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Номер 4, страница 146, часть 2 - 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: 2
ISBN: 978-601-01-3958-9
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Часть 2. Повторение. Серия 3 - номер 4, страница 146.
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" => 1120482 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-3-4" "field_display_title" => "4" "field_outside_task" => null "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" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "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 [ …6] "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 [ …6] "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" => "5" "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 {#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 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 …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" => 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 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 …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 {#1069 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "5" "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 {#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 {#1105 #items: array:3 [ 0 => App\Models\Branch {#1115 #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" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Term {#1116 #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 [ …6] "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 [ …6] "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" => "5" "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 {#1148 #items: array:1 [ 0 => App\Models\Book {#1118 #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 {#1120 #items: array:1 [ 0 => App\Models\Term {#1117 #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 [ …6] "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 [ …6] "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 {#1121 #items: array:1 [ 0 => App\Models\Term {#1122 #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 {#1123 #items: array:1 [ 0 => App\Models\Term {#1124 #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 {#1125 #items: array:3 [ 0 => App\Models\Term {#1126 #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 {#1127 #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 {#1128 #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 {#1129 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1130 #items: array:1 [ 0 => App\Models\Term {#1131 #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 {#1132 #items: array:1 [ 0 => App\Models\Term {#1133 #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 {#1134 #items: array:1 [ 0 => App\Models\Term {#1135 #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 {#1136 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1137 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1138 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1139 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1140 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1141 #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 {#1142 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1143 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1144 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1145 #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 {#1120} "field_class" => Illuminate\Database\Eloquent\Collection {#1121} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1123} "field_author" => Illuminate\Database\Eloquent\Collection {#1125} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1129} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1130} "field_country" => Illuminate\Database\Eloquent\Collection {#1132} "field_city" => Illuminate\Database\Eloquent\Collection {#1134} "field_series" => Illuminate\Database\Eloquent\Collection {#1136} "field_umk" => Illuminate\Database\Eloquent\Collection {#1137} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1138} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1139} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1140} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1141} "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 {#1142} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1143} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1144} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1145} "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 {#1146 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1119} "field_page_start" => "5" "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 {#1148} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1146} ] #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 {#1147 #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" => 1120523 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1149 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "144" "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 {#1150 #items: array:1 [ 0 => App\Models\Book {#1118} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1151 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1120523 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1149} "field_page_start" => "144" "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 {#1150} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1151} ] #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 {#1152 #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" => 1120526 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Серия 3" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1153 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "146" "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 {#1154 #items: array:1 [ 0 => App\Models\Book {#1118} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1163 #items: array:1 [ 0 => App\Models\Branch {#1162 #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" => 1120523 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1164 …2} "field_page_start" => "144" "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 {#1165 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1166 …2} ] #original: array:24 [ "id" => 1120523 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Повторение" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1164 …2} "field_page_start" => "144" "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 {#1165 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1166 …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" => 1120526 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Серия 3" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1153} "field_page_start" => "146" "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 {#1154} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1163} ] #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 {#1070 #items: array:2 [ 0 => App\Models\Element {#1097 #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" => 1301207 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ 0 => App\Models\Edition {#1096 #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 {#1094 …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 {#1093 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092 …2} "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 {#1094 …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 {#1093 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092 …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" => "1120482" "type" => "task" ] "text" => "<p><strong>4.</strong> Решите системы:</p><p><strong>А.</strong> $\begin{cases} \cos x > -\frac{1}{2}; \\ \cos x < \frac{\sqrt{2}}{2}; \end{cases}$</p><p><strong>Б.</strong> $\begin{cases} \sin x - \frac{\sqrt{2}}{2}, \\ \cos x \le \frac{\sqrt{3}}{2}. \end{cases}$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1224" "height" => 376 "path" => "/media/algebra_10/baspasy-u19/0-povtor2-3/4-1.webp?ts=1753535005" ] ] ] #original: array:7 [ "id" => 1301207 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087} "task" => array:2 [ "refs" => "1120482" "type" => "task" ] "text" => "<p><strong>4.</strong> Решите системы:</p><p><strong>А.</strong> $\begin{cases} \cos x > -\frac{1}{2}; \\ \cos x < \frac{\sqrt{2}}{2}; \end{cases}$</p><p><strong>Б.</strong> $\begin{cases} \sin x - \frac{\sqrt{2}}{2}, \\ \cos x \le \frac{\sqrt{3}}{2}. \end{cases}$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1224" "height" => 376 "path" => "/media/algebra_10/baspasy-u19/0-povtor2-3/4-1.webp?ts=1753535005" ] ] ] #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 {#1089 #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" => 1552243 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\Edition {#1090 #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 {#1086 …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 {#1082 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084 …2} "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 {#1086 …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 {#1082 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084 …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" => "1120482" "type" => "task" ] "text" => "<p><strong>А.</strong></p><p>Дана система неравенств:</p><p>$ \begin{cases} \cos x > -\frac{1}{2} \\ \cos x < \frac{\sqrt{2}}{2} \end{cases} $</p><p>Эту систему можно записать в виде двойного неравенства:</p><p>$-\frac{1}{2} < \cos x < \frac{\sqrt{2}}{2}$</p><p>Для решения используем единичную тригонометрическую окружность. Нам нужно найти углы $x$, для которых абсцисса (косинус) точки на окружности находится в интервале $(-\frac{1}{2}, \frac{\sqrt{2}}{2})$.</p><p>1. Найдем углы, для которых $\cos x = \frac{\sqrt{2}}{2}$. На промежутке $[0, 2\pi)$ это $x = \frac{\pi}{4}$ и $x = \frac{7\pi}{4}$.</p><p>2. Найдем углы, для которых $\cos x = -\frac{1}{2}$. На промежутке $[0, 2\pi)$ это $x = \frac{2\pi}{3}$ и $x = \frac{4\pi}{3}$.