{"id":24315,"date":"2021-12-22T20:40:26","date_gmt":"2021-12-22T15:10:26","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=24315"},"modified":"2022-01-04T20:48:02","modified_gmt":"2022-01-04T15:18:02","slug":"model-exam-question-paper-with-solutions-for-engineering-mathematics-1","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=24315","title":{"rendered":"MODEL EXAM QUESTION PAPER WITH SOLUTIONS FOR ENGINEERING MATHEMATICS &#8211; 1"},"content":{"rendered":"\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.\\ Find\\ the\\ cofactor\\ of\\ 4\\ in\\ the\\ matrix\\ \\begin{bmatrix}\n1 &amp; 0 &amp; -1 \\\\\n2 &amp; 3 &amp; 4 \\\\\n7 &amp; 8 &amp; -2 \\\\\n\\end{bmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Example\\ 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/12\/11\/applications-of-matrices-and-determinantstext\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/12\/11\/applications-of-matrices-and-determinantstext\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ Find\\ the\\ rank\\ of\\ the\\ matrix\\ \\begin{bmatrix}\n5 &amp; 2 \\\\\n6 &amp; 3 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Example\\ 1\\ under\\ Rank\\ of\\ a\\ Matrix\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/09\/05\/largecolor-redchapter-1-2-applications-of-matrices-and-determinants-text\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/09\/05\/largecolor-redchapter-1-2-applications-of-matrices-and-determinants-text\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ If\\  Z_1 =  1 + 4i,\\  Z_2 =  -3 + 6i,\\ find\\  3\\ Z_1\\  +\\  2\\ Z_2.\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Example\\ 3\\ under\\ Algebra\\ of\\ Complex\\ Numbers\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/09\/26\/unit-ii-complex-numbers\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/09\/26\/unit-ii-complex-numbers\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4.\\ If \\ \u03c9\\ is\\ the\\ cube\\ roots\\ of unity,\\ what\\ is\\ the\\ value\\ of \\  \u03c9 ^4\\ +\\ \u03c9 ^5\\ +\\  \u03c9 ^6\\ \\hspace{16cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Example\\ 3\\ under\\ Roots\\ of\\ Complex\\ Numbers\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/10\/16\/unit-ii-complex-numbers-3\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/10\/16\/unit-ii-complex-numbers-3\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5.\\ Find\\ \\frac{dy}{dx}\\ if\\  y\\ =\\  8\\ e^x\\ -\\ 4\\ cosec\\ x\\  \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2022\/01\/03\/differentiation-excercise\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2022\/01\/03\/differentiation-excercise\/<\/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\\ -\\ B}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {6.)\\,Prove\\ that}\\ \\begin{bmatrix}\n1 &amp; -1 &amp; 2 \\\\\n2 &amp; -2 &amp; 4 \\\\\n3 &amp; -3 &amp; 6 \\\\\n\\end{bmatrix}\\ is\\ a\\ singular\\ matrix\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\  Example\\ c\\ under\\ Singular\\ and Non-Singular\\ of\\ a\\ Matrix\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/08\/03\/unit-i-algebra\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/08\/03\/unit-i-algebra\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[7.\\ Find\\ the\\ inverse\\ of\\ \\begin{bmatrix}\n1 &amp; -1 \\\\\n-2 &amp; 0 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Example\\ 1\\ under\\ Inverse\\ of\\ a\\ Matrix\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/09\/05\/largecolor-redchapter-1-2-applications-of-matrices-and-determinants-text\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/09\/05\/largecolor-redchapter-1-2-applications-of-matrices-and-determinants-text\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[8.\\ Find\\ the\\ 6^{th}\\ term\\ in\\ the\\ expansion\\ of\\ (x\\ +\\ \\frac{1}{x})^{10}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Example\\ 3\\ under\\ Binomial\\ Theorem\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/09\/16\/unit-i-algebra-2\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/09\/16\/unit-i-algebra-2\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[9.\\ Find\\ the\\ Real\\ and\\ Imaginary\\ parts\\ of\\ \\frac{2+ 3i}{4+ 5i}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/09\/26\/unit-ii-complex-numbers\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/09\/26\/unit-ii-complex-numbers\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[10.