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Номер 21, страница 96, часть 1 - гдз по алгебре 10 класс учебник Пак, Ардакулы
Авторы: Пак О. В., Ардакулы Д., Ескендирова Е. В.
Тип: Учебник
Издательство: Алматыкітап баспасы
Год издания: 2019 - 2026
Часть: 1
ISBN: 978-601-01-3958-9
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Часть 1. Глава 2. Тригонометрические функции. Параграф 2. Графики функций. 2.6. Примеры. Задачи - номер 21, страница 96.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #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" => 1119739 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "96" "field_page_end" => "97" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/2-2-6zad-21" "field_display_title" => "21" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #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" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #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_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 {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Branch {#1034 #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" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #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_page_start" => "6" "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" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #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" => 4295 "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 {#1042 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "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 {#1065 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4295 "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 {#1042} "field_class" => Illuminate\Database\Eloquent\Collection {#1044} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046} "field_author" => Illuminate\Database\Eloquent\Collection {#1048} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053} "field_country" => Illuminate\Database\Eloquent\Collection {#1055} "field_city" => Illuminate\Database\Eloquent\Collection {#1057} "field_series" => Illuminate\Database\Eloquent\Collection {#1059} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "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 {#1065} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #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 {#1069 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "6" "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" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1069} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1186 #items: array:5 [ 0 => App\Models\Branch {#1198 #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" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1202 #items: array:1 [ 0 => App\Models\Term {#1199 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #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_page_start" => "6" "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" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1231 #items: array:1 [ 0 => App\Models\Book {#1201 #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" => 4295 "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 {#1203 #items: array:1 [ 0 => App\Models\Term {#1200 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "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 {#1204 #items: array:1 [ 0 => App\Models\Term {#1205 #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" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #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 {#1206 #items: array:1 [ 0 => App\Models\Term {#1207 #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" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "field_cases" => array:6 [ …6] "field_translit" => "almatykitap baspasy" ] #original: array:6 [ "id" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "field_cases" => array:6 [ …6] "field_translit" => "almatykitap baspasy" ] #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_author" => Illuminate\Database\Eloquent\Collection {#1208 #items: array:3 [ 0 => App\Models\Term {#1209 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Пак" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Олег" "field_patronymic" => "Владимирович" "field_surname_rp" => null ] #original: array:12 [ "id" => 7007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Пак" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Олег" "field_patronymic" => "Владимирович" "field_surname_rp" => null ] #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\Term {#1210 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ардакулы" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дархан" "field_patronymic" => null "field_surname_rp" => null ] #original: array:12 [ "id" => 7008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ардакулы" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дархан" "field_patronymic" => null "field_surname_rp" => null ] #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\Term {#1211 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ескендирова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Елена" "field_patronymic" => "Викторовна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ескендирова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Елена" "field_patronymic" => "Викторовна" "field_surname_rp" => 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 } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1212 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1213 #items: array:1 [ 0 => App\Models\Term {#1214 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #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_country" => Illuminate\Database\Eloquent\Collection {#1215 #items: array:1 [ 0 => App\Models\Term {#1216 #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" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #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_city" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:1 [ 0 => App\Models\Term {#1218 #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" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #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_series" => Illuminate\Database\Eloquent\Collection {#1219 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1220 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1221 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1222 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1223 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1224 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1225 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1226 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1227 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1228 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4295 "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 {#1203} "field_class" => Illuminate\Database\Eloquent\Collection {#1204} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1206} "field_author" => Illuminate\Database\Eloquent\Collection {#1208} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1212} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1213} "field_country" => Illuminate\Database\Eloquent\Collection {#1215} "field_city" => Illuminate\Database\Eloquent\Collection {#1217} "field_series" => Illuminate\Database\Eloquent\Collection {#1219} "field_umk" => Illuminate\Database\Eloquent\Collection {#1220} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1221} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1222} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1223} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1224} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1225} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1226} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1227} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1228} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #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 {#1229 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1202} "field_page_start" => "6" "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" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1231} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1229} ] #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\Branch {#1230 #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" => 1119358 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические функции" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1235 #items: array:1 [ 0 => App\Models\Term {#1232 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => 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 } "field_page_start" => "71" "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 {#1234 #items: array:1 [ 0 => App\Models\Book {#1201} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1233 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119358 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические функции" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1235} "field_page_start" => "71" "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 {#1234} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1233} ] #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\Branch {#1236 #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" => 1119364 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Графики функций" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1240 #items: array:1 [ 0 => App\Models\Term {#1237 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #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_page_start" => "79" "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 {#1239 #items: array:1 [ 0 => App\Models\Book {#1201} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1280 #items: array:1 [ 0 => App\Models\Branch {#1238 #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" => 1119358 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические функции" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1242 