( ! ) Warning: preg_match_all(): Compilation failed: missing closing parenthesis at offset 51 in /home/kbxmjwdb/public_html/wp-content/plugins/seo-by-rank-math/includes/replace-variables/class-post-variables.php on line 543
Call Stack
#TimeMemoryFunctionLocation
10.0001354264{main}( ).../index.php:0
20.0001354624require( '/home/kbxmjwdb/public_html/wp-blog-header.php ).../index.php:17
30.241611781744require_once( '/home/kbxmjwdb/public_html/wp-includes/template-loader.php ).../wp-blog-header.php:19
40.344219510824include( '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/single.php ).../template-loader.php:106
50.344219510824get_header( $name = ???, $args = ??? ).../single.php:12
60.344219516048locate_template( $template_names = [0 => 'header.php'], $load = TRUE, $load_once = TRUE, $args = [] ).../general-template.php:48
70.344319516144load_template( $_template_file = '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/header.php', $load_once = TRUE, $args = [] ).../template.php:745
80.344319516560require_once( '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/header.php ).../template.php:810
90.344419518064wp_head( ).../header.php:16
100.344419518064do_action( $hook_name = 'wp_head' ).../general-template.php:3050
110.344419518440WP_Hook->do_action( $args = [0 => ''] ).../plugin.php:517
120.344419518440WP_Hook->apply_filters( $value = '', $args = [0 => ''] ).../class-wp-hook.php:348
130.364620460744RankMath\Frontend\Head->head( '' ).../class-wp-hook.php:324
140.364620460432RankMath\Frontend\Head->do_action( ...$args = variadic('head') ).../class-head.php:172
150.364620460848do_action_ref_array( $hook_name = 'rank_math/head', $args = [] ).../class-hooker.php:90
160.364620460848WP_Hook->do_action( $args = [] ).../plugin.php:565
170.364620460848WP_Hook->apply_filters( $value = '', $args = [] ).../class-wp-hook.php:348
180.365420545400RankMath\Frontend\Head->metadesc( ).../class-wp-hook.php:324
190.365420545400RankMath\Paper\Paper->get_description( ).../class-head.php:201
200.365420545400RankMath\Paper\Singular->description( ).../class-paper.php:193
210.365420545400RankMath\Paper\Singular->get_post_description( $object = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' } ).../class-singular.php:47
220.365420545456RankMath\Paper\Paper::get_from_options( $id = 'pt_post_description', $object = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' }, $default = ??? ).../class-singular.php:173
230.365420545456RankMath\Helper::replace_vars( $content = '%excerpt%', $args = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' }, $exclude = ??? ).../class-paper.php:477
240.365420545512RankMath\Replace_Variables\Replacer->replace( $string = '%excerpt%', $args = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' }, $exclude = [] ).../class-helper.php:46
250.365420545552RankMath\Replace_Variables\Replacer->set_up_replacements( $string = '%excerpt%' ).../class-replacer.php:104
260.365420547536RankMath\Replace_Variables\Replacer->get_variable_value( $matches = [0 => [0 => '%excerpt%'], 1 => [0 => 'excerpt'], 2 => [0 => ''], 3 => [0 => '']], $index = 0, $id = 'excerpt' ).../class-replacer.php:178
270.365420547536RankMath\Replace_Variables\Variable->run_callback( $var_args = [], $args = class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = ''; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' } ).../class-replacer.php:295
280.365420547536call_user_func:{/home/kbxmjwdb/public_html/wp-content/plugins/seo-by-rank-math/includes/replace-variables/class-variable.php:188}( $callback = [0 => class RankMath\Replace_Variables\Manager { public $post = class WP_Post { ... }; public $args = class stdClass { ... }; private ${RankMath\Replace_Variables\Cache}cache = [...]; protected $counters = [...]; public $is_post_edit = FALSE; public $is_term_edit = FALSE; protected $is_user_edit = FALSE; protected $replacements = [...]; public $remove_non_replaced = TRUE; private $is_setup = TRUE; protected $tmp_args = class stdClass { ... } }, 1 => 'get_excerpt'], $args = [], class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = ''; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' } ).../class-variable.php:188
290.365420547864RankMath\Replace_Variables\Post_Variables->get_excerpt( [], class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = ''; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' } ).../class-variable.php:188
300.365420547864RankMath\Replace_Variables\Post_Variables->get_post_content( $object = class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = ''; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' } ).../class-post-variables.php:365
310.371420613560preg_match_all( $pattern = '/<p>(.*Chapter.1.Sets.(समुच्चय.*)<\\/p>/iu', $subject = '<p>Chapter 1 Sets (समुच्चय).</p>\n\n\nTable of Contents\n\n\n\n\nप्रश्नावली 1.1\nप्रश्नावली 1.2\nप्रश्नावली 1.3\nप्रश्नावली 1.4\nप्रश्नावली 1.5\nप्रश्नावली 1.6\n\n\n\n\nप्रश्नावली 1.1\n<p>प्रश्न 1.निम्नलिखित में कौन से समुच्चय हैं? अपने उत्तर का औचित्य बताइए।(i) '..., $matches = NULL ).../class-post-variables.php:543

( ! ) Warning: preg_match_all(): Compilation failed: missing closing parenthesis at offset 51 in /home/kbxmjwdb/public_html/wp-content/plugins/seo-by-rank-math/includes/replace-variables/class-post-variables.php on line 543
Call Stack
#TimeMemoryFunctionLocation
10.0001354264{main}( ).../index.php:0
20.0001354624require( '/home/kbxmjwdb/public_html/wp-blog-header.php ).../index.php:17
30.241611781744require_once( '/home/kbxmjwdb/public_html/wp-includes/template-loader.php ).../wp-blog-header.php:19
40.344219510824include( '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/single.php ).../template-loader.php:106
50.344219510824get_header( $name = ???, $args = ??? ).../single.php:12
60.344219516048locate_template( $template_names = [0 => 'header.php'], $load = TRUE, $load_once = TRUE, $args = [] ).../general-template.php:48
70.344319516144load_template( $_template_file = '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/header.php', $load_once = TRUE, $args = [] ).../template.php:745
80.344319516560require_once( '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/header.php ).../template.php:810
90.344419518064wp_head( ).../header.php:16
100.344419518064do_action( $hook_name = 'wp_head' ).../general-template.php:3050
110.344419518440WP_Hook->do_action( $args = [0 => ''] ).../plugin.php:517
120.344419518440WP_Hook->apply_filters( $value = '', $args = [0 => ''] ).../class-wp-hook.php:348
130.364620460744RankMath\Frontend\Head->head( '' ).../class-wp-hook.php:324
140.364620460432RankMath\Frontend\Head->do_action( ...$args = variadic('head') ).../class-head.php:172
150.364620460848do_action_ref_array( $hook_name = 'rank_math/head', $args = [] ).../class-hooker.php:90
160.364620460848WP_Hook->do_action( $args = [] ).../plugin.php:565
170.364620460848WP_Hook->apply_filters( $value = '', $args = [] ).../class-wp-hook.php:348
180.376620620720RankMath\Schema\JsonLD->json_ld( ).../class-wp-hook.php:324
190.376620620720RankMath\Schema\JsonLD->get_parts( ).../class-jsonld.php:140
200.376820622208RankMath\Helper::replace_vars( $content = '%excerpt%', $args = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' }, $exclude = ??? ).../class-jsonld.php:751
210.376820622264RankMath\Replace_Variables\Replacer->replace( $string = '%excerpt%', $args = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' }, $exclude = [] ).../class-helper.php:46
220.376820622304RankMath\Replace_Variables\Replacer->set_up_replacements( $string = '%excerpt%' ).../class-replacer.php:104
230.376820622304RankMath\Replace_Variables\Replacer->get_cache( $string = '%excerpt%' ).../class-replacer.php:168
240.376820623440RankMath\Replace_Variables\Variable->run_callback( $var_args = '', $args = class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = ''; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' } ).../class-replacer.php:254
250.376820623440call_user_func:{/home/kbxmjwdb/public_html/wp-content/plugins/seo-by-rank-math/includes/replace-variables/class-variable.php:188}( $callback = [0 => class RankMath\Replace_Variables\Manager { public $post = class WP_Post { ... }; public $args = class stdClass { ... }; private ${RankMath\Replace_Variables\Cache}cache = [...]; protected $counters = [...]; public $is_post_edit = FALSE; public $is_term_edit = FALSE; protected $is_user_edit = FALSE; protected $replacements = [...]; public $remove_non_replaced = TRUE; private $is_setup = TRUE; protected $tmp_args = class stdClass { ... } }, 1 => 'get_excerpt'], $args = '', class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = ''; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' } ).../class-variable.php:188
260.376820623768RankMath\Replace_Variables\Post_Variables->get_excerpt( '', class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = ''; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' } ).../class-variable.php:188
270.376820623768RankMath\Replace_Variables\Post_Variables->get_post_content( $object = class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = ''; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' } ).../class-post-variables.php:365
280.382620689464preg_match_all( $pattern = '/<p>(.*Chapter.1.Sets.(समुच्चय.*)<\\/p>/iu', $subject = '<p>Chapter 1 Sets (समुच्चय).</p>\n\n\nTable of Contents\n\n\n\n\nप्रश्नावली 1.1\nप्रश्नावली 1.2\nप्रश्नावली 1.3\nप्रश्नावली 1.4\nप्रश्नावली 1.5\nप्रश्नावली 1.6\n\n\n\n\nप्रश्नावली 1.1\n<p>प्रश्न 1.निम्नलिखित में कौन से समुच्चय हैं? अपने उत्तर का औचित्य बताइए।(i) '..., $matches = NULL ).../class-post-variables.php:543

Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated

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Call Stack
#TimeMemoryFunctionLocation
10.0001354264{main}( ).../index.php:0
20.0001354624require( '/home/kbxmjwdb/public_html/wp-blog-header.php ).../index.php:17
30.241611781744require_once( '/home/kbxmjwdb/public_html/wp-includes/template-loader.php ).../wp-blog-header.php:19
40.344219510824include( '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/single.php ).../template-loader.php:106
50.431221197136generate_do_template_part( $template = 'single' ).../single.php:29
60.431221197136get_template_part( $slug = 'content', $name = 'single', $args = ??? ).../theme-functions.php:568
70.431221197656locate_template( $template_names = [0 => 'content-single.php', 1 => 'content.php'], $load = TRUE, $load_once = FALSE, $args = [] ).../general-template.php:206
80.431221197768load_template( $_template_file = '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/content-single.php', $load_once = FALSE, $args = [] ).../template.php:745
90.431321198184require( '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/content-single.php ).../template.php:812
100.432621204952the_content( $more_link_text = ???, $strip_teaser = ??? ).../content-single.php:73
110.432721204952apply_filters( $hook_name = 'the_content', $value = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'... ).../post-template.php:256
120.432721205360WP_Hook->apply_filters( $value = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'..., $args = [0 => '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...] ).../plugin.php:205
130.604222897296RankMath\Image_Seo\Add_Attributes->add_img_attributes( $content = '<!-- FeedbackWP Plugin - Results Widget --><div class="rmp-results-widget js-rmp-results-widget js-rmp-results-widget--12034 rmp-results-widget--not-rated" data-post-id="12034"> <div class="rmp-results-widget__visual-rating"> <i class="js-rmp-results-icon rmp-icon rmp-icon--ratings rmp-icon--star "></i><i class="js-rmp-results-icon rmp-icon rmp-icon--ratings rmp-icon--star "></i><i class="js-rmp-results-icon rmp-icon rmp-icon--ratings rmp-icon--star "></i><i class="js-rmp-results-icon rmp-icon rmp-ico'..., $post_id = ??? ).../class-wp-hook.php:324
140.605023007296RankMath\Image_Seo\Add_Attributes->set_image_attribute( $attrs = ['src' => 'https://images-eu.ssl-images-amazon.com/images/I/81faLxLqxFL._AC_UL160_SR160,160_.jpg', 'alt' => '', 'style' => 'max-height:160px'], $attribute = 'alt', $condition = '%excerpt%', $is_dirty = TRUE, $post = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/81faLxLqxFL._AC_UL160_SR160,160_.jpg'; public $alttext = ''; public $titletext = '' } ).../class-add-attributes.php:106
150.605023007296RankMath\Helper::replace_vars( $content = '%excerpt%', $args = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/81faLxLqxFL._AC_UL160_SR160,160_.jpg'; public $alttext = ''; public $titletext = '' }, $exclude = ??? ).../class-add-attributes.php:145
160.605023007352RankMath\Replace_Variables\Replacer->replace( $string = '%excerpt%', $args = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/81faLxLqxFL._AC_UL160_SR160,160_.jpg'; public $alttext = ''; public $titletext = '' }, $exclude = [] ).../class-helper.php:46
170.605023007152RankMath\Replace_Variables\Replacer->set_up_replacements( $string = '%excerpt%' ).../class-replacer.php:104
180.605023007152RankMath\Replace_Variables\Replacer->get_cache( $string = '%excerpt%' ).../class-replacer.php:168
190.605023007592RankMath\Replace_Variables\Variable->run_callback( $var_args = '', $args = class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/81faLxLqxFL._AC_UL160_SR160,160_.jpg'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = ''; public $titletext = '' } ).../class-replacer.php:254
200.605023007592call_user_func:{/home/kbxmjwdb/public_html/wp-content/plugins/seo-by-rank-math/includes/replace-variables/class-variable.php:188}( $callback = [0 => class RankMath\Replace_Variables\Manager { public $post = class WP_Post { ... }; public $args = class stdClass { ... }; private ${RankMath\Replace_Variables\Cache}cache = [...]; protected $counters = [...]; public $is_post_edit = FALSE; public $is_term_edit = FALSE; protected $is_user_edit = FALSE; protected $replacements = [...]; public $remove_non_replaced = TRUE; private $is_setup = TRUE; protected $tmp_args = class stdClass { ... } }, 1 => 'get_excerpt'], $args = '', class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/81faLxLqxFL._AC_UL160_SR160,160_.jpg'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = ''; public $titletext = '' } ).../class-variable.php:188
210.605023007920RankMath\Replace_Variables\Post_Variables->get_excerpt( '', class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/81faLxLqxFL._AC_UL160_SR160,160_.jpg'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = ''; public $titletext = '' } ).../class-variable.php:188
220.605023007920RankMath\Replace_Variables\Post_Variables->get_post_content( $object = class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/81faLxLqxFL._AC_UL160_SR160,160_.jpg'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = ''; public $titletext = '' } ).../class-post-variables.php:365
230.611023073616preg_match_all( $pattern = '/<p>(.*Chapter.1.Sets.(समुच्चय.*)<\\/p>/iu', $subject = '<p>Chapter 1 Sets (समुच्चय).</p>\n\n\nTable of Contents\n\n\n\n\nप्रश्नावली 1.1\nप्रश्नावली 1.2\nप्रश्नावली 1.3\nप्रश्नावली 1.4\nप्रश्नावली 1.5\nप्रश्नावली 1.6\n\n\n\n\nप्रश्नावली 1.1\n<p>प्रश्न 1.निम्नलिखित में कौन से समुच्चय हैं? अपने उत्तर का औचित्य बताइए।(i) '..., $matches = NULL ).../class-post-variables.php:543

