{"id":40204,"date":"2023-04-29T10:02:03","date_gmt":"2023-04-29T04:32:03","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=40204"},"modified":"2024-05-01T12:27:56","modified_gmt":"2024-05-01T06:57:56","slug":"april-2023-tamil-nadu-polytechnic-board-exam-engineering-mathematics-ii40022question-paper-with-solutions","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=40204","title":{"rendered":"APRIL-2023 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}{Find\\ the\\ equation\\ of\\ the\\ circle\\ with\\ centre\\ (0, 0)\\ and\\ radius\\  2\\ units?}\\ \\hspace{15cm}\\]<\/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. 2 under Analytical Geometry-II (Exercise Problems with Solutions)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ \\color{green}{Write\\ the\\ condition\\ for\\ two\\ vectors\\ to\\ be\\ perpendicular}\\ \\hspace{18cm}\\]<\/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. 3 under Scalar Product of  Vectors (Product of Vectors)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ \\color{green}{Evaluate:\\ \\int \\frac{dx}{x^2 + 4}}\\  \\hspace{20cm}\\]<\/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. 1 (STANDARD INTEGRALS)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4.\\ \\color{green}{Evaluate\\ :\\ \\int_0^1( x^2\\ -\\ 2)\\ dx}\\ \\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. 2 DEFINITE INTEGRALS<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5.\\ \\color{green}{Solve:\\ \\frac{dy}{dx}\\ =\\ e^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. 3 (FIRST ORDER DIFFERENTIAL EQUATION)<\/a><\/p>\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\\ centre\\ and\\ radius\\ of\\ the\\ circle\\ x^2\\ +\\ y^2\\  -\\ 8\\ 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\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/02\/24\/1-2-analytical-geometry-ii\/\">Refer Example No. 5 under Equation of Cirrcle with centre and radus (Analytical Geometry-II<\/a><\/p>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/nBvqNJIAneQ\" title=\"Analytical Geometry - 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\">\\[7.\\ \\color {green}{If \\ (2, 5)\\ and\\ (3,\\ &#8211; 2)\\ are\\ the\\ end\\ points\\  of\\ a\\ diameter}\\ , \\hspace{10cm}\\]\\[\\color{green}{find\\ the\\  equation\\ of\\ the\\ circle}\\ \\hspace{5cm}\\]<\/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. 8 under Analytical Geometry-II (Exercise Problems with Solutions)<\/a><\/p>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/fXeKD8h_i5Y\" title=\"Analytical Geometry (Exercise) - 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\">\\[8.\\ \\color{green}{Check\\ whether\\ the\\ conic\\ 2\\ x^2\\ -\\ 16\\ x\\ y\\ +\\   8\\ y^2\\ -\\ y\\ +\\ 3\\ = 0\\ represent\\ a\\ hyperbola}\\  \\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. 3 under General Equation of Conic (Conics)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[9.\\ \\color{green}{Find\\ the\\ area\\ of\\ parallelogram\\ whose\\ adjacent\\ sides\\ are}\\  \\hspace{10cm}\\]\\[ \\color{green} {-\\ \\overrightarrow{i}\\ + 2\\overrightarrow{j}\\ +\\  4\\overrightarrow{k}\\ and\\  \\overrightarrow{i}\\ &#8211; \\overrightarrow{j}\\ -\\ \\overrightarrow{k}}\\ \\hspace{5cm}\\]<\/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. 16 under Vector Product of  Vectors (Product of Vectors)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[10.\\ \\color{green}{Can\\ a\\ vector\\ have\\ direction\\ angles\\ 30^0,\\ 45^0,\\ 60^0}\\  \\hspace{10cm}\\]\n<\/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. 