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_page_start" => "71" "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 {#1243 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1272 #items: array:1 [ …1] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119358 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические функции" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1242} "field_page_start" => "71" "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 {#1243} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1272} ] #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" => 1119364 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Графики функций" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1240} "field_page_start" => "79" "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 {#1239} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1280} ] #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\Branch {#1278 #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" => 1119371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.6. Примеры" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1279 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "92" "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 {#1281 #items: array:1 [ 0 => App\Models\Book {#1201} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1327 #items: array:1 [ 0 => App\Models\Branch {#1282 #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" => 1119364 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Графики функций" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1284 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_page_start" => "79" "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 {#1285 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1314 #items: array:1 [ …1] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119364 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Графики функций" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1284} "field_page_start" => "79" "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 {#1285} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1314} ] #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" => 1119371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.6. Примеры" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1279} "field_page_start" => "92" "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 {#1281} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1327} ] #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\Branch {#1325 #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" => 1119372 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1326 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "93" "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 {#1328 #items: array:1 [ 0 => App\Models\Book {#1201} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1337 #items: array:1 [ 0 => App\Models\Branch {#1336 #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" => 1119371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.6. Примеры" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1338 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "92" "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 {#1339 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1340 #items: array:1 [ …1] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.6. Примеры" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1338} "field_page_start" => "92" "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 {#1339} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1340} ] #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" => 1119372 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1326} "field_page_start" => "93" "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 {#1328} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1337} ] #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 {#1142 #items: array:2 [ 0 => App\Models\Element {#1172 #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" => 1300422 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1162 #items: array:1 [ 0 => App\Models\Edition {#1171 #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" => 4962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1169 #items: array:1 [ …1] #escapeWhenCastingToString: false } "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 {#1168 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1167 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1166 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1169} "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 {#1168} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1167} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1166} "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" => "1119739" "type" => "task" ] "text" => "<p><strong>21. a)</strong>(2) Дана убывающая последовательность чисел $x_1, x_2, x_3, \dots, x_n$, каждое из которых принадлежит отрезку $[-8\pi; -6\pi]$. Что можно сказать о монотонности последовательности $\cos x_1, \cos x_2, \cos x_3, \dots, \cos x_n$?