.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 21\" width=\"231\" height=\"191\" \/><\/p>\n<p>Question 27.<br \/>\nIf (x + iy)<sup>3<\/sup> =u + iv, then show that<br \/>\n\\(\\frac{u}{x}+\\frac{v}{y}=4\\left(x^{2}-y^{2}\\right)\\)<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70021\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-22.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 22\" width=\"340\" height=\"266\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-22.png 340w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-22-300x235.png 300w\" sizes=\"(max-width: 340px) 100vw, 340px\" \/><\/p>\n<p>Question 28.<br \/>\nIf (a + ib)(c + id){e + if) (g + ih) = A + iB then show that (a<sup>2<\/sup>+b<sup>2<\/sup>)(c<sup>2<\/sup>+d<sup>2<\/sup>)(e<sup>2<\/sup>+f<sup>2<\/sup>) (g<sup>2<\/sup>+h<sup>2<\/sup>) = A<sup>2<\/sup>+B<sup>2<\/sup><br \/>\nAnswer:<br \/>\nGiven: A + iB = (a + ib)(c + id)(e + if)(g + ih)<br \/>\n\u2234\u00a0 A-iB = (a- ib)(c &#8211; id)(e &#8211; if)(g &#8211; ih)<br \/>\nBut A<sup>2<\/sup> + B<sup>2<\/sup> = (A + iB)(A &#8211; iB)<br \/>\n= (a<sup>2<\/sup> +b<sup>2<\/sup>)(c<sup>2<\/sup>+d<sup>2<\/sup>)(c<sup>2<\/sup>+f<sup>2<\/sup>)(g<sup>2<\/sup>+h<sup>2<\/sup>)<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 29.<br \/>\nIf \u03b1 and \u03b2 are different complex numbers with |\u03b2|=1 then find<br \/>\n\\(\\left|\\frac{\\beta-\\alpha}{1-\\bar{\\alpha} \\beta}\\right|\\)<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70023\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-23.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 23\" width=\"320\" height=\"312\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-23.png 320w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-23-300x293.png 300w\" sizes=\"(max-width: 320px) 100vw, 320px\" \/><\/p>\n<p>Question 30.<br \/>\nIf \\(\\left(\\frac{1+i}{1-i}\\right)^{m}=1 \\)then find the least positive integral value of m.<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70024\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-24.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 24\" width=\"200\" height=\"47\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70026\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-25.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 25\" width=\"246\" height=\"139\" \/><\/p>\n<p>Question 31.<br \/>\nFind the number of non-zero integral solutions of the equation \\(|1-i|^{x}=2^{x}\\)<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70028\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-26.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 26\" width=\"303\" height=\"173\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-26.png 303w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-26-300x171.png 300w\" sizes=\"(max-width: 303px) 100vw, 303px\" \/><\/p>\n<p>Question 32.<br \/>\nFind real \u03b8 such that \\(\\frac{3+2 i \\sin \\theta}{1-2 i \\sin \\theta}\\) is purely real<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70029\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-27.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 27\" width=\"363\" height=\"283\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-27.png 363w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-27-300x234.png 300w\" sizes=\"(max-width: 363px) 100vw, 363px\" \/><\/p>\n<p>Question 33.<br \/>\nDefine the polar form of complex number.<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70031\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-28.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 28\" width=\"157\" height=\"154\" \/><br \/>\nLet the complex number z = x + iy be represented by the point P(x,y) in the complex plane.<br \/>\nLet \\(| X O P=\\theta \\text { and }|O P|=r&gt;0\\)<br \/>\nThen<br \/>\nP(r, \u03b8) are called the polar coordinates of P.<br \/>\nwhere x = rcos\u03b8<br \/>\ny = rsin\u03b8<br \/>\n\u2234 z = r(cos\u03b8+ i sin \u03b8)<br \/>\nThis is called the polar form or trigonometric form or modulus &#8211; amplitude form of z.<\/p>\n<p>Keen Eye;<\/p>\n<ul>\n<li>\\( r=\\sqrt{x^{2}+y^{2}}=1 z 1 \\) is called the modulus of z and 0 is called the argument (or amplitude) of z, written as arg (z) or amp (z).<\/li>\n<li>The value of such that -\u03c0&lt;\u03b8\u2264n is called principal argument of z.<\/li>\n<li>To find \u03b8<\/li>\n<\/ul>\n<p>Case (i):<br \/>\nWhen z is purely real. Then, it lies on the x-axis.<br \/>\n(i) If x &gt; 0, then \u03b8=0<br \/>\n(ii) If x &lt; 0, then \u03b8 = \u03c0<\/p>\n<p>Case (ii):<br \/>\nWhen z is purely imaginary. Then, it lies on the y-axis<br \/>\n(i) \\(\\text { If } y&gt;0, \\text { then } \\theta=\\frac{\\pi}{2}\\)<br \/>\n(ii) \\(\\text { If } y&lt;0, \\text { then } \\theta=-\\frac{\\pi}{2}\\)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70032\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-29.