</p><p>На единичной окружности неравенству $-\frac{1}{2} < \cos x < \frac{\sqrt{2}}{2}$ соответствуют две открытые дуги:</p><p>Первая дуга находится в первой и второй четвертях. Она соответствует углам, большим чем $\frac{\pi}{4}$ и меньшим чем $\frac{2\pi}{3}$. Таким образом, получаем первый интервал решений: $\frac{\pi}{4} < x < \frac{2\pi}{3}$.</p><p>Вторая дуга находится в третьей и четвертой четвертях. Она соответствует углам, большим чем $\frac{4\pi}{3}$ и меньшим чем $\frac{7\pi}{4}$. Таким образом, получаем второй интервал решений: $\frac{4\pi}{3} < x < \frac{7\pi}{4}$.</p><p>Учитывая периодичность функции косинуса (период $2\pi$), общее решение является объединением этих интервалов с добавлением $2\pi k$ к границам, где $k$ — любое целое число.</p><p><strong>Ответ:</strong> $x \in \left(\frac{\pi}{4} + 2\pi k; \frac{2\pi}{3} + 2\pi k\right) \cup \left(\frac{4\pi}{3} + 2\pi k; \frac{7\pi}{4} + 2\pi k\right), k \in \mathbb{Z}$.</p><p><strong>Б.</strong></p><p>В условии для первого неравенства системы, скорее всего, опечатка. Наиболее вероятный вариант, соответствующий сложности задания, — это система двух неравенств. Предположим, что первое неравенство — это $ \sin x \le \frac{\sqrt{2}}{2} $. Тогда система имеет вид:</p><p>$ \begin{cases} \sin x \le \frac{\sqrt{2}}{2} \\ \cos x \le \frac{\sqrt{3}}{2} \end{cases} $</p><p>Решим каждое неравенство по отдельности, используя единичную тригонометрическую окружность.</p><p>1. Решение неравенства $ \sin x \le \frac{\sqrt{2}}{2} $.</p><p>На единичной окружности ордината (синус) точки должна быть меньше или равна $ \frac{\sqrt{2}}{2} $. Корнями уравнения $ \sin x = \frac{\sqrt{2}}{2} $ являются $x = \frac{\pi}{4} + 2\pi k$ и $x = \frac{3\pi}{4} + 2\pi k$. Неравенству удовлетворяют углы на дуге, идущей от $ \frac{3\pi}{4} $ через $\pi$, $\frac{3\pi}{2}$ до $2\pi + \frac{\pi}{4}$. Таким образом, решение этого неравенства: $x \in \left[\frac{3\pi}{4} + 2\pi k, \frac{9\pi}{4} + 2\pi k\right]$, где $k \in \mathbb{Z}$.</p><p>2. Решение неравенства $ \cos x \le \frac{\sqrt{3}}{2} $.</p><p>На единичной окружности абсцисса (косинус) точки должна быть меньше или равна $ \frac{\sqrt{3}}{2} $. Корнями уравнения $ \cos x = \frac{\sqrt{3}}{2} $ являются $x = \frac{\pi}{6} + 2\pi k$ и $x = -\frac{\pi}{6} + 2\pi k = \frac{11\pi}{6} + 2\pi k$. Неравенству удовлетворяют углы на дуге, идущей от $ \frac{\pi}{6} $ против часовой стрелки до $ \frac{11\pi}{6} $. Решение: $x \in \left[\frac{\pi}{6} + 2\pi k, \frac{11\pi}{6} + 2\pi k\right]$, где $k \in \mathbb{Z}$.</p><p>3. Нахождение пересечения решений.</p><p>Найдем пересечение полученных множеств решений на одном обороте (при $k=0$).</p><p>Первое множество: $x \in [0, \frac{\pi}{4}] \cup [\frac{3\pi}{4}, 2\pi)$.</p><p>Второе множество: $x \in [\frac{\pi}{6}, \frac{11\pi}{6}]$.</p><p>Пересечение этих множеств: $\left([0, \frac{\pi}{4}] \cup [\frac{3\pi}{4}, 2\pi)\right) \cap \left[\frac{\pi}{6}, \frac{11\pi}{6}\right]$.</p><p>Разобьем на два пересечения:</p><p>а) $[0, \frac{\pi}{4}] \cap [\frac{\pi}{6}, \frac{11\pi}{6}] = [\frac{\pi}{6}, \frac{\pi}{4}]$</p><p>б) $[\frac{3\pi}{4}, 2\pi) \cap [\frac{\pi}{6}, \frac{11\pi}{6}] = [\frac{3\pi}{4}, \frac{11\pi}{6}]$</p><p>Объединяя эти результаты и добавляя период $2\pi k$, получаем общее решение системы.</p><p><strong>Ответ:</strong> $x \in \left[\frac{\pi}{6} + 2\pi k; \frac{\pi}{4} + 2\pi k\right] \cup \left[\frac{3\pi}{4} + 2\pi k; \frac{11\pi}{6} + 2\pi k\right], k \in \mathbb{Z}$.</p>" ] #original: array:6 [ "id" => 1552243 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081} "task" => array:2 [ "refs" => "1120482" "type" => "task" ] "text" => "<p><strong>А.</strong></p><p>Дана система неравенств:</p><p>$ \begin{cases} \cos x > -\frac{1}{2} \\ \cos x < \frac{\sqrt{2}}{2} \end{cases} $</p><p>Эту систему можно записать в виде двойного неравенства:</p><p>$-\frac{1}{2} < \cos x < \frac{\sqrt{2}}{2}$</p><p>Для решения используем единичную тригонометрическую окружность. Нам нужно найти углы $x$, для которых абсцисса (косинус) точки на окружности находится в интервале $(-\frac{1}{2}, \frac{\sqrt{2}}{2})$.</p><p>1. Найдем углы, для которых $\cos x = \frac{\sqrt{2}}{2}$. На промежутке $[0, 2\pi)$ это $x = \frac{\pi}{4}$ и $x = \frac{7\pi}{4}$.</p><p>2. Найдем углы, для которых $\cos x = -\frac{1}{2}$. На промежутке $[0, 2\pi)$ это $x = \frac{2\pi}{3}$ и $x = \frac{4\pi}{3}$.</p><p>На единичной окружности неравенству $-\frac{1}{2} < \cos x < \frac{\sqrt{2}}{2}$ соответствуют две открытые дуги:</p><p>Первая дуга находится в первой и второй четвертях. Она соответствует углам, большим чем $\frac{\pi}{4}$ и меньшим чем $\frac{2\pi}{3}$. Таким образом, получаем первый интервал решений: $\frac{\pi}{4} < x < \frac{2\pi}{3}$.</p><p>Вторая дуга находится в третьей и четвертой четвертях. Она соответствует углам, большим чем $\frac{4\pi}{3}$ и меньшим чем $\frac{7\pi}{4}$. Таким образом, получаем второй интервал решений: $\frac{4\pi}{3} < x < \frac{7\pi}{4}$.</p><p>Учитывая периодичность функции косинуса (период $2\pi$), общее решение является объединением этих интервалов с добавлением $2\pi k$ к границам, где $k$ — любое целое число.