</p><p><strong>б)</strong>(2) Дана возрастающая последовательность чисел $x_1, x_2, x_3, \dots, x_n$, каждое из которых принадлежит отрезку $[80,5\pi; 81\pi]$. Что можно сказать о монотонности последовательности $\sin x_1, \sin x_2, \sin x_3, \dots, \sin x_n$?</p><p><strong>в)</strong>(2) Дана убывающая последовательность чисел $x_1, x_2, x_3, \dots, x_n$, каждое из которых принадлежит отрезку $[80,5\pi; 81\pi]$. Что можно сказать о монотонности последовательности $\operatorname{tg} x_1, \operatorname{tg} x_2, \operatorname{tg} x_3, \dots, \operatorname{tg} x_n$?</p>" "img" => array:2 [ 0 => array:5 [ "name" => "21-1.jpg" "alt" => null "width" => "1691" "height" => 574 "path" => "/media/algebra_10/baspasy-u19/0-2-2-6zad/21-1.webp?ts=1753534571" ] 1 => array:5 [ "name" => "21-2.jpg" "alt" => null "width" => "1633" "height" => 319 "path" => "/media/algebra_10/baspasy-u19/0-2-2-6zad/21-2.webp?ts=1753534571" ] ] ] #original: array:7 [ "id" => 1300422 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1162} "task" => array:2 [ "refs" => "1119739" "type" => "task" ] "text" => "<p><strong>21. a)</strong>(2) Дана убывающая последовательность чисел $x_1, x_2, x_3, \dots, x_n$, каждое из которых принадлежит отрезку $[-8\pi; -6\pi]$. Что можно сказать о монотонности последовательности $\cos x_1, \cos x_2, \cos x_3, \dots, \cos x_n$?</p><p><strong>б)</strong>(2) Дана возрастающая последовательность чисел $x_1, x_2, x_3, \dots, x_n$, каждое из которых принадлежит отрезку $[80,5\pi; 81\pi]$. Что можно сказать о монотонности последовательности $\sin x_1, \sin x_2, \sin x_3, \dots, \sin x_n$?</p><p><strong>в)</strong>(2) Дана убывающая последовательность чисел $x_1, x_2, x_3, \dots, x_n$, каждое из которых принадлежит отрезку $[80,5\pi; 81\pi]$. Что можно сказать о монотонности последовательности $\operatorname{tg} x_1, \operatorname{tg} x_2, \operatorname{tg} x_3, \dots, \operatorname{tg} x_n$?</p>" "img" => array:2 [ 0 => array:5 [ "name" => "21-1.jpg" "alt" => null "width" => "1691" "height" => 574 "path" => "/media/algebra_10/baspasy-u19/0-2-2-6zad/21-1.webp?ts=1753534571" ] 1 => array:5 [ "name" => "21-2.jpg" "alt" => null "width" => "1633" "height" => 319 "path" => "/media/algebra_10/baspasy-u19/0-2-2-6zad/21-2.webp?ts=1753534571" ] ] ] #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 {#1165 #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" => 1551458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1156 #items: array:1 [ 0 => App\Models\Edition {#1164 #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" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1160 #items: array:1 [ …1] #escapeWhenCastingToString: false } "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 {#1161 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1158 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1159 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1160} "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 {#1161} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1158} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1159} "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" => "1119739" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p> По условию, дана убывающая последовательность чисел $x_1, x_2, \dots, x_n$, то есть $x_1 > x_2 > \dots > x_n$. Все члены этой последовательности принадлежат отрезку $[-8\pi; -6\pi]$. <br> Нам нужно определить характер монотонности последовательности $y_k = \cos(x_k)$. Для этого исследуем поведение функции $f(x) = \cos(x)$ на отрезке $[-8\pi; -6\pi]$. <br> Функция $f(x) = \cos(x)$ является периодической с периодом $2\pi$. Поведение функции на отрезке $[-8\pi; -6\pi]$ такое же, как и на отрезке $[0; 2\pi]$, так как $[-8\pi; -6\pi] = [0 - 4 \cdot 2\pi; 2\pi - 4 \cdot 2\pi]$. <br> Отрезок $[-8\pi; -6\pi]$ можно разбить на два промежутка монотонности для функции косинуса:</p><p> 1. На отрезке $[-8\pi; -7\pi]$ функция $\cos(x)$ убывает (аналогично поведению на $[0; \pi]$). <br> 2. На отрезке $[-7\pi; -6\pi]$ функция $\cos(x)$ возрастает (аналогично поведению на $[\pi; 2\pi]$).</p><p> Поскольку функция $\cos(x)$ не является монотонной на всем отрезке $[-8\pi; -6\pi]$, характер монотонности последовательности $\cos(x_n)$ будет зависеть от расположения членов последовательности $x_n$ относительно точки $x = -7\pi$.</p><p> Возможны следующие случаи:</p><p> - Если все члены последовательности $x_n$ принадлежат отрезку $[-8\pi; -7\pi]$, то, поскольку $x_n$ убывает, а $\cos(x)$ на этом отрезке также убывает, последовательность $\cos(x_n)$ будет возрастающей (убывающая функция от убывающей последовательности). <br> - Если все члены последовательности $x_n$ принадлежат отрезку $[-7\pi; -6\pi]$, то, поскольку $x_n$ убывает, а $\cos(x)$ на этом отрезке возрастает, последовательность $\cos(x_n)$ будет убывающей (возрастающая функция от убывающей последовательности). <br> - Если последовательность $x_n$ "пересекает" точку $-7\pi$ (то есть существуют такие $x_i$ и $x_j$, что $x_i > -7\pi$ и $x_j < -7\pi$), то последовательность $\cos(x_n)$ не будет монотонной. Например, для последовательности $x_1 = -6.5\pi, x_2 = -7\pi, x_3 = -7.5\pi$ имеем $\cos(x_1) = 0$, $\cos(x_2) = -1$, $\cos(x_3) = 0$. Последовательность $0, -1, 0$ не является монотонной.