{"id":39631,"date":"2023-04-07T13:03:10","date_gmt":"2023-04-07T07:33:10","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=39631"},"modified":"2024-05-01T12:27:19","modified_gmt":"2024-05-01T06:57:19","slug":"october-2022-tamil-nadu-polytechnic-board-exam-engineering-mathematics-ii40022question-paper-with-solutions","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=39631","title":{"rendered":"October-2022 TAMIL NADU POLYTECHNIC BOARD EXAM ENGINEERING MATHEMATICS \u2013 II(40022)QUESTION PAPER WITH SOLUTIONS"},"content":{"rendered":"\n<ol class=\"wp-block-list\">\n<li>Answer all questions in <strong>PART- A. <\/strong>Each question carries one mark.&nbsp;<\/li>\n\n\n\n<li>Answer any ten questions in <strong>PART- B. <\/strong>Each question carries two marks.<\/li>\n\n\n\n<li>Answer all questions by selecting either A or B. Each question carries fifteen marks. (7 + 8)<br>Clarks Table and programmable calculators are not permitted.<\/li>\n<\/ol>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\underline{PART\\ -\\ A}\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ \\color{green}{Write\\ down\\ the\\ formula\\ for\\ center\\ and\\ radius\\ of\\ the\\ circle}\\ \\hspace{7cm}\\]\\[\\color{green}{x^2\\ +\\ y^2\\ +\\ 2\\ g\\ x\\ +\\ 2\\ f\\ y\\ +\\ c\\ =\\ 0}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/05\/10\/1-2-n-analytical-geometry-exercise-problems-with-solutions\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/05\/10\/1-2-n-analytical-geometry-exercise-problems-with-solutions\/\">Refer Problem No. 3 under Analytical Geometry-II (Exercise Problems with Solutions)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ \\color{green}{Show\\ that\\ \\overrightarrow{i}\\ -\\  2\\overrightarrow{j}\\ -\\ 4\\overrightarrow{k}\\ and\\ 3\\overrightarrow{i}\\ -\\ 6\\overrightarrow{j}\\ -\\ 12\\overrightarrow{k}\\  are\\ parallel}.\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/03\/01\/2-2-scalar-product-of-vectors\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/03\/01\/2-2-scalar-product-of-vectors\/\">Refer Example No. 15 under Product of Vectors<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/EKWTwXmfPrM\" title=\"Product of Vectors - Part - 15\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ \\color{green}{What\\ is\\ the\\ value\\ of\\ \\int\\ \\frac{f^! (x)}{f(x)} dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p>Soln:<a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/12\/4-2-integration-by-substitution\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/12\/4-2-integration-by-substitution\/\"> Refer Example No. 4 METHODS OF INTEGRATION &#8211; INTEGRATION BY SUBSTITUTION<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/QDXb5sT9XBk\" title=\"Integration - Substitution - Part - 4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4.\\ \\color{green}{Evaluate: \\int_1^2 \\frac{dx}{x}}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/21\/definite-integrals\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/21\/definite-integrals\/\">Refer Example No. 5 DEFINITE INTEGRALS<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/kEdkXGdvc6s\" title=\"Definite Integrals - Part - 5\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5.\\ \\color{green}{Find\\ the\\ Area\\ bounded\\ by\\ the\\ curve\\ y\\ =\\ f(x)}\\ \\hspace{12cm}\\]\\[\\color{green} {the\\ X-axis\\ and\\ ordinates\\ x\\ =\\ a\\ and\\ x\\ =\\ b}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2022\/05\/15\/area-and-volume\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2022\/05\/15\/area-and-volume\/\">Refer Example No. 1  (AREA AND VOLUME)<\/a><\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"autorelaxed\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"6771622407\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\underline{PART\\ -\\ B}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[6.\\ \\color {green}{Find\\ the\\ equation\\ of\\ the\\ circle\\ with\\ centre\\ (0, -3)\\ and\\ radius\\  2\\ units.