\\ If\\  a =  cos\u2061\\ x +  i sin\u2061\\ x,\\ b =  cos\u2061\\ y +  i sin\u2061\\ y,\\ find\\ ab\\ and\\ \\frac{1}{ab}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Example\\ 3\\ under\\ De-Moivre&#8217;s\\ Theorem\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<pre class=\"wp-block-code\"><code><a href=\"https:\/\/yanamtakshashila.com\/2021\/10\/13\/unit-ii-complex-numbers-2\/(opens in a new tab)\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/10\/13\/unit-ii-complex-numbers-2\/(opens in a new tab)\">https:\/\/yanamtakshashila.com\/2021\/10\/13\/unit-ii-complex-numbers-2<\/a><\/code><\/pre>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[11.\\ Find\\ the\\ value\\ of\\  \\frac{Tan\\ {20}^0\\  +\\ Tan\\ {25}^0}{1\\ -\\ Tan\\ {20}^0\\ Tan\\ {25}^0}\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Example\\ 3\\ under\\ Compound\\ Angles\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/10\/21\/unit-ii-complex-numbers-4\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/10\/21\/unit-ii-complex-numbers-4\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[12:\\  Prove\\ that\\ Cos^4\\ A\\ -\\ Sin^4\\ A\\ =\\ Cos\\ 2A\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Example\\ 5\\ under\\ Multiple\\ and\\ Sub-multiple\\ angles\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<pre class=\"wp-block-code\"><code><a href=\"https:\/\/yanamtakshashila.com\/2021\/10\/27\/3-2-multiple-and-sub-multiple-angles\/\" data-type=\"URL\" data-id=\"https:\/\/yanamtakshashila.com\/2021\/10\/27\/3-2-multiple-and-sub-multiple-angles\/\">https:\/\/yanamtakshashila.com\/2021\/10\/27\/3-2-multiple-and-sub-multiple-angles\/<\/a><\/code><\/pre>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[13.\\  Express\\ Sin\\ 5A\\ -\\ Sin\\ 3A\\ as\\ product\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Example\\ 2\\ under\\ Sum\\ and\\ Product\\ Formulae\\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/10\/31\/3-3-sum-and-product-formulae\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/10\/31\/3-3-sum-and-product-formulae\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[14.\\  Evaluate:\\ \\lim\\ _{x\\ \\to\\ 2}\\ \\frac{x^4\\ -\\ 2^4}{x\\ -\\ 2}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Example\\ 2\\ under\\ Limits\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/11\/08\/limits\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/11\/08\/limits\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[15.\\  Find\\ \\frac{dy}{dx}\\ if\\  y\\ =\\ e^x\\ log\\ x\\  \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Example\\ 6\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/11\/13\/differentiation\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/11\/13\/differentiation\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[16.\\ Find\\ \\frac{dy}{dx}\\  if\\  y\\ =\\ (x^2\\ +\\ 5)\\ Cos\\ x\\ e^{-\\ 2x}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2022\/01\/03\/differentiation-excercise\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2022\/01\/03\/differentiation-excercise\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[17.\\ If\\ y\\ =\\ log(Sin\\ x),\\ find\\ \\frac{dy}{dx}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Example\\ 3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a href=\"https:\/\/yanamtakshashila.com\/2021\/11\/18\/differentiation-methods\/\">https:\/\/yanamtakshashila.com\/2021\/11\/18\/differentiation-methods\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[18.\\ Form\\ the\\ differential\\ equation\\ of\\ y^2\\ =\\ 4\\ a\\ x\\ by\\ eliminating\\ the\\ constant\\ &#8216;a&#8217;\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2022\/01\/03\/successive-differentiation-excercise\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2022\/01\/03\/successive-differentiation-excercise\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[19.\\ If\\ u\\ = 2\\ x^3\\ +\\ 4\\ \\ y^3\\ +\\ 2\\ x\\ y\\ ,\\ find\\ \\frac{\u2202u}{\u2202x}\\ and\\ \\frac{\u2202u}{\u2202y}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Example\\ 3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/12\/03\/partial-differentiation\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/12\/03\/partial-differentiation\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[20.\\ If\\ u\\ =\\ x^3\\  +\\ y^3\\ ,\\ find\\ \\frac{\u2202^2u}{\u2202x^2}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2022\/01\/03\/partial-differentiation-excercise\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2022\/01\/03\/partial-differentiation-excercise\/<\/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: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\">\\[\\color {black} {21.