</p><p> Таким образом, в общем случае нельзя однозначно определить монотонность последовательности.</p><p> <strong>Ответ:</strong> Последовательность в общем случае не является монотонной. Она может быть возрастающей, убывающей или немонотонной в зависимости от расположения членов последовательности $x_n$ относительно точки $-7\pi$.</p><p><strong>б)</strong></p><p> По условию, дана возрастающая последовательность чисел $x_1, x_2, \dots, x_n$, то есть $x_1 < x_2 < \dots < x_n$. Все члены этой последовательности принадлежат отрезку $[80,5\pi; 81\pi]$. <br> Нам нужно определить характер монотонности последовательности $y_k = \sin(x_k)$. Для этого исследуем поведение функции $f(x) = \sin(x)$ на отрезке $[80,5\pi; 81\pi]$. <br> Функция $f(x) = \sin(x)$ является периодической с периодом $2\pi$. Преобразуем границы отрезка: <br> $80,5\pi = 80\pi + 0,5\pi = 40 \cdot 2\pi + \frac{\pi}{2}$ <br> $81\pi = 80\pi + \pi = 40 \cdot 2\pi + \pi$ <br> Следовательно, поведение функции $\sin(x)$ на отрезке $[80,5\pi; 81\pi]$ такое же, как и на отрезке $[\frac{\pi}{2}; \pi]$. <br> На отрезке $[\frac{\pi}{2}; \pi]$ функция $\sin(x)$ является убывающей (ее значения изменяются от $\sin(\frac{\pi}{2})=1$ до $\sin(\pi)=0$). Производная $(\sin x)' = \cos x$ на этом интервале отрицательна. <br> Таким образом, функция $\sin(x)$ убывает на всем отрезке $[80,5\pi; 81\pi]$. <br> Поскольку последовательность $x_n$ возрастает, а функция $\sin(x)$ на области значений $x_n$ убывает, то результирующая последовательность $\sin(x_n)$ будет убывающей. Если $x_k < x_{k+1}$ и $f$ — убывающая функция, то $f(x_k) > f(x_{k+1})$.</p><p> <strong>Ответ:</strong> Последовательность является убывающей.</p><p><strong>в)</strong></p><p> По условию, дана убывающая последовательность чисел $x_1, x_2, \dots, x_n$, то есть $x_1 > x_2 > \dots > x_n$. Все члены этой последовательности принадлежат отрезку $[80,5\pi; 81\pi]$. <br> Нам нужно определить характер монотонности последовательности $y_k = \tan(x_k)$. <br> Функция $f(x) = \tan(x)$ не определена в точках вида $x = \frac{\pi}{2} + k\pi$, где $k$ — целое число. Левая граница данного отрезка, $x = 80,5\pi = \frac{\pi}{2} + 80\pi$, является одной из таких точек. Поэтому, чтобы последовательность $\tan(x_n)$ была определена для всех ее членов, необходимо, чтобы $x_n \neq 80,5\pi$ для всех $n$. Будем исходить из этого предположения, то есть все $x_n \in (80,5\pi; 81\pi]$. <br> Исследуем поведение функции $f(x) = \tan(x)$ на интервале $(80,5\pi; 81\pi]$. <br> Функция $\tan(x)$ имеет период $\pi$. Поведение функции на $(80,5\pi; 81\pi]$ такое же, как на интервале $(\frac{\pi}{2}; \pi]$, так как $80,5\pi = 80\pi + 0,5\pi$ и $81\pi = 80\pi + \pi$. <br> Производная функции $(\tan x)' = \frac{1}{\cos^2 x}$. Так как $\cos^2 x > 0$ на любом интервале, где тангенс определен, производная всегда положительна. Это означает, что функция $\tan(x)$ является строго возрастающей на каждом интервале своей непрерывности, в том числе и на $(80,5\pi; 81\pi]$. <br> Поскольку последовательность $x_n$ убывает, а функция $\tan(x)$ на области значений $x_n$ возрастает, то результирующая последовательность $\tan(x_n)$ будет убывающей. Если $x_k > x_{k+1}$ и $f$ — возрастающая функция, то $f(x_k) > f(x_{k+1})$.</p><p> <strong>Ответ:</strong> При условии, что все члены последовательности $x_n$ отличны от $80,5\pi$ (то есть $\tan(x_n)$ определен для всех $n$), последовательность является убывающей.</p>" ] #original: array:6 [ "id" => 1551458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1156} "task" => array:2 [ "refs" => "1119739" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p> По условию, дана убывающая последовательность чисел $x_1, x_2, \dots, x_n$, то есть $x_1 > x_2 > \dots > x_n$. Все члены этой последовательности принадлежат отрезку $[-8\pi; -6\pi]$. <br> Нам нужно определить характер монотонности последовательности $y_k = \cos(x_k)$. Для этого исследуем поведение функции $f(x) = \cos(x)$ на отрезке $[-8\pi; -6\pi]$. <br> Функция $f(x) = \cos(x)$ является периодической с периодом $2\pi$. Поведение функции на отрезке $[-8\pi; -6\pi]$ такое же, как и на отрезке $[0; 2\pi]$, так как $[-8\pi; -6\pi] = [0 - 4 \cdot 2\pi; 2\pi - 4 \cdot 2\pi]$. <br> Отрезок $[-8\pi; -6\pi]$ можно разбить на два промежутка монотонности для функции косинуса:</p><p> 1. На отрезке $[-8\pi; -7\pi]$ функция $\cos(x)$ убывает (аналогично поведению на $[0; \pi]$). <br> 2. На отрезке $[-7\pi; -6\pi]$ функция $\cos(x)$ возрастает (аналогично поведению на $[\pi; 2\pi]$).</p><p> Поскольку функция $\cos(x)$ не является монотонной на всем отрезке $[-8\pi; -6\pi]$, характер монотонности последовательности $\cos(x_n)$ будет зависеть от расположения членов последовательности $x_n$ относительно точки $x = -7\pi$.