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 29\" width=\"181\" height=\"182\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-29.png 181w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-29-150x150.png 150w\" sizes=\"(max-width: 181px) 100vw, 181px\" \/><br \/>\nCase(iii):<br \/>\nLet tan \u03b1 =|tan \u03b8|<br \/>\n\\(\\text { where } 0&lt;\\alpha&lt;\\frac{\\pi}{2}\\)<\/p>\n<p>(i) \u03b8 = \u03b1, when z lies in I quadrant<br \/>\n(ii) \u03b8 &#8211; \u03c0 &#8211; \u03b1, when z lies in II quadrant<br \/>\n(iii) \u03b8 = \u03b1 &#8211; \u03c0, when z lies in III quadrant<br \/>\n(iv) \u03b8 = -\u03b1, when z lies in IV quadrant<\/p>\n<p>Question 34.<br \/>\nFind the modulus and the arguments of the following:<br \/>\n(i) 1<br \/>\n(ii) -3<br \/>\n(iii) i<br \/>\n(iv) -8i<br \/>\n(v) 1+i<br \/>\n(vi) \\(\\sqrt{3}-i\\)<br \/>\n(vii)\\(-\\sqrt{3}+i\\)<br \/>\n(viii)\\(-1-i \\sqrt{3}\\)<br \/>\n(ix)\\(\\frac{1+2 i}{1-3 i}\\)<br \/>\n(x)\\(\\frac{1+i}{1-i}\\)<br \/>\n(xi) \\(\\frac{1}{1+i}\\)<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70033\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-30.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 30\" width=\"355\" height=\"246\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-30.png 355w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-30-300x208.png 300w\" sizes=\"(max-width: 355px) 100vw, 355px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70035\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-31.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 31\" width=\"320\" height=\"552\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-31.png 320w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-31-174x300.png 174w\" sizes=\"(max-width: 320px) 100vw, 320px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70036\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-32.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 32\" width=\"276\" height=\"559\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-32.png 276w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-32-148x300.png 148w\" sizes=\"(max-width: 276px) 100vw, 276px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70041\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-33.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 33\" width=\"349\" height=\"697\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-33.png 349w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-33-150x300.png 150w\" sizes=\"(max-width: 349px) 100vw, 349px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70043\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-34.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 34\" width=\"289\" height=\"357\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-34.png 289w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-34-243x300.png 243w\" sizes=\"(max-width: 289px) 100vw, 289px\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 35.<br \/>\nConvert the Following complex numbers in the polar form:<br \/>\n(i) 1+i<br \/>\n(ii) 1-i<br \/>\n(iii) -1+i<br \/>\n(iv) -1-i<br \/>\n(v) \\(\\sqrt{3}+i\\)<br \/>\n(vi) \\(\\sqrt{3}-i\\)<br \/>\n(vii) \\(-\\sqrt{3}-i\\)<br \/>\n(viii) \\(-\\sqrt{3}+i\\)<br \/>\n(ix) \\(1+i \\sqrt{3}\\)<br \/>\n(x) \\(1-i \\sqrt{3}\\)<br \/>\n(xi) \\(-1+i \\sqrt{3}\\)<br \/>\n(xii) \\(-1-i \\sqrt{3}\\)<br \/>\n(xiii) -3<br \/>\n(xiv) i<br \/>\n(xv) \\(\\frac{-16}{1+i \\sqrt{3}}\\)<br \/>\n(xvi) \\(\\frac{1+7 i}{(2-i)^{2}}\\)<br \/>\n(xvii) \\(\\frac{1+3 i}{1-2 i}\\)<br \/>\n(xviii)<br \/>\n\\(\\frac{i-1}{\\cos \\frac{\\pi}{3}+i \\sin \\frac{\\pi}{3}}\\)<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70045\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-35.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 35\" width=\"353\" height=\"702\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-35.png 353w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-35-151x300.png 151w\" sizes=\"(max-width: 353px) 100vw, 353px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70046\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-36.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 36\" width=\"331\" height=\"656\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-36.png 331w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-36-151x300.png 151w\" sizes=\"(max-width: 331px) 100vw, 331px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70048\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-37.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 37\" width=\"363\" height=\"442\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-37.png 363w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-37-246x300.png 246w\" sizes=\"(max-width: 363px) 100vw, 363px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70050\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-38.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 38\" width=\"328\" height=\"571\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-38.png 328w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-38-172x300.png 172w\" sizes=\"(max-width: 328px) 100vw, 328px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70051\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-39.