}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/02\/24\/1-2-analytical-geometry-ii\/\"><\/a><a href=\"https:\/\/yanamtakshashila.com\/2021\/05\/10\/1-2-n-analytical-geometry-exercise-problems-with-solutions\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/05\/10\/1-2-n-analytical-geometry-exercise-problems-with-solutions\/\">Refer Problem No. 2 under Analytical Geometry-II (Exercise Problems with Solutions)<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/Gp61asMPrF4\" title=\"Analytical Geometry (Exercise) - Part - 2\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[7.\\ \\color {green}{Find\\ the\\ equation\\ of\\ the\\ circle\\ concentric\\ with\\ the\\ circle\\ x^2\\ +\\ y^2\\ -\\ 4\\ x\\ -\\ 6\\ y\\ -\\ 9\\ =\\ 0}\\ \\hspace{7cm}\\]\\[\\color {green}{and\\ passing\\ through\\ the\\ point\\ (-4 ,-5)}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/02\/24\/1-2-analytical-geometry-ii\/\"><\/a><a href=\"https:\/\/yanamtakshashila.com\/2021\/05\/10\/1-2-n-analytical-geometry-exercise-problems-with-solutions\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/05\/10\/1-2-n-analytical-geometry-exercise-problems-with-solutions\/\">Refer Problem No. 10 under Analytical Geometry-II (Exercise Problems with Solutions)<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/qer34XSYxaE\" title=\"Analytical Geometry (Exercise) - Part - 10\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[8.\\ \\color{green}{What\\ is\\ the\\ condition\\ for\\ the\\ conic\\ a\\ x^2\\ +\\ 2\\ h\\ x\\ y\\ +\\ b\\ y^2\\ +\\ 2\\ g\\ x\\ +\\ 2\\ f\\ y\\ +\\ c\\ =\\ 0}\\ \\hspace{7cm}\\]\\[\\color {green} {represents\\ a\\ pair\\ of\\ straight\\ lines}\\  \\hspace{5cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/05\/12\/1-3-n-conics-exercise-problems-with-solutions\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/05\/12\/1-3-n-conics-exercise-problems-with-solutions\/\">Refer Problem No. 6 under Conics (Exercise Problems with Solutions)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[9.\\ \\color{green}{If\\ the\\ vectors\\ 2\\overrightarrow{i}\\ -\\  3\\overrightarrow{j}\\ and\\ 6\\overrightarrow{i}\\ -\\  m\\overrightarrow{j}\\ are\\ collinear,}\\ \\hspace{10cm}\\]\\[\\color {green} {find\\ the\\ value\\ of\\ m}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/02\/27\/2-1-vector-introduction\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/02\/27\/2-1-vector-introduction\/\">Refer Example No. 5 under Vector Introduction<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/sWwUuCCFTnU\" title=\"Vector Introduction - Part - 5\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[10.\\ \\color{green}{If\\ |\\overrightarrow{a}| = 2,\\ |\\overrightarrow{b}|= 7\\ and\\ |\\overrightarrow{a} \u00d7 \\overrightarrow{b}|=7,\\ find\\ the\\ angle\\ between\\ \\overrightarrow{a}\\ and\\ \\overrightarrow{b}}.\\ \\hspace {3cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/03\/01\/2-2-scalar-product-of-vectors\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/03\/01\/2-2-scalar-product-of-vectors\/\">Refer Example No. 18 under Product of Vectors<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/Uhw5Qp4T7wY\" title=\"Product of Vectors - Part - 18\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[11.\\ \\color{green}{Find\\ the\\ work\\ done\\ by\\ the\\ force\\  \\overrightarrow{i}\\ -\\  7\\overrightarrow{j}\\ &#8211; 2\\overrightarrow{k},}\\ \\hspace{8cm}\\]\\[\\color{green}{when\\ the\\ displacement\\ is\\  3\\overrightarrow{i}\\ -\\ 5\\overrightarrow{j}\\  -\\ 4\\overrightarrow{k}}\\ \\hspace{7cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/05\/16\/n-2-2-product-of-two-vectors-exercise-problems-with-solutions\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/05\/16\/n-2-2-product-of-two-vectors-exercise-problems-with-solutions\/\">Refer Problem No. 12 under Product of two  Vectors (Exercise Problems with solutions)<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/cg_80ccLZwY\" title=\"Product of vectors (Exercise) - Part - 12\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[12.\\ \\color {green}{Evaluate: \\int cos^2 x \\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/09\/4-1-integration-decomposition-method\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/09\/4-1-integration-decomposition-method\/\">Refer Example No. 12  INTEGRATION DECOMPOSITON METHOD (TEXT)<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/WJgjtziGMlo\" title=\"Integration - decomposition - method - Part - 12\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[13.