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Номер 2, страница 150, 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 7 - номер 2, страница 150.
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" => 1120512 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "150" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-7-2" "field_display_title" => "2" "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 [ "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" => "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 #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 …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} "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 [ …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 {#1142 #items: array:3 [ 0 => 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" => 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 {#1156 #items: array:1 [ 0 => App\Models\Term {#1153 #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 {#1185 #items: array:1 [ 0 => App\Models\Book {#1155 #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 {#1157 #items: array:1 [ 0 => App\Models\Term {#1154 #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 {#1158 #items: array:1 [ 0 => App\Models\Term {#1159 #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 {#1160 #items: array:1 [ 0 => App\Models\Term {#1161 #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 {#1162 #items: array:3 [ 0 => App\Models\Term {#1163 #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 {#1164 #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 {#1165 #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 {#1166 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1167 #items: array:1 [ 0 => App\Models\Term {#1168 #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 {#1169 #items: array:1 [ 0 => App\Models\Term {#1170 #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 {#1171 #items: array:1 [ 0 => App\Models\Term {#1172 #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 {#1173 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1174 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1175 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1176 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1177 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1178 #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 {#1179 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1180 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1181 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1182 #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 {#1157} "field_class" => Illuminate\Database\Eloquent\Collection {#1158} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1160} "field_author" => Illuminate\Database\Eloquent\Collection {#1162} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1166} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1167} "field_country" => Illuminate\Database\Eloquent\Collection {#1169} "field_city" => Illuminate\Database\Eloquent\Collection {#1171} "field_series" => Illuminate\Database\Eloquent\Collection {#1173} "field_umk" => Illuminate\Database\Eloquent\Collection {#1174} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1175} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1176} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1177} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1178} "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 {#1179} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1180} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1181} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1182} "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 {#1183 #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 {#1156} "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 {#1185} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1183} ] #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 {#1184 #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 {#1186 #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 {#1187 #items: array:1 [ 0 => App\Models\Book {#1155} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1188 #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 {#1186} "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 {#1187} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1188} ] #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 {#1189 #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" => 1120530 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Серия 7" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1190 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "150" "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 {#1191 #items: array:1 [ 