2 under Direction cosines and ratios (Vector Introduction)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[11.\\ \\color{green}{Find\\  the\\  Dot\\  product\\  of\\  \\overrightarrow{i}+\\overrightarrow{j}\\ +\\ \\overrightarrow{k}\\  ,\\  \\overrightarrow{i}\\ + 3\\overrightarrow{k}}\\ \\hspace{10cm}\\]\n<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/03\/01\/2-2-scalar-product-of-vectors\/\">Refer Example No. 2 Product of Vectors<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[12.\\ \\color {green}{Evaluate: \\int(3\\ x^2\\ -\\ 5\\ sec^2\\ x\\ +\\ \\frac{7}{x})\\ dx}\\ \\hspace{15cm}\\] \n<\/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. 9  INTEGRATION DECOMPOSITON METHOD (TEXT)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[13.\\ \\color {green}{Evaluate: \\int\\ \\frac{3x^2}{x^3\\ +\\ 1}\\ 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. 7 METHODS OF INTEGRATION &#8211; INTEGRATION BY SUBSTITUTION<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[14.\\ \\color{green}{Evaluate:\\ \\int \\frac{dx}{9\\ -\\ 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. 9 (STANDARD INTEGRALS)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[15.\\ \\color{green}{Evaluate:\\ \\int x\\ cos\\ 5x\\ 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. 6 INTEGRATION BY PARTS<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[16.\\ \\color{green}{Evaluate:\\ \\int x^2\\ e^{-2x}\\ dx}\\ \\hspace{18cm}\\]\n<\/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. 5 (ii)  (BERNOULLI&#8217;S FORMULA &#8211; Excercise Problems with solutions)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[17.\\ \\color {green}{Evaluate: \\int_0^\\frac{\\pi}{4} sec^2 x\\ dx}\\ \\hspace{18cm}\\] <\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2022\/05\/06\/definite-integrals-excercise-problems-with-solutions\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2022\/05\/06\/definite-integrals-excercise-problems-with-solutions\/\">Refer Problem No. 4  (DEFINITE INTEGRALS &#8211; Excercise Problems with solutions)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[18.\\ \\color {green}{Find\\ the\\ Order\\ and\\ degree\\ of\\ the\\ differential\\ equation}\\ \\hspace{12cm}\\]\\[\\color {green}{x^2\\ y^{II}\\ +\\ x(y^I)^3\\ +\\ y\\ =\\ 0}\\ \\hspace{7cm}\\] <\/div>\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\n<div class=\"wp-block-mathml-mathmlblock\">\\[20.\\ \\color{green}{Solve:\\ \\frac{dy}{dx}\\ =\\ \\frac{x}{1\\ +\\ x^2}}\\ \\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. 5 (FIRST ORDER DIFFERENTIAL EQUATION)<\/a><\/p>\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\\ \\ passing\\ through\\ (2, 4)\\ and\\ whose\\ diameters\\ are}\\ \\hspace{15cm}\\]\\[ \\color{green} {x\\ -\\ y\\ =\\ 4\\ and\\ 2x\\ +\\ 3y=\\ -\\ 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. 11 under Equation of circle joining two end points as diameter  (Analytical Geometry-II<\/a><\/p>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/oXJEh0DJOdM\" title=\"Analytical Geometry - 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}{Prove\\ that\\ the\\ circles\\ x^2\\ +\\ y^2\\ -\\ 2x\\ -\\ 2y\\ +\\ 1\\ = 0}\\ \\hspace{10cm}\\]\\[ \\color{green} {and\\ x^2\\ +\\  y^2\\ -\\ 4x\\ -\\ 2y\\ -\\ 4\\ = 0\\ touch\\ each\\ other.}\\  \\hspace{5cm}\\]<\/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. 17 under Analytical Geometry-II (Exercise Problems with Solutions)<\/a><\/p>\n\n\n\n<p>                  This question was wrong one.<\/p>\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\\ (2,\\ 3)\\ and\\ directrix\\ x\\ =\\ 7\\ and\\ e\\ =\\ \\frac{1}{2}}\\  \\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. 