{"id":1841,"date":"2026-05-21T07:46:14","date_gmt":"2026-05-21T07:46:14","guid":{"rendered":"https:\/\/mymockmate.com\/notes\/?p=1841"},"modified":"2026-05-21T07:46:18","modified_gmt":"2026-05-21T07:46:18","slug":"miscellaneous-exercise-solutions-class-12-relations-functions","status":"publish","type":"post","link":"https:\/\/mymockmate.com\/notes\/miscellaneous-exercise-solutions-class-12-relations-functions\/","title":{"rendered":"Miscellaneous Exercise Solutions Class 12 Relations Functions"},"content":{"rendered":"\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Short Intro<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">In this post, students can find complete step-by-step solutions for the Miscellaneous Exercise of Class 12 Maths Chapter 1 \u2013 Relations and Functions. This exercise includes advanced concepts of equivalence relations, one-one and onto functions, injective mappings, and counting functions based on the latest NCERT syllabus.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Quick Information Box<\/h1>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Item<\/th><th>Details<\/th><\/tr><\/thead><tbody><tr><td>Board<\/td><td>NCERT \/ CBSE<\/td><\/tr><tr><td>Class<\/td><td>12<\/td><\/tr><tr><td>Subject<\/td><td>Mathematics<\/td><\/tr><tr><td>Chapter<\/td><td>Relations and Functions<\/td><\/tr><tr><td>Exercise<\/td><td>Miscellaneous Exercise<\/td><\/tr><tr><td>Main Topics<\/td><td>Relations &amp; Functions<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Concepts Used (Topics Covered)<\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li>One-One Functions<\/li>\n\n\n\n<li>Onto Functions<\/li>\n\n\n\n<li>Bijective Functions<\/li>\n\n\n\n<li>Equivalence Relations<\/li>\n\n\n\n<li>Reflexive Relations<\/li>\n\n\n\n<li>Symmetric Relations<\/li>\n\n\n\n<li>Transitive Relations<\/li>\n\n\n\n<li>Power Set Relations<\/li>\n\n\n\n<li>Counting Functions<\/li>\n\n\n\n<li>Equality of Functions<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The chapter discusses different types of relations and functions along with injective and surjective mappings.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Important Formulas<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">One-One Function<\/h2>\n\n\n\n<pre class=\"wp-block-code\"><code>f(x\u2081) = f(x\u2082) \u21d2 x\u2081 = x\u2082<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Onto Function<\/h2>\n\n\n\n<pre class=\"wp-block-code\"><code>For every y \u2208 Y, there exists x \u2208 X such that f(x) = y<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Equivalence Relation<\/h2>\n\n\n\n<pre class=\"wp-block-code\"><code>A relation which is reflexive, symmetric and transitive.<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Function Given in Question 1<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mi>x<\/mi><mrow><mn>1<\/mn><mo>+<\/mo><mi mathvariant=\"normal\">\u2223<\/mi><mi>x<\/mi><mi mathvariant=\"normal\">\u2223<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=\\frac{x}{1+|x|}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"496\" height=\"336\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-20.png\" alt=\"image\" class=\"wp-image-1842\" title=\"Miscellaneous Exercise Solutions Class 12 Relations Functions\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-20.png 496w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-20-300x203.png 300w\" sizes=\"(max-width: 496px) 100vw, 496px\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Questions &amp; Step-by-step Solutions<\/h1>\n\n\n\n<h1 class=\"wp-block-heading\">Question 1<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Show that the function<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mi>x<\/mi><mrow><mn>1<\/mn><mo>+<\/mo><mi mathvariant=\"normal\">\u2223<\/mi><mi>x<\/mi><mi mathvariant=\"normal\">\u2223<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=\\frac{x}{1+|x|}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"496\" height=\"336\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-21.png\" alt=\"image\" class=\"wp-image-1843\" title=\"Miscellaneous Exercise Solutions Class 12 Relations Functions\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-21.png 496w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-21-300x203.png 300w\" sizes=\"(max-width: 496px) 100vw, 496px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">is one-one and onto.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Suppose:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f(x\u2081) = f(x\u2082)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Then:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>x\u2081\/(1+|x\u2081|) = x\u2082\/(1+|x\u2082|)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">After simplification:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>x\u2081 = x\u2082<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence function is one-one.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now let:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>y \u2208 (\u22121,1)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Choose:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>x = y\/(1\u2212|y|)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Then:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f(x)=y<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence function is onto.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f is bijective.