\\ \\color {green}{Evaluate: \\int\\ cot\\ x\\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p>Soln:<a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/12\/4-2-integration-by-substitution\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/12\/4-2-integration-by-substitution\/\"> Refer Example No. 5 METHODS OF INTEGRATION &#8211; INTEGRATION BY SUBSTITUTION<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/-RDw7rayq5I\" title=\"Integration - Substitution - Part - 5\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[14.\\ \\color{green}{Evaluate:\\ \\int \\frac{dx}{49\\ +\\ x^2}}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/16\/4-3-standard-integrals\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/16\/4-3-standard-integrals\/\">Refer Example No. 2 (STANDARD INTEGRALS)<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/PZS5D6w2IqA\" title=\"Integration - Standard Integrals - Part - 2\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[15.\\ \\color{green}{Evaluate:\\ \\int\\ x\\ log\\ x\\ dx}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/18\/5-1-integration-by-parts\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/18\/5-1-integration-by-parts\/\">Refer Example No. 8 INTEGRATION BY PARTS<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/AlQOEMfkMqk\" title=\"Integration By Parts - Part - 8\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[16.\\ \\color{green}{\\int x^2\\ e^x\\ dx}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2022\/05\/02\/bernoullis-formulaexcercise-problems-with-solutions\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2022\/05\/02\/bernoullis-formulaexcercise-problems-with-solutions\/\">Refer Problem No. 2   (BERNOULLI&#8217;S FORMULA &#8211; Excercise Problems with solutions)<\/a> <\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/-sOGL4EsFms\" title=\"Bernoulli's Formula (Exercise Problems) - Part - 2\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[17.\\ \\color {green}{Evaluate: \\int_0^1 \\frac {dx}{{\\sqrt{1 &#8211; x^2}}}}\\ \\hspace{18cm}\\] <\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/21\/definite-integrals\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/21\/definite-integrals\/\">Refer Example No. 3  DEFINITE INTEGRALS<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/1cWsk7TSR_0\" title=\"Definite Integrals - Part - 3\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[18.\\ \\color {green}{Find\\ the\\ Area\\ bounded\\ by\\ the\\ curve\\ y\\ =\\ x^2\\ +\\ x}\\ \\hspace{12cm}\\]\\[\\color {green} {the\\ X-axis\\ between\\ x\\ =\\ 0\\ and\\ x\\ =\\ 4}\\ \\hspace{5cm}\\] <\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2022\/05\/15\/area-and-volume\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2022\/05\/15\/area-and-volume\/\">Refer Example No. 3  (AREA AND VOLUME)<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/laSgzXltdn4\" title=\"Area and Volume - Part - 3\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[19.\\ \\color{green}{Find\\ the\\ Volume\\ of\\ the\\ solid\\ formed\\ when\\ the\\ area\\ bounded\\ by\\ the\\ curve\\ y^2\\ =\\ 4\\  x}\\ \\hspace{8cm}\\]\\[\\color{green}{between\\ x\\ =\\ 0\\ and\\ x\\ =\\ 1\\ is\\ rotated\\ about\\ the\\ X\\ -\\ axis.}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2022\/05\/15\/area-and-volume\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2022\/05\/15\/area-and-volume\/\">Refer Example No. 4  (AREA AND VOLUME)<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/oEJrdP5-WU4\" title=\"Area and Volume (Application of Integration) - Part - 4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[20.\\ \\color{green}{Solve\\ :\\ \\frac{y^2\\ dy}{1\\ +\\ y^3}\\   =\\ \\frac{dx}{x}}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2022\/05\/21\/first-order-differential-equation\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2022\/05\/21\/first-order-differential-equation\/\">Refer Example No. 6 (FIRST ORDER DIFFERENTIAL EQUATION)<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/7WOAM24TFvo\" title=\"First Order Differential Equation - Part - 6\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Leader board 1 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8769628924\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\underline{PART\\ -\\ C}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[21.