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 39\" width=\"323\" height=\"519\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-39.png 323w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-39-187x300.png 187w\" sizes=\"(max-width: 323px) 100vw, 323px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70053\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-40.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 40\" width=\"325\" height=\"577\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-40.png 325w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-40-169x300.png 169w\" sizes=\"(max-width: 325px) 100vw, 325px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70054\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-41.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 41\" width=\"317\" height=\"508\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-41.png 317w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-41-187x300.png 187w\" sizes=\"(max-width: 317px) 100vw, 317px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70056\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-42.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 42\" width=\"314\" height=\"478\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-42.png 314w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-42-197x300.png 197w\" sizes=\"(max-width: 314px) 100vw, 314px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70057\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-43.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 43\" width=\"377\" height=\"598\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-43.png 377w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-43-189x300.png 189w\" sizes=\"(max-width: 377px) 100vw, 377px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 36.<br \/>\nKeen Eye:<\/p>\n<ul>\n<li>A polynomial equation has at least one root.<\/li>\n<li>A polynomial equation of degree V has n roots. Solve each of the following equations:<\/li>\n<\/ul>\n<p>(i) x<sup>2<\/sup> + 2 = 0<br \/>\n(ii) x<sup>2<\/sup> + 3 = 0<br \/>\n(iii) x<sup>2<\/sup> + x +1 = 0<br \/>\n(iv) 2x<sup>2<\/sup> + x +1 = 0<br \/>\n(v) x<sup>2<\/sup> + 3x + 9 = 0<br \/>\n(vi) -x<sup>2<\/sup> + x &#8211; 2 = 0<br \/>\n(vii) x<sup>2<\/sup> + 3x + 5 = 0<br \/>\n(viii) x<sup>2<\/sup> &#8211; x + 2 = 0<br \/>\n(ix)\\(3 x^{2}-4 x+\\frac{20}{3}=0\\)<br \/>\n(x) \\(x^{2}-2 x+\\frac{3}{2}=0\\)<br \/>\n(xi) 27x<sup>2<\/sup> -10x + 1 = 0<br \/>\n(xii) \\(\\sqrt{2} x^{2}+x+\\sqrt{2}=0\\)<br \/>\n(xiii) \\(\\sqrt{3} x^{2}-\\sqrt{2} x+3 \\sqrt{3}=0\\)<br \/>\n(xiv)\\( x^{2}+x+\\frac{1}{\\sqrt{2}}=0\\)<br \/>\n(xv) \\(x^{2}+\\frac{x}{\\sqrt{2}}+1=0\\)<br \/>\nAnswer:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70059\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-44.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 44\" width=\"316\" height=\"575\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-44.png 316w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-44-165x300.png 165w\" sizes=\"(max-width: 316px) 100vw, 316px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70066\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-45.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 45\" width=\"344\" height=\"468\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-45.png 344w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-45-221x300.png 221w\" sizes=\"(max-width: 344px) 100vw, 344px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70068\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-46.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 46\" width=\"312\" height=\"219\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-46.png 312w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-46-300x211.png 300w\" sizes=\"(max-width: 312px) 100vw, 312px\" \/><br \/>\n(viii) x<sup>2<\/sup> -x+2=0<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70069\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-47.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 47\" width=\"259\" height=\"159\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70178\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-62.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 62\" width=\"102\" height=\"50\" \/><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/live.staticflickr.com\/65535\/49246064682_d500087490_o.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 49\" width=\"325\" height=\"259\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70073\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-49.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 49\" width=\"357\" height=\"416\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-49.png 357w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-49-257x300.png 257w\" sizes=\"(max-width: 357px) 100vw, 357px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70075\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-50.