\\ A)\\ i.}\\ Solve\\ by\\ using\\ Cramers\\ Rule\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3x &#8211; y + 2z=8,\\ x &#8211; y + z = 2\\ and\\ 2x + y &#8211; z = 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/12\/11\/applications-of-matrices-and-determinantstext\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/12\/11\/applications-of-matrices-and-determinantstext\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ii.\\ Find\\ the\\ inverse\\ of\\ \\begin{bmatrix}\n3 &amp; 1 &amp; -1\\\\ \n2 &amp; -2 &amp; 0 \\\\\n1 &amp; 2 &amp; -1 \\\\ \\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 6\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/12\/11\/applications-of-matrices-and-determinantstext\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/12\/11\/applications-of-matrices-and-determinantstext\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {B)\\ i.}\\ Find\\ the\\ rank\\ of\\ the\\ matrix\\ \\begin{bmatrix}\n1 &amp; 2 &amp; 3  &amp; 2 \\\\\n2 &amp; 3 &amp; 5 &amp; 1 \\\\\n1 &amp; 3 &amp; 4 &amp; 5 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Example\\ 4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/09\/05\/largecolor-redchapter-1-2-applications-of-matrices-and-determinants-text\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/09\/05\/largecolor-redchapter-1-2-applications-of-matrices-and-determinants-text\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ii.\\ Find\\ the\\ middle\\ terms\\ in\\ the\\ expansion\\ of\\ (x^3\\ +\\ \\frac{2}{x^3})^{11}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/12\/15\/binomial-theorem-revision\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/12\/15\/binomial-theorem-revision\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {22.\\ A)\\ i.}\\ Find\\ the\\ modulus\\ and\\ amplitude\\ of\\ \\frac{-\\ 3\\ +\\  i}{-\\ 1\\ +\\ i}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a href=\"https:\/\/yanamtakshashila.com\/2022\/01\/02\/algebra-of-complex-numbers-excersice\/\">https:\/\/yanamtakshashila.com\/2022\/01\/02\/algebra-of-complex-numbers-excersice\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ii.\\  Show\\ that\\ the\\ complex\\ numbers\\ 2+i,\\ 4+3i,\\ 2+5i\\ and\\ 3i\\ form\\ a\\ square\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 11\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/09\/26\/unit-ii-complex-numbers\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/09\/26\/unit-ii-complex-number<\/a>\/<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {B)\\ i.}\\ Simplify\\ using\\ DeMoivre\u2019s\\ theorem:\\ \\frac{(cos\u2061\\  2\u03b8 &#8211;  i sin\u2061\\  2\u03b8)^7\\ (cos\u2061\\  3\u03b8 +  i sin\u2061\\  3\u03b8)^{-5}} {(cos\u2061\\  4\u03b8 + i sin\u2061\\  4\u03b8)^2\\ (cos\u2061\\  5\u03b8 &#8211;  i sin\u2061\\  5\u03b8)^{-6}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/12\/16\/de-movires-theorem-revision\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/12\/16\/de-movires-theorem-revision\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ii.\\  Solve\\  x^5 +\\  x^3\\ +\\ x^2\\ +\\ 1\\  =\\ 0\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 6\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/10\/16\/unit-ii-complex-numbers-3\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/10\/16\/unit-ii-complex-numbers-3\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {23.\\ A)\\ i.}\\ If\\ A\\ and\\ B\\ are\\ acute\\ angles\\ and\\ Sin\\ A\\ =\\ \\frac{1}{\\sqrt{10}},\\  Sin\\ B\\ =\\ \\frac{1}{\\sqrt{5}},\\ \\hspace{15cm}\\]\\[ Prove\\ that\\  (A\\ +\\ B)\\ =\\ \\frac{\u03c0}{4}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 6\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/10\/21\/unit-ii-complex-numbers-4\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/10\/21\/unit-ii-complex-numbers-4\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ii.\\ If\\ A\\ +\\ B\\ =\\ {45}^0 \\  Prove\\ that\\ (1\\ +\\ Tan\\ A)\\ (1\\ +\\ Tan\\ B)\\ =\\ 2.\\ \\hspace{15cm}\\]\\[Hence\\ deduce\\ the\\ value\\ of\\ Tan\\ \\ 22\\ {\\frac{1}{2}}^0\\   \\hspace{13cm}\\]<\/div>\n\n\n\n<p><a href=\"https:\/\/yanamtakshashila.com\/2021\/10\/21\/unit-ii-complex-numbers-4\/\">https:\/\/yanamtakshashila.com\/2021\/10\/21\/unit-ii-complex-numbers-4\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {B)\\ i.