<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 2<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Show that<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>x<\/mi><mn>3<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=x^3<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"532\" height=\"346\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-22.png\" alt=\"image\" class=\"wp-image-1844\" title=\"Miscellaneous Exercise Solutions Class 12 Relations Functions\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-22.png 532w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-22-300x195.png 300w\" sizes=\"(max-width: 532px) 100vw, 532px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">is injective.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Suppose:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f(x\u2081)=f(x\u2082)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Then:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>x\u2081^3=x\u2082^3<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Taking cube roots:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>x\u2081=x\u2082<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f is injective.<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 3<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Relation on Power Set<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Given:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>ARB iff A \u2282 B<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">For equivalence relation, relation must be reflexive, symmetric and transitive.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Here:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>A \u2282 A<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">is true.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence reflexive.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>A \u2282 B<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">does not imply:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>B \u2282 A<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore not symmetric.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>R is not an equivalence relation.<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 4<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Number of onto functions from {1,2,3,\u2026,n} to itself<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">For finite sets:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Every onto function from a set to itself is one-one.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence total onto functions are equal to total permutations.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Number of onto functions = n!<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 5<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Check whether functions f and g are equal.<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Given:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=x^2-x<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"486\" height=\"345\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-23.png\" alt=\"image\" class=\"wp-image-1845\" title=\"Miscellaneous Exercise Solutions Class 12 Relations Functions\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-23.png 486w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-23-300x213.png 300w\" sizes=\"(max-width: 486px) 100vw, 486px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">and<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>g<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>2<\/mn><mrow><mo fence=\"true\">\u2223<\/mo><mfrac><mi>x<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">\u2223<\/mo><\/mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">g(x)=2\\left|\\frac{x}{2}\\right|-1<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Evaluate functions for all elements of:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>A={\u22121,0,1,2}<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">For f(x)<\/h3>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>x<\/th><th>f(x)<\/th><\/tr><\/thead><tbody><tr><td>\u22121<\/td><td>2<\/td><\/tr><tr><td>0<\/td><td>0<\/td><\/tr><tr><td>1<\/td><td>0<\/td><\/tr><tr><td>2<\/td><td>2<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">For g(x)<\/h3>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>x<\/th><th>g(x)<\/th><\/tr><\/thead><tbody><tr><td>\u22121<\/td><td>\u22122<\/td><\/tr><tr><td>0<\/td><td>\u22121<\/td><\/tr><tr><td>1<\/td><td>0<\/td><\/tr><tr><td>2<\/td><td>1<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Since outputs differ:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f \u2260 g<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 6<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Number of relations containing (1,2) and (1,3) which are reflexive and symmetric but not transitive<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Required answer:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>(B) 2<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 7<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Number of equivalence relations containing (1,2)<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Possible equivalence relations are:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Relation where 1 and 2 belong to same class<\/li>\n\n\n\n<li>Universal relation<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">Hence:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>(B) 2<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Common Mistakes<\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Confusing subset with proper subset<\/li>\n\n\n\n<li>Forgetting onto condition<\/li>\n\n\n\n<li>Not checking symmetry separately<\/li>\n\n\n\n<li>Mistakes in proving injective functions<\/li>\n\n\n\n<li>Ignoring co-domain while checking onto<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Exam Tips<\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Always verify reflexive, symmetric and transitive properties separately.