</p><p><strong>Ответ:</strong> $x \in \left(\frac{\pi}{4} + 2\pi k; \frac{2\pi}{3} + 2\pi k\right) \cup \left(\frac{4\pi}{3} + 2\pi k; \frac{7\pi}{4} + 2\pi k\right), k \in \mathbb{Z}$.</p><p><strong>Б.</strong></p><p>В условии для первого неравенства системы, скорее всего, опечатка. Наиболее вероятный вариант, соответствующий сложности задания, — это система двух неравенств. Предположим, что первое неравенство — это $ \sin x \le \frac{\sqrt{2}}{2} $. Тогда система имеет вид:</p><p>$ \begin{cases} \sin x \le \frac{\sqrt{2}}{2} \\ \cos x \le \frac{\sqrt{3}}{2} \end{cases} $</p><p>Решим каждое неравенство по отдельности, используя единичную тригонометрическую окружность.</p><p>1. Решение неравенства $ \sin x \le \frac{\sqrt{2}}{2} $.</p><p>На единичной окружности ордината (синус) точки должна быть меньше или равна $ \frac{\sqrt{2}}{2} $. Корнями уравнения $ \sin x = \frac{\sqrt{2}}{2} $ являются $x = \frac{\pi}{4} + 2\pi k$ и $x = \frac{3\pi}{4} + 2\pi k$. Неравенству удовлетворяют углы на дуге, идущей от $ \frac{3\pi}{4} $ через $\pi$, $\frac{3\pi}{2}$ до $2\pi + \frac{\pi}{4}$. Таким образом, решение этого неравенства: $x \in \left[\frac{3\pi}{4} + 2\pi k, \frac{9\pi}{4} + 2\pi k\right]$, где $k \in \mathbb{Z}$.</p><p>2. Решение неравенства $ \cos x \le \frac{\sqrt{3}}{2} $.</p><p>На единичной окружности абсцисса (косинус) точки должна быть меньше или равна $ \frac{\sqrt{3}}{2} $. Корнями уравнения $ \cos x = \frac{\sqrt{3}}{2} $ являются $x = \frac{\pi}{6} + 2\pi k$ и $x = -\frac{\pi}{6} + 2\pi k = \frac{11\pi}{6} + 2\pi k$. Неравенству удовлетворяют углы на дуге, идущей от $ \frac{\pi}{6} $ против часовой стрелки до $ \frac{11\pi}{6} $. Решение: $x \in \left[\frac{\pi}{6} + 2\pi k, \frac{11\pi}{6} + 2\pi k\right]$, где $k \in \mathbb{Z}$.</p><p>3. Нахождение пересечения решений.</p><p>Найдем пересечение полученных множеств решений на одном обороте (при $k=0$).</p><p>Первое множество: $x \in [0, \frac{\pi}{4}] \cup [\frac{3\pi}{4}, 2\pi)$.</p><p>Второе множество: $x \in [\frac{\pi}{6}, \frac{11\pi}{6}]$.</p><p>Пересечение этих множеств: $\left([0, \frac{\pi}{4}] \cup [\frac{3\pi}{4}, 2\pi)\right) \cap \left[\frac{\pi}{6}, \frac{11\pi}{6}\right]$.</p><p>Разобьем на два пересечения:</p><p>а) $[0, \frac{\pi}{4}] \cap [\frac{\pi}{6}, \frac{11\pi}{6}] = [\frac{\pi}{6}, \frac{\pi}{4}]$</p><p>б) $[\frac{3\pi}{4}, 2\pi) \cap [\frac{\pi}{6}, \frac{11\pi}{6}] = [\frac{3\pi}{4}, \frac{11\pi}{6}]$</p><p>Объединяя эти результаты и добавляя период $2\pi k$, получаем общее решение системы.</p><p><strong>Ответ:</strong> $x \in \left[\frac{\pi}{6} + 2\pi k; \frac{\pi}{4} + 2\pi k\right] \cup \left[\frac{3\pi}{4} + 2\pi k; \frac{11\pi}{6} + 2\pi k\right], k \in \mathbb{Z}$.</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" => "1120483" "type" => "task" ] "previous" => array:2 [ "refs" => "1120481" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Book {#1118} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1075 #items: array:1 [ 0 => App\Models\BookPage {#1101 #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" => 1099299 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/page-2-146" "field_display_title" => "146" "field_folder" => "2" "field_image_name" => "146" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1083 #items: array:1 [ 0 => App\Models\Branch {#1115} ] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1080 #items: array:1 [ 0 => App\Models\Book {#1118} ] #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 {#1078 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1109 #items: array:1 [ 0 => App\Models\Element {#1113 #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" => 1262724 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1262724 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107 …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" => "1099300" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1099298" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1255 #items: array:9 [ 0 => App\Models\Task {#1271 #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" => 1120478 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-2-8" "field_display_title" => "8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1272 …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 {#1273 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1274 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1374 …2} "content" => Illuminate\Database\Eloquent\Collection {#1308 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1371 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120478 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-2-8" "field_display_title" => "8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1272 …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 {#1273 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1274 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1374 …2} "content" => Illuminate\Database\Eloquent\Collection {#1308 