\\ A)\\ i.\\ \\color{green}{Find\\ the\\ equation\\ of\\ the\\ circle}\\ \\hspace{10cm}\\]\\[\\color{green}{having\\ centre\\ (2,-1)\\ and\\ passing\\ through\\ the\\ point\\ (8,7)}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/02\/24\/1-2-analytical-geometry-ii\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/02\/24\/1-2-analytical-geometry-ii\/\">Refer Example No. 8 under Analytical Geometry-II<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/3WUxXxmFh6I\" title=\"Analytical Geometry - Part - 8\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Find\\ the\\ equation\\ of\\ the\\ circle\\ passing\\ through\\ the\\ origin\\ and\\ cuts\\ orthogonally}\\ \\hspace{3cm}\\]\\[\\color{green}{each\\ of\\ the\\ circles\\ x^2\\ +\\ y^2\\ -\\ 8\\ y\\ +\\ 12\\ =\\ 0\\ and\\ x^2\\ +\\ y^2\\ -\\ 4\\ x\\ -\\ 6\\ y\\ -\\ 3\\ =\\ 0}\\  \\hspace{5cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/02\/24\/1-2-analytical-geometry-ii\/\"><\/a><a href=\"https:\/\/yanamtakshashila.com\/2021\/05\/10\/1-2-n-analytical-geometry-exercise-problems-with-solutions\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/05\/10\/1-2-n-analytical-geometry-exercise-problems-with-solutions\/\">Refer Problem No. 14 under Analytical Geometry-II (Exercise Problems with Solutions)<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/3WUxXxmFh6I\" title=\"Analytical Geometry - Part - 8\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(OR)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{0.5cm}\\ B)\\ i.\\ \\color{green}{Find\\ the\\ equation\\ of\\ the\\ Ellipse\\ with\\ focus\\ (-1,\\ -3)\\ and\\ directrix\\ x\\ -\\ 2y\\ =\\ 0\\ and\\ e\\ =\\ \\frac{4}{5}}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/05\/12\/1-3-n-conics-exercise-problems-with-solutions\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/05\/12\/1-3-n-conics-exercise-problems-with-solutions\/\">Refer Problem No. 7 under Conics (Exercise Problems with Solutions)<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/2B2l5nTNiEo?list=PLQIom4Rz29vxOLGeuJJ4oSrruBu7DMOPS\" title=\"CONICS (Exercise) - PART - 7\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Find\\ the\\ value\\ of\\ \u03b1\\ if\\ \\ \u03b1\\ x^2\\ -\\ 10\\ x\\ y\\ +\n \\ 5\\ x\\ +\\ 12\\ y^2\\ -\\ 16y\\ -\\ 3\\ =\\ 0}\\ \\hspace{7cm}\\]\\[\\color {green} {represents\\ a\\ pair\\ of\\ straight\\ lines}\\  \\hspace{5cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/02\/26\/1-3-conics\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/02\/26\/1-3-conics\/\">Refer Example No. 9 under Conics<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/go_dfwkv1ig\" title=\"CONICS - PART - 9\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[22.\\ A)\\ i.\\ \\color{green}{Prove\\ that\\ the\\ points}\\ \\hspace{15cm}\\]\\[\\color{green}{2\\overrightarrow{i}\\ + 3\\overrightarrow{j}+ 4\\overrightarrow{k},  3\\overrightarrow{i}\\ + 4\\overrightarrow{j}\\ + 2\\overrightarrow{k} and\\ 4\\overrightarrow{i}\\ +\\ 2 \\overrightarrow{j}+ 3\\overrightarrow{k}\\  form\\  an\\ equilateral\\  triangle}\\]<\/div>\n\n\n\n<p><a href=\"https:\/\/yanamtakshashila.com\/2021\/05\/12\/n-2-1-vector-introduction-exercise-problems-with-solutions\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/05\/12\/n-2-1-vector-introduction-exercise-problems-with-solutions\/\">Soln: Refer Problem No. 8 under Vector Introduction (Exercise Problems with Solutions)<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/na_cn2SxPYI\" title=\"Vector Introduction (Exercise) - Part - 08\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Show\\ that\\ the\\ points\\ whose\\ position\\ vectors}\\ \\hspace{15cm}\\]\\[\\color{green}{\\overrightarrow{i}\\ -\\  2\\overrightarrow{j}\\ -\\  \\overrightarrow{k},\\  2\\overrightarrow{i}\\ +\\ 3\\overrightarrow{j}\\ +\\  3\\overrightarrow{k}\\ and\\  -\\ 7 \\overrightarrow{j}\\ -\\ 5\\overrightarrow{k}\\ are\\ collinear}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/05\/12\/n-2-1-vector-introduction-exercise-problems-with-solutions\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/05\/12\/n-2-1-vector-introduction-exercise-problems-with-solutions\/\">Refer Problem No. 7 under Vector Introduction (Exercise Problems with Solutions)<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/L9korbYjiAo\" title=\"Vector Introduction (Exercise) - Part - 07\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(OR)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{0.5cm}\\ B)\\ i.