}\\  Prove\\ that\\  \\frac{Sin\\ 3\\ \\theta}{Sin\\ \\theta}\\ -\\ \\frac{Cos\\ 3\\ \\theta}{Cos\\ \\theta}\\ =\\  2\\  \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2022\/01\/02\/multiple-and-sub-multiple-angles-excercise\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2022\/01\/02\/multiple-and-sub-multiple-angles-excercise\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {24.\\ A)\\ i.}\\  Evaluate:\\ (i)\\ \\lim\\ _{x\\ \\to\\ 3}\\ \\frac{x^6\\ -\\ 3^6}{x\\ -\\ 3}\\ \\hspace{1cm}\\ (ii)\\ \\lim\\ _{\u03b8\\ \\to\\ 0}\\ \\frac{Sin\\ 7\\ \u03b8}{Sin\\ 2\\ \u03b8}\\ \\hspace{7cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/12\/23\/limits-revision\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/12\/23\/limits-revision\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ii.\\ Find\\ \\frac{dy}{dx}\\ (i)\\ if\\  y\\ =\\ e^x\\ log\\ x\\ Cos\\ x\\ (ii)\\ y\\ =\\ \\frac{x^2\\ +\\ Tan\\ x}{x\\ -\\ Sin\\ x}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 9\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/11\/13\/differentiation\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/11\/13\/differentiation\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {B)\\ i.}\\ \\frac{dy}{dx}\\ (i)\\ if\\  y\\ =\\ (2\\ x\\ +\\ 1)(3\\ x\\ -\\ 7)(4\\ -\\ 9\\ x)\\  (ii)\\ y\\ =\\ \\frac{e^x\\ +\\ Sin\\ x}{1\\ -\\ Cos\\ x}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2022\/01\/03\/differentiation-excercise\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2022\/01\/03\/differentiation-excercise\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ii..\\ Find\\ \\frac{dy}{dx}\\ if (i)\\ \\ x^2\\ +\\ y^2\\ +\\ 2\\ x\\ +\\ 3\\ y\\ =\\ 0\\ \\hspace{2cm}\\ and\\ (ii)\\ y\\ =\\ a\\ +\\ x\\ e^y\\ \\hspace{2cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Q.\\ No.\n 14th\\ and\\ 15th\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<p><a href=\"https:\/\/yanamtakshashila.com\/2021\/11\/18\/differentiation-methods\/\">https:\/\/yanamtakshashila.com\/2021\/11\/18\/differentiation-methods\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {25.\\ A)\\ i.}\\  If\\ y\\ =\\ x^2\\ Sin\\  x,\\ prove\\ that\\ x^2\\ y_2\\ -\\ 4\\ x\\ y_1\\ +\\ (x^2\\ +\\ 6)y\\ =\\ 0\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 6\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a href=\"https:\/\/yanamtakshashila.com\/2021\/11\/23\/successive-differentiation\/\">https:\/\/yanamtakshashila.com\/2021\/11\/23\/successive-differentiation\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ii.\\ If\\ u\\ = \\frac{x^3\\ y^3}{x^3\\ +\\ y^3} ,\\ Show\\ that\\ x\\ \\frac{\u2202u}{\u2202x}\\ +\\ y\\ \\frac{\u2202u}{\u2202y}\\ =\\ 3\\ u\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 5\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/12\/03\/partial-differentiation\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/12\/03\/partial-differentiation\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {B)\\ i.}\\  If\\ y\\ =\\ a\\ Cos(log\\  x)\\ +\\ b\\ Sin\\ (log\\ x),\\ prove\\ that\\ x^2\\ y_2\\ +\\ x\\ y_1\\ +\\ y\\ =\\ 0\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 7\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a href=\"https:\/\/yanamtakshashila.com\/2021\/11\/23\/successive-differentiation\/\">https:\/\/yanamtakshashila.com\/2021\/11\/23\/successive-differentiation\/<\/a><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ii.\\ If\\ u\\ =\\ x^3\\ -\\ 2\\ x^2\\ y\\ +\\ 3\\ x\\ y^2\\ +\\ y^3\\ ,\\ find\\ \\frac{\u2202^2u}{\u2202x^2}\\ and\\ \\frac{\u2202^2u}{\u2202y^2}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Refer\\ Problem\\ 6\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p><a rel=\"noreferrer noopener\" href=\"https:\/\/yanamtakshashila.com\/2021\/12\/03\/partial-differentiation\/\" target=\"_blank\">https:\/\/yanamtakshashila.com\/2021\/12\/03\/partial-differentiation\/<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p><a