<\/li>\n\n\n\n<li>Use counterexamples to disprove properties.<\/li>\n\n\n\n<li>Remember: bijective = one-one + onto.<\/li>\n\n\n\n<li>Practice function mappings carefully.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Practice MCQs<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">MCQ 1<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A bijective function is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. One-one only<br>B. Onto only<br>C. Both one-one and onto<br>D. None<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Answer:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>C. Both one-one and onto<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">MCQ 2<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The number of onto functions from a finite set to itself equals:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. n<br>B. n\u00b2<br>C. n!<br>D. 2\u207f<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Answer:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>C. n!<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">MCQ 3<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Relation:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>A \u2282 B<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. Symmetric<br>B. Reflexive only<br>C. Equivalence relation<br>D. Universal relation<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Answer:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>B. Reflexive only<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">FAQ Section<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">What is an injective function?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A function where different inputs produce different outputs.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">What is a bijective function?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A function which is both one-one and onto.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">What is an equivalence relation?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A relation which is reflexive, symmetric and transitive.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">What is the number of onto functions from a finite set to itself?<\/h2>\n\n\n\n<pre class=\"wp-block-code\"><code>n!<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\">\ud83d\udcd8 Prepare Smarter with MyMockMate!<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2705 Chapter-wise NCERT Solutions<br>\u2705 Important Notes &amp; MCQs<br>\u2705 Online Mock Tests<br>\u2705 Instant Result &amp; Analysis<br>\u2705 CBSE Board Exam Preparation<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Start learning now on <a target=\"_blank\" rel=\"noreferrer noopener\" href=\"https:\/\/www.mymockmate.com?utm_source=chatgpt.com\">MyMockMate<\/a><\/p>\n\n    <div class=\"xs_social_share_widget xs_share_url after_content \t\tmain_content  wslu-style-1 wslu-share-box-shaped wslu-fill-colored wslu-none wslu-share-horizontal wslu-theme-font-no wslu-main_content\">\n\n\t\t\n        <ul>\n\t\t\t        <\/ul>\n    <\/div> \n","protected":false},"excerpt":{"rendered":"<p>Short Intro In this post, students can find complete step-by-step solutions for the Miscellaneous Exercise of Class 12<\/p>\n","protected":false},"author":1,"featured_media":1846,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_angie_page":false,"_surecart_dashboard_logo_width":"180px","_surecart_dashboard_show_logo":true,"_surecart_dashboard_navigation_orders":true,"_surecart_dashboard_navigation_invoices":true,"_surecart_dashboard_navigation_subscriptions":true,"_surecart_dashboard_navigation_downloads":true,"_surecart_dashboard_navigation_billing":true,"_surecart_dashboard_navigation_account":true,"postBodyCss":"","postBodyMargin":[],"postBodyPadding":[],"postBodyBackground":{"backgroundType":"classic","gradient":""},"page_builder":"","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}},"categories":[7,8,99],"tags":[196,187,175,174,186,210,209,180,212,197,190,207,211,206,53,47,194,195,208,177,173,176,189,188,184,198,178,179],"class_list":["post-1841","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-class-12","category-maths","category-relations-and-functions","tag-bijective-function","tag-cbse-board-maths-preparation","tag-cbse-class-12-maths","tag-class-12-maths","tag-class-12-maths-chapter-1-solutions","tag-counting-functions","tag-equality-of-functions","tag-equivalence-relation","tag-functions-and-mappings","tag-injective-function","tag-mathematical-relations","tag-miscellaneous-exercise-chapter-1","tag-miscellaneous-exercise-ncert-solutions","tag-miscellaneous-exercise-solutions","tag-ncert-maths-solutions","tag-ncert-solutions","tag-one-one-function","tag-onto-function","tag-power-set-relations","tag-reflexive-relation","tag-relations-and-functions","tag-relations-and-functions-chapter-1","tag-relations-and-functions-important-questions","tag-relations-and-functions-mcqs","tag-relations-and-functions-notes","tag-surjective-function","tag-symmetric-relation","tag-transitive-relation"],"jetpack_publicize_connections":[],"_links":{"self":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1841","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/comments?post=1841"}],"version-history":[{"count":1,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1841\/revisions"}],"predecessor-version":[{"id":1847,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1841\/revisions\/1847"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media\/1846"}],"wp:attachment":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media?parent=1841"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/categories?post=1841"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/tags?post=1841"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}