\\ \\color{green}{Prove\\ that\\ the\\ vectors\\ \\overrightarrow{i}+2\\overrightarrow{j}+ \\overrightarrow{k},\\  \\overrightarrow{i}  + \\overrightarrow{j}- 3\\overrightarrow{k}\\ and\\ 7\\overrightarrow{i}-4\\overrightarrow{j}+\\overrightarrow{k}}\\ \\hspace{15cm}\\]\\[\\color{green}{are\\ perpendicular\\ to\\ each\\ other.}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/05\/16\/n-2-2-product-of-two-vectors-exercise-problems-with-solutions\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/05\/16\/n-2-2-product-of-two-vectors-exercise-problems-with-solutions\/\">Refer Problem No.17 Product of Two vectors Excercise Problems with solutions<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/5m31gltcYek\" title=\"Product of Vectors (Exercise) - Part - 17\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Find\\ the\\ torque\\ of\\ the\\ force\\ 3\\overrightarrow{i}+4\\overrightarrow{j}+5\\overrightarrow{k}\\ acting\\ through\\ the\\ point}\\      \\hspace{10cm}\\]\\[\\color{green}{\\overrightarrow{i}-2\\overrightarrow{j}+3\\overrightarrow{k} about\\ the\\ point\\  4\\overrightarrow{i}- 3\\overrightarrow{j}+\\overrightarrow{k}.}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/05\/16\/n-2-2-product-of-two-vectors-exercise-problems-with-solutions\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/05\/16\/n-2-2-product-of-two-vectors-exercise-problems-with-solutions\/\">Refer Problem No.25 Product of Two vectors Excercise Problems with solutions<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/nTwRHUzi1P8\" title=\"Product of Vectors (Exercise) - 25\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Ad1 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8240817448\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[23.\\ A)\\ i.\\ \\color{green}{Evaluate:\\ \\hspace{2cm}\\ (i)\\ \\int ( tan\\ x\\ +\\ cot\\ x ) ^ 2\\ dx\\ \\hspace{2cm}\\ (ii)\\ \\int \\sqrt{1\\ +\\ sin\\ 2x}\\ dx}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/09\/4-1-integration-decomposition-method\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/09\/4-1-integration-decomposition-method\/\">For (i)  Refer Example No. 17  INTEGRATION DECOMPOSITON METHOD (TEXT)<\/a><\/p>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/O_i59TP8w0c\" title=\"Integration - decomposition - Method - Part 17\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/09\/4-1-integration-decomposition-method\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/09\/4-1-integration-decomposition-method\/\">For (ii) Refer Example No. 10  INTEGRATION DECOMPOSITON METHOD (TEXT)<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/mbYOVjBmu2w\" title=\"Integration - decomposition - method - Part - 10\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Evaluate:\\ \\hspace{2cm}\\ (i)\\ \\int\\ e^{sin^2\\ x}\\ sin\\ 2x\\ dx\\ \\hspace{2cm}\\ (ii)\\ \\int\\ (2x^2\\ -\\ 8 x\\ +\\ 5)^{11}\\ (x\\ -\\ 2) dx}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<p>Soln:<a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/12\/4-2-integration-by-substitution\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/12\/4-2-integration-by-substitution\/\"> For (i) Refer Example No. 14 METHODS OF INTEGRATION &#8211; INTEGRATION BY SUBSTITUTION<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/3oWx2jodyyc\" title=\"Integration - Substitution - Part - 14\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<p>Soln:<a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/12\/4-2-integration-by-substitution\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/12\/4-2-integration-by-substitution\/\"> For (ii) Refer Example No. 3 METHODS OF INTEGRATION &#8211; INTEGRATION BY SUBSTITUTION<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/ks7YJbumyw8\" title=\"Integration - Substitution - Part - 3\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(OR)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{0.5cm}\\ B)\\ i.