0 => App\Models\Book {#1155} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1231 #items: array:1 [ 0 => App\Models\Branch {#1192 #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 {#1194 …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 {#1193 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1223 …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 {#1194 …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 {#1193 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1223 …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" => 1120530 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Серия 7" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1190} "field_page_start" => "150" "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 {#1191} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1231} ] #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 {#1104 #items: array:2 [ 0 => App\Models\Element {#1131 #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" => 1301237 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ 0 => App\Models\Edition {#1130 #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 {#1128 …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 {#1127 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1125 …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 {#1128 …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 {#1127 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1125 …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" => "1120512" "type" => "task" ] "text" => "<p><strong>2.</strong> Построй графики функций:</p><p><strong>А.</strong> $y = \operatorname{tg}\left(\frac{\pi}{3}-2x\right)$;</p><p><strong>Б.</strong> $y=2\cos\frac{x}{2}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1072" "height" => 232 "path" => "/media/algebra_10/baspasy-u19/0-povtor2-7/2-1.webp?ts=1753535019" ] ] ] #original: array:7 [ "id" => 1301237 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1121} "task" => array:2 [ "refs" => "1120512" "type" => "task" ] "text" => "<p><strong>2.</strong> Построй графики функций:</p><p><strong>А.</strong> $y = \operatorname{tg}\left(\frac{\pi}{3}-2x\right)$;</p><p><strong>Б.</strong> $y=2\cos\frac{x}{2}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1072" "height" => 232 "path" => "/media/algebra_10/baspasy-u19/0-povtor2-7/2-1.webp?ts=1753535019" ] ] ] #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 {#1123 #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" => 1552273 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1115 #items: array:1 [ 0 => App\Models\Edition {#1124 #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 {#1119 …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 {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …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 {#1119 …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 {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …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" => "1120512" "type" => "task" ] "text" => "<p><strong>А. $y = \text{tg}(\frac{\pi}{3} - 2x)$</strong></p><p>Для построения графика данной функции выполним анализ и последовательные преобразования, исходя из базового графика функции $y = \text{tg}(x)$.</p><p>1. <em>Преобразуем исходное выражение.</em><br>Используя свойство нечетности тангенса ($\text{tg}(-u) = -\text{tg}(u)$), можем записать:<br>$y = \text{tg}(\frac{\pi}{3} - 2x) = \text{tg}(-(2x - \frac{\pi}{3})) = -\text{tg}(2x - \frac{\pi}{3})$.<br>Вынесем коэффициент 2 за скобки в аргументе тангенса:<br>$y = -\text{tg}(2(x - \frac{\pi}{6}))$.<br>Эта форма удобна для анализа преобразований.</p><p>2. <em>Определим последовательность преобразований графика $y = \text{tg}(x)$.</em><br> а) Сжатие графика $y = \text{tg}(x)$ вдоль оси OX в 2 раза. Получаем функцию $y = \text{tg}(2x)$. Ее период $T = \frac{\pi}{2}$.<br> б) Сдвиг полученного графика вправо вдоль оси OX на $\frac{\pi}{6}$. Получаем функцию $y = \text{tg}(2(x - \frac{\pi}{6}))$.<br> в) Симметричное отражение последнего графика относительно оси OX. Получаем искомую функцию $y = -\text{tg}(2(x - \frac{\pi}{6}))$.</p><p>3. <em>Найдем основные свойства функции для построения.</em><br> - <em>Период:</em> $T = \frac{\pi}{|k|} = \frac{\pi}{|-2|} = \frac{\pi}{2}$.<br> - <em>Вертикальные асимптоты:</em> Аргумент тангенса должен быть не равен $\frac{\pi}{2} + \pi n$, где $n \in \mathbb{Z}$.<br> $\frac{\pi}{3} - 2x = \frac{\pi}{2} + \pi n$<br> $-2x = \frac{\pi}{2} - \frac{\pi}{3} + \pi n$<br> $-2x = \frac{\pi}{6} + \pi n$<br> $x = -\frac{\pi}{12} - \frac{\pi n}{2}$.<br> Асимптоты: $x = -\frac{\pi}{12} + \frac{\pi k}{2}, k \in \mathbb{Z}$. (например, $x=...-\frac{7\pi}{12}, -\frac{\pi}{12}, \frac{5\pi}{12}, \frac{11\pi}{12}...