{"id":68039,"date":"2023-09-11T15:54:00","date_gmt":"2023-09-11T10:24:00","guid":{"rendered":"https:\/\/www.kopykitab.com\/blog\/?p=68039"},"modified":"2023-11-09T12:05:40","modified_gmt":"2023-11-09T06:35:40","slug":"rd-sharma-solutions-class-11-maths-chapter-23-exercise-23-12","status":"publish","type":"post","link":"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-23-exercise-23-12\/","title":{"rendered":"RD Sharma Class 11 Solutions Chapter 23 Exercise 23.12 (Updated for 2024)"},"content":{"rendered":"\n<p><img class=\"alignnone size-full wp-image-123440\" src=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/01\/RD-Sharma-Class-11-Solutions-Chapter-23-Exercise-23.12.jpg\" alt=\"RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.12\" width=\"1200\" height=\"675\" srcset=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/01\/RD-Sharma-Class-11-Solutions-Chapter-23-Exercise-23.12.jpg 1200w, https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/01\/RD-Sharma-Class-11-Solutions-Chapter-23-Exercise-23.12-768x432.jpg 768w\" sizes=\"(max-width: 1200px) 100vw, 1200px\" \/><\/p>\n<p><strong>RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.12:&nbsp;<\/strong>Practicing textbook questions will assist you in determining your level of preparation and understanding of the concept. From the links provided below, you can quickly get the pdf of <a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-23\/\"><strong>RD Sharma Solutions Class 11 Maths Chapter 23<\/strong><\/a> Exercise 23.12.<\/p>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_47_1 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"ez-toc-toggle-icon-1\"><label for=\"item-69e2eb6713cf8\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;direction:ltr;\"><svg style=\"fill: #000000;color:#000000\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #000000;color:#000000\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input  type=\"checkbox\" id=\"item-69e2eb6713cf8\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-visibility-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-23-exercise-23-12\/#download-rd-sharma-class-11-solutions-chapter-23-exercise-2312-free-pdf\" title=\"Download RD Sharma Class 11 Solutions Chapter 23 Exercise 23.12 Free PDF\">Download RD Sharma Class 11 Solutions Chapter 23 Exercise 23.12 Free PDF<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-23-exercise-23-12\/#access-answers-to-rd-sharma-solutions-class-11-maths-chapter-23-exercise-2312-important-question-with-answers\" title=\"Access answers to RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.12- Important Question with Answers\">Access answers to RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.12- Important Question with Answers<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-23-exercise-23-12\/#faqs-on-rd-sharma-class-11-solutions-chapter-23-exercise-2312\" title=\"FAQs on RD Sharma Class 11 Solutions Chapter 23 Exercise 23.12\">FAQs on RD Sharma Class 11 Solutions Chapter 23 Exercise 23.12<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-23-exercise-23-12\/#where-can-i-download-rd-sharma-class-11-solutions-chapter-23-exercise-2312-free-pdf\" title=\"Where can I download RD Sharma Class 11 Solutions Chapter 23 Exercise 23.12 free PDF?\">Where can I download RD Sharma Class 11 Solutions Chapter 23 Exercise 23.12 free PDF?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-23-exercise-23-12\/#why-is-kopykitab%e2%80%99s-rd-sharma-solutions-class-11-maths-chapter-23-exercise-2312-the-best-study-material\" title=\"Why is Kopykitab\u2019s RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.12 the best study material?\">Why is Kopykitab\u2019s RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.12 the best study material?