\\ \\color{green}{Evaluate:\\ \\hspace{2cm}\\ (i)\\ \\int cos^3 x \\ dx\\ \\hspace{2cm}\\ (ii)\\ \\int\\frac{(sin^{-1}\\ x)^4}{\\sqrt{1\\ -\\ x^2}}\\ dx}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/09\/4-1-integration-decomposition-method\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/09\/4-1-integration-decomposition-method\/\">For (i)  Refer Example No. 18  INTEGRATION DECOMPOSITON METHOD (TEXT)<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/FKhThbhP1_I\" title=\"Integration - decomposition - method - Part - 18\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<p>Soln:<a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/12\/4-2-integration-by-substitution\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/12\/4-2-integration-by-substitution\/\"> For (ii) Refer Example No. 17 METHODS OF INTEGRATION &#8211; INTEGRATION BY SUBSTITUTION<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/1yzc_XCiawI?list=PLQIom4Rz29vwqvlNhNHEtjSZXFI8H9BC8\" title=\"Integration - Substitution - Part - 17\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Evaluate:\\ \\hspace{2cm}\\ (i)\\ \\int\\frac{dx}{(2x + 3)^2\\ +\\ 49}\\ \\hspace{2cm}\\ (ii)\\ \\int \\frac{dx}{(x\\ +\\ 1)^2\\ -\\ 9}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2022\/04\/26\/standard-integrals-excercise-problems-with-solutions\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2022\/04\/26\/standard-integrals-excercise-problems-with-solutions\/\">For (i) Refer Problem No. 8  (STANDARD INTEGRALS- Excercise Problems with Solutions)<\/a><\/p>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/NbKQ21JcRX0\" title=\"Standard Integrals (Exercise Problems) - Part - 8\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<p>Son: <a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/16\/4-3-standard-integrals\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/16\/4-3-standard-integrals\/\">For (ii) Refer Example No. 7 (STANDARD INTEGRALS)<\/a><\/p>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/Ef8URt50vN4\" title=\"Integration - Standard Integrals - Part - 7\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[24.\\ A)\\ i.\\ \\color{green}{Evaluate:\\ \\hspace{2cm}\\ (a)\\ \\int x^n\\ log\\ x\\ dx\\ \\hspace{2cm}\\ (b)\\ \\int x^3\\ sin\\ x\\ dx}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/18\/5-1-integration-by-parts\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/18\/5-1-integration-by-parts\/\">For (i) Refer Example No. 10 INTEGRATION BY PARTS<\/a><\/p>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/uziinmdHiWs?list=PLQIom4Rz29vwGCxEmPNzYjFAnLwfikslQ\" title=\"Integration By Parts - Part - 10\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/18\/bernoulllis-formula\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/18\/bernoulllis-formula\/\">For (ii) Refer Example No. 1 (BERNOULLI&#8217;S FORMULA)<\/a><\/p>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/kjAvXO9O8tU\" title=\"Bernoulli's Formula - Part - 1\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Evaluate:\\ \\hspace{2cm}\\ (i)\\ \\int x^2\\ cos\\ 2x\\ dx\\ \\hspace{2cm}\\ (ii)\\ \\int x^3\\ e^{-x}\\ dx}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/18\/bernoulllis-formula\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/18\/bernoulllis-formula\/\">For (i) Refer Example No. 4 (BERNOULLI&#8217;S FORMULA)<\/a><\/p>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/2a_3ipMvWsQ\" title=\"Bernoulli's Formula - Part - 4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/18\/bernoulllis-formula\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/18\/bernoulllis-formula\/\">For (ii) Refer Example No. 9 (BERNOULLI&#8217;S FORMULA)<\/a><\/p>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/Uspi2j2iaro?list=PLQIom4Rz29vwGCxEmPNzYjFAnLwfikslQ\" title=\"Bernoulli's Formula - Part - 9\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(OR)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{0.5cm}\\ B)\\ i.