</p><p> Возможны следующие случаи:</p><p> - Если все члены последовательности $x_n$ принадлежат отрезку $[-8\pi; -7\pi]$, то, поскольку $x_n$ убывает, а $\cos(x)$ на этом отрезке также убывает, последовательность $\cos(x_n)$ будет возрастающей (убывающая функция от убывающей последовательности). <br> - Если все члены последовательности $x_n$ принадлежат отрезку $[-7\pi; -6\pi]$, то, поскольку $x_n$ убывает, а $\cos(x)$ на этом отрезке возрастает, последовательность $\cos(x_n)$ будет убывающей (возрастающая функция от убывающей последовательности). <br> - Если последовательность $x_n$ "пересекает" точку $-7\pi$ (то есть существуют такие $x_i$ и $x_j$, что $x_i > -7\pi$ и $x_j < -7\pi$), то последовательность $\cos(x_n)$ не будет монотонной. Например, для последовательности $x_1 = -6.5\pi, x_2 = -7\pi, x_3 = -7.5\pi$ имеем $\cos(x_1) = 0$, $\cos(x_2) = -1$, $\cos(x_3) = 0$. Последовательность $0, -1, 0$ не является монотонной.</p><p> Таким образом, в общем случае нельзя однозначно определить монотонность последовательности.</p><p> <strong>Ответ:</strong> Последовательность в общем случае не является монотонной. Она может быть возрастающей, убывающей или немонотонной в зависимости от расположения членов последовательности $x_n$ относительно точки $-7\pi$.</p><p><strong>б)</strong></p><p> По условию, дана возрастающая последовательность чисел $x_1, x_2, \dots, x_n$, то есть $x_1 < x_2 < \dots < x_n$. Все члены этой последовательности принадлежат отрезку $[80,5\pi; 81\pi]$. <br> Нам нужно определить характер монотонности последовательности $y_k = \sin(x_k)$. Для этого исследуем поведение функции $f(x) = \sin(x)$ на отрезке $[80,5\pi; 81\pi]$. <br> Функция $f(x) = \sin(x)$ является периодической с периодом $2\pi$. Преобразуем границы отрезка: <br> $80,5\pi = 80\pi + 0,5\pi = 40 \cdot 2\pi + \frac{\pi}{2}$ <br> $81\pi = 80\pi + \pi = 40 \cdot 2\pi + \pi$ <br> Следовательно, поведение функции $\sin(x)$ на отрезке $[80,5\pi; 81\pi]$ такое же, как и на отрезке $[\frac{\pi}{2}; \pi]$. <br> На отрезке $[\frac{\pi}{2}; \pi]$ функция $\sin(x)$ является убывающей (ее значения изменяются от $\sin(\frac{\pi}{2})=1$ до $\sin(\pi)=0$). Производная $(\sin x)' = \cos x$ на этом интервале отрицательна. <br> Таким образом, функция $\sin(x)$ убывает на всем отрезке $[80,5\pi; 81\pi]$. <br> Поскольку последовательность $x_n$ возрастает, а функция $\sin(x)$ на области значений $x_n$ убывает, то результирующая последовательность $\sin(x_n)$ будет убывающей. Если $x_k < x_{k+1}$ и $f$ — убывающая функция, то $f(x_k) > f(x_{k+1})$.</p><p> <strong>Ответ:</strong> Последовательность является убывающей.</p><p><strong>в)</strong></p><p> По условию, дана убывающая последовательность чисел $x_1, x_2, \dots, x_n$, то есть $x_1 > x_2 > \dots > x_n$. Все члены этой последовательности принадлежат отрезку $[80,5\pi; 81\pi]$. <br> Нам нужно определить характер монотонности последовательности $y_k = \tan(x_k)$. <br> Функция $f(x) = \tan(x)$ не определена в точках вида $x = \frac{\pi}{2} + k\pi$, где $k$ — целое число. Левая граница данного отрезка, $x = 80,5\pi = \frac{\pi}{2} + 80\pi$, является одной из таких точек. Поэтому, чтобы последовательность $\tan(x_n)$ была определена для всех ее членов, необходимо, чтобы $x_n \neq 80,5\pi$ для всех $n$. Будем исходить из этого предположения, то есть все $x_n \in (80,5\pi; 81\pi]$. <br> Исследуем поведение функции $f(x) = \tan(x)$ на интервале $(80,5\pi; 81\pi]$. <br> Функция $\tan(x)$ имеет период $\pi$. Поведение функции на $(80,5\pi; 81\pi]$ такое же, как на интервале $(\frac{\pi}{2}; \pi]$, так как $80,5\pi = 80\pi + 0,5\pi$ и $81\pi = 80\pi + \pi$. <br> Производная функции $(\tan x)' = \frac{1}{\cos^2 x}$. Так как $\cos^2 x > 0$ на любом интервале, где тангенс определен, производная всегда положительна. Это означает, что функция $\tan(x)$ является строго возрастающей на каждом интервале своей непрерывности, в том числе и на $(80,5\pi; 81\pi]$. <br> Поскольку последовательность $x_n$ убывает, а функция $\tan(x)$ на области значений $x_n$ возрастает, то результирующая последовательность $\tan(x_n)$ будет убывающей. Если $x_k > x_{k+1}$ и $f$ — возрастающая функция, то $f(x_k) > f(x_{k+1})$.</p><p> <strong>Ответ:</strong> При условии, что все члены последовательности $x_n$ отличны от $80,5\pi$ (то есть $\tan(x_n)$ определен для всех $n$), последовательность является убывающей.</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" => "1119740" "type" => "task" ] "previous" => array:2 [ "refs" => "1119738" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1183 #items: array:1 [ 0 => App\Models\Book {#1201} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1179 #items: array:1 [ 0 => App\Models\BookPage {#1184 #connection: "mysql" #table: 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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 => "*" ] }
№21 (с. 96)
Условие. №21 (с. 96)
скриншот условия
21. a)(2) Дана убывающая последовательность чисел $x_1, x_2, x_3, \dots, x_n$, каждое из которых принадлежит отрезку $[-8\pi; -6\pi]$. Что можно сказать о монотонности последовательности $\cos x_1, \cos x_2, \cos x_3, \dots, \cos x_n$?
