{"id":30278,"date":"2022-06-01T19:11:41","date_gmt":"2022-06-01T13:41:41","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=30278"},"modified":"2022-07-24T19:07:20","modified_gmt":"2022-07-24T13:37:20","slug":"solutions-to-model-exam-for-engineering-mathematics-ii","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=30278","title":{"rendered":"SOLUTIONS TO MODEL EXAM FOR ENGINEERING MATHEMATICS-II"},"content":{"rendered":"\n<h2 class=\"has-text-align-center wp-block-heading\">DR.B.R.AMBEDKAR POLTECHNIC COLLEGE, YANAM<\/h2>\n\n\n\n<h3 class=\"has-text-align-center wp-block-heading\">MODEL EXAM<\/h3>\n\n\n\n<h3 class=\"has-text-align-center wp-block-heading\">ENGINEERING MATHEMATICS &#8211; II<\/h3>\n\n\n\n<h5 class=\"wp-block-heading\">Time &#8211; 3 hours                                                                                                                                        (Maximum Marks &#8211; 100)            <\/h5>\n\n\n\n<h5 class=\"wp-block-heading\">      Note:<\/h5>\n\n\n\n<ol class=\"wp-block-list\" type=\"1\"><li>Answer all questions in <strong>PART- A. <\/strong>Each question carries one mark.&nbsp;<\/li><li>Answer any ten questions in <strong>PART- B. <\/strong>Each question carries two marks.<\/li><li>Answer all questions by selecting either A or B. Each question carries fifteen marks.<\/li><li>Clarks Table and programmable calculators are not permitted.<\/li><\/ol>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\underline{PART\\ -\\ A}\\ (5\u00d7 1\\ =\\ 5)\\]<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\\ centre\\ and\\ radius\\ of\\ the\\ circle\\ x^2\\ +\\ y^2\\ +\\ 4\\ x\\ +\\ 4\\ y\\ -\\ 1\\ =\\ 0}\\ \\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.3 under Analytical Geometry Excercise problems with solutions<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ \\color{green}{Find\\ the\\  Unit\\ vector\\ along\\ the\\ vector\\  3\\overrightarrow{i}\\ + 4\\overrightarrow{j}- 5\\overrightarrow{k}}\\ \\hspace{15cm}\\]<\/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. 1 under Unit Vector (Vector Introduction)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ \\color{green}{Evaluate: \\int(x^2 -x + 5)\\ 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. 1 INTEGRATION DECOMPOSITON METHOD (TEXT)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4.\\ \\color{green}{Evaluate\\ :\\ \\int\\ \\frac{2x}{1 + x^2}\\ 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. 3 METHODS OF INTEGRATION &#8211; INTEGRATION BY SUBSTITUTION<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5.\\ \\color{green}{Evaluate\\ :\\ \\int_1^2 x^2\\ dx}\\ \\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. 1 DEFINITE INTEGRALS<\/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}\\ (10\u00d7 2\\ =\\ 20)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[6.\\ \\color{green}{Prove\\ that\\ the\\ circles\\ x^2\\ +\\  y^2\\  -\\ 4\\ x\\ +\\  6y\\  +\\ 4\\ =\\ 0\\ and}\\ \\hspace{7cm}\\]\\[\\color{green}{and\\ x^2\\ +\\  y^2\\  +\\ 2\\ x\\ +\\  4y\\  +\\ 4\\ =\\ 0\\ cut\\ orthogonally}\\ \\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. 1 under Orthogonal Circles (Analytical Geometry-II<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[7.\\ \\color{green}{Find\\ the\\ equation\\ of\\ the\\ circle\\ passing\\ though\\ the\\ point\\ (5,4)}\\ \\hspace{7cm}\\]\\[ \\color{green} {and\\ concentric\\ to\\ the\\ circle\\ x^2\\ +\\  y^2\\  -\\ 8\\ x\\ +\\  12y\\  +\\ 15\\ =\\ 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. 2 under Family of  Circles (Analytical Geometry-II<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[8.\\ \\color{green}{Prove\\ that\\ the\\ equation\\  x^2\\ +\\ 6\\ x\\ y\\ +\\ 9y^2\\ +\\ 4\\ x\\ +\\ 12\\ y\\ -\\ 5\\ =\\ 0}\\ \\hspace{7cm}\\]\\[ \\color{green} {is\\ a\\ parobala}\\  \\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. 1 under General Equation of Conic (Conics)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[9.\\ \\color{green}{Find\\ the\\ equation\\ of\\ the\\ parabola\\ with\\ focus\\ at\\ (1,\\ -1)}\\ \\hspace{8cm}\\]\\[ \\color{green} {and\\ directrix\\ x\\ -\\ y\\ =\\ 0.