\\ \\color{green}{Evaluate:\\ \\hspace{2cm}\\ (i)\\ \\int\\ x^2\\ sin\\ 4x\\ dx\\ \\hspace{2cm}\\ (ii)\\ \\int x^2\\ e^{5x}\\ dx}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2022\/05\/02\/bernoullis-formulaexcercise-problems-with-solutions\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2022\/05\/02\/bernoullis-formulaexcercise-problems-with-solutions\/\">For (i) Refer Problem No. 4 (i)   (BERNOULLI&#8217;S FORMULA &#8211; Excercise Problems with solutions)<\/a><\/p>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/16uJcdTEcW8\" title=\"Bernoulli's Formula (Exercise Problems) - Part - 4 (i)\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/18\/bernoulllis-formula\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/18\/bernoulllis-formula\/\">For (ii) Refer Example No. 5 (BERNOULLI&#8217;S FORMULA)<\/a><\/p>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/pVezf87Yz2Q\" title=\"Bernoulli's Formula - Part - 5\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Evaluate: \\int_0^\\frac{\\pi}{2} \\frac{sin\\ x}{sin\\ x + cos\\ x}\\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/21\/definite-integrals\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/04\/21\/definite-integrals\/\">Refer Example No. 13  DEFINITE INTEGRALS<\/a><\/p>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/lIt3dBKsUc4\" title=\"Definite Integrals - Part - 13\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[25.\\ A)\\ i.\\ \\color{green}{Find\\ the\\ volume\\ of\\  sphere\\ having\\ radius\\ &#8216;a&#8217;\\ by\\ using\\ integration}\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2022\/05\/15\/area-and-volume\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2022\/05\/15\/area-and-volume\/\">Refer Example No. 6  (AREA AND VOLUME)<\/a><\/p>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/rrf6BsCQF_Y?list=PLQIom4Rz29vwGCxEmPNzYjFAnLwfikslQ\" title=\"Area and Volume (Application of Integration) Part - 6\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Solve\\ :\\ x\\ \\sqrt{1\\ -\\ y^2}\\ dx\\ +\\ y\\ \\sqrt{1\\ -\\ x^2}\\ dy\\ =\\ 0}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2022\/05\/21\/first-order-differential-equation\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2022\/05\/21\/first-order-differential-equation\/\">Refer Example No. 9 (FIRST ORDER DIFFERENTIAL EQUATION)<\/a><\/p>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/eRRTSZum_Qk?list=PLQIom4Rz29vwGCxEmPNzYjFAnLwfikslQ\" title=\"First Order Differential Equation - Part - 9\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(OR)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{0.5cm}\\ B)\\ i.\\ \\color{green}{Solve:\\ (1\\ +\\ e^x)\\ cos\\ y\\ dy\\ +\\ e^x\\ sin\\ y\\ dx\\ =\\ 0}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2022\/05\/21\/first-order-differential-equation\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2022\/05\/21\/first-order-differential-equation\/\">Refer Example No. 11 (FIRST ORDER DIFFERENTIAL EQUATION)<\/a><\/p>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/Qtalp6UPgJQ\" title=\"First Order Differential Equation - Part - 11\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Solve:\\ 2\\ cos\\ x\\ \\frac{dy}{dx}\\ +\\ 4\\ y\\ sin\\ x\\ =\\ sin\\ 2\\ x}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2022\/05\/23\/linear-type-differential-equation\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2022\/05\/23\/linear-type-differential-equation\/\">Refer Example No. 9 (LINEAR TYPE DIFFERENTIAL EQUATION)<\/a><\/p>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/dN31Udujj4U\" title=\"Linear Differential Equation - Part - 9\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Soln: Refer Problem No. 3 under Analytical Geometry-II (Exercise Problems with Solutions) Soln: Refer Example No. 15 under Product of Vectors Soln: Refer Example No. 4 METHODS OF INTEGRATION &#8211; INTEGRATION BY SUBSTITUTION Soln: Refer Example No. 5 DEFINITE INTEGRALS Soln: Refer Example No. 1 (AREA AND VOLUME) Soln: Refer Problem No. 2 under Analytical [&hellip;]<\/p>\n","protected":false},"author":187055548,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"_wpas_customize_per_network":false,"jetpack_post_was_ever_published":false},"categories":[711788462],"tags":[],"class_list":["post-39631","post","type-post","status-publish","format-standard","hentry","category-n-em-ii-board-exams"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - 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