б)(2) Дана возрастающая последовательность чисел $x_1, x_2, x_3, \dots, x_n$, каждое из которых принадлежит отрезку $[80,5\pi; 81\pi]$. Что можно сказать о монотонности последовательности $\sin x_1, \sin x_2, \sin x_3, \dots, \sin x_n$?
в)(2) Дана убывающая последовательность чисел $x_1, x_2, x_3, \dots, x_n$, каждое из которых принадлежит отрезку $[80,5\pi; 81\pi]$. Что можно сказать о монотонности последовательности $\operatorname{tg} x_1, \operatorname{tg} x_2, \operatorname{tg} x_3, \dots, \operatorname{tg} x_n$?
Решение 2 (rus). №21 (с. 96)
а)
По условию, дана убывающая последовательность чисел $x_1, x_2, \dots, x_n$, то есть $x_1 > x_2 > \dots > x_n$. Все члены этой последовательности принадлежат отрезку $[-8\pi; -6\pi]$.
Нам нужно определить характер монотонности последовательности $y_k = \cos(x_k)$. Для этого исследуем поведение функции $f(x) = \cos(x)$ на отрезке $[-8\pi; -6\pi]$.
Функция $f(x) = \cos(x)$ является периодической с периодом $2\pi$. Поведение функции на отрезке $[-8\pi; -6\pi]$ такое же, как и на отрезке $[0; 2\pi]$, так как $[-8\pi; -6\pi] = [0 - 4 \cdot 2\pi; 2\pi - 4 \cdot 2\pi]$.
Отрезок $[-8\pi; -6\pi]$ можно разбить на два промежутка монотонности для функции косинуса:
1. На отрезке $[-8\pi; -7\pi]$ функция $\cos(x)$ убывает (аналогично поведению на $[0; \pi]$).
2. На отрезке $[-7\pi; -6\pi]$ функция $\cos(x)$ возрастает (аналогично поведению на $[\pi; 2\pi]$).
Поскольку функция $\cos(x)$ не является монотонной на всем отрезке $[-8\pi; -6\pi]$, характер монотонности последовательности $\cos(x_n)$ будет зависеть от расположения членов последовательности $x_n$ относительно точки $x = -7\pi$.
Возможны следующие случаи:
- Если все члены последовательности $x_n$ принадлежат отрезку $[-8\pi; -7\pi]$, то, поскольку $x_n$ убывает, а $\cos(x)$ на этом отрезке также убывает, последовательность $\cos(x_n)$ будет возрастающей (убывающая функция от убывающей последовательности).
- Если все члены последовательности $x_n$ принадлежат отрезку $[-7\pi; -6\pi]$, то, поскольку $x_n$ убывает, а $\cos(x)$ на этом отрезке возрастает, последовательность $\cos(x_n)$ будет убывающей (возрастающая функция от убывающей последовательности).
- Если последовательность $x_n$ "пересекает" точку $-7\pi$ (то есть существуют такие $x_i$ и $x_j$, что $x_i > -7\pi$ и $x_j < -7\pi$), то последовательность $\cos(x_n)$ не будет монотонной. Например, для последовательности $x_1 = -6.5\pi, x_2 = -7\pi, x_3 = -7.5\pi$ имеем $\cos(x_1) = 0$, $\cos(x_2) = -1$, $\cos(x_3) = 0$. Последовательность $0, -1, 0$ не является монотонной.