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Call Stack
#TimeMemoryFunctionLocation
10.0001354264{main}( ).../index.php:0
20.0001354624require( '/home/kbxmjwdb/public_html/wp-blog-header.php ).../index.php:17
30.241611781744require_once( '/home/kbxmjwdb/public_html/wp-includes/template-loader.php ).../wp-blog-header.php:19
40.344219510824include( '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/single.php ).../template-loader.php:106
50.431221197136generate_do_template_part( $template = 'single' ).../single.php:29
60.431221197136get_template_part( $slug = 'content', $name = 'single', $args = ??? ).../theme-functions.php:568
70.431221197656locate_template( $template_names = [0 => 'content-single.php', 1 => 'content.php'], $load = TRUE, $load_once = FALSE, $args = [] ).../general-template.php:206
80.431221197768load_template( $_template_file = '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/content-single.php', $load_once = FALSE, $args = [] ).../template.php:745
90.431321198184require( '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/content-single.php ).../template.php:812
100.432621204952the_content( $more_link_text = ???, $strip_teaser = ??? ).../content-single.php:73
110.432721204952apply_filters( $hook_name = 'the_content', $value = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'... ).../post-template.php:256
120.432721205360WP_Hook->apply_filters( $value = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'..., $args = [0 => '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...] ).../plugin.php:205
130.604222897296RankMath\Image_Seo\Add_Attributes->add_img_attributes( $content = '<!-- FeedbackWP Plugin - Results Widget --><div class="rmp-results-widget js-rmp-results-widget js-rmp-results-widget--12034 rmp-results-widget--not-rated" data-post-id="12034"> <div class="rmp-results-widget__visual-rating"> <i class="js-rmp-results-icon rmp-icon rmp-icon--ratings rmp-icon--star "></i><i class="js-rmp-results-icon rmp-icon rmp-icon--ratings rmp-icon--star "></i><i class="js-rmp-results-icon rmp-icon rmp-icon--ratings rmp-icon--star "></i><i class="js-rmp-results-icon rmp-icon rmp-ico'..., $post_id = ??? ).../class-wp-hook.php:324
140.611923040176RankMath\Image_Seo\Add_Attributes->set_image_attribute( $attrs = ['src' => 'https://images-eu.ssl-images-amazon.com/images/I/911b573N8JL._AC_UL160_SR160,160_.jpg', 'alt' => '', 'style' => 'max-height:160px'], $attribute = 'alt', $condition = '%excerpt%', $is_dirty = TRUE, $post = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/911b573N8JL._AC_UL160_SR160,160_.jpg'; public $alttext = ''; public $titletext = '' } ).../class-add-attributes.php:106
150.611923040176RankMath\Helper::replace_vars( $content = '%excerpt%', $args = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/911b573N8JL._AC_UL160_SR160,160_.jpg'; public $alttext = ''; public $titletext = '' }, $exclude = ??? ).../class-add-attributes.php:145
160.611923040232RankMath\Replace_Variables\Replacer->replace( $string = '%excerpt%', $args = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/911b573N8JL._AC_UL160_SR160,160_.jpg'; public $alttext = ''; public $titletext = '' }, $exclude = [] ).../class-helper.php:46
170.611923040160RankMath\Replace_Variables\Replacer->set_up_replacements( $string = '%excerpt%' ).../class-replacer.php:104
180.611923040160RankMath\Replace_Variables\Replacer->get_cache( $string = '%excerpt%' ).../class-replacer.php:168
190.611923040600RankMath\Replace_Variables\Variable->run_callback( $var_args = '', $args = class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/911b573N8JL._AC_UL160_SR160,160_.jpg'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = ''; public $titletext = '' } ).../class-replacer.php:254
200.611923040600call_user_func:{/home/kbxmjwdb/public_html/wp-content/plugins/seo-by-rank-math/includes/replace-variables/class-variable.php:188}( $callback = [0 => class RankMath\Replace_Variables\Manager { public $post = class WP_Post { ... }; public $args = class stdClass { ... }; private ${RankMath\Replace_Variables\Cache}cache = [...]; protected $counters = [...]; public $is_post_edit = FALSE; public $is_term_edit = FALSE; protected $is_user_edit = FALSE; protected $replacements = [...]; public $remove_non_replaced = TRUE; private $is_setup = TRUE; protected $tmp_args = class stdClass { ... } }, 1 => 'get_excerpt'], $args = '', class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/911b573N8JL._AC_UL160_SR160,160_.jpg'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = ''; public $titletext = '' } ).../class-variable.php:188
210.611923040928RankMath\Replace_Variables\Post_Variables->get_excerpt( '', class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/911b573N8JL._AC_UL160_SR160,160_.jpg'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = ''; public $titletext = '' } ).../class-variable.php:188
220.611923040928RankMath\Replace_Variables\Post_Variables->get_post_content( $object = class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/911b573N8JL._AC_UL160_SR160,160_.jpg'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = ''; public $titletext = '' } ).../class-post-variables.php:365
230.617723106576preg_match_all( $pattern = '/<p>(.*Chapter.1.Sets.(समुच्चय.*)<\\/p>/iu', $subject = '<p>Chapter 1 Sets (समुच्चय).</p>\n\n\nTable of Contents\n\n\n\n\nप्रश्नावली 1.1\nप्रश्नावली 1.2\nप्रश्नावली 1.3\nप्रश्नावली 1.4\nप्रश्नावली 1.5\nप्रश्नावली 1.6\n\n\n\n\nप्रश्नावली 1.1\n<p>प्रश्न 1.निम्नलिखित में कौन से समुच्चय हैं? अपने उत्तर का औचित्य बताइए।(i) '..., $matches = NULL ).../class-post-variables.php:543

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Call Stack
#TimeMemoryFunctionLocation
10.0001354264{main}( ).../index.php:0
20.0001354624require( '/home/kbxmjwdb/public_html/wp-blog-header.php ).../index.php:17
30.241611781744require_once( '/home/kbxmjwdb/public_html/wp-includes/template-loader.php ).../wp-blog-header.php:19
40.344219510824include( '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/single.php ).../template-loader.php:106
50.431221197136generate_do_template_part( $template = 'single' ).../single.php:29
60.431221197136get_template_part( $slug = 'content', $name = 'single', $args = ??? ).../theme-functions.php:568
70.431221197656locate_template( $template_names = [0 => 'content-single.php', 1 => 'content.php'], $load = TRUE, $load_once = FALSE, $args = [] ).../general-template.php:206
80.431221197768load_template( $_template_file = '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/content-single.php', $load_once = FALSE, $args = [] ).../template.php:745
90.431321198184require( '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/content-single.php ).../template.php:812
100.432621204952the_content( $more_link_text = ???, $strip_teaser = ??? ).../content-single.php:73
110.432721204952apply_filters( $hook_name = 'the_content', $value = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'... ).../post-template.php:256
120.432721205360WP_Hook->apply_filters( $value = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'..., $args = [0 => '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...] ).../plugin.php:205
130.604222897296RankMath\Image_Seo\Add_Attributes->add_img_attributes( $content = '<!-- FeedbackWP Plugin - Results Widget --><div class="rmp-results-widget js-rmp-results-widget js-rmp-results-widget--12034 rmp-results-widget--not-rated" data-post-id="12034"> <div class="rmp-results-widget__visual-rating"> <i class="js-rmp-results-icon rmp-icon rmp-icon--ratings rmp-icon--star "></i><i class="js-rmp-results-icon rmp-icon rmp-icon--ratings rmp-icon--star "></i><i class="js-rmp-results-icon rmp-icon rmp-icon--ratings rmp-icon--star "></i><i class="js-rmp-results-icon rmp-icon rmp-ico'..., $post_id = ??? ).../class-wp-hook.php:324
140.618723072944RankMath\Image_Seo\Add_Attributes->set_image_attribute( $attrs = ['src' => 'https://images-eu.ssl-images-amazon.com/images/I/61BXe8wgQUL._AC_UL160_SR160,160_.jpg', 'alt' => '', 'style' => 'max-height:160px'], $attribute = 'alt', $condition = '%excerpt%', $is_dirty = TRUE, $post = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/61BXe8wgQUL._AC_UL160_SR160,160_.jpg'; public $alttext = ''; public $titletext = '' } ).../class-add-attributes.php:106
150.618723072944RankMath\Helper::replace_vars( $content = '%excerpt%', $args = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/61BXe8wgQUL._AC_UL160_SR160,160_.jpg'; public $alttext = ''; public $titletext = '' }, $exclude = ??? ).../class-add-attributes.php:145
160.618723073000RankMath\Replace_Variables\Replacer->replace( $string = '%excerpt%', $args = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/61BXe8wgQUL._AC_UL160_SR160,160_.jpg'; public $alttext = ''; public $titletext = '' }, $exclude = [] ).../class-helper.php:46
170.618723072928RankMath\Replace_Variables\Replacer->set_up_replacements( $string = '%excerpt%' ).../class-replacer.php:104
180.618723072928RankMath\Replace_Variables\Replacer->get_cache( $string = '%excerpt%' ).../class-replacer.php:168
190.618723073368RankMath\Replace_Variables\Variable->run_callback( $var_args = '', $args = class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/61BXe8wgQUL._AC_UL160_SR160,160_.jpg'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = ''; public $titletext = '' } ).../class-replacer.php:254
200.618723073368call_user_func:{/home/kbxmjwdb/public_html/wp-content/plugins/seo-by-rank-math/includes/replace-variables/class-variable.php:188}( $callback = [0 => class RankMath\Replace_Variables\Manager { public $post = class WP_Post { ... }; public $args = class stdClass { ... }; private ${RankMath\Replace_Variables\Cache}cache = [...]; protected $counters = [...]; public $is_post_edit = FALSE; public $is_term_edit = FALSE; protected $is_user_edit = FALSE; protected $replacements = [...]; public $remove_non_replaced = TRUE; private $is_setup = TRUE; protected $tmp_args = class stdClass { ... } }, 1 => 'get_excerpt'], $args = '', class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/61BXe8wgQUL._AC_UL160_SR160,160_.jpg'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = ''; public $titletext = '' } ).../class-variable.php:188
210.618723073696RankMath\Replace_Variables\Post_Variables->get_excerpt( '', class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/61BXe8wgQUL._AC_UL160_SR160,160_.jpg'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = ''; public $titletext = '' } ).../class-variable.php:188
220.618723073696RankMath\Replace_Variables\Post_Variables->get_post_content( $object = class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/61BXe8wgQUL._AC_UL160_SR160,160_.jpg'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = ''; public $titletext = '' } ).../class-post-variables.php:365
230.624623139344preg_match_all( $pattern = '/<p>(.*Chapter.1.Sets.(समुच्चय.*)<\\/p>/iu', $subject = '<p>Chapter 1 Sets (समुच्चय).</p>\n\n\nTable of Contents\n\n\n\n\nप्रश्नावली 1.1\nप्रश्नावली 1.2\nप्रश्नावली 1.3\nप्रश्नावली 1.4\nप्रश्नावली 1.5\nप्रश्नावली 1.6\n\n\n\n\nप्रश्नावली 1.1\n<p>प्रश्न 1.निम्नलिखित में कौन से समुच्चय हैं? अपने उत्तर का औचित्य बताइए।(i) '..., $matches = NULL ).../class-post-variables.php:543