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-23-exercise-23-12\/#is-rd-sharma-solutions-class-11-maths-chapter-23-exercise-2312-for-free\" title=\"Is RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.12 for free?\">Is RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.12 for free?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"download-rd-sharma-class-11-solutions-chapter-23-exercise-2312-free-pdf\"><\/span>Download RD Sharma Class 11 Solutions Chapter 23 Exercise 23.12 Free PDF<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><a href=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/01\/RD-Sharma-Solutions-Class-11-Maths-Chapter-23-Ex-23.12-1.pdf\">RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.12<\/a><\/p>\n<div id=\"example1\" style=\"text-align: justify;\">&nbsp;<\/div>\n<p style=\"text-align: justify;\"><style>\n.pdfobject-container { height: 800px;}<br \/>\n.pdfobject { border: 1px solid #666; }<br \/>\n<\/style><\/p>\n<p style=\"text-align: justify;\"><script src=\"https:\/\/www.kopykitab.com\/_utility\/js\/pdfobject.min.js\"><\/script><br><script>PDFObject.embed(\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/01\/RD-Sharma-Solutions-Class-11-Maths-Chapter-23-Ex-23.12-1.pdf\", \"#example1\");<\/script><\/p>\n<p>We will cover the notions of lines parallel and perpendicular to a given line in this part, with examples to help students understand the concepts. Students who want to do well in their board exams should refer to <a href=\"https:\/\/www.kopykitab.com\/blog\/cbse-class-11-maths-rd-sharma-solutions\/\"><strong>RD Sharma Class 11 Solutions<\/strong><\/a> on a daily basis to practise the solutions.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"access-answers-to-rd-sharma-solutions-class-11-maths-chapter-23-exercise-2312-important-question-with-answers\"><\/span>Access answers to RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.12- Important Question with Answers<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>1. Find the equation of a line passing through the point (2, 3) and parallel to the line 3x\u2013 4y + 5 = 0.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given:&nbsp;<\/p>\n<p>The equation is parallel to 3x \u2212 4y + 5 = 0 and pass-through (2, 3)<\/p>\n<p>The equation of the line parallel to 3x&nbsp;\u2212&nbsp;4y + 5 = 0 is<\/p>\n<p>3x \u2013 4y +&nbsp;\u03bb&nbsp;= 0,<\/p>\n<p>Where,&nbsp;\u03bb&nbsp;is a constant.<\/p>\n<p>It passes through (2, 3).<\/p>\n<p>Substitute the values in above equation, we get<br><br>3 (2) \u2013 4 (3) + \u03bb = 0<\/p>\n<p>6 \u2013 12 +&nbsp;\u03bb&nbsp;= 0<\/p>\n<p>\u03bb&nbsp;= 6<\/p>\n<p>Now, substitute the value of \u03bb&nbsp;=&nbsp;6 in 3x \u2013 4y +&nbsp;\u03bb&nbsp;= 0, we get<\/p>\n<p>3x&nbsp;\u2212&nbsp;4y + 6<\/p>\n<p>\u2234 The required line is 3x&nbsp;\u2212&nbsp;4y + 6 = 0.<\/p>\n<p><strong>2. Find the equation of a line passing through (3, -2) and perpendicular to the line x \u2013 3y + 5 = 0.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given:&nbsp;<\/p>\n<p>The equation is perpendicular to x \u2013 3y + 5 = 0 and passes through (3,-2)<\/p>\n<p>The equation of the line perpendicular to x&nbsp;\u2212&nbsp;3y + 5 = 0 is<\/p>\n<p>3x + y +&nbsp;\u03bb&nbsp;= 0,<\/p>\n<p>Where,&nbsp;\u03bb&nbsp;is a constant.<\/p>\n<p>It passes through (3,&nbsp;\u2212&nbsp;2).<\/p>\n<p>Substitute the values in above equation, we get<br><br>3 (3) + (-2) + \u03bb&nbsp;= 0<\/p>\n<p>9 \u2013 2 +&nbsp;\u03bb&nbsp;= 0<\/p>\n<p>\u03bb&nbsp;= \u2013 7<\/p>\n<p>Now, substitute the value of \u03bb&nbsp;=&nbsp;\u2212&nbsp;7 in 3x + y +&nbsp;\u03bb&nbsp;= 0, we get<\/p>\n<p>3x + y \u2013 7 = 0<\/p>\n<p>\u2234 The required line is 3x + y \u2013 7 = 0.<\/p>\n<p><strong>3. Find the equation of the perpendicular bisector of the line joining the points (1, 3) and (3, 1).<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given:&nbsp;<\/p>\n<p>A (1, 3) and B (3, 1) be the points joining the perpendicular bisector<\/p>\n<p>Let C be the midpoint of AB.