}\\  \\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. 1 under Equation of Parobola (Conics)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[10.\\ \\color{green}{Show\\ that\\ the\\ points\\ whose\\ position\\ vectors}\\ \\hspace{12cm}\\]\\[ \\color{green} {2\\overrightarrow{i}\\ +\\  3\\overrightarrow{j}\\ -\\  5\\overrightarrow{k},\\  3\\overrightarrow{i}\\ +\\ \\overrightarrow{j}\\ -\\  2\\overrightarrow{k}\\ and\\ 6\\overrightarrow{i}\\ -\\ 5 \\overrightarrow{j}\\ +\\ 7\\overrightarrow{k}\\ are\\ collinear}\\  \\hspace{3cm}\\]<\/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. 1 under Conditions for Vectors (Vector Introduction)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[11.\\ \\color{green}{Find\\ the\\ value\\ of\\ m\\ if\\ the\\ vectors\\ 3\\overrightarrow{i} -\\overrightarrow{j} + 5\\overrightarrow{k}}\\ \\hspace{10cm}\\]\\[ \\color{green} {and\\  -6\\overrightarrow{i}+ m\\overrightarrow{j}+4\\overrightarrow{k} are\\ perpendicular}\\  \\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. 3 under Scalar Product of  Vectors (Product of Vectors)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[12.\\ \\color{green}{Find\\ the\\ area\\ of\\ parallelogram\\ whose\\ adjacent\\ sides\\ are}\\  \\hspace{10cm}\\]\\[ \\color{green} {3\\overrightarrow{i}- \\overrightarrow{k} and\\  \\overrightarrow{i}+ \\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. 2 under Vector Product of  Vectors (Product of Vectors)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[13.\\ \\color{green}{Find\\ the\\ moment\\ of\\ the\\ force\\ 3\\overrightarrow{i}+\\overrightarrow{k}\\ acting\\ through\\ the\\ point}\\  \\hspace{10cm}\\]\\[ \\color{green} {\\overrightarrow{i}+2\\overrightarrow{j}-\\overrightarrow{k} about\\ the\\ point\\  2\\overrightarrow{i}+ \\overrightarrow{j}-2\\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. 1 under Application of Vector Product  (Product of Vectors)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[14.\\ \\color{green}{Evaluate: \\int cos^3 x \\ dx}\\ \\hspace{18cm}\\]<\/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. 7 under Trigonometry Related Formulae (INTEGRATION DECOMPOSITON METHOD (TEXT)<\/a>)<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[15.\\ \\color{green}{Evaluate:\\ \\int \\frac{dx}{4x^2 + 9}}\\ \\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\n<div class=\"wp-block-mathml-mathmlblock\">\\[16.\\ \\color{green}{Evaluate:\\ \\int x\\ sin\\ 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. 1 INTEGRATION BY PARTS<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[17.\\ \\color{green}{\\int x^3\\ cos\\ x\\ dx}\\ \\hspace{18cm}\\]<\/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\/\">Refer Example No. 2 (BERNOULLI&#8217;S FORMULA)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[18.\\ \\color{green}{Find\\ the\\ Area\\ bounded\\ by\\ the\\ curve\\ y\\ =\\ x^2}\\ \\hspace{12cm}\\]\\[\\color{green}{the\\ X-axis\\ and\\ ordinates\\ x\\ =\\ 0\\ and\\ x\\ =\\ 2}\\ \\hspace{7cm}\\]<\/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<div class=\"wp-block-mathml-mathmlblock\">\\[19.\\ \\color{green}{Solve:\\ \\frac{dy}{dx}\\ =\\ \\frac{1\\ +\\ y^2}{1\\ +\\ x^2}}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-group is-vertical is-layout-flex wp-container-core-group-is-layout-fe9cc265 wp-block-group-is-layout-flex\">\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. 4 (FIRST ORDER DIFFERENTIAL EQUATION)<\/a><\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[20.\\ \\color{green}{Find\\ the\\ integrating\\ factor\\ of\\ \\frac{dy}{dx}\\ -\\ \\frac{3}{x}\\ y\\ =\\ x^2}\\ \\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. 2 (LINEAR TYPE DIFFERENTIAL EQUATION)<\/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<!-- 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\">\\[\\underline{PART\\ -\\ C}\\ (5\u00d7 15\\ =\\ 75)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[21.\\ A)\\ i.\\ \\color{green}{Find\\ the\\ equation\\ of\\ the\\ circle\\ passing\\ through\\ the\\ point\\ A(2.