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Call Stack
#TimeMemoryFunctionLocation
10.0001354264{main}( ).../index.php:0
20.0001354624require( '/home/kbxmjwdb/public_html/wp-blog-header.php ).../index.php:17
30.241611781744require_once( '/home/kbxmjwdb/public_html/wp-includes/template-loader.php ).../wp-blog-header.php:19
40.344219510824include( '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/single.php ).../template-loader.php:106
50.431221197136generate_do_template_part( $template = 'single' ).../single.php:29
60.431221197136get_template_part( $slug = 'content', $name = 'single', $args = ??? ).../theme-functions.php:568
70.431221197656locate_template( $template_names = [0 => 'content-single.php', 1 => 'content.php'], $load = TRUE, $load_once = FALSE, $args = [] ).../general-template.php:206
80.431221197768load_template( $_template_file = '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/content-single.php', $load_once = FALSE, $args = [] ).../template.php:745
90.431321198184require( '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/content-single.php ).../template.php:812
100.432621204952the_content( $more_link_text = ???, $strip_teaser = ??? ).../content-single.php:73
110.432721204952apply_filters( $hook_name = 'the_content', $value = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'... ).../post-template.php:256
120.432721205360WP_Hook->apply_filters( $value = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'..., $args = [0 => '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...] ).../plugin.php:205
130.604222897296RankMath\Image_Seo\Add_Attributes->add_img_attributes( $content = '<!-- FeedbackWP Plugin - Results Widget --><div class="rmp-results-widget js-rmp-results-widget js-rmp-results-widget--12034 rmp-results-widget--not-rated" data-post-id="12034"> <div class="rmp-results-widget__visual-rating"> <i class="js-rmp-results-icon rmp-icon rmp-icon--ratings rmp-icon--star "></i><i class="js-rmp-results-icon rmp-icon rmp-icon--ratings rmp-icon--star "></i><i class="js-rmp-results-icon rmp-icon rmp-icon--ratings rmp-icon--star "></i><i class="js-rmp-results-icon rmp-icon rmp-ico'..., $post_id = ??? ).../class-wp-hook.php:324
140.625423101616RankMath\Image_Seo\Add_Attributes->set_image_attribute( $attrs = ['src' => 'https://images-eu.ssl-images-amazon.com/images/I/71uSXJRRwSL._AC_UL160_SR160,160_.jpg', 'alt' => '', 'style' => 'max-height:160px'], $attribute = 'alt', $condition = '%excerpt%', $is_dirty = TRUE, $post = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/71uSXJRRwSL._AC_UL160_SR160,160_.jpg'; public $alttext = ''; public $titletext = '' } ).../class-add-attributes.php:106
150.625423101616RankMath\Helper::replace_vars( $content = '%excerpt%', $args = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/71uSXJRRwSL._AC_UL160_SR160,160_.jpg'; public $alttext = ''; public $titletext = '' }, $exclude = ??? ).../class-add-attributes.php:145
160.625423101672RankMath\Replace_Variables\Replacer->replace( $string = '%excerpt%', $args = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/71uSXJRRwSL._AC_UL160_SR160,160_.jpg'; public $alttext = ''; public $titletext = '' }, $exclude = [] ).../class-helper.php:46
170.625423101600RankMath\Replace_Variables\Replacer->set_up_replacements( $string = '%excerpt%' ).../class-replacer.php:104
180.625423101600RankMath\Replace_Variables\Replacer->get_cache( $string = '%excerpt%' ).../class-replacer.php:168
190.625423102040RankMath\Replace_Variables\Variable->run_callback( $var_args = '', $args = class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/71uSXJRRwSL._AC_UL160_SR160,160_.jpg'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = ''; public $titletext = '' } ).../class-replacer.php:254
200.625423102040call_user_func:{/home/kbxmjwdb/public_html/wp-content/plugins/seo-by-rank-math/includes/replace-variables/class-variable.php:188}( $callback = [0 => class RankMath\Replace_Variables\Manager { public $post = class WP_Post { ... }; public $args = class stdClass { ... }; private ${RankMath\Replace_Variables\Cache}cache = [...]; protected $counters = [...]; public $is_post_edit = FALSE; public $is_term_edit = FALSE; protected $is_user_edit = FALSE; protected $replacements = [...]; public $remove_non_replaced = TRUE; private $is_setup = TRUE; protected $tmp_args = class stdClass { ... } }, 1 => 'get_excerpt'], $args = '', class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/71uSXJRRwSL._AC_UL160_SR160,160_.jpg'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = ''; public $titletext = '' } ).../class-variable.php:188
210.625423102368RankMath\Replace_Variables\Post_Variables->get_excerpt( '', class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/71uSXJRRwSL._AC_UL160_SR160,160_.jpg'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = ''; public $titletext = '' } ).../class-variable.php:188
220.625423102368RankMath\Replace_Variables\Post_Variables->get_post_content( $object = class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/71uSXJRRwSL._AC_UL160_SR160,160_.jpg'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = ''; public $titletext = '' } ).../class-post-variables.php:365
230.631223168016preg_match_all( $pattern = '/<p>(.*Chapter.1.Sets.(समुच्चय.*)<\\/p>/iu', $subject = '<p>Chapter 1 Sets (समुच्चय).</p>\n\n\nTable of Contents\n\n\n\n\nप्रश्नावली 1.1\nप्रश्नावली 1.2\nप्रश्नावली 1.3\nप्रश्नावली 1.4\nप्रश्नावली 1.5\nप्रश्नावली 1.6\n\n\n\n\nप्रश्नावली 1.1\n<p>प्रश्न 1.निम्नलिखित में कौन से समुच्चय हैं? अपने उत्तर का औचित्य बताइए।(i) '..., $matches = NULL ).../class-post-variables.php:543

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Call Stack
#TimeMemoryFunctionLocation
10.0001354264{main}( ).../index.php:0
20.0001354624require( '/home/kbxmjwdb/public_html/wp-blog-header.php ).../index.php:17
30.241611781744require_once( '/home/kbxmjwdb/public_html/wp-includes/template-loader.php ).../wp-blog-header.php:19
40.344219510824include( '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/single.php ).../template-loader.php:106
50.431221197136generate_do_template_part( $template = 'single' ).../single.php:29
60.431221197136get_template_part( $slug = 'content', $name = 'single', $args = ??? ).../theme-functions.php:568
70.431221197656locate_template( $template_names = [0 => 'content-single.php', 1 => 'content.php'], $load = TRUE, $load_once = FALSE, $args = [] ).../general-template.php:206
80.431221197768load_template( $_template_file = '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/content-single.php', $load_once = FALSE, $args = [] ).../template.php:745
90.431321198184require( '/home/kbxmjwdb/public_html/wp-content/themes/generatepress/content-single.php ).../template.php:812
100.432621204952the_content( $more_link_text = ???, $strip_teaser = ??? ).../content-single.php:73
110.432721204952apply_filters( $hook_name = 'the_content', $value = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'... ).../post-template.php:256
120.432721205360WP_Hook->apply_filters( $value = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'..., $args = [0 => '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...] ).../plugin.php:205
130.604222897296RankMath\Image_Seo\Add_Attributes->add_img_attributes( $content = '<!-- FeedbackWP Plugin - Results Widget --><div class="rmp-results-widget js-rmp-results-widget js-rmp-results-widget--12034 rmp-results-widget--not-rated" data-post-id="12034"> <div class="rmp-results-widget__visual-rating"> <i class="js-rmp-results-icon rmp-icon rmp-icon--ratings rmp-icon--star "></i><i class="js-rmp-results-icon rmp-icon rmp-icon--ratings rmp-icon--star "></i><i class="js-rmp-results-icon rmp-icon rmp-icon--ratings rmp-icon--star "></i><i class="js-rmp-results-icon rmp-icon rmp-ico'..., $post_id = ??? ).../class-wp-hook.php:324
140.632023134384RankMath\Image_Seo\Add_Attributes->set_image_attribute( $attrs = ['src' => 'https://images-eu.ssl-images-amazon.com/images/I/71ZVZaDkCuL._AC_UL160_SR160,160_.jpg', 'alt' => '', 'style' => 'max-height:160px'], $attribute = 'alt', $condition = '%excerpt%', $is_dirty = TRUE, $post = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/71ZVZaDkCuL._AC_UL160_SR160,160_.jpg'; public $alttext = ''; public $titletext = '' } ).../class-add-attributes.php:106
150.632023134384RankMath\Helper::replace_vars( $content = '%excerpt%', $args = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/71ZVZaDkCuL._AC_UL160_SR160,160_.jpg'; public $alttext = ''; public $titletext = '' }, $exclude = ??? ).../class-add-attributes.php:145
160.632023134440RankMath\Replace_Variables\Replacer->replace( $string = '%excerpt%', $args = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $filename = 'https://images-eu.ssl-images-amazon.com/images/I/71ZVZaDkCuL._AC_UL160_SR160,160_.jpg'; public $alttext = ''; public $titletext = '' }, $exclude = [] ).../class-helper.php:46
170.632023134368RankMath\Replace_Variables\Replacer->set_up_replacements( $string = '%excerpt%' ).../class-replacer.php:104
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Chapter 1 Sets (समुच्चय).

Table of Contents

प्रश्नावली 1.1

प्रश्न 1.
निम्नलिखित में कौन से समुच्चय हैं? अपने उत्तर का औचित्य बताइए।
(i) j अक्षर से प्रारम्भ होने वाले वर्ष के सभी महीनों का संग्रह।
(ii) भारत के दस सबसे अधिक प्रतिभाशाली लेखकों का संग्रह।
(iii) विश्व के सर्वश्रेष्ठ ग्यारह बल्लबाजों का संग्रह।
(iv) आप की कक्षा के सभी बालकों का संग्रह।
(v) 100 से कम सभी प्राकृत संख्याओं का संग्रह।
(vi) लेखक प्रेमचन्द द्वारा लिखित उपन्यासों का संग्रह।
(vii) सर्भीसम पूर्णाकों का संग्रह।
(viii) इस अध्याय में आने वाले प्रश्नों का संग्रह।
(ix) विश्व में सबसे अधिक खतरनाक जानवरों का संग्रह।
हल:
(i) j से शुरु होने वाले महीनों के नाम : जनवरी, जून व जुलाई। अतः यह एक समुच्चय है।
(ii) प्रतिभाशाली लेखक को परिभाषित नहीं किया जा सकता। इसीलिए यह एक समुच्चय नहीं है।
(iii) सर्वश्रेष्ठ बल्लेबाज को परिभाषित नहीं कर सकते। अतः यह एक समुच्चय नहीं है।
(iv) कक्षा के सभी विद्यार्थियों की संख्या निश्चित होती है। अतः यह एक समुच्चय है।
(v) 100 से कम प्राकृत संख्याएँ 1, 2, 3, …….. 99 हैं। अतः यह एक समुच्चय है।
(vi) लेखक प्रेमचन्द्र द्वारा लिखित उपन्यासों का संग्रह गबन, गोदान आदि द्वारा परिभाषित हैं। अतः यह एक समुच्चय है।
(vii) समपूर्णांक {….. -6, -4. 2, 4, 6, ….} हैं। इसलिए यह एक समुच्चय है।
(viii) इस अध्याय के प्रश्न परिभाषित हैं। अतः यह एक समुच्चय है।
(ix) संसार के सबसे अधिक खतरनाक पशुओं के संग्रह को परिभाषित नही किया जा सकता। इसलिए यह एक समुच्चय नहीं है।

प्रश्न 2.
मान लीजिए A = {1, 2, 3, 4, 5, 6}. रिक्त स्थानों में उपयुक्त प्रतीक ∈अथवा ∉ भरिए।
(i) 5 …… A
(ii) 8 …….. A
(iii) 0 ……… A
(iv) 4 ……. A
(v) 2 …….. A
(vi) 10 ……. A
हल:
(i) 5 ∈ A
(ii) 8 ∉ A
(iii) 0 ∉ 4
(iv) 4 ∈ A
(v) 2 ∈ A
(vi) 10 ∉ A