5 under Equation of Conic (Conics)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Show\\ that\\ the\\ second\\ degree\\ equation}\\ \\hspace{15cm}\\]\\[\\color{green}{12\\ x^2\\ +\\ 7\\ x\\ y\\ -\\ 10\\ y^2\\ +\\ 13\\ x\\ +\\ 45\\ y\\ -\\ 35\\ =\\ 0}\\ \\hspace{7cm}\\]\\[\\color{green}{in\\ x\\ and\\ y\\ 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. 7 under  Conics<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[22.\\ A)\\ i.\\ \\color{green}{{Prove\\ that\\ the\\ points}\\ \\hspace{15cm}}\\ \\hspace{7cm}\\]\\[\\color{green} {2\\ \\overrightarrow{j}\\  +\\ 10 \\overrightarrow{k}\\ ,\\  7\\overrightarrow{i}\\ +\\  6\\overrightarrow{j}\\ +\\  6\\overrightarrow{k} and\\ -\\ 4\\overrightarrow{i}\\  +\\ 9 \\overrightarrow{j}\\  +\\ 6 \\overrightarrow{k}\\  form\\  an\\ isosceles\\ triangle}\\]<\/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. 8 under Vector Introduction<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Find\\ the\\ value\\ of\\ &#8216;P\\ such\\ the\\ that\\  the\\ vectors\\ 2\\overrightarrow{i}\\ &#8211; 3\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k},\\  p\\overrightarrow{i}\\ +\\ 2\\overrightarrow{j}\\ -\\ \\overrightarrow{k}\\ and}\\  \\hspace{7cm}\\]\\[ \\color{green} {3\\overrightarrow{i}\\ -\\ \\overrightarrow{j}\\ +\\ 4\\overrightarrow{k}\\ lie\\ on\\ the\\ same\\ plane}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/03\/01\/2-2-scalar-product-of-vectors\/\">Refer Example No. 22 under Product of Vectors<\/a><\/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\\ area\\ of\\ the\\ triangle\\ formed\\ by\\ the\\ points\\ whose\\ position\\ vectors}\\ \\hspace{15cm}\\]\\[\\color{green}{3\\overrightarrow{i}\\ -\\ 2\\overrightarrow{j}\\ +\\ \\overrightarrow{k}\\ ,\\  \\overrightarrow{i}\\ -\\ 3\\overrightarrow{j}\\ +\\ 5\\overrightarrow{k}\\ and\\  2\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\ 4\\overrightarrow{k}}\\  \\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.20  Product of Two vectors Excercise Problems with solutions<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Find\\ the\\ Magnitude\\ of\\ the\\ moment\\ of\\ the\\ force\\ 6\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\ \\overrightarrow{k}}\\ \\hspace{7cm}\\]\\[ \\color{green} {acting\\ along\\ the\\ point\\ (0,\\ 1,\\ -\\ 1)\\ about\\ the\\ point\\ (4,\\ 3,\\ -\\ 1)}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/03\/01\/2-2-scalar-product-of-vectors\/\">Refer Example No. 21 under Product of two Vectors<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[23.\\ A)\\ i.\\ \\color{green}{Evaluate:\\ \\hspace{2cm}\\ (a)\\ \\int (3x + 2 ) ( x + 1) \\ dx\\ \\hspace{2cm}\\ (b)\\ \\int sin 3x\\  sin x\\ 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\/\">Refer Example No. 6  and Example No. 20  under  (INTEGRATION DECOMPOSION METHOD (TEXT)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Evaluate:\\ \\hspace{2cm}\\ (a)\\ \\int\\ sec^7\\ x\\ tan\\ x\\ dx\\ \\hspace{2cm}\\ (b)\\ \\int\\ \\frac{e^x}{e^x\\ +\\ 5}\\ 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\/\"> Refer Example No. 15 and Example No.10  METHODS OF INTEGRATION &#8211; INTEGRATION BY SUBSTITUTION<\/a><\/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}\\ (a)\\ \\int\\ \\frac{2ax + b}{{\\sqrt{(ax^2 + bx + c)}}}\\ dx\\ \\hspace{2cm}\\ (b)\\ \\int\\ \\frac{dx}{(7x + 1)^2 + 25}}\\ \\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\/\"> Refer Example No. 16 in  METHODS OF INTEGRATION &#8211; INTEGRATION BY SUBSTITUTION<\/a><\/p>\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\/\">Refer Example No. 6 (i) (STANDARD INTEGRALS- Excercise Problems with Solutions)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Evaluate:\\ \\hspace{2cm}\\ (a)\\ \\int \\frac{dx}{(4x\\ +\\ 1)^2 &#8211; 36}\\ dx\\ \\hspace{2cm}\\ (b)\\ \\int \\frac{dx}{{\\sqrt{4\\ -\\ 81x^2}}}}\\ \\hspace{10cm}\\]<\/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. 