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Занимательные задачи 2, страница 69 - гдз по алгебре 9 класс учебник Солтан, Солтан
Авторы: Солтан Г. Н., Солтан А. Е., Жумадилова А. Ж.
Тип: Учебник
Издательство: Кокшетау
Год издания: 2019 - 2026
ISBN: 978-601-317-424-2
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 9 классе
II. Элементы комбинаторики. 7. Основные понятия и правила комбинаторики - страница 69.
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Элементы комбинаторики" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1038 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "62" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Book {#1039 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1050 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1051 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1057 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } 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#casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107646 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "II. 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"field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1100 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107646 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "II. 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"field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1107 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1109 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1111 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132 #items: array:1 [ 0 => App\Models\Branch {#1133 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107646 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "II. Элементы комбинаторики" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_page_start" => "62" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1135 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1136 …2} ] #original: array:24 [ "id" => 1107646 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "II. Элементы комбинаторики" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_page_start" => "62" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1135 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1136 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107647 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "7. Основные понятия и правила комбинаторики" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1102} "field_page_start" => "63" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1103} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1140 #items: array:3 [ 0 => App\Models\Element {#1149 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1277743 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1159 #items: array:1 [ 0 => App\Models\Edition {#1158 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1169 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1166 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1162 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1160 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1169 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1166 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1162 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1160 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1107922" "type" => "task" ] "text" => "<p>2) Кусок ткани разрезали на 4 части, затем одну из них – еще на 4 части. Одну из вновь полученных частей снова разрезали на 4 части и так далее. Могло ли получиться всех кусков: а) 36; б) 35?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1571" "height" => 283 "path" => "/media/algebra_09/soltan-u19/0-zad69/2-1.webp?ts=1752835192" ] ] ] #original: array:7 [ "id" => 1277743 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1159} "task" => array:2 [ "refs" => "1107922" "type" => "task" ] "text" => "<p>2) Кусок ткани разрезали на 4 части, затем одну из них – еще на 4 части. Одну из вновь полученных частей снова разрезали на 4 части и так далее. Могло ли получиться всех кусков: а) 36; б) 35?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1571" "height" => 283 "path" => "/media/algebra_09/soltan-u19/0-zad69/2-1.webp?ts=1752835192" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1151 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1278909 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1164 #items: array:1 [ 0 => App\Models\Edition {#1152 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1155 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1156 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1154 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1155 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1156 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1154 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1107922" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1371" "height" => 1465 "path" => "/media/algebra_09/soltan-u19/1-zad69/2-1.webp?ts=1752832066" ] ] ] #original: array:6 [ "id" => 1278909 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1164} "task" => array:2 [ "refs" => "1107922" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1371" "height" => 1465 "path" => "/media/algebra_09/soltan-u19/1-zad69/2-1.webp?ts=1752832066" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1161 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1554885 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1177 #items: array:1 [ 0 => App\Models\Edition {#1165 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1107922" "type" => "task" ] "text" => "<p><strong>2)</strong> Для того чтобы определить, существует ли такое значение $x$, при котором верно данное равенство, необходимо упростить обе его части и проверить, может ли оно выполняться.</p><p>Сначала преобразуем левую часть равенства: $(1 + \text{tg}^2x)^{-1}$. Используя основное тригонометрическое тождество $1 + \text{tg}^2x = \frac{1}{\cos^2x}$, которое справедливо для всех $x$, при которых $\text{tg}x$ определен ($x \neq \frac{\pi}{2} + \pi k$, где $k \in \mathbb{Z}$), получаем:</p><p>$(1 + \text{tg}^2x)^{-1} = \left(\frac{1}{\cos^2x}\right)^{-1} = \cos^2x$.</p><p>Далее преобразуем правую часть равенства: $\sqrt{2^{-4} + 2^{-3} + 2^{-2} + 2^{-1} + 2^0}$. Вычислим значение выражения под корнем, представив степени с отрицательными показателями в виде дробей:</p><p>$2^{-4} + 2^{-3} + 2^{-2} + 2^{-1} + 2^0 = \frac{1}{2^4} + \frac{1}{2^3} + \frac{1}{2^2} + \frac{1}{2} + 1 = \frac{1}{16} + \frac{1}{8} + \frac{1}{4} + \frac{1}{2} + 1$.</p><p>Приведем все слагаемые к общему знаменателю 16:</p><p>$\frac{1}{16} + \frac{2}{16} + \frac{4}{16} + \frac{8}{16} + \frac{16}{16} = \frac{1 + 2 + 4 + 8 + 16}{16} = \frac{31}{16}$.