प्रश्न 3.
निम्नलिखित समुच्चयों को रोस्टर रूप में लिखिए:
(i) A = {x : x एक पूर्णाक है और -3 < x < 7}
(ii) B = {x : x संख्या 6 से कम एक प्राकृत संख्या है।}
(iii) C = {x : x दो अंको की ऐसी प्राकृत संख्या है जिसके अंकों का योगफल 8 है।}
(iv) D = {x : x एक अभाज्य संख्या है जो 60 की भाजक है।
(v) E = TRIGONOMETRY शब्द के सभी अक्षरों का समुच्चय
(vi) F = BETTER शब्द के सभी अक्षरों क़ा समुच्च्य
हल:
(i) A = {-2, -1, 0, 1, 2, 3, 4, 5, 6}
(ii) B = {1, 2, 3, 4, 5}
(iii) C = {17, 26, 35, 44, 53, 62, 71, 80}
(iv) D = {2, 3, 5}
(v) E = {T, R, I, G, O, N, M, E, Y}
(vi) F = {B, E, T, R}

प्रश्न 4.
निम्नलिखित समुच्चयों को समुच्चय निर्माण रूप में व्यक्त कीजिए:
(i) {3, 6, 9, 12}
(i) {2, 4, 8, 16, 32}
(iii) {5, 25, 125, 625}
(iv) {2, 4, 6, ….}
(v) {1, 4, 9, ……100}
हल:
(i) {x : x = 3n और 1 ≤ n ≤ 4}
(ii) {x : x = 2n और 1 ≤ n ≤ 5}
(iii) {x : x = 5 और 1 ≤ n ≤ 4}
(iv) {x : x एक सम प्राकृत संख्या है।}
(v) {x : x = n², 1 ≤ n ≤ 10}

प्रश्न 5.
निम्नलिखित समुच्चयों के सभी अवयवों (सदस्यों) को सूचीबद्ध कीजिए।
(i) A = {x : x एक विषम प्राकृत संख्या है।
(i) B = x : x एक पूर्णाक है,\frac { -1 }{ 2 }

< x <\frac { 9 }{ 2 }}
(iii) C = {x : x एक पूर्णाक है, x² ≤ 4}
(iv) D = {x : x, LOYAL शब्द का एक अक्षर है।}
(v) F = {x : x वर्ष का एक ऐसा महीना है, जिसमें 31 दिन नहीं होते हैं।}
(vi) F = {x : x अंग्रेजी वर्णमाला का एक व्यंजन है, जो k से पहले आता है।}
हल:
(i) A = {1, 3, 5, 7, ……….}
(ii) B = { 0, 1, 2, 3, 4}
(iii) C = {-2, – 1, 0, 1, 2}
(iv) D = {L, O, Y, A}
(v) E = {फरवरी, अप्रैल, जून, सितम्बर, नवम्बर}
(vi) F = {b, c, d, f, g, h, j}

प्रश्न 6.
बाई ओर रोस्टर रूप में लिखित और दाईं ओर समुच्चय निर्माण रूप में वर्णित समुच्चयों का सही मिलान कीजिए।
(i) {1, 2, 3, 6} (a) {x : x एक अभाज्य संख्या है और 6 की भाजक है।
(ii) {2, 3} (b) {x : x संख्या 10 से कम एक विषम प्राकृत संख्या है।
(iii) {M, A, T, H, E, I, C, S} (c) {x : x एक प्राकृत संख्या है और 6 की भाजक है।
(iv) {1, 3, 5, 7, 9} (d) {x : x MATHEMATICS शब्द का एक अक्षर है:
हल:
(i) → (c)
(ii) → (a)
(iii) → (d)
(iv) → (b)

प्रश्नावली 1.2

प्रश्न 1.
निम्नलिखित में से कौन से रिक्त समुच्चय के उदाहरण है?
(i) 2 से भाज्यं विषम प्राकृत संख्याओं का समुच्चय
(ii) सम अभाज्य संख्याओं का समुच्चय
(iii) {x : x एक प्राकृत संख्या है, x < 5 और साथ ही साथ x > 7}
(iv) {y : y किन्हीं भी दो समांतर रेखाओं का उभयनिष्ठ बिन्दु है।}
हल:
(i) 2 से भाज्य कोई भी विषम प्राकृत संख्याएँ नहीं हैं। अत: यह एक रिक्त समुच्चय है।
(ii) सम अभाज्य संख्या का समुच्चय {2} है। यह एक रिक्त समुच्चय नहीं है।
(iii) x < 5 और x > 7 कोई प्राकृत संख्या नहीं है। अत: यह एक रिक्त समुच्चय है।
(iv) समांतर रेखाएँ कहीं भी नहीं मिलती हैं। अत: यह एक रिक्त समुच्चय है।

प्रश्न 2.
निम्नलिखित समुच्च्यों में से कौन परिमित और कौन अपरिमित हैं?
(i) वर्ष के महीनों का समुच्चय।
(ii) {1, 2, 3, …..}
(ii) {1, 2, 3, ….. 99, 100}
(iv) 100 से बड़े धन पूर्णाकों का समुच्चय
(v) 99 से छोटे अभाज्य पूर्णाकों का समुच्चय
हल:
(i) वर्ष में 12 महीने होते हैं।
अतः यह एक परिमित समुच्चय है।
(ii) समुच्चय {1, 2, 3, ……} में अनंत अवयव हैं।
अंत: यह एक अपरिमित समुच्चय है।
(iii) समुच्चय {1, 2, 3, …… 99, 100} में कुल 100 अवयव हैं।
अत: यह एक परिमित समुच्चय है।
(iv) 100 से बड़े पूर्णाकों का समुच्चैय {101, 102, 103, ….} है जिसमें अनंत अवयव हैं।
अत: यह एक अपरिमित समुच्चय है।
(v) 99 से छोटे अभाज्य पूर्णांकों का समुच्चय {2, 3, 5, 7, …… 97} है जिसमें अवयवों की संख्या निश्चित है।
अत: यह एक परिमित समुच्चय है।

प्रश्न 3.
निम्नलिखित समुच्चयों में से प्रत्येक के लिए बताइए कि कौन परिमित है और कौन अपरिमित है?
(i) x-अक्ष के समांतर रेखाओं का समुच्चय।
(ii) अंग्रेजी वर्णमाला के अक्षरों का समुच्चय।
(iii) उन संख्याओं का समुच्चय जो 5 के गुणज हैं।
(iv) पृथ्वी पर रहने वाले जानवरों का समुच्चय
(v) मूल बिन्दु (0, 0) से होकर जाने वाले वृत्तों का समुच्चय।
हल:
(i) x-अक्ष के समांतर अनंत रेखाएँ खींची जा सकती हैं। अत: यह एक अपरिमित समुच्चय है।
(ii) अंग्रेजी वर्णमाला में कुल 26 अक्षर होते हैं। इन अक्षरों से बनने वाला समुच्चय परिमित होगा।
(iii) 5 से विभाजित होने वाली संख्याओं का समुच्चय {5, 10, 15, 20, ….} है, जिसमें अनंत अवयव हैं। अतः यह एक अपरिमित समुच्चय है।
(iv) पृथ्वी पर रहने वाले जानवरों का समुच्चय परिमित होगा।
(v) मूल बिन्दु को केन्द्र मानकर अनन्त वृत्त चे जा सकते हैं। अत: यह अपरिमित होगा।

प्रश्न 4.
निम्नलिखित में बताइए कि A = B है अथवा नहीं है।
(i) A = {a, b, c, a}, B = {a, c, b, a}
(ii) A = {4, 8, 12, 16}, B = {8, 4, 16, 18}
(iii) A = {2, 4, 6, 8, 10}, B = {x : x सम धन पूर्णाक है और x ≤ 10}
(iv) A = {x : x संख्या 10 का एक गुणज है}, B = {10, 15,20, 25, 30, …}
हल:
(i) A और B दोनों समुच्चयों के अवयव a, b, c, d हैं अतः A = B.
(ii) A में अवयव 12 है परन्तु B में नहीं है अतः A ≠ B.
(iii) A और B दोनों समुच्चयों में अवयव 2, 4, 6, 8 और 10 हैं। अतः A = B.
(iv) A = {10, 20, 30, 40, …..}, B = {10, 15, 25, 30, ….}
10 के गुणजों में 5, 15, 25 नहीं आता है। अतः A ≠ B.

प्रश्न 5.
क्या निम्नलिखित समुच्चय युग्म समान हैं ? कारण सहित बताइए।
(i) A = {2, 3}
B = {x : x समीकरण x² + 5x + 6 = 0 का एक हल है।}
(ii) A = {k : x शब्द ‘FOLLOW’ का एक अक्षर है।}
B = {y : y शब्द ‘WOLF का एक अक्षर है।}
हल:
(i) A = {2, 3}, B = x : x समीकरण x² + 5x + 6 = 0} = {-2, -3}
स्पष्ट है कि समुच्चय A और B के अवयव भिन्न हैं।
अत: A ≠ B.
(ii) A = {F, O, L, W}, B = {W, O, L, F}
समुच्च्य A और B के अवयव समान हैं। अत: A = B.

प्रश्न 6.
नीचे दिए गए समुच्चयों में से समान समुच्चयों का चयन कीजिए:
A = {2, 4, 8, 12}
B = {1, 2, 3, 4}
C = {4, 8, 12, 14}
D = {3, 1, 4, 2}
E = {- 1, 1}
F = {0, a}
G = {1, -1}
H = {0, 1}
हल:
यहाँ समुच्चय B और D के अवयव 1, 2, 3, 4, हैं।
B = D
तथा समुच्चय E और G में -1, 1 अवयव समान हैं।
E = G

प्रश्नावली 1.3

प्रश्न 1.
रिक्त स्थानों में प्रतीक ⊂या ⊄ को भर कर सही कथन बनाइए:
(i) {2, 3, 4} …. {1, 2, 3, 4, 5}
(ii) {a, b, c}….. {b, c, d}
(iii) {x : x आपके विद्यालय की कक्षा XI का एक विद्यार्थी है } …. {x : x आपके विद्यालय का एक विद्यार्थी है।}
(iv) {x : x किसी समतल में स्थित एक वृत है} ….. {x : x एक समान समतल में एक वृत्त है। जिसकी त्रिज्या 1 इकाई है।}
(v) {x : x किसी समतल में स्थित एक त्रिभुज है} …. {x : x किसी समतल में स्थित एक आयत है।}
(vi) {x : x किसी संमतल में स्थित एक समबाहु त्रिभुज है} …… {x : x किसी समतल में स्थित एक त्रिभुज है।}
(vii) {x : x एक सम प्राकृत संख्या है} ……. {x : x एक पूर्णाक है}
हल:
(i) अवयव 2, 3, 4 ∈ {1, 2, 3, 4, 5}
अतः {2, 3, 4} ⊂ {1, 2, 3, 4, 5}
(ii) {a, b, c} का अवयव a ∉ {b, c, d}
अत: {a, b, c} ⊄ {b, c, d}
(iii) जो विद्यार्थी विद्यालय की कक्षा XI में हैं वे विद्यालय में भी हैं।
अतः {x : x विद्यालय की कक्षा XI का विद्यार्थी} ⊂ {x : x आपके विद्यालय का विद्यार्थी}
(iv) समुच्चय {x : x समतल में एक वृत्त} के एक अवयव वृत्त की त्रिज्या 1 से भिन्न हो सकती है।
अतः {x : x समतल में वृत्त} ⊄ {x : x वृत्त की त्रिज्या 1 इकाई है }
(v) त्रिभुजों का समुच्चय आयतों के समुच्चय से बिल्कुल भिन्न है।
अतः {x : x समतल में एक त्रिभुज} ⊄ {x : x समतल में एक आयत}
(vi) प्रत्येक समबाहु त्रिभुज एक त्रिभुज है।
अतः {x : x समतल में एक समबाहु त्रिभुज} ⊂ {x : x समतल में एक त्रिभुज}
(vii) प्रत्येक सम प्राकृत संख्या एक पूर्णाक है।
अतः {x : x एक सम प्राकृत संख्या} ⊂ {x : x एक पूर्णाक}