8 (STANDARD INTEGRALS)<\/a><\/p>\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\/\">Refer Example No. 6 (ii) (STANDARD INTEGRALS- Excercise Problems with Solutions)<\/a><\/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\\ e^{-\\ 7x}\\ 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\/\">Refer Example No. 10 and No. 3 INTEGRATION BY PARTS<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Evaluate:\\ \\hspace{2cm}\\ (a)\\ \\int x^2\\ cos\\ 6x\\ dx\\ \\hspace{2cm}\\ (b)\\ \\int x^2\\ e^{-9x}\\ 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\/\">Refer Problem No. 6  (BERNOULLI&#8217;S FORMULA &#8211; Excercise Problems with solutions)<\/a><\/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\/\">Refer Example No. 6 (BERNOULLI&#8217;S FORMULA)<\/a><\/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: \\int_0^\\frac{\\pi}{2} cos\\ 3 x\\  cos\\ x\\ dx}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p>Soln: <a href=\"https:\/\/yanamtakshashila.com\/2021\/04\/21\/definite-integrals\/\">Refer Example No. 9 DEFINITE INTEGRALS<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Evaluate:\\ \\hspace{2cm}\\ (a)\\ \\int (x\\ +\\ 3)\\ sin\\ 7x\\ dx\\ \\hspace{2cm}\\ (b)\\ \\int x^3\\ e^{-3x}\\ 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 (a) Refer Example No. 11 INTEGRATION BY PARTS<\/a><\/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 (b) Refer Example No. 8 (BERNOULLI&#8217;S FORMULA)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[25.\\ A)\\ i.\\ \\color{green}{Find\\ the\\ Volume\\ of\\ the\\ solid\\ generated\\ by\\ the\\ area\\ enclosed\\ by\\ the\\ curve\\ y^2\\ =\\ x(x\\ -\\ 1)^2}\\ \\hspace{10cm}\\]\\[\\color{green}{and\\ the\\ X-axis\\ when\\ rotated\\ about\\ X- axis}\\ \\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. 8  (AREA AND VOLUME)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Solve:\\ \\frac{dy}{dx}\\ +\\ y\\ cot\\ x\\ =\\ sin^3\\ 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. 6 (LINEAR TYPE DIFFERENTIAL EQUATION)<\/a><\/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^y)\\ sec^2\\ x\\ dx\\ +\\ 5\\ e^y\\ tan\\ x\\ 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. 13 (FIRST ORDER DIFFERENTIAL EQUATION)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Solve:\\ \\frac{dy}{dx}\\ +\\ 2y\\ tan\\ x\\ =\\ e^{tan\\ 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. 5 (LINEAR TYPE DIFFERENTIAL EQUATION)<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Soln: Refer Problem No. 2 under Analytical Geometry-II (Exercise Problems with Solutions) Soln: Refer Example No. 3 under Scalar Product of Vectors (Product of Vectors) Soln: Refer Example No. 1 (STANDARD INTEGRALS) Soln: Refer Example No. 2 DEFINITE INTEGRALS Soln: Refer Example No. 3 (FIRST ORDER DIFFERENTIAL EQUATION) Soln: Refer Example No. 5 under Equation [&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 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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":false,"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-40204","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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