<\/p>\n<p>So, coordinates of C = [(1+3)\/2, (3+1)\/2]&nbsp;<\/p>\n<p>= (2, 2)<br><br>Slope of AB = [(1-3) \/ (3-1)]&nbsp;<\/p>\n<p>= -1<br><br>The slope of the perpendicular bisector of AB = 1<\/p>\n<p>Thus, the equation of the perpendicular bisector of AB is given as,<\/p>\n<p>y \u2013 2 = 1(x \u2013 2)<\/p>\n<p>y = x<\/p>\n<p>x \u2013 y = 0<\/p>\n<p>\u2234 The required equation is y = x.<\/p>\n<p><strong>4. Find the equations of the altitudes of a \u0394ABC whose vertices are A (1, 4), B (-3, 2), and C (-5, -3).<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given:&nbsp;<\/p>\n<p>The vertices of \u2206ABC are A (1, 4), B (\u2212 3, 2), and C (\u2212 5, \u2212 3).<\/p>\n<p>Now let us find the slopes of \u2206ABC.<\/p>\n<p><img class=\"wp-image-123582 aligncenter\" src=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/01\/a1-1.png\" alt=\"a1\" width=\"628\" height=\"294\"><\/p>\n<p>Slope of AB = [(2 \u2013 4) \/ (-3-1)]&nbsp;<\/p>\n<p>= \u00bd<\/p>\n<p>Slope of BC = [(-3 \u2013 2) \/ (-5+3)]&nbsp;<\/p>\n<p>= 5\/2<\/p>\n<p><br>Slope of CA = [(4 + 3) \/ (1 + 5)]&nbsp;<\/p>\n<p>= 7\/6&nbsp;<\/p>\n<p>Thus, we have:<\/p>\n<p>Slope of CF = -2<\/p>\n<p>Slope of AD = -2\/5&nbsp;<\/p>\n<p>Slope of BE = -6\/7&nbsp;<\/p>\n<p>Hence,<\/p>\n<p>Equation of CF is:<\/p>\n<p>y + 3 = -2(x + 5)<\/p>\n<p>y + 3 = -2x \u2013 10<\/p>\n<p>2x + y + 13 = 0<\/p>\n<p>Equation of AD is:<\/p>\n<p>y \u2013 4 = (-2\/5) (x \u2013 1)<\/p>\n<p>5y \u2013 20 = -2x + 2<\/p>\n<p>2x + 5y \u2013 22 = 0<\/p>\n<p><br>Equation of BE is:<\/p>\n<p>y \u2013 2 = (-6\/7) (x + 3)<\/p>\n<p>7y \u2013 14 = -6x \u2013 18<\/p>\n<p>6x + 7y + 4 = 0<\/p>\n<p>\u2234 The required equations are 2x + y + 13 = 0, 2x + 5y \u2013 22 = 0, 6x + 7y + 4 = 0.<\/p>\n<p><strong>5. Find the equation of a line that is perpendicular to the line\u221a3x \u2013 y + 5 = 0 and which cuts off an intercept of 4 units with the negative direction of the y-axis.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given:&nbsp;<\/p>\n<p>The equation is perpendicular to \u221athe 3x \u2013 y + 5 = 0 equation and cuts off an intercept of 4 units with the negative direction of the y-axis.<\/p>\n<p>The line perpendicular to&nbsp;\u221a3x \u2013 y + 5 = 0&nbsp;is x + \u221a3y + \u03bb = 0<\/p>\n<p>It is given that the line&nbsp;x + \u221a3y + \u03bb = 0&nbsp;cuts off an intercept of 4 units with the negative direction of the y-axis.<\/p>\n<p>This means that the line passes through (0,-4).<\/p>\n<p>So,<\/p>\n<p>Let us substitute the values in the equation x + \u221a3y + \u03bb = 0, we get<\/p>\n<p>0 \u2013 \u221a3 (4) + \u03bb = 0&nbsp;<br><br>\u03bb = 4\u221a3<\/p>\n<p>Now, substitute the value of&nbsp;\u03bb back, we get<\/p>\n<p>x + \u221a3y + 4\u221a3 = 0<\/p>\n<p>\u2234 The required equation of line is x + \u221a3y + 4\u221a3 = 0.<\/p>\n<p><strong>Straight lines Ex 23.12 Q6<\/strong><br><img src=\"https:\/\/farm8.staticflickr.com\/7507\/15958800258_d264fbb2ef_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-23\" width=\"411\" height=\"479\"><br><strong>Straight lines Ex 23.12 Q7<\/strong><br><img src=\"https:\/\/farm9.staticflickr.com\/8592\/15526519813_99e5b9742a_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-7\" width=\"470\" height=\"284\"><br><strong>Straight lines Ex 23.12 Q8<\/strong><br><img src=\"https:\/\/farm8.staticflickr.com\/7490\/15958802968_23b4397aec_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-8\" width=\"477\" height=\"437\"><br><strong>Straight lines Ex 23.12 Q9<\/strong><br><img src=\"https:\/\/farm8.staticflickr.com\/7502\/15958796878_7c33ff9490_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-The-Straight-Lines-Ex-23.13-Q-9\" width=\"419\" height=\"301\"><br><strong>Straight lines Ex 23.12 Q10<\/strong><br><img src=\"https:\/\/farm8.staticflickr.com\/7489\/15958941240_4463c7cd37_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-10\" width=\"557\" height=\"332\"><br><strong>Straight