प्रश्न 2.
जाँचिए कि निम्नलिखित कथन सत्य हैं अथवा असत्य हैं:
(i) {a, b} ⊄ {b, c, a}
(ii) {a, e} ⊂ {x : x अंग्रेजी वर्णमाला का एक स्वर है।
(iii) {1, 2, 3} ⊂ {1, 3, 5}
(iv) {a} ⊂ {a, b, c}
(v) {a} ⊂ {a, b, c}
(vi) {x : x संख्या 6 से कम एक सम प्राकृत संख्या है। ⊂ {x : x एक प्राकृत संख्या है, जो संख्या 36 को विभाजित करती है।
हल:
(i) समुच्चय {a, b} के अवयव a, b दासमुच्चय {b, c, a} में है।
{a, b} ⊄ {b, c, a}
अतः उपरोक्त कथन असत्य है।
(ii) a, e दोनों ही स्वर हैं।
{a, e} = {x : x, अंग्रेजी वर्णमाला का एक स्वर है।
अतः यह कथन सत्य है।
(iii) समुच्चय {1, 2, 3} और {1, 3, 5} में अवयव 2 समुच्चय {1, 3, 5} नहीं है।
{1, 2, 3} ⊂ {1, 3, 5} कथने असत्य है।
(iv) a ∈ {a, b, c}
{a} ⊂ {a, b, c} यह कथन सत्य है।
(v) {4} समुच्चय है, अवयव नही है।
{a} } ∈ {a, b, c} कथन असत्य है।
(vi) सम प्राकृत संख्या 2, 4 संख्या 6 से कम है तथा 36 को विभाजित करती है।
{x : x एक सम प्राकृत संख्या है जो 6 से कम है} ⊂ {x : x एक सम प्राकृत संख्या 36 को विभाजित करती है। अतः यह कथन सत्य है।}

प्रश्न 3.
मान लीजिए कि A = {1, 2, 3, 4, 5}, निम्नलिखित में से कौन सा कथन सही नहीं है और क्यों?
(i) {3, 4} ⊂ A
(i) {3, 4} ∈ A
(iii) {{3, 4}} ⊂ A
(iv) 1 ∈ A
(v) 1 ⊂ A
(vi) {1, 2, 5} ⊂ A
(vii) {1, 2, 5} ∈ A
(viii) {1, 2, 3} ⊂ A
(ix) Φ ∈ A
(x) Φ ⊂A
(xi) {Φ} ⊂ A
हल:
(i) सही नहीं है। समुच्चय {3, 4} एक अवयव है।
(ii) सही है। क्योंकि {3, 4} समुच्चय A का एक अवयव है।
(iii) सही है। A के अवयव {3, 4} का एक उपसमुच्चय है।
(iv) 1 ∈ A, सही है।
(v) 1 ⊂ A सही नहीं है क्योंकि 1 एक समुच्चय नहीं है।
(vi) {1, 2, 5} ⊂ A सही है। समुच्चय {1, 2,5} के अवयव 1, 2, 5 समुच्चय A में है।
(vii) {1, 2, 5} ∈ सही नहीं है। {12, 5} अवयव नहीं है। यह एक समुच्चय है।
(viii) {1, 2, 3} ⊂ A सही नहीं है। अवयव 3 समुच्चय में नही है।
(ix) Φ ∈ A, सही नहीं है। Φ एक समुच्चय है, अवयव नहीं है।
(x) {Φ} ⊂ A सही है। सभी समुच्चयों का उपसमुच्चय है।
(xi) {Φ} ⊂ A सही नहीं है। {Φ} समुच्चय का समुच्चय है।

प्रश्न 4.
निम्नलिखित समुच्चयों के सभी उपसमुच्चय लिखिए।
(i) {a}
(ii) {a, b}
(iii) {1, 2, 3}
(iv) Φ
हल:
(i) Φ, {a}
(ii) Φ, {a}, {b}, {a, b}
(iii) Φ, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}
(iv) Φ

प्रश्न 5.
P (A) के कितने अवयव हैं, यदि A = Φ
हल:
A = Φ, P(A) = Φ इस प्रकार P (A) को 2° = 1 अवयव है।

प्रश्न 6.
निम्नलिखित को अंतराल रूप में लिखिए:
(i) {x : x ∈R, -4 < x ≤ 6}
(ii) {x : x ∈R, -12 < x < -10}
(iii) {x : x ∈ R, 0 ≤ x < 7}
(iv) {x : x ∈ R, 3 ≤ x ≤ 4}
हल:
वांछित अंतराल इस प्रकार हैं।
(i) (-4, 6]
(ii) (-12, – 10)
(iii) [0, 7)
(iv) [3, 4]

प्रश्न 7.
निम्नलिखित अंतरालों को समुच्चय निर्माण रूप में लिखिए:
(i) (-3, 0)
(ii) [6, 12]
(iii) (6, 12]
(iv) [-23, 5]
हल:
(i) (-3, 0) = {x : x ∈ R, -3 < x < 0}
(ii) [6, 12] = {x : x ∈ R, 6 ≤ x ≤ 12}
(iii) (6,12] = {x : x ∈ R, 6 < x ≤ 12}।
(iv) [-23, 5] = {x : x ∈ R, -23 ≤ x ≤ 5}

प्रश्न 8.
निम्नलिखित में से प्रत्येक के लिए आप कौन सा सार्वत्रिक समुच्चय प्रस्तावित करेंगे?
(i) समकोण त्रिभुजों का समुच्चय
(ii) समद्विबाहु त्रिभुजों का समुच्चय
हल:
दोनों समुच्चयों के लिए सार्वत्रिक समुच्चय :
{x : x समतल में स्थित एक त्रिभुज}

प्रश्न 9.
समुच्चय A = {1, 3, 5}, B = {2, 4, 6} और C = {0, 2, 4, 6, 8} प्रदत्त हैं। इन तीनों समुच्चयों A, B और C के लिए निम्नलिखित में से कौन सा (से) सार्वत्रिक समुच्चय लिए जा सकते हैं?
(i) {0,1, 2, 3, 4, 5, 6}
(ii) Φ
(iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(iv) {1, 2, 3, 4, 5, 6, 7, 8}
हल:
समुच्चय (iii),
तीनों समुच्चय A, B, C के लिए {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} सार्वत्रिक समुच्चय हैं।

प्रश्नावली 1.4

प्रश्न 1.
निम्नलिखित में से प्रत्येक समुच्चय युग्म का सम्मिलन ज्ञात कीजिए:
(i) X = {1, 3, 5}, Y = {1, 2, 3}
(i) A = {a, e, i, 0, u}, B = {a, b, c}
(iii) A = {x : एक प्राकृत संख्या है और 3 का गुणज है।
B = {x : x संख्या 6 से कम एक प्राकृत संख्या है।
(iv) A = {x : x एक प्राकृत संख्या है और 1 < x ≤ 6}
B = {x : x एक प्राकृत संख्या है और 6 < x < 10}
(v) A = {1, 2, 3}, B = Φ
हल:
(i) X ∪ Y = {1, 3, 5} ∪ {1, 2, 3} = {1, 2, 3, 5}
(ii) A ∪ B = {a, e, i, 0, u} ∪ {a, b, c} = {a, b, c, e, i, 0, u}
(iii) A ∪ B = {3, 6, 9….} ∪ {1, 2, 3, 4, 5} = {1, 2, 4, 5 या संख्या 3 का गुणज}
(iv) A = {2, 3, 4, 5, 6}, B = {7, 8, 9}
A ∪ B = {2, 3, 4, 5, 6, 7, 8, 9} अर्थात् {x : 1 < x < 10, x ∈ N}
(v) A ∪B = {1, 2, 3} ∪ Φ = {1, 2, 3}

प्रश्न 2.
मान लीजिए कि A = {a, b}, B = {a, b, c} क्या A ⊂ B? A ∪ B ज्ञात कीजिए।
हल:
A = {a, b}, B = {a, b, c}। समुच्चय A के अवयव a, b समुच्चय B में भी है। A ⊂ B = A ∪ B = B और A ∪ B = {a, b} ∪ {a, b, c} = {a, b, c}

प्रश्न 3.
यदि A और B दो ऐसे समुच्चय हैं कि A ⊂ B, तो A ∪ B क्या है?
हल:
A ⊂ B समुच्चय A के सभी अवयव समुच्चय B में हैं। A ⊂ B = B.

प्रश्न 4.
यदि A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} और D = {7, 8, 9, 10}, तो निम्नलिखित ज्ञात कीजिए:
(i) A ∪ B
(ii) A ∪ C
(ii) B ∪ C
(iv) B ∪ D
(v) A ∪ B ∪ C
(vi) A ∪ B ∪D
(vii) B ∪ C ∪ D
हल:
(i) A ∪ B = {1, 2, 3, 4} ∪ {3, 4, 5, 6} = {1, 2, 3, 4, 5, 6}
(ii) A ∪ C = {1, 2, 3, 4} ∪ {5, 6, 7, 8} = {1, 2, 3, 4, 5, 6, 7, 8}
(iii) B ∪ C = {3, 4, 5, 6} ∪ {5, 6, 7, 8} = {3, 4, 5, 6, 7, 8}
(iv) B ∪ D = {3, 4, 5, 6} ∪ {7, 8, 9, 10} = {3,4, 5, 6, 7, 8, 9, 10}
(v) A ∪ B ∪C = ({1, 2, 3, 4} {3, 4, 5, 6}) ∪ {5, 6, 7, 8} = {1, 2, 3, 4, 5, 6} ∪ {5, 6, 7, 8} = {1, 2, 3, 4, 5, 6, 7, 8}.
(vi) A ∪ B ∪ D = ({1, 2, 3, 4} ∪ {3, 4, 5, 6}) ∪ {7, 8, 9, 10} = {1, 2, 3, 4, 5, 6} ∪ {7, 8, 9, 10} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(vii) B ∪ C ∪ D = ({3, 4, 5, 6} ∪ {5, 6, 7, 8}) ∪ {7, 8, 9, 10} = {3, 4, 5, 6, 7, 8} ∪ {7, 8, 9, 10) = {3, 4, 5, 6, 7, 8, 9, 10}

प्रश्न 5.
प्रश्न 1 में दिए प्रत्येक समुच्चय युग्म का सर्वनिष्ठ समुच्चय ज्ञात कीजिए:
हल:
(i) X ∩ Y= {1, 3, 5} ∩ {1, 2, 3} = {1, 3}
(ii) A ∩ B = {a, e, i, o, u} ∩ {a,b,c} = {a}.
(iii) A ∩ B = {3, 6, 9 …..} ∩ {1, 2, 3, 4, 5} = {3}.
(iv) A ∩ B = {2, 3, 4, 5, 6} ∩{7, 8, 9} = Φ
(v) A ∩ B = {1, 2, 3} ∩ Φ = 0

प्रश्न 6.
यदि A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} और D = {15, 17}; तो निम्नलिखित ज्ञात कीजिए:
(i) A ∩ B
(ii) B ∩C
(iii) A ∩ C ∩D
(iv) A ∩C
(v) B ∩ D
(vi) Á ∩ (B ∪ C)
(vii) A ∩ D
(viii) A ∩(B ∪D)
(ix) (A ∩ B) ∩ (B ∪ C)
(x) (A ∪ D) ∩ (B ∪ C)
हल:
(i) A ∩ B = {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13} = {7, 9, 11}
(ii) B ∩ C = {7, 9, 11, 13} ∩ {11, 13, 15} = {11, 13}
(iii) Á ∩ C ∩ D = ({3, 5, 7, 9, 11} ∩ {11, 13, 15}) ∩ {15, 17} = {11} ∩{15, 17} = Φ
(iv) A ∩ C = {3, 5, 7, 9, 11} ∩ {11, 13, 15} = {11}.
(v) B ∩ D = {7, 9, 11, 13} ∩ {15, 17} = Φ
(vi) A ∩ (B ∪ C) = {3, 5, 7, 9, 11} ∩ ({7, 9, 11, 13} ∪ {11, 13, 15}) = {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15} = {7, 9, 11}.
(vii) A ∩ D = {3, 5, 7, 9, 11} ∩ {15, 17} = Φ
(viii) A ∩(B ∪D) = {3, 5, 7, 9, 11} ∩{7, 9, 11, 13} ∪{15, 17}) = {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15, 17} = {7, 9, 11).
(ix) A ∩ B = {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13} = {7, 9, 11}
B ∪ C = {7, 9, 11, 13} ∪ {11, 13, 15) = {7, 9, 11, 13, 15).
(A ∩ B) ∩ (B ∪ C) = {7, 9, 11} ∩ {7, 9, 11, 13, 15} = {7, 9, 11}.
(x) A ∪ D = {3, 5, 7, 9, 11} ∪ {15, 17} = {3, 5, 7, 9, 11, 15, 17}
B ∪ C = {7, 9, 11, 13} ∪ {11, 13, 15} = {7, 9, 11, 13, 15}
(A ∪ D) ∩ (B ∪ C) = {3, 5, 7, 9, 11, 15, 17} ∩ {7, 9, 11, 13, 15} = {7, 9, 11, 15}