-3)}\\ \\hspace{7cm}\\]\\[ \\color{green} {and\\ having\\ its\\ centre\\ at\\ C(- 5,1)}\\  \\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. 3 under General Equation of Circle (Analytical Geometry-II<\/a><\/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\\ -\\ 4y\\ -\\ 3\\ = 0\\ and}\\  \\hspace{7cm}\\]\\[ \\color{green} {x^2\\ +\\ y^2\\ -\\ 8x\\ +\\ 6y\\ +\\ 7\\ = 0\\ touch\\ each\\ other.}\\  \\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. <\/a><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\/(\">1 Under Contact of Circles <\/a><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\/\">(Analytical Geometry-II<\/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}{Prove\\ that\\ equation\\ 6\\ x^2\\ +\\ 13\\ x\\ y\\ +\\ 6\\ y^2\\ +\\ 8\\ x\\ +\\ 7\\ y\\ +\\ 2\\ =\\ 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. 1 under Condition for General Equation of Conic to represent Pair of Straight lines (Conics)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Find\\ c\\ if\\ \\ 2\\ x^2\\ +\\ 3\\ x\\ y\\ -\\ 2\\ y^2\\ -\\ 5\\ x\\ +\\ 5\\ 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\/02\/26\/1-3-conics\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/02\/26\/1-3-conics\/\">Refer Example No. 2 under Condition for General Equation of Conic to represent Pair of Straight lines (Conics)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[22.\\ A)\\ i.\\ \\color{green}{Prove\\ that\\ the\\ points\\ whose\\ position\\ vectors\\ are}\\ \\hspace{7cm}\\]\\[ \\color{green} {3\\overrightarrow{i}\\ &#8211; \\overrightarrow{j}+ 6\\overrightarrow{k},  5\\overrightarrow{i}\\ &#8211; 2\\overrightarrow{j}+ 7\\overrightarrow{k} and\\ 6\\overrightarrow{i}\\ -5 \\overrightarrow{j}+ 2\\overrightarrow{k}\\  form\\  a\\ right\\   angled\\ 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. 3 under Conditions for Vectors (Vector Introduction)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Find\\ the\\ projection\\ of\\ the\\ vector\\ 3\\overrightarrow{i}+ \\overrightarrow{j}- 2\\overrightarrow{k} on\\  7\\overrightarrow{i}+ \\overrightarrow{j}+2\\overrightarrow{k}.}\\  \\hspace{7cm}\\]\\[ \\color{green} {Also\\ find\\ the\\ angle\\ between\\ them}\\  \\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. 2 under Angle between vectors using scalar product (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\\ unit\\ vector\\ perpendicular\\ to\\ each\\ of\\ the\\ vectors\\   2\\overrightarrow{i} -\\overrightarrow{j}+\\overrightarrow{k}   and\\  3\\overrightarrow{i}+ 4\\overrightarrow{j} -\\overrightarrow{k}}\\ \\hspace{7cm}\\]\\[ \\color{green} {Also\\ find\\ the\\ sine\\ of\\ the\\ angle\\ between\\ the\\ vectors .}\\  \\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. 5 under Vector Product of  Vectors (Product of Vectors)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{A\\ particle\\ acted\\ on\\  by\\ the\\ forces\\ 3\\overrightarrow{i}+ 2\\overrightarrow{j}- 3\\overrightarrow{k} and\\  \\overrightarrow{i}+ 7\\overrightarrow{j}+7\\overrightarrow{k} acting\\ on\\ the\\ particle}\\]\\[ \\color{green} {displaces\\ the\\ particle\\ from\\ the\\  point\\  \\overrightarrow{i}+ 2\\overrightarrow{j}+ 3\\overrightarrow{k} to\\  the\\ point\\  3\\overrightarrow{i}- 5\\overrightarrow{j}+4\\overrightarrow{k}.}\\  \\hspace{5cm}\\]\\[\\color{green}{Find\\ the\\ total\\ work\\ done\\ by\\ the\\ forces.}\\ \\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. 2 under Application of Scalar Product  (Product of Vectors)<\/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<!-- Leader board 1 -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:728px;height:90px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8769628924\"><\/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 \\frac{sin^2 x}{ 1- cos\\ x} \\ dx\\ \\hspace{2cm}\\ (ii)\\ \\int sin 5x\\  cos 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\/\">Refer Example No. 4 and Example No. 8  under Trigonometry Related Formulae (INTEGRATION DECOMPOSITON METHOD (TEXT)<\/a>)<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Evaluate:\\ \\hspace{2cm}\\ (i)\\ \\int\\ \\frac{(2x + 1)}{ x^2 + x + 1}\\ dx\\ \\hspace{2cm}\\ (ii)\\ \\int\\ \\frac{sec^2(log x)}{ x}\\ 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. 