$)<br> - <em>Нули функции (точки пересечения с осью OX):</em> $y=0$ при $\text{tg}(\frac{\pi}{3} - 2x)=0$.<br> $\frac{\pi}{3} - 2x = \pi n, n \in \mathbb{Z}$<br> $-2x = -\frac{\pi}{3} + \pi n$<br> $x = \frac{\pi}{6} - \frac{\pi n}{2}$.<br> Нули функции: $x = \frac{\pi}{6} + \frac{\pi k}{2}, k \in \mathbb{Z}$. (например, $x=...-\frac{\pi}{3}, \frac{\pi}{6}, \frac{2\pi}{3}...$)<br> - <em>Точка пересечения с осью OY:</em> При $x=0$, $y = \text{tg}(\frac{\pi}{3}) = \sqrt{3}$. Точка $(0, \sqrt{3})$.<br> - <em>Поведение функции:</em> Из-за знака "-" перед тангенсом (или знака "-" перед $x$ в аргументе), функция является убывающей на каждом интервале между асимптотами.</p><p><em>Построение:</em><br>1. Начертить координатную плоскость.<br>2. Провести вертикальные асимптоты $x = -\frac{\pi}{12}$, $x = \frac{5\pi}{12}$ и т.д. с шагом $\frac{\pi}{2}$.<br>3. Отметить на оси OX нули функции: $x=\frac{\pi}{6}$, $x=\frac{2\pi}{3}$ и т.д.<br>4. Отметить точку пересечения с осью OY $(0, \sqrt{3})$.<br>5. На каждом интервале между асимптотами построить ветвь тангенсоиды, которая убывает от $+\infty$ до $-\infty$, проходя через соответствующий нуль функции. Например, на интервале $(-\frac{\pi}{12}, \frac{5\pi}{12})$ кривая пройдет через точки $(0, \sqrt{3})$ и $(\frac{\pi}{6}, 0)$.<br>6. Повторить построение ветви на других интервалах с учетом периода $T = \frac{\pi}{2}$.</p><p><strong>Ответ:</strong> График функции $y = \text{tg}(\frac{\pi}{3} - 2x)$ представляет собой тангенсоиду с периодом $\frac{\pi}{2}$, убывающую на каждом из интервалов области определения. Асимптоты графика задаются уравнениями $x = -\frac{\pi}{12} + \frac{\pi k}{2}, k \in \mathbb{Z}$. Нули функции находятся в точках $x = \frac{\pi}{6} + \frac{\pi k}{2}, k \in \mathbb{Z}$. График пересекает ось ординат в точке $(0, \sqrt{3})$.</p><hr><p><strong>Б. $y = 2\cos\frac{x}{2}$</strong></p><p>Для построения графика данной функции выполним анализ и последовательные преобразования, исходя из базового графика функции $y = \cos(x)$.</p><p>1. <em>Определим последовательность преобразований графика $y = \cos(x)$.</em><br> а) Растяжение графика $y = \cos(x)$ вдоль оси OX в 2 раза (горизонтальное растяжение). Это преобразование задается коэффициентом $\frac{1}{2}$ при $x$. Получаем функцию $y = \cos(\frac{x}{2})$. Период функции увеличивается в 2 раза и становится $T = 2\pi \cdot 2 = 4\pi$.<br> б) Растяжение полученного графика вдоль оси OY в 2 раза (вертикальное растяжение). Это преобразование задается множителем 2 перед функцией. Получаем искомую функцию $y = 2\cos(\frac{x}{2})$. Амплитуда функции увеличивается в 2 раза и становится равной 2.</p><p>2. <em>Найдем основные свойства функции для построения.</em><br> - <em>Период:</em> $T = \frac{2\pi}{|k|} = \frac{2\pi}{|1/2|} = 4\pi$.<br> - <em>Амплитуда:</em> $A=2$.<br> - <em>Область значений:</em> $[-2, 2]$.<br> - <em>Четность:</em> Функция четная, так как $y(-x) = 2\cos(\frac{-x}{2}) = 2\cos(\frac{x}{2}) = y(x)$. График симметричен относительно оси OY.<br> - <em>Ключевые точки на одном периоде, например, на отрезке $[0, 4\pi]$:</em><br> - При $x=0$, $y = 2\cos(0) = 2 \cdot 1 = 2$ (точка максимума).<br> - При $x=\pi$, $y = 2\cos(\frac{\pi}{2}) = 2 \cdot 0 = 0$ (точка пересечения с осью OX).<br> - При $x=2\pi$, $y = 2\cos(\frac{2\pi}{2}) = 2\cos(\pi) = 2 \cdot (-1) = -2$ (точка минимума).<br> - При $x=3\pi$, $y = 2\cos(\frac{3\pi}{2}) = 2 \cdot 0 = 0$ (точка пересечения с осью OX).<br> - При $x=4\pi$, $y = 2\cos(\frac{4\pi}{2}) = 2\cos(2\pi) = 2 \cdot 1 = 2$ (точка максимума).</p><p><em>Построение:</em><br>1. Начертить координатную плоскость.<br>2. Отметить на осях ключевые точки, найденные выше: $(0, 2), (\pi, 0), (2\pi, -2), (3\pi, 0), (4\pi, 2)$.<br>3. Соединить эти точки плавной линией, имеющей форму косинусоиды.<br>4. Используя свойство периодичности ($T=4\pi$) и четности (симметрия относительно оси OY), продолжить график влево и вправо.</p><p><strong>Ответ:</strong> График функции $y = 2\cos\frac{x}{2}$ является косинусоидой с периодом $4\pi$ и амплитудой 2. Область значений функции — отрезок $[-2, 2]$. Максимумы, равные 2, достигаются в точках $x=4\pi n$, а минимумы, равные -2, в точках $x=2\pi+4\pi n$, где $n \in \mathbb{Z}$. График пересекает ось абсцисс в точках $x=\pi+2\pi n, n \in \mathbb{Z}$.</p>" ] #original: array:6 [ "id" => 1552273 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1115} "task" => array:2 [ "refs" => "1120512" "type" => "task" ] "text" => "<p><strong>А. $y = \text{tg}(\frac{\pi}{3} - 2x)$</strong></p><p>Для построения графика данной функции выполним анализ и последовательные преобразования, исходя из базового графика функции $y = \text{tg}(x)$.</p><p>1. <em>Преобразуем исходное выражение.</em><br>Используя свойство нечетности тангенса ($\text{tg}(-u) = -\text{tg}(u)$), можем записать:<br>$y = \text{tg}(\frac{\pi}{3} - 2x) = \text{tg}(-(2x - \frac{\pi}{3})) = -\text{tg}(2x - \frac{\pi}{3})$.