प्रश्न 7.
यदि A = {x : x एक प्राकृत संख्या है}, B = {x : x एक सम प्राकृत संख्या है} C = {x : x एक विषम प्राकृत संख्या है}, D = {x : एक अभाज्य संख्या है} तो निम्नलिखित को ज्ञात कीजिए:
(i) A ∩ B
(ii) A ∩C
(iii) A ∩ D
(iv) B ∩C
(v) B ∩D
(vi) C ∩ D
हल:
A = {x : x एक प्राकृत संख्या है} = {1, 2, 3, 4……} B = {x : x एक समं प्राकृत संख्या है} = {2, 4, 6, 8…} C = {x : x एक विषम प्राकृत संख्या है} = {1, 3, 5, 7…} D = {x : x एक अभाज्य संख्या है} = {2, 3, 5, 7, 11….}
(i) A ∩ B = {1, 2, 3, 4….} ∩{2, 4, 6, 8….} = {2, 4, 6, 8….} = B
(ii) A ∩C = {1, 2, 3, 4……} ∩ {1, 3, 5, 7….} = {1, 3, 5, 7….} = C
(iii) A ∩D = {1, 2, 3, 4…} ∩ {2, 3, 5, 7…..} = {2, 3, 5, 7……} = D
(iv) B ∩ C = {2, 4, 6, 8…} ∩ {1, 3, 5, 7……} = Φ
(v) B ∩D = {2, 4, 6, 8…..} ∩ {2, 3, 5, 7…..} = {2}
(vi) C ∩D = {1, 3, 5, 7…..} ∩{2, 3, 5, 7, 11…….} = {3, 5, 7, 11, 13……} = {x : x एक विषम अभाज्य संख्या}

प्रश्न 8.
निम्नलिखित समुच्चय युग्मों में से कौन से युग्म असंयुक्त हैं?
(i) {1, 2, 3, 4} तथा {x : x एक प्राकृत संख्या है और 4 ≤ x ≤ 6}
(ii) {a, e, i, 0, u} तथा {c, d, e, f}
(iii) {x : x एक सम पूर्णांक है। और {x : x एक विषम पूर्णाक है।
हल:
(i) मान लीजिए E = {1, 2, 3, 4} F = {x : x एक प्राकृत संख्या और 4 ≤ x ≤ 6} = {4, 5, 6} अवयव 4, E और F दोनों समुच्चयों में है। अत: दोनों युग्म असंयुक्त नहीं हैं।
(ii) दिये हुए समुच्चयों में अवयव उभयनिष्ठ है। अत: यह असंयुक्त समुच्चय नहीं है।
(iii) मान लीजिए A = {x : x एक सम पूर्णांक हैं। = {….-4, -2, 0, 2, 4…} B = {x : x एक विषम पूर्णांक है} = {….-5, -3, -1, 1, 3, 5…..} A और B समुच्चयों में कोई भी अवयव उभयनिष्ठ नहीं है। अत: यह समुच्चय असंयुक्त है।

प्रश्न 9.
यदि A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}, तो निम्नलिखित को ज्ञात कीजिए:
(i) A – B
(ii) A – C
(iii) A – D
(iv) B – A
(v) C – A
(vi) D – A
(vii) B – C
(viii) B – D
(ix) C – B
(x) D – B
(xi) C – D
(xii) D – C
हल:
(i) A – B = {3, 6, 9, 12, 15, 18, 21} – {4, 8, 12, 16, 20} = {3, 6, 9, 15, 18, 21}
(ii) A – C = {3, 6, 9, 12, 15, 18, 21} – {2, 4, 6, 8, 10, 12, 14, 16} = {3, 9, 15, 18, 21}
(iii) A – D = {3, 6, 9, 12, 15, 18, 21} – {5, 10, 15, 20} = {3, 6, 9, 12, 18, 21}
(iv) B – A = {4, 8, 12, 16, 20} – {3, 6, 9, 12, 15, 18, 21} = {4, 8, 16, 20}
(v) C – A = {2, 4, 6, 8, 10, 12, 14, 16} – {3, 6, 9, 12, 15, 18, 21} = {2, 4, 8, 10, 14, 16}
(vi) D – A = {5, 10, 15, 20} – {3, 6, 9, 12, 15, 18, 21} = {5, 10, 20}
(vii) B – C = {4, 8, 12, 16, 20} – {2, 4, 6, 8, 10, 12, 14, 16} = {20}
(viii) B – D = {4, 8, 12, 16, 20} – {5, 10, 15, 20} = {4, 8, 12, 16}
(ix) C – B = {2, 4, 6, 8, 10, 12, 14, 16} – {4, 8, 12, 16, 20} = {2, 6, 10, 14}
(x) D – B = {5, 10, 15, 20} – {4, 8, 12, 16, 20} = {5, 10, 15}
(xi) C – D = {2, 4, 6, 8, 10, 12, 14, 16} – {5, 10, 15, 20} = {2, 4, 6, 8, 12, 14, 16}
(xii) D – C = {5, 10, 15, 20} – {2, 4, 6, 8, 10, 12, 14, 16} = {5, 15, 20}

प्रश्न 10.
यदि X = {a, b, c, d} औरै Y = {f, b, d, g} तो निम्नलिखित को ज्ञात कीजिए:
(i) X – Y
(ii) Y – X
(iii) X ∩ Y
हल:
(i) X – Y= {a, b, c, d} – {f, b, d, g} = {a, c}
(ii) Y – X = {f, b, d, g} – {a, b, c, d} = {f, g}
(iii) X ∩Y= {a, b, c, d} ∩ {f, b, d, g} = {b, d}

प्रश्न 11.
यदि R वास्तविक संख्याओं और Q परिमेय संख्याओं के समुच्चय हैं, तो R – Q क्या होगा?
हल:
R = {x : x एक वास्तविक संख्या है।} p= {x : x एक परिमेय संख्या है।}
R – Q = {x : x एक अपरिमेय संख्या है।} अत: यह अपरिमेय संख्याओं का समुच्चय है।

प्रश्न 12.
बताइए कि निम्नलिखित कथनों में से प्रत्येक सत्य है या असत्य? अपने उत्तर का औचित्य भी बताइए।
(i) {2, 3, 4, 5} तथा {3, 6} असंयुक्त समुच्चय हैं।
(ii) {a, e, i, 0, u} तथा {a, b, c, 4} असंयुक्त समुच्चय हैं।
(iii) {2, 6, 10, 14} तथा {3, 7, 11, 15} असंयुक्त समुच्चय हैं।
(iv) {2, 6, 10} तथा {3, 7, 11} असंयुक्त समुच्चय हैं।
हल:
(i) यह कथन सत्य नहीं है क्योंकि समुच्चय {2, 3, 4, 5} और {3, 6} में अवयव 3 उभयनिष्ठ है।
(ii) यह कथन सत्य नहीं है क्योंकि समुच्चय {a, e, i, o, u} और {a, b, c, d} में अवयव a उभयनिष्ठ है।
(iii) यह कथन सत्य है क्योंकि समुच्चय {2, 6, 10, 14} और {3, 7, 11, 15} में कोई अवयव उभयनिष्ठ नहीं है। अत: यह समुच्चय असंयुक्त है।
(iv) यह कथन सत्य है क्योंकि समुच्चय {2, 6, 10} और {3, 7, 11} में कोई अवयव उभयनिष्ठ नहीं है। अतः यह समुच्चय असंयुक्त है।

प्रश्नावली 1.5

प्रश्न 1.
मान लीजिए कि U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} और C = {3, 4, 5, 6} तो निम्नलिखित को ज्ञात कीजिए:
(i) A’
(ii) B’
(iii) (A ∪ C)’
(iv) (A ∪ B)’
(v) (A’)’
(vi) (B – C)’
हल:
(i) A’ = U – A = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4} = {5, 6, 7, 8, 9}
(ii) B’ = U – B = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 4, 6, 8} = {1, 3, 5, 7, 9)
(iii) A ∪ C = {1, 2, 3, 4} ∪ {3, 4, 5, 6} = {1, 2, 3, 4, 5, 6} (A ∪ C)’ = U – (A ∪ C) = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4, 5, 6} = {7, 8, 9}
(iv) A ∪ B = {1, 2, 3, 4} ∪ {2, 4, 6, 8} = {1, 2, 3, 4, 6, 8} (A ∪ B)’ = U – (A ∪B) = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4, 6, 8} = {5, 7, 9} (v) (A)’ = U – A = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4} = {5, 6, 7, 8, 9} (A’)’ = U- A’ = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {5, 6, 7, 8, 9} = {1, 2, 3, 4}
(vi) B – C = {2, 4, 6, 8} – {3, 4, 5, 6} = {2, 8} (B – C)’ = U – (B – C) = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2,8} = {1, 3, 4, 5, 6, 7, 9}

प्रश्न 2.
यदि U = {a, b, c, d, e, f, g, h}, तो निम्नलिखित समुच्चयों के पूरक ज्ञात कीजिये:
(i) A = {a, b, c}
(ii) B = {d, e, f, g}
(iii) C = {a, c, e, g}
(iv) D = {f, g, h, a}
हल:
(i) A’ = U – A = {a, b, c, d, e, f, g, h} – {a, b,c} = {d, e, f, g, h}
(ii) B’ = U – B = {a, b, c, d, e, f, g, h} – {d, e, f, g} = {a, b, c, h}
(iii) C = U – C = {a, b, c, d, e, f, g, h} – {a, c, e, g} = {b, d, f, h}
(iv) D’ = U – D = {a, b, c, d, e, f, g, h} – {f, g, h, a} = {b, c, d, e}.

प्रश्न 3.
प्राकृत संख्याओं के समुच्चय को सार्वत्रिक समुच्चय मानते हुए, निम्नलिखित समुच्चयों के पूरक लिखिए:
(i) {x : x एक प्राकृत सम संख्या है।}
(ii) {x : x एक प्राकृत विषम संख्या है।}
(iii) {x : x संख्या 3 को एक धन गुणज है।}
(iv) {x : x एक अभाज्य संख्या है।}
(v) {x : x, 3 और 5 से विभाजित होने वाली एक संख्या है।}
(vi) {x : x एक पूर्ण वर्ग संख्या है।}
(vii) {x : x एक पूर्ण घन संख्या है।}
(viii){x : x + 5 = 8}
(ix) {x : 2x + 5 = 9}
(x) {x : x ≥ 7}
(xi) {x : x ∈ N और 2x + 1 > 10}
हल:
(i) {x : x एक विषम प्राकृत संख्या है।}
(ii) {x : एक सम संख्या है।}
(iii) {x : x ∈ N और x संख्या 3 का धन गुणज नहीं है।}
(iv) {x : x = 1 और x एक धन भाज्य संख्या है।}
(v) {x : x ∈ N और x, संख्या 3 व 5 किसी से भी विभाजित नहीं होती।}
(vi) {x : x ∈ N तथा x एक पूण वर्ग संख्या नहीं है।}
(vii) {x : x ∈ N तथा x एक पूर्ण वर्ग घन संख्या नहीं है।}
(viii) {x : x ∈ N तथा x ≠ 3}
(ix) {x : x ∈ N तथा x ≠ 2}
(x) {x : x ∈ N तथा x < 7}
(xi) {x : x ∈ N तथा x <\frac { 9 }{ 2 }}

प्रश्न 4.
यदि U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} और B = {2, 3, 5, 7}, तो सत्यापित कीजिए कि:
(i) (A ∪ B)’ = A’ ∩ B’
(ii) (A ∩ B)’ = A’ ∪ B’
हल:
(i) A ∪ B = {2, 4, 6, 8} ∪{2, 3, 5, 7} = {2, 3, 4, 5, 6, 7, 8}
बायाँ पक्ष = (A ∪B)’ = U – (A ∪ B) = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 3, 4, 5, 6, 7, 8} = {1, 9}
A’ = U – A = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 4, 6, 8} = {1, 3, 5, 7, 9}
B’ = U – B = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 3, 5, 7} = {1, 4, 6, 8, 9}
दायाँ पक्ष = A’ ∩ B’ = {1, 3, 5, 7, 9} ∩ {1, 4, 6, 8, 9} = {1, 9}
अतः (A ∪ B)’ = A’ ∩ B’.
(ii) बायाँ पक्ष = (A ∩B)’
(A ∩B) = {2, 4, 6, 8} ∩ {2, 3, 5, 7} = {2}
(A ∩ B)’ = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2} = {1, 3, 4, 5, 6, 7, 8, 9}
दायाँ पक्ष : A’ ∪ B’ = {1, 3, 5, 7, 9} ∪ {1, 4, 6, 8, 9} = {1, 3, 4, 5, 6, 7, 8, 9}
अत: (A ∩ B)’ = A’ ∪ B’.