Таким образом, в общем случае нельзя однозначно определить монотонность последовательности.
Ответ: Последовательность в общем случае не является монотонной. Она может быть возрастающей, убывающей или немонотонной в зависимости от расположения членов последовательности $x_n$ относительно точки $-7\pi$.
б)
По условию, дана возрастающая последовательность чисел $x_1, x_2, \dots, x_n$, то есть $x_1 < x_2 < \dots < x_n$. Все члены этой последовательности принадлежат отрезку $[80,5\pi; 81\pi]$.
Нам нужно определить характер монотонности последовательности $y_k = \sin(x_k)$. Для этого исследуем поведение функции $f(x) = \sin(x)$ на отрезке $[80,5\pi; 81\pi]$.
Функция $f(x) = \sin(x)$ является периодической с периодом $2\pi$. Преобразуем границы отрезка:
$80,5\pi = 80\pi + 0,5\pi = 40 \cdot 2\pi + \frac{\pi}{2}$
$81\pi = 80\pi + \pi = 40 \cdot 2\pi + \pi$
Следовательно, поведение функции $\sin(x)$ на отрезке $[80,5\pi; 81\pi]$ такое же, как и на отрезке $[\frac{\pi}{2}; \pi]$.
На отрезке $[\frac{\pi}{2}; \pi]$ функция $\sin(x)$ является убывающей (ее значения изменяются от $\sin(\frac{\pi}{2})=1$ до $\sin(\pi)=0$). Производная $(\sin x)' = \cos x$ на этом интервале отрицательна.
Таким образом, функция $\sin(x)$ убывает на всем отрезке $[80,5\pi; 81\pi]$.
Поскольку последовательность $x_n$ возрастает, а функция $\sin(x)$ на области значений $x_n$ убывает, то результирующая последовательность $\sin(x_n)$ будет убывающей. Если $x_k < x_{k+1}$ и $f$ — убывающая функция, то $f(x_k) > f(x_{k+1})$.
Ответ: Последовательность является убывающей.
в)
По условию, дана убывающая последовательность чисел $x_1, x_2, \dots, x_n$, то есть $x_1 > x_2 > \dots > x_n$. Все члены этой последовательности принадлежат отрезку $[80,5\pi; 81\pi]$.
Нам нужно определить характер монотонности последовательности $y_k = \tan(x_k)$.
Функция $f(x) = \tan(x)$ не определена в точках вида $x = \frac{\pi}{2} + k\pi$, где $k$ — целое число. Левая граница данного отрезка, $x = 80,5\pi = \frac{\pi}{2} + 80\pi$, является одной из таких точек. Поэтому, чтобы последовательность $\tan(x_n)$ была определена для всех ее членов, необходимо, чтобы $x_n \neq 80,5\pi$ для всех $n$. Будем исходить из этого предположения, то есть все $x_n \in (80,5\pi; 81\pi]$.
Исследуем поведение функции $f(x) = \tan(x)$ на интервале $(80,5\pi; 81\pi]$.
Функция $\tan(x)$ имеет период $\pi$. Поведение функции на $(80,5\pi; 81\pi]$ такое же, как на интервале $(\frac{\pi}{2}; \pi]$, так как $80,5\pi = 80\pi + 0,5\pi$ и $81\pi = 80\pi + \pi$.
Производная функции $(\tan x)' = \frac{1}{\cos^2 x}$. Так как $\cos^2 x > 0$ на любом интервале, где тангенс определен, производная всегда положительна. Это означает, что функция $\tan(x)$ является строго возрастающей на каждом интервале своей непрерывности, в том числе и на $(80,5\pi; 81\pi]$.
Поскольку последовательность $x_n$ убывает, а функция $\tan(x)$ на области значений $x_n$ возрастает, то результирующая последовательность $\tan(x_n)$ будет убывающей. Если $x_k > x_{k+1}$ и $f$ — возрастающая функция, то $f(x_k) > f(x_{k+1})$.
Ответ: При условии, что все члены последовательности $x_n$ отличны от $80,5\pi$ (то есть $\tan(x_n)$ определен для всех $n$), последовательность является убывающей.
Другие задания:
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