</p><p>Теперь извлечем квадратный корень:</p><p>$\sqrt{\frac{31}{16}} = \frac{\sqrt{31}}{\sqrt{16}} = \frac{\sqrt{31}}{4}$.</p><p>После упрощения обеих частей исходное равенство принимает вид:</p><p>$\cos^2x = \frac{\sqrt{31}}{4}$.</p><p>Область значений функции $y = \cos^2x$ есть отрезок $[0, 1]$, так как для любого действительного $x$ выполняется неравенство $-1 \le \cos x \le 1$, и, следовательно, $0 \le \cos^2x \le 1$.</p><p>Для того чтобы равенство имело решение, значение в правой части должно принадлежать этой области значений. Оценим значение выражения $\frac{\sqrt{31}}{4}$. Поскольку $5^2 = 25$ и $6^2 = 36$, то справедливо неравенство $5 < \sqrt{31} < 6$.</p><p>Разделим все части этого двойного неравенства на 4:</p><p>$\frac{5}{4} < \frac{\sqrt{31}}{4} < \frac{6}{4}$</p><p>$1.25 < \frac{\sqrt{31}}{4} < 1.5$</p><p>Таким образом, значение $\frac{\sqrt{31}}{4}$ больше 1. Поскольку $\cos^2x$ не может принимать значения, большие 1, уравнение $\cos^2x = \frac{\sqrt{31}}{4}$ не имеет действительных решений.</p><p><strong>Ответ:</strong> нет, такого значения $x$ не существует.</p>" ] #original: array:6 [ "id" => 1554885 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1177} "task" => array:2 [ "refs" => "1107922" "type" => "task" ] "text" => "<p><strong>2)</strong> Для того чтобы определить, существует ли такое значение $x$, при котором верно данное равенство, необходимо упростить обе его части и проверить, может ли оно выполняться.</p><p>Сначала преобразуем левую часть равенства: $(1 + \text{tg}^2x)^{-1}$. Используя основное тригонометрическое тождество $1 + \text{tg}^2x = \frac{1}{\cos^2x}$, которое справедливо для всех $x$, при которых $\text{tg}x$ определен ($x \neq \frac{\pi}{2} + \pi k$, где $k \in \mathbb{Z}$), получаем:</p><p>$(1 + \text{tg}^2x)^{-1} = \left(\frac{1}{\cos^2x}\right)^{-1} = \cos^2x$.</p><p>Далее преобразуем правую часть равенства: $\sqrt{2^{-4} + 2^{-3} + 2^{-2} + 2^{-1} + 2^0}$. Вычислим значение выражения под корнем, представив степени с отрицательными показателями в виде дробей:</p><p>$2^{-4} + 2^{-3} + 2^{-2} + 2^{-1} + 2^0 = \frac{1}{2^4} + \frac{1}{2^3} + \frac{1}{2^2} + \frac{1}{2} + 1 = \frac{1}{16} + \frac{1}{8} + \frac{1}{4} + \frac{1}{2} + 1$.</p><p>Приведем все слагаемые к общему знаменателю 16:</p><p>$\frac{1}{16} + \frac{2}{16} + \frac{4}{16} + \frac{8}{16} + \frac{16}{16} = \frac{1 + 2 + 4 + 8 + 16}{16} = \frac{31}{16}$.</p><p>Теперь извлечем квадратный корень:</p><p>$\sqrt{\frac{31}{16}} = \frac{\sqrt{31}}{\sqrt{16}} = \frac{\sqrt{31}}{4}$.</p><p>После упрощения обеих частей исходное равенство принимает вид:</p><p>$\cos^2x = \frac{\sqrt{31}}{4}$.</p><p>Область значений функции $y = \cos^2x$ есть отрезок $[0, 1]$, так как для любого действительного $x$ выполняется неравенство $-1 \le \cos x \le 1$, и, следовательно, $0 \le \cos^2x \le 1$.</p><p>Для того чтобы равенство имело решение, значение в правой части должно принадлежать этой области значений. Оценим значение выражения $\frac{\sqrt{31}}{4}$. Поскольку $5^2 = 25$ и $6^2 = 36$, то справедливо неравенство $5 < \sqrt{31} < 6$.</p><p>Разделим все части этого двойного неравенства на 4:</p><p>$\frac{5}{4} < \frac{\sqrt{31}}{4} < \frac{6}{4}$</p><p>$1.25 < \frac{\sqrt{31}}{4} < 1.5$</p><p>Таким образом, значение $\frac{\sqrt{31}}{4}$ больше 1. Поскольку $\cos^2x$ не может принимать значения, большие 1, уравнение $\cos^2x = \frac{\sqrt{31}}{4}$ не имеет действительных решений.</p><p><strong>Ответ:</strong> нет, такого значения $x$ не существует.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1107923" "type" => "task" ] "previous" => array:2 [ "refs" => "1107921" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1197 #items: array:1 [ 0 => App\Models\Book {#1146 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: 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"field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1145 #items: array:1 [ 0 => App\Models\Term {#1147 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1143 #items: array:1 [ 0 => App\Models\Term {#1141 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #original: 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array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:1 [ 0 => App\Models\BookPage {#1210 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1097811 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "69" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-69" "field_display_title" => "69" "field_folder" => "folder1" "field_image_name" => "69" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1212 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1213 #items: array:1 [ 0 => App\Models\Book {#1146} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1214 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1224 #items: array:1 [ 0 => App\Models\Element {#1223 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1260962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1225 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1260962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1225 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1097812" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097810" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1380 #items: array:6 [ 0 => App\Models\Task {#1392 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107920 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "69" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-217" "field_display_title" => "217" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1396 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1395 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1394 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1397 …2} "content" => Illuminate\Database\Eloquent\Collection {#1400 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1406 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107920 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "69" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-217" "field_display_title" => "217" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1396 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1395 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1394 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1397 …2} "content" => Illuminate\Database\Eloquent\Collection {#1400 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1406 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1399 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107921 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "69" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/zad69-1" "field_display_title" => "Занимательные задачи 1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1398 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1405 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1407 