प्रश्न 5.
निम्नलिखित में से प्रत्येक के लिए उपयुक्त वेन आरेख खींचिए।
(i) (A ∪ B)’
(ii) A’ ∩ B’
(iii) (A ∩B)’
(iv) (A’ ∪ B’)
हल:
छायांकित क्षेत्र को निम्नलिखित समुच्चयों द्वारा दर्शाते हैं:
UP Board Solutions for Class 11 Maths Chapter 1 Sets 1.5 5

प्रश्न 6.
मान लीजिए कि किसी समतल में स्थित सभी त्रिभुजों का समुच्चय सार्वत्रिक समुच्चय U है। यदि A उन सभी त्रिभुजों का समुच्चय हैं जिनमें कम से कम एक कोण 60° से भिन्न है, तो A’ क्या है?
हल:
U = {x : समतल में एक त्रिभुज है।}
A = {x : x एक त्रिभुज जिसका कम से कम एक कोण 60° का न हो।}
A’ = {सभी समबाहु त्रिभुजों का समुच्चय है।}

प्रश्न 7.
निम्नलिखित कथनों को सत्य बनाने के लिए रिक्त स्थान भरिए:
(i) A ∪ A’ = ………..
(ii) Φ’ ∩ A = ………..
(iii) A ∩A’ = ………….
(iv) U’ ∩ A = …………
हल:
(i) A ∪ A’ = U
(ii) Φ’ ∩ A = U ∩ A = A
(iii) A ∩A’ = Φ
(iv) U’ ∩ A = Φ ∩ A = Φ

प्रश्नावली 1.6

प्रश्न 1.
यदि X और Y दो ऐसे समुच्चय हैं कि n(X) = 17, n(Y) = 23 तथा n(X ∪ Y) = 38, तो n(X ∩Y) ज्ञात कीजिए।
हल:
दिया है। n(X) = 17, n(Y) = 23
n(X ∪ Y) = 38
n(X ∪ Y) = n(X) + n(Y) – n(X ∩ Y)
38 = 17 + 23 – n(K ∩ Y) = 40 – n (X ∩ Y).
n(X ∩ Y) = 40 – 38 = 2.

प्रश्न 2.
यदि X और Yदो ऐसे समुच्चय हैं कि X ∪ Y में 18, X में 8 और Y में 15 अवयवे हों तो X ∩Y में कितने अवयव होंगे?
हलः
n(X) = 8, n(Y) = 15, n(X ∪ Y) = 18
हम जानते हैं कि,
n(X ∪ Y) = n(X) + n(Y) – n (X ∩ Y)
18 = 8 + 15 – n(X ∩ Y) = 23 – n(X ∩ Y)
n(X ∩ Y) = 23 – 18 = 5.

प्रश्न 3.
400 व्यक्तियों के समूह में, 250 हिन्दी तथा 200 अंग्रेजी बोल सकते हैं। कितने व्यक्ति हिन्दी तथा अंग्रेजी दोनों बोल सकते हैं?
हल:
मान लीजिए कि H और E क्रमशः हिन्दी व अंग्रेजी बोलने वालों के समुच्चय हों, तब
n(H) = 250, n(E) = 200 और
n(H ∪ E) = 400
n(H ∪E) = n(H) + n(E) – n(H ∩E)
400 = 250 + 200 – n(H ∩E) = 450 – n(H ∩E)
n (H ∩E) = 450 – 400 = 50.

प्रश्न 4.
यदि S और T दो ऐसे समुच्चय हैं कि S में 21, T में 32 और S ∩ T में 11 अवयव हों तो S ∪T में कितने अवयव होंगे?
हल:
यहाँ n(S) = 21, n (T) = 32, n(S ∩T) = 11
n(S∪T) = n(S) + n(T) – n(S ∩ T) = 21 + 32 – 11 = 53 – 11 = 42.

प्रश्न 5.
यदि X और दो ऐसे समुच्चय हैं कि X में 40, X ∪Y में 60, और X ∩ Y में 10 अवयव हों, तो ? में कितने अवयव होंगे?
हल:
n(X) = 40, n(X ∪Y) = 60, n(X ∩ Y) = 10, n(Y) = ?
n(X ∪ Y) = n(X) + n(Y) – n (X ∩Y)
60 = 40 + n (Y) – 10
n(Y) = 60 – 40 + 10 = 30.

प्रश्न 6.
70 व्यक्तियों के समूह में 37 कॉफी, 52 चाय पसंद करते हैं और प्रत्येक व्यक्ति दोनों में से कम से कम एक पेय पसंद करता है, तो कितने व्यक्ति कॉफी और चाय दोनों पसंद करते हैं?
हल:
मान लिया C, कॉफी पीने वाले लोगों के समुच्चय को और T, चाय पीने वाले लोगों के समुच्चय हों, तब
n(C ∪T) = 70, n(C) = 37, n(T) = 52
n(C ∪T) = n (C) +n(T) – n(C ∩ T)
70 = 37 + 52 – n(C ∩T)
n(C ∩ T) = 37 + 52 -70 = 89 – 70 = 19.

प्रश्न 7.
65 व्यक्तियों के समूह में, 40 व्यक्ति क्रिकेट और 10 व्यक्ति क्रिकेट तथा टेनिस दोनों को पंसद करते हैं, तो कितने व्यक्ति केवल टेनिस को पंसद करते हैं किंतु क्रिकेट को नहीं? कितने व्यक्ति टेनिस को पंसद करते हैं?
हल:
मान लीजिए C, क्रिकेट पंसद करने वाले लोगों का समुच्चय है और T टेनिस पंसद करने वालों का समुच्चय हो, तब
n(C ∪T) = 65, n(C) = 40, n(C ∩T) = 10
हम जानते हैं कि
n(C ∪ T) = n(C) + n(T) – n(C ∩ T)
65 = 40 + n(T) – 10 = 30 + n(T)
n(T) = 65 – 30 = 35
केवल टेनिस पंसद करने वालो की संख्या = n(T) – n(C ∩T) = 35 – 10 = 25.
इस प्रकार टेनिस पंसद करने वालों की संख्या जो क्रिकेट पंसद नहीं करते = 25
अत: टेनिस पंसद करने वाले लोगों की संख्या = 35.

प्रश्न 8.
एक कमेटी में, 50 व्यक्ति फ्रेंच 20 व्यक्ति स्पेनिश और 10 व्यक्ति स्पेनिश और फ्रेंच दोनों ही भाषाओं को बोल सकते हैं। कितने व्यक्ति इन दोनों ही भाषाओं में से कम से कम एक भाषा बोल सकते हैं?
हल:
मान लीजिए फ्रांसीसी बोलने वाले लोगों के समुच्चय को F से तथा स्पैनिश बोलने वाले लोगों के समुच्चय का S से निरुपित किया हो, तब
n(F) = 50, n(S) = 20, n(F ∩S) = 10
अब, n(F ∪S) = n(F) + n (S) – n (F ∩S) = 50 + 20 – 10 = 60
कम से कम एक भाषा बोलने वाले लोगों की संख्या = 60.

अध्याय 1 पर विविध प्रश्नावली

प्रश्न 1.
निम्नलिखित समुच्चयों में से कौन किसका उपसमुच्चय है, इसका निर्णय कीजिए:
A = x : x ∈R तथा x² – 8x + 12 = 0 को संतुष्ट करने वाली सभी वास्तविक संख्याएं = x}, B= {2, 4, 6}, C = {2, 4, 6, 8….}, D = {6}.
हल:
A = {x : x ∈ R, x समीकरण x² – 8x + 12 = 0 को संतुष्ट करता है। अर्थात्
A = {2, 6}
B = {2, 4, 6} ।
C = {2, 4, 6, 8….}
D = {6}
(i) समुच्चय A के अवयव 2, 6 समुच्चय B में भी हैं।
A ⊂ B.
(ii) इस प्रकार समुच्चय A के अवयव 2, 6 समुच्चय C में भी है।
A ⊂ C.
(iii) समुच्चय B के अवयव 2, 4, 6 समुच्चय C में हैं।
B ⊂C.
(iv) समुच्चय D का अवयव 6, समुच्चय A, B और C तीनों में हैं,
D ⊂ A, D ⊂ B, D ⊂C.

प्रश्न 2.
ज्ञात कीजिए कि निम्नलिखित में से प्रत्येक कथन सत्य है या असत्य है। यदि सत्य है, तो उसे सिद्ध कीजिए। यदि असत्य है तो एक उदाहरण दीजिए।
(i) यदि x ∈ A तथा A ∈ B, तो x ∈ B
(ii) यदि A ⊂B तथा B ∈ C, तो A ∈ C
(iii) यदि A ⊂ B तथा B ⊂C, तो A ⊂C
(iv) यदि A ⊄ B B ⊄ C, तो A ⊄ C
(v) यदि x ∈ A तथा A ⊄ B, तो x ∈ B
(vi) यदि A ⊂ B तथा x ∉ B, तो x ∉ A
हल:
(i) असत्य : मान लीजिए A = {1}, B = {{1}, 2}
स्पष्ट है कि 1 ∈ A, A ∈ B परंतु 1 ∉ समुच्चय B क्योंकि 1 B में नहीं है। इस प्रकार दिया हुआ कथन सत्य नहीं है
(ii) असत्य : मान लीजिए A = {1}, B = {1, 2} और C = {{1, 2}, 3}
समुच्चय A का अवयव समुच्चय B में हैं A ∈ B
अवयव {1, 2} समुच्चय C में हैं B ∈ C
पंरतु A = {1} समुच्चय C में नहीं है।
कथन A ∈ C सत्य नहीं है।
(iii) सत्य : A ⊂ B ⇒ यदि x ∈ A तथा x ∈ B
परंतु B ⊂ C ⇒ यदि x ∈ B तब x ∈ C
यदि x ∈ A तब x ∈ A तब x ∈ C ⇒ A ⊂ C
(iv) असत्य : मान लीजिए A = {1, 2}, B = {2, 3}, C = {1, 2, 5}
समुच्चय A के सभी अवयव 1, 2 समुच्चय B में नहीं हैं।
A ⊄ D
समुच्चय B के सभी अवयव 2, 3 समुच्चय C में नहीं हैं।
A ⊂ C
पंरतु समुच्चय A के सभी अवयव 1, 2 समुच्चय C में हैं।
A ⊂ C
इस प्रकार दिया कथन सत्य नहीं है।
(v) समुच्चय A = {1, 2}, B = {2, 3, 4, 5}
समुच्चय A का अवयव 1, 2 समुच्चय B में नहीं है ।
A ⊄ B
समुच्चय A का अवयव 1 समुच्चय B में नहीं हैं।
x ∉ B
इस प्रकार दिया गया कथन सत्य नहीं है।
(vi) सत्य : A ⊂ B = यदि x ∈ A तब x ∈ B यदि x ∉ B तथा x ∉ A
इस प्रकार कथन A ⊂ B, x ∉ B तब x ∉ A सत्य हैं।

प्रश्न 3.
मान लीजिए A, B और Cऐसे समुच्चय हैं कि A ∪ B = A ∪ C तथा A ∩ B = A ∩ C, तो दर्शाइए कि B = C
हल:
दिया है:
A ∪ B = A ∪ C
(A ∪ B) ∩ C = (A ∪ C) ∩ C = C [(A ∪ C) ∩ C = C ]
(A ∩ C) ∩ (B ∩ C) = C
(A ∩ B) ∪ (B ∩ C) = C ….(i) [A ∩ C = A ∩ B = दिया है।]
A ∪ B = A ∪ C
(A ∪ B) ∩ B = (A ∪ C) ∩ B
B = (A ∪C) ∩ B = (A ∩ B) ∪ (C ∩ B)
या (A ∩ ) ∪ (B ∩ C) = B ……(ii)
(i) और (ii) से B = C प्राप्त होता है।

प्रश्न 4.
दिखाइए कि निम्नलिखित चार प्रतिबन्ध तुल्य हैं:
(i) A ⊂ B
(ii) A – B = Φ
(iii) A ∪ B = B
हल:
(i) A ⊂ B अर्थात् समुच्चय A के सभी अवयव B में हैं।
A – B = Φ अर्थात (i) ⇔ (ii)
(ii) A – B = Φ ⇔ समुच्चय A के सभी अवयव B में हैं।
A ∪ B = B
अर्थात (ii) ⇔ (iii)
(iii) A ∪ B = B ⇔ समुच्चय A के सभी अवयव B में है।
समुच्चय A और B मे समुच्चय A के अवयव उभयनिष्ठ है।
A ∩B = A
इससे स्पष्ट है सभी कथन समान हैं।

प्रश्न 5.
दिखाइए कि यदि A ⊂ B तो C – B ⊂ C – A.
हल:
मान लीजिए x ∈ C – B ⇒ x ∈ C पंरतु x ∈ B
दिया है: A ⊂ B ⇒ यदि x ∉ B ⇒ x ∉ A
अर्थात, x ∈ C और x ∉ A ⇒ x ∈ C – A
यहाँ हम पाते हैं कि
यदि x ∈ C – B तब x ∈ C – A
⇒ C – B ⊂ C – A.