lines Ex 23.12 Q11<\/strong><br><img src=\"https:\/\/farm9.staticflickr.com\/8663\/15958940980_f83881a1a9_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-11\" width=\"619\" height=\"338\"><br><strong>Straight lines Ex 23.12 Q12<\/strong><br><img src=\"https:\/\/farm8.staticflickr.com\/7545\/15960165889_c4f110dde8_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-12\" width=\"548\" height=\"431\"><br><strong>Straight lines Ex 23.12 Q13<\/strong><br><img src=\"https:\/\/farm8.staticflickr.com\/7547\/15526518583_3dfc9af084_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-13\" width=\"610\" height=\"474\"><br><strong>Straight lines Ex 23.12 Q14<\/strong><br><img src=\"https:\/\/farm8.staticflickr.com\/7552\/15526518473_b9d25f9b89_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-14\" width=\"613\" height=\"335\"><br><strong>Straight lines Ex 23.12 Q15<\/strong><br><img src=\"https:\/\/farm8.staticflickr.com\/7495\/16120456796_bc53c8e242_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-15\" width=\"571\" height=\"304\"><br><strong>Straight lines Ex 23.12 Q16<\/strong><br><img src=\"https:\/\/farm9.staticflickr.com\/8621\/15523909254_a79cf2a221_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-16\" width=\"514\" height=\"285\"><br><strong>Straight lines Ex 23.12 Q17<\/strong><br><img src=\"https:\/\/farm8.staticflickr.com\/7505\/15526518083_8f94a08f8f_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-17\" width=\"376\" height=\"283\"><br><strong>Straight lines Ex 23.12 Q18<\/strong><br><img src=\"https:\/\/farm8.staticflickr.com\/7552\/15958939760_4b0852beda_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-18\" width=\"578\" height=\"324\"><br><strong>Straight lines Ex 23.12 Q19<\/strong><br><img src=\"https:\/\/farm8.staticflickr.com\/7511\/16145494302_5e1a2c966a_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-19\" width=\"517\" height=\"305\"><br><strong>Straight lines Ex 23.12 Q20<\/strong><br><img src=\"https:\/\/farm8.staticflickr.com\/7572\/15960489947_00a9bd5b5a_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-20\" width=\"438\" height=\"448\"><br><strong>Straight lines Ex 23.12 Q21<\/strong><br><img src=\"https:\/\/farm8.staticflickr.com\/7466\/15958939280_6c12678804_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-21\" width=\"592\" height=\"349\"><br><strong>Straight lines Ex 23.12 Q22<\/strong><br><img src=\"https:\/\/farm9.staticflickr.com\/8609\/15526517453_7b250c4a91_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-22\" width=\"449\" height=\"670\"><br><strong>Straight lines Ex 23.12 Q23<\/strong><br><img src=\"https:\/\/farm8.staticflickr.com\/7583\/15523911064_cea48fb380_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-6\" width=\"338\" height=\"390\"><br><strong>Straight lines Ex 23.12 Q24<\/strong><br><img src=\"https:\/\/farm8.staticflickr.com\/7528\/15523908314_f8976ff8e3_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-24\" width=\"525\" height=\"651\"><br><strong>Straight lines Ex 23.12 Q25<\/strong><br><img src=\"https:\/\/farm8.staticflickr.com\/7512\/15526516523_ed0abcff99_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-25\" width=\"512\" height=\"532\"><br><strong>Straight lines Ex 23.12 Q26<\/strong><br><img src=\"https:\/\/farm9.staticflickr.com\/8639\/15958938180_7b7d6dc519_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-26\" width=\"451\" height=\"661\"><br><img src=\"https:\/\/farm8.staticflickr.com\/7531\/16120454526_b23b815a8c_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-26-i\" width=\"435\" height=\"622\"><br><strong>Straight lines Ex 23.12 Q27<\/strong><br><img src=\"https:\/\/farm9.staticflickr.com\/8621\/16144335561_3600704499_o.png\" alt=\"RD-Sharma-class-11-Solutions-Chapter-23-Straight-Lines-Ex-23.12-Q-27\" width=\"538\" height=\"517\"><\/p>\n<p>We have provided complete details of RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.12. If you have any queries related to <a href=\"https:\/\/www.cbse.gov.in\/\" target=\"_blank\" rel=\"noopener\"><strong>CBSE<\/strong><\/a>&nbsp;Class 11, feel free to ask us in the comment section below.