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Номер 3, страница 118, часть 1 - гдз по алгебре 10 класс учебник Пак, Ардакулы
Авторы: Пак О. В., Ардакулы Д., Ескендирова Е. В.
Тип: Учебник
Издательство: Алматыкітап баспасы
Год издания: 2019 - 2026
Часть: 1
ISBN: 978-601-01-3958-9
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Часть 1. Глава 2. Тригонометрические функции. Параграф 5. Уравнения и неравенства с аркфункциями. Задачи - номер 3, страница 118.
App\Models\Task {#1024 // 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" => 1119786 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "118" "field_page_end" => "119" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/2-5zad-3" "field_display_title" => "3" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #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 {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: array:1 [ 0 => App\Models\Term {#1033 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1030 #items: array:1 [ 0 => App\Models\Book {#1032 #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 {#1034 #items: array:1 [ 0 => App\Models\Term {#1035 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1036 #items: array:1 [ 0 => App\Models\Term {#1037 #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 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "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 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "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 {#1038 #items: array:1 [ 0 => App\Models\Term {#1039 #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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 {#1040 #items: array:3 [ 0 => App\Models\Term {#1041 #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 {#1042 #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 {#1043 #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 {#1044 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1045 #items: array:1 [ 0 => App\Models\Term {#1046 #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 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "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 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "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 {#1047 #items: array:1 [ 0 => App\Models\Term {#1048 #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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 {#1049 #items: array:1 [ 0 => App\Models\Term {#1050 #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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 {#1051 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1056 #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 {#1057 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1060 #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 {#1034} "field_class" => Illuminate\Database\Eloquent\Collection {#1036} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1038} "field_author" => Illuminate\Database\Eloquent\Collection {#1040} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1044} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1045} "field_country" => Illuminate\Database\Eloquent\Collection {#1047} "field_city" => Illuminate\Database\Eloquent\Collection {#1049} "field_series" => Illuminate\Database\Eloquent\Collection {#1051} "field_umk" => Illuminate\Database\Eloquent\Collection {#1052} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1053} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1054} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1055} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1056} "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 {#1057} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1058} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1059} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1060} "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 {#1061 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1030} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1061} ] #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 {#1062 #items: array:4 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1063 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119358 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические функции" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "71" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1066 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1067 #items: array:1 [ 0 => App\Models\Branch {#1068 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1069 #items: array:1 [ 0 => App\Models\Term {#1033} ] #escapeWhenCastingToString: false } "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1069} "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1070} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1071} ] #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" => 1119358 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические функции" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1064} "field_page_start" => "71" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1066} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1067} ] #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 {#1072 #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" => 1119380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Уравнения и неравенства с аркфункциями" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Term {#1074 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "117" "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 {#1075 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ 0 => App\Models\Branch {#1077 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119358 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические функции" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1078 #items: array:1 [ 0 => App\Models\Term {#1065} ] #escapeWhenCastingToString: false } "field_page_start" => "71" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1080 #items: array:1 [ 0 => App\Models\Branch {#1081 #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 [ …24] #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:24 [ "id" => 1119358 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические функции" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1078} "field_page_start" => "71" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1079} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1080} ] #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" => 