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 50\" width=\"359\" height=\"507\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-50.png 359w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-50-212x300.png 212w\" sizes=\"(max-width: 359px) 100vw, 359px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70172\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-61.png\" alt=\"\" width=\"201\" height=\"87\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70076\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-51.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 51\" width=\"275\" height=\"163\" \/><\/p>\n<p>Question 37.<br \/>\nFind the equation roots of the following:<br \/>\n(i) i<br \/>\n(ii) i<br \/>\n(iii) 1+i<br \/>\n(iv) 1-i<br \/>\n(v) -8-6i<br \/>\n(vi)-15-8i<br \/>\n(vii) -7-24i<br \/>\nAnswer:<br \/>\n(i)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70078\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-52.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 52\" width=\"340\" height=\"677\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-52.png 340w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-52-151x300.png 151w\" sizes=\"(max-width: 340px) 100vw, 340px\" \/><br \/>\n(ii)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70079\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-53.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 53\" width=\"279\" height=\"136\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70081\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-54.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 54\" width=\"309\" height=\"495\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-54.png 309w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-54-187x300.png 187w\" sizes=\"(max-width: 309px) 100vw, 309px\" \/><br \/>\n(iii)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70082\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-55.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 55\" width=\"282\" height=\"598\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-55.png 282w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-55-141x300.png 141w\" sizes=\"(max-width: 282px) 100vw, 282px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70089\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-56.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 56\" width=\"278\" height=\"68\" \/><\/p>\n<p>(iv)<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70091\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-57.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 57\" width=\"290\" height=\"452\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-57.png 290w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-57-192x300.png 192w\" sizes=\"(max-width: 290px) 100vw, 290px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70092\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-58.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 58\" width=\"357\" height=\"565\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-58.png 357w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-58-190x300.png 190w\" sizes=\"(max-width: 357px) 100vw, 357px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70094\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-59.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 59\" width=\"339\" height=\"622\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-59.png 339w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-59-164x300.png 164w\" sizes=\"(max-width: 339px) 100vw, 339px\" \/><br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-70095\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-60.png\" alt=\"1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations 60\" width=\"354\" height=\"452\" srcset=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-60.png 354w, https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/12\/1st-PUC-Maths-Question-Bank-Chapter-5-Complex-Numbers-and-Quadratic-Equations-60-235x300.png 235w\" sizes=\"(max-width: 354px) 100vw, 354px\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Students can Download Maths Chapter 5 Complex Numbers and Quadratic Equations Questions and Answers, Notes Pdf, 1st PUC Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations. Karnataka 1st PUC Maths Question Bank Chapter 5 Complex Numbers and Quadratic Equations Question 1. &hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[81],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/13526"}],"collection":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/comments?post=13526"}],"version-history":[{"count":0,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/13526\/revisions"}],"wp:attachment":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/media?parent=13526"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/categories?post=13526"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/tags?post=13526"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}