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1371 …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 {#1339 #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" => 1120479 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-3-1" "field_display_title" => "1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1381 …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 {#1369 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1343 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1340 …2} "content" => Illuminate\Database\Eloquent\Collection {#1372 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1358 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120479 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-3-1" "field_display_title" => "1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1381 …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 {#1369 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1343 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1340 …2} "content" => Illuminate\Database\Eloquent\Collection {#1372 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1358 …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 {#1364 #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" => 1120480 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-3-2" "field_display_title" => "2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1344 …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 {#1356 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1360 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1363 …2} "content" => Illuminate\Database\Eloquent\Collection {#1362 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1348 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120480 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-3-2" "field_display_title" => "2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1344 …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 {#1356 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1360 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1363 …2} "content" => Illuminate\Database\Eloquent\Collection {#1362 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1348 …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 {#1353 #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" => 1120481 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-3-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1361 …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 {#1346 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1350 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1352 …2} "content" => Illuminate\Database\Eloquent\Collection {#1349 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1333 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120481 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-3-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1361 …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 {#1346 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1350 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1352 …2} "content" => Illuminate\Database\Eloquent\Collection {#1349 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1333 …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 {#1338 #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" => 1120482 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-3-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1347 …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 {#1331 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1335 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1337 …2} "content" => Illuminate\Database\Eloquent\Collection {#1334 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1336 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120482 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-3-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1347 …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 {#1331 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1335 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1337 …2} "content" => Illuminate\Database\Eloquent\Collection {#1334 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1336 …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 {#1325 #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" => 1120483 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-3-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1326 …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 {#1323 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1324 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1322 …2} "content" => Illuminate\Database\Eloquent\Collection {#1321 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1315 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120483 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-3-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1326 …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 {#1323 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1324 