{"id":619118,"date":"2023-03-04T06:27:26","date_gmt":"2023-03-04T06:27:26","guid":{"rendered":"https:\/\/www.indcareer.com\/schools\/?p=619118"},"modified":"2024-05-01T13:14:32","modified_gmt":"2024-05-01T13:14:32","slug":"ncert-exemplar-class-11-maths-chapter-5-complex-numbers-and-quadratic-equations","status":"publish","type":"post","link":"https:\/\/www.indcareer.com\/schools\/ncert-exemplar-class-11-maths-chapter-5-complex-numbers-and-quadratic-equations\/","title":{"rendered":"NCERT Exemplar Class 11 Maths\u00a0Chapter 5: Complex Numbers and Quadratic Equations"},"content":{"rendered":"\n<p>NCERT Exemplar Class 11 Maths&nbsp;Chapter 5: Complex Numbers and Quadratic Equations. NCERT Exemplar Solutions for Class 11 Maths&nbsp;Chapter 5 Complex Numbers and Quadratic Equations prepare students for their Class 11 exams thoroughly. <\/p>\n\n\n\n<p>Maths&nbsp;problems and solutions for the Class 11 pdf are provided here which are similar to the questions being asked in the previous year&#8217;s board. <\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-ncert-exemplar-class-11-maths-nbsp-chapter-5-complex-numbers-and-quadratic-equations\"><strong>NCERT Exemplar Class 11 Maths&nbsp;Chapter 5: Complex Numbers and Quadratic Equations<\/strong><\/h2>\n\n\n\n<p><strong>Class 11: Maths\u00a0Chapter 5 solutions. Complete Class 11 Maths\u00a0Chapter 5 Notes.<\/strong><\/p>\n\n\n\n<p><strong>5.1 Overview<\/strong><br>We know that the square of a real number is always non-negative e.g. &nbsp;4<sup>2&nbsp;<\/sup>= 16 and&nbsp; (\u2013 4)<sup>2<\/sup>&nbsp;= 16. Therefore, square root of 16 is \u00b1 4. What about the square root of a negative number? It is clear that a negative number can not have a real square root. So we need to extend the system of real numbers to a system in which we can find out the square roots of negative numbers. Euler (1707 \u2013 1783) was the first mathematician to introduce the symbol i (iota) for positive square root of \u2013 1 i.e., i =<\/p>\n\n\n\n<p><strong>5.1.1 Imaginary numbers<br><\/strong>Square root of a negative number is called an imaginary number., for example,<strong><\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053378487_5427bc76d7_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>5.1.2 Integral powers of i<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053381327_308a69c997_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>5.1.3 Complex numbers<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053364067_00a4b23f82_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p>a) A number which can be written in the form a + ib, where a, b are real numbers<br>and i = &nbsp;is called a complex number.<br>(b) If z = a + ib is the complex number, then a and b are called real and imaginary parts, respectively, of the complex number and written as Re (z) = a, Im (z) = b.<br>(c) Order relations \u201cgreater than\u201d and \u201cless than\u201d are not defined for complex numbers.<br>(d) If the imaginary part of a complex number is zero, then the complex number is known as purely real number and if real part is zero, then it is called purely imaginary number, for example, 2 is a purely real number because its imaginary part is zero and 3i is a purely imaginary number because its real part<br>is zero.<\/p>\n\n\n\n<p><strong>5.1.4 Algebra of complex numbers<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49052660633_125e1117c7_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>5.1.5 Addition of complex numbers satisfies the following properties<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053392227_c4ceee1ed5_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>5.1.6 Multiplication of complex numbers<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49052672993_afddd0963a_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053185496_f609695019_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053400267_4678d0cbf5_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>5.1.7<\/strong><strong><\/strong><\/p>\n\n\n\n<p><strong>5.1.8 Conjugate of a complex number<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053185436_7f0de482f6_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>5.1.9 Modulus of a complex number<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053198256_67aeb694fb_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49052685828_24212d4450_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>5.1.10 Properties of modulus of a complex number<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053416807_b2d4b7d7e7_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>5.2 Argand Plane<br><\/strong>A complex number z = a + ib can be represented by a unique point P (a, b) in the cartesian plane referred to a pair of rectangular axes. The complex number 0 + 0i represent the origin 0 ( 0, 0). A purely real number a, i.e., (a + 0i) is represented by the point (a, 0) on x \u2013 axis. Therefore, x-axis is called real axis. A purely imaginary number ib, i.e., (0 + ib) is represented by the point (0, b) on y-axis. Therefore, y-axis is called imaginary axis.<br>Similarly, the representation of complex numbers as points in the plane is known as Argand diagram. The plane representing complex numbers as points is called complex plane or Argand plane or Gaussian plane.<strong><\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053422462_1b6dc231e6_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>5.2.1 Polar form of a complex number<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49052698903_d340dcb262_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49052704113_004db0fbd6_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>5.2.2 Solution of a quadratic equation<br><\/strong>The equations ax 2 + bx + c = 0, where a, b and c are numbers (real or complex, a \u2260 0) is called the general quadratic equation in variable x. The values of the variable satisfying the given equation are called roots of the equation.<br>The quadratic equation ax<sup>2<\/sup>&nbsp;+ bx + c = 0 with real coefficients has two roots given by&nbsp; &nbsp; , where D = b<sup>2&nbsp;<\/sup>\u2013 4ac, called the discriminant of the equation.<\/p>\n\n\n\n<p><strong>Notes:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>When D = 0, roots of the quadratic equation are real and equal. When D > 0, roots are real and unequal. Further, if a, b, c \u2208 Q and D is a perfect square, then the roots of the equation are rational and unequal, and if a, b, c \u2208Q and D is not a perfect square, then the roots are irrational and occur in pair.<\/li>\n\n\n\n<li>When D &lt; 0, roots of the quadratic equation are non real (or complex).