<br>Вынесем коэффициент 2 за скобки в аргументе тангенса:<br>$y = -\text{tg}(2(x - \frac{\pi}{6}))$.<br>Эта форма удобна для анализа преобразований.</p><p>2. <em>Определим последовательность преобразований графика $y = \text{tg}(x)$.</em><br> а) Сжатие графика $y = \text{tg}(x)$ вдоль оси OX в 2 раза. Получаем функцию $y = \text{tg}(2x)$. Ее период $T = \frac{\pi}{2}$.<br> б) Сдвиг полученного графика вправо вдоль оси OX на $\frac{\pi}{6}$. Получаем функцию $y = \text{tg}(2(x - \frac{\pi}{6}))$.<br> в) Симметричное отражение последнего графика относительно оси OX. Получаем искомую функцию $y = -\text{tg}(2(x - \frac{\pi}{6}))$.</p><p>3. <em>Найдем основные свойства функции для построения.</em><br> - <em>Период:</em> $T = \frac{\pi}{|k|} = \frac{\pi}{|-2|} = \frac{\pi}{2}$.<br> - <em>Вертикальные асимптоты:</em> Аргумент тангенса должен быть не равен $\frac{\pi}{2} + \pi n$, где $n \in \mathbb{Z}$.<br> $\frac{\pi}{3} - 2x = \frac{\pi}{2} + \pi n$<br> $-2x = \frac{\pi}{2} - \frac{\pi}{3} + \pi n$<br> $-2x = \frac{\pi}{6} + \pi n$<br> $x = -\frac{\pi}{12} - \frac{\pi n}{2}$.<br> Асимптоты: $x = -\frac{\pi}{12} + \frac{\pi k}{2}, k \in \mathbb{Z}$. (например, $x=...-\frac{7\pi}{12}, -\frac{\pi}{12}, \frac{5\pi}{12}, \frac{11\pi}{12}...$)<br> - <em>Нули функции (точки пересечения с осью OX):</em> $y=0$ при $\text{tg}(\frac{\pi}{3} - 2x)=0$.<br> $\frac{\pi}{3} - 2x = \pi n, n \in \mathbb{Z}$<br> $-2x = -\frac{\pi}{3} + \pi n$<br> $x = \frac{\pi}{6} - \frac{\pi n}{2}$.<br> Нули функции: $x = \frac{\pi}{6} + \frac{\pi k}{2}, k \in \mathbb{Z}$. (например, $x=...-\frac{\pi}{3}, \frac{\pi}{6}, \frac{2\pi}{3}...$)<br> - <em>Точка пересечения с осью OY:</em> При $x=0$, $y = \text{tg}(\frac{\pi}{3}) = \sqrt{3}$. Точка $(0, \sqrt{3})$.<br> - <em>Поведение функции:</em> Из-за знака "-" перед тангенсом (или знака "-" перед $x$ в аргументе), функция является убывающей на каждом интервале между асимптотами.</p><p><em>Построение:</em><br>1. Начертить координатную плоскость.<br>2. Провести вертикальные асимптоты $x = -\frac{\pi}{12}$, $x = \frac{5\pi}{12}$ и т.д. с шагом $\frac{\pi}{2}$.<br>3. Отметить на оси OX нули функции: $x=\frac{\pi}{6}$, $x=\frac{2\pi}{3}$ и т.д.<br>4. Отметить точку пересечения с осью OY $(0, \sqrt{3})$.<br>5. На каждом интервале между асимптотами построить ветвь тангенсоиды, которая убывает от $+\infty$ до $-\infty$, проходя через соответствующий нуль функции. Например, на интервале $(-\frac{\pi}{12}, \frac{5\pi}{12})$ кривая пройдет через точки $(0, \sqrt{3})$ и $(\frac{\pi}{6}, 0)$.<br>6. Повторить построение ветви на других интервалах с учетом периода $T = \frac{\pi}{2}$.</p><p><strong>Ответ:</strong> График функции $y = \text{tg}(\frac{\pi}{3} - 2x)$ представляет собой тангенсоиду с периодом $\frac{\pi}{2}$, убывающую на каждом из интервалов области определения. Асимптоты графика задаются уравнениями $x = -\frac{\pi}{12} + \frac{\pi k}{2}, k \in \mathbb{Z}$. Нули функции находятся в точках $x = \frac{\pi}{6} + \frac{\pi k}{2}, k \in \mathbb{Z}$. График пересекает ось ординат в точке $(0, \sqrt{3})$.</p><hr><p><strong>Б. $y = 2\cos\frac{x}{2}$</strong></p><p>Для построения графика данной функции выполним анализ и последовательные преобразования, исходя из базового графика функции $y = \cos(x)$.</p><p>1. <em>Определим последовательность преобразований графика $y = \cos(x)$.</em><br> а) Растяжение графика $y = \cos(x)$ вдоль оси OX в 2 раза (горизонтальное растяжение). Это преобразование задается коэффициентом $\frac{1}{2}$ при $x$. Получаем функцию $y = \cos(\frac{x}{2})$. Период функции увеличивается в 2 раза и становится $T = 2\pi \cdot 2 = 4\pi$.<br> б) Растяжение полученного графика вдоль оси OY в 2 раза (вертикальное растяжение). Это преобразование задается множителем 2 перед функцией. Получаем искомую функцию $y = 2\cos(\frac{x}{2})$. Амплитуда функции увеличивается в 2 раза и становится равной 2.</p><p>2. <em>Найдем основные свойства функции для построения.</em><br> - <em>Период:</em> $T = \frac{2\pi}{|k|} = \frac{2\pi}{|1/2|} = 4\pi$.<br> - <em>Амплитуда:</em> $A=2$.<br> - <em>Область значений:</em> $[-2, 2]$.<br> - <em>Четность:</em> Функция четная, так как $y(-x) = 2\cos(\frac{-x}{2}) = 2\cos(\frac{x}{2}) = y(x)$. График симметричен относительно оси OY.<br> - <em>Ключевые точки на одном периоде, например, на отрезке $[0, 4\pi]$:</em><br> - При $x=0$, $y = 2\cos(0) = 2 \cdot 1 = 2$ (точка максимума).<br> - При $x=\pi$, $y = 2\cos(\frac{\pi}{2}) = 2 \cdot 0 = 0$ (точка пересечения с осью OX).<br> - При $x=2\pi$, $y = 2\cos(\frac{2\pi}{2}) = 2\cos(\pi) = 2 \cdot (-1) = -2$ (точка минимума).<br> - При $x=3\pi$, $y = 2\cos(\frac{3\pi}{2}) = 2 \cdot 0 = 0$ (точка пересечения с осью OX).