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1403 …2} "content" => Illuminate\Database\Eloquent\Collection {#1402 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1420 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107921 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "69" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/zad69-1" "field_display_title" => "Занимательные задачи 1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1398 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1405 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1407 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1403 …2} "content" => Illuminate\Database\Eloquent\Collection {#1402 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1420 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1404 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107922 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "69" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/zad69-2" "field_display_title" => "Занимательные задачи 2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1401 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1419 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1421 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1417 …2} "content" => Illuminate\Database\Eloquent\Collection {#1415 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1418 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107922 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "69" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/zad69-2" "field_display_title" => "Занимательные задачи 2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1401 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1419 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1421 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1417 …2} "content" => Illuminate\Database\Eloquent\Collection {#1415 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1418 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1416 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107923 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "69" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/zad69-3" "field_display_title" => "Занимательные задачи 3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1429 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1430 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1431 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1432 …2} "content" => Illuminate\Database\Eloquent\Collection {#1435 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1441 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107923 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "69" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/zad69-3" "field_display_title" => "Занимательные задачи 3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1429 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1430 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1431 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1432 …2} "content" => Illuminate\Database\Eloquent\Collection {#1435 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1441 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1434 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107924 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "69" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/zad69-4" "field_display_title" => "Занимательные задачи 4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1433 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1440 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1442 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1438 …2} "content" => Illuminate\Database\Eloquent\Collection {#1437 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1455 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107924 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "69" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/zad69-4" "field_display_title" => "Занимательные задачи 4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1433 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1440 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1442 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1438 …2} "content" => Illuminate\Database\Eloquent\Collection {#1437 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1455 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Task {#1439 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107925 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "69" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/zad69-5" "field_display_title" => "Занимательные задачи 5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1436 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1454 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1456 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1452 …2} "content" => Illuminate\Database\Eloquent\Collection {#1451 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1469 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107925 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "69" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/zad69-5" "field_display_title" => "Занимательные задачи 5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1436 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1454 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1456 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1452 …2} "content" => Illuminate\Database\Eloquent\Collection {#1451 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1469 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] 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#attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107922 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "69" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/zad69-2" "field_display_title" => "Занимательные задачи 2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1036} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1034} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1070} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1137} "content" => Illuminate\Database\Eloquent\Collection {#1140} "next" => array:2 [ "refs" => "1107923" "type" => "task" ] "previous" => array:2 [ "refs" => "1107921" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1197} "page" => array:2 [ "refs" => "1097811" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
Занимательные задачи 2 (с. 69)
Условие. Занимательные задачи 2 (с. 69)
скриншот условия
2) Кусок ткани разрезали на 4 части, затем одну из них – еще на 4 части. Одну из вновь полученных частей снова разрезали на 4 части и так далее. Могло ли получиться всех кусков: а) 36; б) 35?