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1322 …2} "content" => Illuminate\Database\Eloquent\Collection {#1321 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1315 …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 => "*" ] } 6 => App\Models\Task {#1320 #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" => 1120484 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-3-6" "field_display_title" => "6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1318 …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 {#1313 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1317 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1319 …2} "content" => Illuminate\Database\Eloquent\Collection {#1316 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1302 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120484 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-3-6" "field_display_title" => "6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1318 …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 {#1313 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1317 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1319 …2} "content" => Illuminate\Database\Eloquent\Collection {#1316 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1302 …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 => "*" ] } 7 => App\Models\Task {#1279 #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" => 1120485 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-3-7" "field_display_title" => "7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1314 …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 {#1300 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1304 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1306 …2} "content" => Illuminate\Database\Eloquent\Collection {#1305 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1290 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120485 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-3-7" "field_display_title" => "7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1314 …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 {#1300 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1304 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1306 …2} "content" => Illuminate\Database\Eloquent\Collection {#1305 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1290 …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 => "*" ] } 8 => App\Models\Task {#1296 #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" => 1120486 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_page_end" => "147" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-3-8" "field_display_title" => "8" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1303 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null …12 ] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1099299 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "146" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/page-2-146" "field_display_title" => "146" "field_folder" => "2" "field_image_name" => "146" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1083} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1080} "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 {#1078} "content" => Illuminate\Database\Eloquent\Collection {#1109} "next" => array:2 [ "refs" => "1099300" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1099298" "type" => "book_page" ] 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№4 (с. 146)
Условие. №4 (с. 146)
скриншот условия
4. Решите системы:
А. $\begin{cases} \cos x > -\frac{1}{2}; \\ \cos x < \frac{\sqrt{2}}{2}; \end{cases}$
Б. $\begin{cases} \sin x - \frac{\sqrt{2}}{2}, \\ \cos x \le \frac{\sqrt{3}}{2}. \end{cases}$
Решение 2 (rus). №4 (с. 146)
А.
Дана система неравенств:
$ \begin{cases} \cos x > -\frac{1}{2} \\ \cos x < \frac{\sqrt{2}}{2} \end{cases} $
Эту систему можно записать в виде двойного неравенства:
$-\frac{1}{2} < \cos x < \frac{\sqrt{2}}{2}$
Для решения используем единичную тригонометрическую окружность. Нам нужно найти углы $x$, для которых абсцисса (косинус) точки на окружности находится в интервале $(-\frac{1}{2}, \frac{\sqrt{2}}{2})$.
1. Найдем углы, для которых $\cos x = \frac{\sqrt{2}}{2}$. На промежутке $[0, 2\pi)$ это $x = \frac{\pi}{4}$ и $x = \frac{7\pi}{4}$.
2. Найдем углы, для которых $\cos x = -\frac{1}{2}$. На промежутке $[0, 2\pi)$ это $x = \frac{2\pi}{3}$ и $x = \frac{4\pi}{3}$.
На единичной окружности неравенству $-\frac{1}{2} < \cos x < \frac{\sqrt{2}}{2}$ соответствуют две открытые дуги:
Первая дуга находится в первой и второй четвертях. Она соответствует углам, большим чем $\frac{\pi}{4}$ и меньшим чем $\frac{2\pi}{3}$. Таким образом, получаем первый интервал решений: $\frac{\pi}{4} < x < \frac{2\pi}{3}$.
Вторая дуга находится в третьей и четвертой четвертях. Она соответствует углам, большим чем $\frac{4\pi}{3}$ и меньшим чем $\frac{7\pi}{4}$. Таким образом, получаем второй интервал решений: $\frac{4\pi}{3} < x < \frac{7\pi}{4}$.