href=\"https:\/\/yanamtakshashila.com\/2021\/12\/11\/applications-of-matrices-and-determinantstext\/\">https:\/\/yanamtakshashila.com\/2021\/12\/11\/applications-of-matrices-and-determinantstext\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/09\/05\/largecolor-redchapter-1-2-applications-of-matrices-and-determinants-text\/\">https:\/\/yanamtakshashila.com\/2021\/09\/05\/largecolor-redchapter-1-2-applications-of-matrices-and-determinants-text\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/09\/26\/unit-ii-complex-numbers\/\">https:\/\/yanamtakshashila.com\/2021\/09\/26\/unit-ii-complex-numbers\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/10\/16\/unit-ii-complex-numbers-3\/\">https:\/\/yanamtakshashila.com\/2021\/10\/16\/unit-ii-complex-numbers-3\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2022\/01\/03\/differentiation-excercise\/\">https:\/\/yanamtakshashila.com\/2022\/01\/03\/differentiation-excercise\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/08\/03\/unit-i-algebra\/\">https:\/\/yanamtakshashila.com\/2021\/08\/03\/unit-i-algebra\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/09\/05\/largecolor-redchapter-1-2-applications-of-matrices-and-determinants-text\/\">https:\/\/yanamtakshashila.com\/2021\/09\/05\/largecolor-redchapter-1-2-applications-of-matrices-and-determinants-text\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/09\/16\/unit-i-algebra-2\/\">https:\/\/yanamtakshashila.com\/2021\/09\/16\/unit-i-algebra-2\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/09\/26\/unit-ii-complex-numbers\/\">https:\/\/yanamtakshashila.com\/2021\/09\/26\/unit-ii-complex-numbers\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/10\/21\/unit-ii-complex-numbers-4\/\">https:\/\/yanamtakshashila.com\/2021\/10\/21\/unit-ii-complex-numbers-4\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/10\/31\/3-3-sum-and-product-formulae\/\">https:\/\/yanamtakshashila.com\/2021\/10\/31\/3-3-sum-and-product-formulae\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/11\/08\/limits\/\">https:\/\/yanamtakshashila.com\/2021\/11\/08\/limits\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/11\/13\/differentiation\/\">https:\/\/yanamtakshashila.com\/2021\/11\/13\/differentiation\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2022\/01\/03\/differentiation-excercise\/\">https:\/\/yanamtakshashila.com\/2022\/01\/03\/differentiation-excercise\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/11\/18\/differentiation-methods\/\">https:\/\/yanamtakshashila.com\/2021\/11\/18\/differentiation-methods\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2022\/01\/03\/successive-differentiation-excercise\/\">https:\/\/yanamtakshashila.com\/2022\/01\/03\/successive-differentiation-excercise\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/12\/03\/partial-differentiation\/\">https:\/\/yanamtakshashila.com\/2021\/12\/03\/partial-differentiation\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2022\/01\/03\/partial-differentiation-excercise\/\">https:\/\/yanamtakshashila.com\/2022\/01\/03\/partial-differentiation-excercise\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/12\/11\/applications-of-matrices-and-determinantstext\/\">https:\/\/yanamtakshashila.com\/2021\/12\/11\/applications-of-matrices-and-determinantstext\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/12\/11\/applications-of-matrices-and-determinantstext\/\">https:\/\/yanamtakshashila.com\/2021\/12\/11\/applications-of-matrices-and-determinantstext\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/09\/05\/largecolor-redchapter-1-2-applications-of-matrices-and-determinants-text\/\">https:\/\/yanamtakshashila.com\/2021\/09\/05\/largecolor-redchapter-1-2-applications-of-matrices-and-determinants-text\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/12\/15\/binomial-theorem-revision\/\">https:\/\/yanamtakshashila.com\/2021\/12\/15\/binomial-theorem-revision\/<\/a> <a 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href=\"https:\/\/yanamtakshashila.com\/2021\/11\/23\/successive-differentiation\/\">https:\/\/yanamtakshashila.com\/2021\/11\/23\/successive-differentiation\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/12\/03\/partial-differentiation\/\">https:\/\/yanamtakshashila.com\/2021\/12\/03\/partial-differentiation\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/11\/23\/successive-differentiation\/\">https:\/\/yanamtakshashila.com\/2021\/11\/23\/successive-differentiation\/<\/a> <a href=\"https:\/\/yanamtakshashila.com\/2021\/12\/03\/partial-differentiation\/\">https:\/\/yanamtakshashila.com\/2021\/12\/03\/partial-differentiation\/<\/a><\/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 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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":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":[711788288],"tags":[],"class_list":["post-24315","post","type-post","status-publish","format-standard","hentry","category-n-em-1-model-exam"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - 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