5 and Example No.7  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}\\ (i)\\ \\int\\ \\frac{2x + 9}{ x^2 + 9x + 30}\\ dx\\ \\hspace{2cm}\\ (ii)\\ \\int\\ sin^3x\\ cos x\\ 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. 6 and Example No.8  METHODS OF INTEGRATION &#8211; INTEGRATION BY SUBSTITUTION<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Evaluate:\\ \\hspace{2cm}\\ (i)\\ \\int \\frac{dx}{x^2 &#8211; 36}\\ dx\\ \\hspace{2cm}\\ (ii)\\ \\int \\frac{dx}{(3x + 2)^2 + 16}}\\ \\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. 5 and Example No. 4 (STANDARD INTEGRALS)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[24.\\ A)\\ i.\\ \\color{green}{Evaluate:\\ \\hspace{2cm}\\ (i)\\ \\int x^2\\ log\\ x\\ dx\\ \\hspace{2cm}\\ (ii)\\ \\int x\\ cos\\ 5x\\ 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. 7 and No. 6 INTEGRATION BY PARTS<\/a><\/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^{2x}\\ 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\/\">Refer Example No. 4 and 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:\\ \\hspace{2cm}\\ (i)\\ \\int x^2\\ sin\\ 2x\\ dx\\ \\hspace{2cm}\\ (ii)\\ \\int x^2\\ e^{5x}\\ 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\/\">Refer Example No. 3 and Example No. 5 (BERNOULLI&#8217;S FORMULA)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Evaluate: \\int_0^\\frac{\\pi}{2} (2 + sin x)^3 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. 9 DEFINITE INTEGRALS<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[25.\\ A)\\ i.\\ \\color{green}{Find\\ the\\ volume\\ of\\ right\\ circular\\ cone\\ \\ of\\ height\\ &#8216;h&#8217;\\ and\\ base\\ radius\\ &#8216;r&#8217;\\ by\\ 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\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ ii.\\ \\color{green}{Find\\ the\\ Area\\ bounded\\ by\\ the\\ curve\\ y\\ =\\ x^2\\ -\\ 6x\\ +\\ 8}\\ \\hspace{10cm}\\]\\[\\color{red}{and\\ 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\">\\[(OR)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{0.5cm}\\ B)\\ i.\\ \\color{green}{Solve:\\ (1\\ +\\ e^x)\\ sec^2\\ y\\ dy\\ -\\ e^x\\ tan\\ 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. 7 (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}\\ -\\ \\frac{2x}{1\\ +\\ x^2}\\ y\\ =\\ (1\\ +\\ x^2)}\\ \\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. 7 (LINEAR TYPE DIFFERENTIAL EQUATION)<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<h5 class=\"wp-block-heading\">                       <\/h5>\n","protected":false},"excerpt":{"rendered":"<p>DR.B.R.AMBEDKAR POLTECHNIC COLLEGE, YANAM MODEL EXAM ENGINEERING MATHEMATICS &#8211; II Time &#8211; 3 hours (Maximum Marks &#8211; 100) Note: Answer all questions in PART- A. Each question carries one mark.&nbsp; Answer any ten questions in PART- B. Each question carries two marks. Answer all questions by selecting either A or B. Each question carries fifteen [&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":"default","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":[711788417,711788460],"tags":[],"class_list":["post-30278","post","type-post","status-publish","format-standard","hentry","category-n-em-ii-internals","category-n-em-ii-model-exam-solutions"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>SOLUTIONS TO MODEL EXAM FOR ENGINEERING MATHEMATICS-II - YANAMTAKSHASHILA<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/yanamtakshashila.com\/?p=30278\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"SOLUTIONS TO MODEL EXAM FOR ENGINEERING MATHEMATICS-II - YANAMTAKSHASHILA\" \/>\n<meta property=\"og:description\" content=\"DR.B.R.AMBEDKAR POLTECHNIC COLLEGE, YANAM MODEL EXAM ENGINEERING MATHEMATICS &#8211; II Time &#8211; 3 hours (Maximum Marks &#8211; 100) Note: Answer all questions in PART- A. 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Each question carries one mark.&nbsp; Answer any ten questions in PART- B. Each question carries two marks. Answer all questions by selecting either A or B. 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