Учитывая периодичность функции косинуса (период $2\pi$), общее решение является объединением этих интервалов с добавлением $2\pi k$ к границам, где $k$ — любое целое число.
Ответ: $x \in \left(\frac{\pi}{4} + 2\pi k; \frac{2\pi}{3} + 2\pi k\right) \cup \left(\frac{4\pi}{3} + 2\pi k; \frac{7\pi}{4} + 2\pi k\right), k \in \mathbb{Z}$.
Б.
В условии для первого неравенства системы, скорее всего, опечатка. Наиболее вероятный вариант, соответствующий сложности задания, — это система двух неравенств. Предположим, что первое неравенство — это $ \sin x \le \frac{\sqrt{2}}{2} $. Тогда система имеет вид:
$ \begin{cases} \sin x \le \frac{\sqrt{2}}{2} \\ \cos x \le \frac{\sqrt{3}}{2} \end{cases} $
Решим каждое неравенство по отдельности, используя единичную тригонометрическую окружность.
1. Решение неравенства $ \sin x \le \frac{\sqrt{2}}{2} $.
На единичной окружности ордината (синус) точки должна быть меньше или равна $ \frac{\sqrt{2}}{2} $. Корнями уравнения $ \sin x = \frac{\sqrt{2}}{2} $ являются $x = \frac{\pi}{4} + 2\pi k$ и $x = \frac{3\pi}{4} + 2\pi k$. Неравенству удовлетворяют углы на дуге, идущей от $ \frac{3\pi}{4} $ через $\pi$, $\frac{3\pi}{2}$ до $2\pi + \frac{\pi}{4}$. Таким образом, решение этого неравенства: $x \in \left[\frac{3\pi}{4} + 2\pi k, \frac{9\pi}{4} + 2\pi k\right]$, где $k \in \mathbb{Z}$.
2. Решение неравенства $ \cos x \le \frac{\sqrt{3}}{2} $.
На единичной окружности абсцисса (косинус) точки должна быть меньше или равна $ \frac{\sqrt{3}}{2} $. Корнями уравнения $ \cos x = \frac{\sqrt{3}}{2} $ являются $x = \frac{\pi}{6} + 2\pi k$ и $x = -\frac{\pi}{6} + 2\pi k = \frac{11\pi}{6} + 2\pi k$. Неравенству удовлетворяют углы на дуге, идущей от $ \frac{\pi}{6} $ против часовой стрелки до $ \frac{11\pi}{6} $. Решение: $x \in \left[\frac{\pi}{6} + 2\pi k, \frac{11\pi}{6} + 2\pi k\right]$, где $k \in \mathbb{Z}$.
3. Нахождение пересечения решений.
Найдем пересечение полученных множеств решений на одном обороте (при $k=0$).
Первое множество: $x \in [0, \frac{\pi}{4}] \cup [\frac{3\pi}{4}, 2\pi)$.
Второе множество: $x \in [\frac{\pi}{6}, \frac{11\pi}{6}]$.
Пересечение этих множеств: $\left([0, \frac{\pi}{4}] \cup [\frac{3\pi}{4}, 2\pi)\right) \cap \left[\frac{\pi}{6}, \frac{11\pi}{6}\right]$.
Разобьем на два пересечения:
а) $[0, \frac{\pi}{4}] \cap [\frac{\pi}{6}, \frac{11\pi}{6}] = [\frac{\pi}{6}, \frac{\pi}{4}]$
б) $[\frac{3\pi}{4}, 2\pi) \cap [\frac{\pi}{6}, \frac{11\pi}{6}] = [\frac{3\pi}{4}, \frac{11\pi}{6}]$
Объединяя эти результаты и добавляя период $2\pi k$, получаем общее решение системы.
Ответ: $x \in \left[\frac{\pi}{6} + 2\pi k; \frac{\pi}{4} + 2\pi k\right] \cup \left[\frac{3\pi}{4} + 2\pi k; \frac{11\pi}{6} + 2\pi k\right], k \in \mathbb{Z}$.
Другие задания:
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