प्रश्न 6.
मान लीजिए कि P(A) = P(B), सिद्ध कीजिए कि A = B.
हल:
मान लीजिए x, समुच्चय A का कोई अवयव है।
तब एक उपसमुच्चय X (मान लो) ऐसा होगा जिसमे x ∈ A जिसके अनुसार
X ⊂ A ⇒ X ∈ P(A)
X ∈ P(B) [P(A) = P(B)]
X ⊂ B या x ∈ B
अर्थात यदि
x ∈ A तब x ∈ B ⇒ A ∈ B …..(i)
y समुच्चय B का कोई अवयव हो, तब
समुच्चय B का कोई उपसमुच्चय Y (मान लो) होगा जिससे y ∈ Y
Y ⊂ B ⇒ Y ∈P(B)
Y ∈P(A) [P(A) = P(B)]
Y ⊂ A यदि y ∈ B तब y ∈ A
B ⊂ A ………(ii)
समीकरण (i) और (ii) से, हम पाते हैं।
A = B.

प्रश्न 7.
किन्हीं भी समुच्चयों A तक B के लिए क्या यह सत्य है कि P(A) ∪ P(B) = P(A ∪ B) ? अपने उत्तर का औचित्य बताइए।
हल:
मान लीजिए।
A = {a}, B = {b}, और A ∪ B = {a, b}
P(A) = {Φ, {a}}, P(B) = {Φ, {b}}
P(A) ∪ P(B) = {Φ, {4}, {5}} …(i)
अब A ∪ B = {a, b}
P(A ∪ B) = {Φ, {a}, {b}, {a, b}}
समी. (i) और (ii) से हम देखते हैं कि
अतः P(A) ∪ P(B) ≠ P(A ∪ B)

प्रश्न 8.
किन्हीं दो समुच्चयों A तथा B के लिए सिद्ध कीजिए कि
A = (A ∩ B) ∪ (A – B) और A ∪ (B – A) = A ∪ B.
हल:
(i) दायाँ पक्ष = (A ∩ B) ∪ (A – B)
= (A ∩ B) ∪ (A – B) [A – B = A ∩ B’]
= (A ∩ (B ∪ B’) (वितरण गुण से)
= A ∩ U (यहाँ U सार्वत्रिक समुच्चय)
= A
अतः (A ∩ B) ∪ (A – B) = A.
(ii) बायाँ पक्ष = A ∪ (B – A)
= A ∪ (B ∩ A’) [B – A = B ∩ A’]
= (A ∪ B) ∩ (A ∪ A’) (वितरण गुण से)
= (A ∪ B) ∩ U [यहाँ U सार्वत्रिक समुच्चय]
= A ∪ B
अतः : A ∪ (B – A) = A ∪ B

प्रश्न 9.
समुच्चयों के गुणधर्मों का प्रयोग करके सिद्ध कीजिए कि
(i) A ∪ (A ∩ B) = A
(ii) A ∩ (A ∪ B) = A.
हल:
(i) बायाँ पक्ष = A ∪ (A ∩ B)
= (A ∪ A) ∩ (A ∪ B) (वितरण गुण से)
= A ∩ (A ∪ B) (A ∪ A = A)
= A [A ⊂ A ∪ B]
A ∪ (A ∩ B) = A.
(ii) बायाँ पक्षु = A ∩ (A ∪ B)
= (A ∩ A) ∪ (A ∩ B) [वितरण गुण से]
= A ∪ (A ∩ B) [A ∩ A = A]
= A [A ∩ B ⊂ A]
अतः A ∩(A ∪ B) = A.

प्रश्न 10.
दिखलाइए कि A ∩ B = A ∩ C का तात्पर्य B = C आवश्यक रूप से नहीं होता।
हल:
मान लीजिए A = {1, 2}, B = {1, 7} तथा C = {1, 4} हो, तब
A ∩ B = {1, 2} ∩ {1, 7} = {1}
A ∩ C = {1, 2} ∩ {1, 4} = {1}
A ∩ B = A ∩ C
B ≠ C
यदि A ∩ B = A ∩ C तो आवश्यक नूह है कि B = C.

प्रश्न 11.
मान लीजिए कि A और B समुच्चय हैं। यदि किसी समुच्चय X के लिए A∪ X = B ∪ X = Φ तथा A ∪ X = B ∪ X तो सिद्ध कीजिए कि A = B.
हल:
दिया है A ∪ X = B ∪ X, जब कि X कोई समुच्चय है।
A ∩ (A ∪ X) = A ∩ (B ∪ X) [A ⊂ A ∪X, A ∩ (A ∪ X) = A]
A = A ∩ (B ∪ X)
= (A ∩ B) ∪ (A ∩ X) [वितरण गुण से]
= (A ∩ B) ∪ Φ (दिया है, A ∩ X = Φ
= A ∩ B
A ⊂ B ……(i)
A ∪ X = B ∪ X
B ∩(A ∪ X) = B ∩ (B ∪ X)
B ∩(A ∪ X) = B [B ⊂ B ∪ X]
(B ∩ A) ∪ (B ∩ X) = B [वितरण गुण से]
(B ∩ A) ∪ Φ = B [दिया है: B ∩ X = Φ]
(B ∩ A) = B
B ⊂ A …..(ii)
समी. (i) और (ii) से, हम पाते हैं कि A = B.

प्रश्न 12.
ऐसे समुच्चय A, B और C ज्ञात कीजिए ताकि A ∩ B, B ∩ C तथा A ∩ C आरिक्त समुच्चय हों और A ∩ B ∩ C = Φ.
हल:
मान लीजिए। A = {1, 2}, B = {2, 3}, C = {1, 3}
A ∩ B = {1, 2} ∩ {2, 3} = {2},
B ∩ C = {2, 3} ∩ {1, 3} = {3}
C ∩ A = {1, 3} ∩ {1, 2} = {1}
अतः A ∩ B, B ∩ C, C ∩ A रिक्त समुच्चय नहीं हैं।
A ∩ B ∩ C = (A ∩ B) ∩ C = {2} ∩ {1, 3} = Φ
इति सिद्धम्

प्रश्न 13.
किसी विद्यालय के 600 विद्यार्थियों के सर्वेक्षण से ज्ञात हुआ कि 150 विद्यार्थी चाय, 225 विद्यार्थी कॉफी तथा 100 विद्यार्थी चाय और कॉफी दोनों पीते हैं। ज्ञात कीजिए कि कितने विद्यार्थी न तो चाय पीते हैं और न कॉफी पीते हैं।
हल:
मान लीजिए 7 और C चाय तथा कॉफी पीने वाले विद्यार्थियों के समुच्चय हों, तब
n(T) = 150, n(C) = 225, n(T ∩ C) = 100
n(T ∪ C) = n(T) + n(C) – n(T ∩ C) = 150 + 225 – 100 = 275
= उन विद्यार्थियों की संख्या जो चाय या कॉफी पीते हैं या चाय और कॉफी दोनों पीते हैं।
विद्यार्थियों की कुल संख्या = 600
उन विद्यार्थियों की संख्या जो चाय या कॉफी कुछ भी नहीं पीते = 600 – 275 = 325.

प्रश्न 14.
विद्यार्थियों के समूह में, 100 विद्यार्थी हिन्दी, 50 विद्यार्थी अंग्रेजी तथा 25 विद्यार्थी दोनों भाषाओं को जानते हैं। विद्यार्थियों में से प्रत्येक या तो हिन्दी या अंग्रेजी जानता है। समूह में कुल कितने विद्यार्थी हैं?
हल:
पाना तथा क्रमशः हिन्दी और अंग्रेजी जानने वालों के समुच्चय हों, तब
n(H) = 100, n(E) = 50, n(H ∩ E) = 25
n(H ∪ E) = n(H) + n(E) – n(H ∩E) = 100 + 50 – 25 = 125
उन विद्यार्थियों की संख्या जो हिन्दी या अंग्रेजी जानते हैं = 125.

प्रश्न 15.
60 लोगों के सर्वेक्षण में पाया गया कि 25 लोग समाचार पत्र H, 26 लोग समाचार पत्र T, 26 लोग समाचार पत्र I, 9 लोग H तथा I दोनों, 11 लोग H तथा T दोनों, 8 लोग T तथा I दोनों और 3 लोग तीनों ही समाचार पत्र पढ़ते हैं, तो निम्नलिखित ज्ञात कीजिए:
(i) कम से कम एक समाचार पत्र पढ़ने वालों की संख्या।
(ii) ठीक ठीक केवल एक समाचार पत्र पढ़ने वालों की संख्या।
हल:
कुल लोगों की संख्या जिनका सर्वेक्षण किया गया = 60
H समाचार पत्र पढ़ने वालों की संख्या, n (H) = 25
T समाचार पत्र पढ़ने वालों की संख्या, n (T) = 26
I समाचार पत्र पढ़ने वालों की संख्या, n (I) = 26
UP Board Solutions for Class 11 Maths Chapter 1 Sets 15
H और I समाचार पत्र पढ़ने वालों की संख्या, n (H ∩ I) = 9
H और I समाचार पत्र पढ़ने वालों की संख्या, n (H ∩ T) = 11
T और I समाचार पत्र पढ़ने वालों की संख्या, n (T ∩ I) = 8
तीनों समाचार पत्र पढ़ने वालों की संख्या, n (H ∩ T ∩ I) = 3
H और I समाचार पत्र पढ़ने वाले तथा T समाचार पत्र न पढ़ने वालों की संख्या = 9 – 3 = 6
H और T समाचार पत्र पढ़ने वाले तथा I समाचार पत्र न पढ़ने वालों की संख्या = 11 – 3 = 8
T और I समाचार पत्र पढ़ने वाले तथा H समाचार पत्र न पढ़ने वालों की संख्या = 8 – 3 = 5
केवल H समाचार पत्र पढ़ने वालों की संख्या = 25 – 8 – 6 – 3 = 8
केवल T समाचार पत्र पढ़ने वालों की संख्या = 26 – 8 – 3 – 5 = 10
केवल I समाचार पत्रं पढ़ने वालों की संख्या = 26 – 6 – 3 – 5 = 12
कम से कम एकं समाचार पत्र पढ़ने वालों की संख्या = केवेल एक समाचार पत्र पढ़ने वालों की संख्या + केवल दो समाचार पत्र पढ़ने वालों की संख्या + तीनों समाचार पत्र पढ़ने वालों की संख्या = (8 + 10 + 12) + (8 + 6 + 5) + 3 = 30 + 19 + 3 = 52
वैकल्पिक विधि :
n(H ∪ T ∪ I) = n(H) + n(T) + n(I) – n(H ∩ T) – n(T ∩ I) – n(H∩I) + n(H ∩ T ∩ I)
= 25 + 26 + 26 – 11 – 8 – 9 + 3 = 77 – 28 + 3 = 80 – 28 = 52

(ii) केवल H और T समाचार पत्र पढ़ने वालों की संख्या = 11 – 3 = 8
केवल T और 1 समाचार पत्र पढ़ने वालों की संख्या = 8 – 3 = 5
केवल 1 और H समाचार पत्र पढ़ने वालों की संख्या = 9 – 3 = 6
तीनों समाचार पत्र पढ़ने वालों की संख्या = 3
केवल एक समाचार पत्र पढ़ने वालों की संख्या = 52 – (8 + 5 + 6 + 3) = 52 – 22 = 30.

प्रश्न 16.
एक सर्वेक्षण में पाया गया कि 21 लोग उत्पाद A, 26 लोग उत्पाद B, 29 लोग उत्पाद C पसंद करते हैं। यदि 14 लोग उत्पाद A तथा B, 12 लोग उत्पाद C तथा A, 14 लोग उत्पाद B तथा C और 8 लोग तीनों ही उत्पादों को पसंद करते हैं। ज्ञात कीजिए कि कितने लोग केवल उत्पाद C को पसंद करते हैं?
हल:
दिया है:
n(A) = 21,
n(B) = 26,
n(C) = 29
n(A ∩ B) = 14,
n(A ∩ C) = 12
n(B ∩ C) = 14,
n(A ∩ B ∩ C) = 8
n(A ∩ C) = 12,
P(A ∩ B ∩ C) = 8
UP Board Solutions for Class 11 Maths Chapter 1 Sets 16
n(केवल A और C) = 12 – 8 = 4
n(केवल B और C) = 14 – 8 = 6
n(केवल C) = n(C) – n (केवल A और C) – n (केवल B और C) – n(A ∩ B ∩ C) = 29 – 4 – 6 – 8 = 29 – 18 = 11

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