Решение. Занимательные задачи 2 (с. 69)
Решение 2 (rus). Занимательные задачи 2 (с. 69)
2) Для того чтобы определить, существует ли такое значение $x$, при котором верно данное равенство, необходимо упростить обе его части и проверить, может ли оно выполняться.
Сначала преобразуем левую часть равенства: $(1 + \text{tg}^2x)^{-1}$. Используя основное тригонометрическое тождество $1 + \text{tg}^2x = \frac{1}{\cos^2x}$, которое справедливо для всех $x$, при которых $\text{tg}x$ определен ($x \neq \frac{\pi}{2} + \pi k$, где $k \in \mathbb{Z}$), получаем:
$(1 + \text{tg}^2x)^{-1} = \left(\frac{1}{\cos^2x}\right)^{-1} = \cos^2x$.
Далее преобразуем правую часть равенства: $\sqrt{2^{-4} + 2^{-3} + 2^{-2} + 2^{-1} + 2^0}$. Вычислим значение выражения под корнем, представив степени с отрицательными показателями в виде дробей:
$2^{-4} + 2^{-3} + 2^{-2} + 2^{-1} + 2^0 = \frac{1}{2^4} + \frac{1}{2^3} + \frac{1}{2^2} + \frac{1}{2} + 1 = \frac{1}{16} + \frac{1}{8} + \frac{1}{4} + \frac{1}{2} + 1$.
Приведем все слагаемые к общему знаменателю 16:
$\frac{1}{16} + \frac{2}{16} + \frac{4}{16} + \frac{8}{16} + \frac{16}{16} = \frac{1 + 2 + 4 + 8 + 16}{16} = \frac{31}{16}$.
Теперь извлечем квадратный корень:
$\sqrt{\frac{31}{16}} = \frac{\sqrt{31}}{\sqrt{16}} = \frac{\sqrt{31}}{4}$.
После упрощения обеих частей исходное равенство принимает вид:
$\cos^2x = \frac{\sqrt{31}}{4}$.
Область значений функции $y = \cos^2x$ есть отрезок $[0, 1]$, так как для любого действительного $x$ выполняется неравенство $-1 \le \cos x \le 1$, и, следовательно, $0 \le \cos^2x \le 1$.
Для того чтобы равенство имело решение, значение в правой части должно принадлежать этой области значений. Оценим значение выражения $\frac{\sqrt{31}}{4}$. Поскольку $5^2 = 25$ и $6^2 = 36$, то справедливо неравенство $5 < \sqrt{31} < 6$.
Разделим все части этого двойного неравенства на 4:
$\frac{5}{4} < \frac{\sqrt{31}}{4} < \frac{6}{4}$
$1.25 < \frac{\sqrt{31}}{4} < 1.5$
Таким образом, значение $\frac{\sqrt{31}}{4}$ больше 1. Поскольку $\cos^2x$ не может принимать значения, большие 1, уравнение $\cos^2x = \frac{\sqrt{31}}{4}$ не имеет действительных решений.
Ответ: нет, такого значения $x$ не существует.
Другие задания:
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