<br> - При $x=4\pi$, $y = 2\cos(\frac{4\pi}{2}) = 2\cos(2\pi) = 2 \cdot 1 = 2$ (точка максимума).</p><p><em>Построение:</em><br>1. Начертить координатную плоскость.<br>2. Отметить на осях ключевые точки, найденные выше: $(0, 2), (\pi, 0), (2\pi, -2), (3\pi, 0), (4\pi, 2)$.<br>3. Соединить эти точки плавной линией, имеющей форму косинусоиды.<br>4. Используя свойство периодичности ($T=4\pi$) и четности (симметрия относительно оси OY), продолжить график влево и вправо.</p><p><strong>Ответ:</strong> График функции $y = 2\cos\frac{x}{2}$ является косинусоидой с периодом $4\pi$ и амплитудой 2. Область значений функции — отрезок $[-2, 2]$. Максимумы, равные 2, достигаются в точках $x=4\pi n$, а минимумы, равные -2, в точках $x=2\pi+4\pi n$, где $n \in \mathbb{Z}$. График пересекает ось абсцисс в точках $x=\pi+2\pi n, n \in \mathbb{Z}$.</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" => "1120513" "type" => "task" ] "previous" => array:2 [ "refs" => "1120511" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1134 #items: array:1 [ 0 => App\Models\Book {#1155} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1140 #items: array:1 [ 0 => App\Models\BookPage {#1107 #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" => 1099303 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "150" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/page-2-150" "field_display_title" => "150" "field_folder" => "2" "field_image_name" => "150" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1117 #items: array:1 [ 0 => App\Models\Branch {#1152} ] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Book {#1155} ] #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 {#1112 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ 0 => App\Models\Element {#1105 #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" => 1262728 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1103 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1262728 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1103 …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" => "1099304" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1099302" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1321 #items: array:8 [ 0 => App\Models\Task {#1335 #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" => 1120511 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "150" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-7-1" "field_display_title" => "1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1336 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] 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#observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1354 #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" => 1120513 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "150" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-7-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1355 …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" => 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#observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1373 #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" => 1120515 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "150" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-7-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1365 …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 {#1380 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1376 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1374 …2} "content" => Illuminate\Database\Eloquent\Collection {#1375 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1390 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120515 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "150" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-7-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1365 …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" => 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"2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "150" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/povtor2-7-6" "field_display_title" => "6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1377 …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 {#1392 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1388 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1386 …2} "content" => Illuminate\Database\Eloquent\Collection {#1387 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1402 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120516 "created_at" => "2026-04-10 13:58:26" "updated_at" 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№2 (с. 150)
Условие. №2 (с. 150)
скриншот условия
2. Построй графики функций:
А. $y = \operatorname{tg}\left(\frac{\pi}{3}-2x\right)$;
Б. $y=2\cos\frac{x}{2}$.