<br>Let \u03b1, \u03b2 be the roots of the quadratic equation ax<sup>2<\/sup>\u00a0+ bx + c = 0, then sum of the roots (\u03b1 + \u03b2) =<br>\u2212 b\/a and the product of the roots ( \u03b1 . \u03b2) =\u00a0 c\/a<\/li>\n\n\n\n<li>Let S and P be the sum of roots and product of roots, respectively, of a quadratic equation. Then the quadratic equation is given by x<sup>2<\/sup>\u00a0\u2013 Sx + P = 0.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Short Answer Type Questions (Solved Examples)<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053239781_e5e1b4c814_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49052727723_5e50191238_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49052727688_af705df909_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053455102_cdc725dfef_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49052727638_5d3468e45b_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49052727613_e73d8fd55f_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053454972_43e6f3ca47_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053454937_1063e091e7_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Long Answer Type Questions (Solved Examples)<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053256751_835caef3ae_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49052744728_58207e9b7f_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Objective Type Questions (Solved Examples)<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053482242_c1d3ea1ac7_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053266581_cfff8b481c_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053482152_473e3342bf_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053266521_e934a00058_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053482132_552762159f_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053289231_6baaaca955_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49052777878_0f94b9e30f_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053505492_0cc1827226_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49052777823_71eed500e1_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053505477_c5a8e0333d_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49052783628_776a7acddd_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49052783658_25cfe4d116_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49052783613_aa5cec1868_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053511397_b7becd1534_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053511352_af7d1acd95_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053511302_919f1a969d_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Short Answer Type Questions (Exercise)<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/15695585970_8b33161577_o.jpg\" alt=\"Complex Numbers and Quadratic Equations\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Long Answer Type Questions (Exercise)<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/15260626024_edac054dea_o.jpg\" alt=\"Complex Numbers and Quadratic Equations\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Fill in the Blank Type Questions (Exercise)<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/15695377488_fbc29f5ddd_o.jpg\" alt=\"Complex Numbers and Quadratic Equations\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">True or False Type Questions (Exercise)<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/15880907221_e51a3af424_o.jpg\" alt=\"Complex Numbers and Quadratic Equations\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Match Type Questions (Exercise)<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/15260625564_44684e160d_o.jpg\" alt=\"Complex Numbers and Quadratic Equations\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Other Questions (Exercise)<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/15697126547_897e0a05e6_o.jpg\" alt=\"Complex Numbers and Quadratic Equations\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Objective Type Questions (Exercise)<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/15260625524_48b78433c8_o.jpg\" alt=\"Complex Numbers and Quadratic Equations\"\/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Answers<\/h2>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49052798988_eaff966748_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49053526957_3f504002b0_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<div class=\"wp-block-buttons is-layout-flex wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link has-background wp-element-button\" href=\"https:\/\/www.indcareer.com\/schools\/ncert-exemplar-problems-solutions\/\" style=\"background-color:#f06d6d\" target=\"_blank\" rel=\"noreferrer noopener\">NCERT Exemplar Solutions<\/a><\/div>\n\n\n\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link has-background wp-element-button\" href=\"https:\/\/www.indcareer.com\/schools\/ncert-exemplar-class-11-Maths-solutions\/\" style=\"background-color:#f06d6d\" target=\"_blank\" rel=\"noreferrer noopener\">NCERT Class 11 Exemplar Solutions<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>NCERT Exemplar Class 11 Maths&nbsp;Chapter 5: Complex Numbers and Quadratic Equations. NCERT Exemplar Solutions for Class 11 Maths&nbsp;Chapter 5 Complex Numbers and Quadratic Equations prepare students for their Class 11 exams thoroughly. Maths&nbsp;problems and solutions for the Class 11 pdf are provided here which are similar to the questions being asked in the previous year&#8217;s [&hellip;]<\/p>\n","protected":false},"author":302,"featured_media":619125,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"newspack_featured_image_position":"","newspack_post_subtitle":"","newspack_article_summary_title":"Overview:","newspack_article_summary":"","newspack_hide_updated_date":false,"newspack_show_updated_date":false,"footnotes":""},"categories":[1411,919],"tags":[],"boards":[],"class_list":["post-619118","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-book-solutions","category-class-11","entry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v27.0 (Yoast SEO v27.1.1) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>NCERT Exemplar for Class 11, Maths\u00a0Chapter 5 - IndCareer Schools<\/title>\n<meta name=\"description\" content=\"NCERT Exemplar Class 11 Maths\u00a0Chapter 5: Complex Numbers and Quadratic Equations | Browse all Class 11 Maths\u00a0Chapters NCERT Exemplar books - IndCareer Schools\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.indcareer.com\/schools\/ncert-exemplar-class-11-maths-chapter-5-complex-numbers-and-quadratic-equations\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"NCERT Exemplar Class 11 Maths\u00a0Chapter 5: Complex Numbers and Quadratic Equations\" \/>\n<meta property=\"og:description\" content=\"NCERT Exemplar Class 11 Maths&nbsp;Chapter 5: Complex Numbers and Quadratic Equations. 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