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"faqs-on-rd-sharma-class-11-solutions-chapter-23-exercise-2312\"><\/span>FAQs on RD Sharma Class 11 Solutions Chapter 23 Exercise 23.12<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n<div id=\"rank-math-faq\" class=\"rank-math-block\">\n<div class=\"rank-math-list \">\n<div id=\"faq-question-1630580221187\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"where-can-i-download-rd-sharma-class-11-solutions-chapter-23-exercise-2312-free-pdf\"><\/span>Where can I download RD Sharma Class 11 Solutions Chapter 23 Exercise 23.12 free PDF?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>You can download RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.12 free PDF from the above article.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1630580441644\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"why-is-kopykitab%e2%80%99s-rd-sharma-solutions-class-11-maths-chapter-23-exercise-2312-the-best-study-material\"><\/span>Why is Kopykitab\u2019s RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.12 the best study material?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>The RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.12 available on Kopykitab\u2019s website has been created by highly qualified experts to assist students in achieving high scores on the board exam. The solutions are well-organized and logical, giving pupils a clear picture of the most important questions.\u00a0<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1630580650446\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"is-rd-sharma-solutions-class-11-maths-chapter-23-exercise-2312-for-free\"><\/span>Is RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.12 for free?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>Yes, You can get RD Sharma Solutions Class 12 Maths Chapter 23 Exercise 23.12 for free.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.12:&nbsp;Practicing textbook questions will assist you in determining your level of preparation and understanding of the concept. From the links provided below, you can quickly get the pdf of RD Sharma Solutions Class 11 Maths Chapter 23 Exercise 23.12. Download RD Sharma Class 11 Solutions Chapter &#8230; <a title=\"RD Sharma Class 11 Solutions Chapter 23 Exercise 23.12 (Updated for 2024)\" class=\"read-more\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-23-exercise-23-12\/\" aria-label=\"More on RD Sharma Class 11 Solutions Chapter 23 Exercise 23.12 (Updated for 2024)\">Read more<\/a><\/p>\n","protected":false},"author":238,"featured_media":123440,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"fifu_image_url":"","fifu_image_alt":""},"categories":[73411,2985,73410],"tags":[3428,73334,73564,4388],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/68039"}],"collection":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/users\/238"}],"replies":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/comments?post=68039"}],"version-history":[{"count":5,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/68039\/revisions"}],"predecessor-version":[{"id":504843,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/68039\/revisions\/504843"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media\/123440"}],"wp:attachment":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media?parent=68039"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/categories?post=68039"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/tags?post=68039"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}