Решение 2 (rus). №2 (с. 150)
А. $y = \text{tg}(\frac{\pi}{3} - 2x)$
Для построения графика данной функции выполним анализ и последовательные преобразования, исходя из базового графика функции $y = \text{tg}(x)$.
1. Преобразуем исходное выражение.
Используя свойство нечетности тангенса ($\text{tg}(-u) = -\text{tg}(u)$), можем записать:
$y = \text{tg}(\frac{\pi}{3} - 2x) = \text{tg}(-(2x - \frac{\pi}{3})) = -\text{tg}(2x - \frac{\pi}{3})$.
Вынесем коэффициент 2 за скобки в аргументе тангенса:
$y = -\text{tg}(2(x - \frac{\pi}{6}))$.
Эта форма удобна для анализа преобразований.
2. Определим последовательность преобразований графика $y = \text{tg}(x)$.
а) Сжатие графика $y = \text{tg}(x)$ вдоль оси OX в 2 раза. Получаем функцию $y = \text{tg}(2x)$. Ее период $T = \frac{\pi}{2}$.
б) Сдвиг полученного графика вправо вдоль оси OX на $\frac{\pi}{6}$. Получаем функцию $y = \text{tg}(2(x - \frac{\pi}{6}))$.
в) Симметричное отражение последнего графика относительно оси OX. Получаем искомую функцию $y = -\text{tg}(2(x - \frac{\pi}{6}))$.
3. Найдем основные свойства функции для построения.
- Период: $T = \frac{\pi}{|k|} = \frac{\pi}{|-2|} = \frac{\pi}{2}$.
- Вертикальные асимптоты: Аргумент тангенса должен быть не равен $\frac{\pi}{2} + \pi n$, где $n \in \mathbb{Z}$.
$\frac{\pi}{3} - 2x = \frac{\pi}{2} + \pi n$
$-2x = \frac{\pi}{2} - \frac{\pi}{3} + \pi n$
$-2x = \frac{\pi}{6} + \pi n$
$x = -\frac{\pi}{12} - \frac{\pi n}{2}$.
Асимптоты: $x = -\frac{\pi}{12} + \frac{\pi k}{2}, k \in \mathbb{Z}$. (например, $x=...-\frac{7\pi}{12}, -\frac{\pi}{12}, \frac{5\pi}{12}, \frac{11\pi}{12}...$)
- Нули функции (точки пересечения с осью OX): $y=0$ при $\text{tg}(\frac{\pi}{3} - 2x)=0$.
$\frac{\pi}{3} - 2x = \pi n, n \in \mathbb{Z}$
$-2x = -\frac{\pi}{3} + \pi n$
$x = \frac{\pi}{6} - \frac{\pi n}{2}$.
Нули функции: $x = \frac{\pi}{6} + \frac{\pi k}{2}, k \in \mathbb{Z}$. (например, $x=...-\frac{\pi}{3}, \frac{\pi}{6}, \frac{2\pi}{3}...$)
- Точка пересечения с осью OY: При $x=0$, $y = \text{tg}(\frac{\pi}{3}) = \sqrt{3}$. Точка $(0, \sqrt{3})$.
- Поведение функции: Из-за знака "-" перед тангенсом (или знака "-" перед $x$ в аргументе), функция является убывающей на каждом интервале между асимптотами.
Построение:
1. Начертить координатную плоскость.
2. Провести вертикальные асимптоты $x = -\frac{\pi}{12}$, $x = \frac{5\pi}{12}$ и т.д. с шагом $\frac{\pi}{2}$.
3. Отметить на оси OX нули функции: $x=\frac{\pi}{6}$, $x=\frac{2\pi}{3}$ и т.д.
4. Отметить точку пересечения с осью OY $(0, \sqrt{3})$.
5. На каждом интервале между асимптотами построить ветвь тангенсоиды, которая убывает от $+\infty$ до $-\infty$, проходя через соответствующий нуль функции. Например, на интервале $(-\frac{\pi}{12}, \frac{5\pi}{12})$ кривая пройдет через точки $(0, \sqrt{3})$ и $(\frac{\pi}{6}, 0)$.
6. Повторить построение ветви на других интервалах с учетом периода $T = \frac{\pi}{2}$.
Ответ: График функции $y = \text{tg}(\frac{\pi}{3} - 2x)$ представляет собой тангенсоиду с периодом $\frac{\pi}{2}$, убывающую на каждом из интервалов области определения. Асимптоты графика задаются уравнениями $x = -\frac{\pi}{12} + \frac{\pi k}{2}, k \in \mathbb{Z}$. Нули функции находятся в точках $x = \frac{\pi}{6} + \frac{\pi k}{2}, k \in \mathbb{Z}$. График пересекает ось ординат в точке $(0, \sqrt{3})$.
Б. $y = 2\cos\frac{x}{2}$
Для построения графика данной функции выполним анализ и последовательные преобразования, исходя из базового графика функции $y = \cos(x)$.
1. Определим последовательность преобразований графика $y = \cos(x)$.
а) Растяжение графика $y = \cos(x)$ вдоль оси OX в 2 раза (горизонтальное растяжение). Это преобразование задается коэффициентом $\frac{1}{2}$ при $x$. Получаем функцию $y = \cos(\frac{x}{2})$. Период функции увеличивается в 2 раза и становится $T = 2\pi \cdot 2 = 4\pi$.
б) Растяжение полученного графика вдоль оси OY в 2 раза (вертикальное растяжение). Это преобразование задается множителем 2 перед функцией. Получаем искомую функцию $y = 2\cos(\frac{x}{2})$. Амплитуда функции увеличивается в 2 раза и становится равной 2.
2. Найдем основные свойства функции для построения.
- Период: $T = \frac{2\pi}{|k|} = \frac{2\pi}{|1/2|} = 4\pi$.
- Амплитуда: $A=2$.
- Область значений: $[-2, 2]$.
- Четность: Функция четная, так как $y(-x) = 2\cos(\frac{-x}{2}) = 2\cos(\frac{x}{2}) = y(x)$. График симметричен относительно оси OY.
- Ключевые точки на одном периоде, например, на отрезке $[0, 4\pi]$:
- При $x=0$, $y = 2\cos(0) = 2 \cdot 1 = 2$ (точка максимума).
- При $x=\pi$, $y = 2\cos(\frac{\pi}{2}) = 2 \cdot 0 = 0$ (точка пересечения с осью OX).
- При $x=2\pi$, $y = 2\cos(\frac{2\pi}{2}) = 2\cos(\pi) = 2 \cdot (-1) = -2$ (точка минимума).
- При $x=3\pi$, $y = 2\cos(\frac{3\pi}{2}) = 2 \cdot 0 = 0$ (точка пересечения с осью OX).
- При $x=4\pi$, $y = 2\cos(\frac{4\pi}{2}) = 2\cos(2\pi) = 2 \cdot 1 = 2$ (точка максимума).
Построение:
1. Начертить координатную плоскость.
2. Отметить на осях ключевые точки, найденные выше: $(0, 2), (\pi, 0), (2\pi, -2), (3\pi, 0), (4\pi, 2)$.
3. Соединить эти точки плавной линией, имеющей форму косинусоиды.
4. Используя свойство периодичности ($T=4\pi$) и четности (симметрия относительно оси OY), продолжить график влево и вправо.
Ответ: График функции $y = 2\cos\frac{x}{2}$ является косинусоидой с периодом $4\pi$ и амплитудой 2. Область значений функции — отрезок $[-2, 2]$. Максимумы, равные 2, достигаются в точках $x=4\pi n$, а минимумы, равные -2, в точках $x=2\pi+4\pi n$, где $n \in \mathbb{Z}$. График пересекает ось абсцисс в точках $x=\pi+2\pi n, n \in \mathbb{Z}$.
Другие задания:
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