1119380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Уравнения и неравенства с аркфункциями" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1073} "field_page_start" => "117" "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 {#1075} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Branch {#1105 #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" => 1119381 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1106 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "118" "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 {#1107 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Branch {#1109 #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" => 1119380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Уравнения и неравенства с аркфункциями" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Term {#1074} ] #escapeWhenCastingToString: false } "field_page_start" => "117" "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 {#1111 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1112 #items: array:1 [ 0 => App\Models\Branch {#1077} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Уравнения и неравенства с аркфункциями" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1110} "field_page_start" => "117" 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"field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1106} "field_page_start" => "118" "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 {#1107} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1108} ] #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 {#1113 #items: array:2 [ 0 => App\Models\Element {#1114 #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" => 1300471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1115 #items: array:1 [ 0 => App\Models\Edition {#1116 #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 {#1117 #items: array:1 [ 0 => App\Models\Term {#1039} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1118 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1119 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1120 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1117} "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 {#1118} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1119} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1120} "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" => "1119786" "type" => "task" ] "text" => "<p><strong>3. a)(1) Решите уравнение</strong> $arccos\\left(x^2 - \\frac{5}{2}x + 2\\right) = \\frac{\\pi}{3}$;</p><p><strong>б)(2) Решите неравенство</strong> $arccos\\left(x^2 - \\frac{5}{2}x + 2\\right) < \\frac{\\pi}{3}$;</p><p><strong>в)(3) Решите неравенство</strong> $arccos\\left(x^2 - \\frac{5}{2}x + 2\\right) > \\frac{\\pi}{3}$.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "3-1.jpg" "alt" => null "width" => "1237" "height" => 240 "path" => "/media/algebra_10/baspasy-u19/0-2-5zad/3-1.webp?ts=1753534599" ] 1 => array:5 [ "name" => "3-2.jpg" "alt" => null "width" => "1171" "height" => 347 "path" => "/media/algebra_10/baspasy-u19/0-2-5zad/3-2.webp?ts=1753534599" ] ] ] #original: array:7 [ "id" => 1300471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1115} "task" => array:2 [ "refs" => "1119786" "type" => "task" ] "text" => "<p><strong>3. a)(1) Решите уравнение</strong> $arccos\\left(x^2 - \\frac{5}{2}x + 2\\right) = \\frac{\\pi}{3}$;</p><p><strong>б)(2) Решите неравенство</strong> $arccos\\left(x^2 - \\frac{5}{2}x + 2\\right) < \\frac{\\pi}{3}$;</p><p><strong>в)(3) Решите неравенство</strong> $arccos\\left(x^2 - \\frac{5}{2}x + 2\\right) > \\frac{\\pi}{3}$.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "3-1.jpg" "alt" => null "width" => "1237" "height" => 240 "path" => "/media/algebra_10/baspasy-u19/0-2-5zad/3-1.webp?ts=1753534599" ] 1 => array:5 [ "name" => "3-2.jpg" "alt" => null "width" => "1171" "height" => 347 "path" => "/media/algebra_10/baspasy-u19/0-2-5zad/3-2.webp?ts=1753534599" ] ] ] #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 {#1121 #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" => 1551507 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => App\Models\Edition {#1123 #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 {#1124 #items: array:1 [ 0 => App\Models\Term {#1125 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1126 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1127 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1128 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1124} "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 {#1126} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1127} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1128} "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" => "1119786" "type" => "task" ] "text" => "<p><strong>а) (1)</strong> Решите уравнение $\arccos(x^2 - \frac{5}{2}x + 2) = \frac{\pi}{3}$</p><p> По определению арккосинуса, если $\arccos(y) = a$, то $y = \cos(a)$. <br> Применим это к нашему уравнению: <br> $x^2 - \frac{5}{2}x + 2 = \cos(\frac{\pi}{3})$ <br> Так как $\cos(\frac{\pi}{3}) = \frac{1}{2}$, получаем: <br> $x^2 - \frac{5}{2}x + 2 = \frac{1}{2}$ <br> Перенесем все в левую часть и приведем подобные слагаемые: <br> $x^2 - \frac{5}{2}x + 2 - \frac{1}{2} = 0$ <br> $x^2 - \frac{5}{2}x + \frac{3}{2} = 0$ <br> Умножим обе части уравнения на 2, чтобы избавиться от дробей: <br> $2x^2 - 5x + 3 = 0$ <br> Это квадратное уравнение. Найдем его корни через дискриминант: <br> $D = b^2 - 4ac = (-5)^2 - 4 \cdot 2 \cdot 3 = 25 - 24 = 1$ <br> $x_{1,2} = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{1}}{2 \cdot 2} = \frac{5 \pm 1}{4}$ <br> $x_1 = \frac{5 - 1}{4} = \frac{4}{4} = 1$ <br> $x_2 = \frac{5 + 1}{4} = \frac{6}{4} = \frac{3}{2}$ <br> При решении уравнения с арккосинусом необходимо, чтобы его аргумент принадлежал отрезку $[-1, 1]$. В нашем случае аргумент равен $\cos(\frac{\pi}{3}) = \frac{1}{2}$, что удовлетворяет этому условию. Следовательно, найденные действительные корни являются решениями. <br> Ответ: $x=1, x=\frac{3}{2}$.</p><p><strong>б) (2)</strong> Решите неравенство $\arccos(x^2 - \frac{5}{2}x + 2) < \frac{\pi}{3}$</p><p> Область значений функции арккосинус - это отрезок $[0, \pi]$. Таким образом, данное неравенство равносильно двойному неравенству: <br> $0 \leq \arccos(x^2 - \frac{5}{2}x + 2) < \frac{\pi}{3}$ <br> Функция $y = \arccos(t)$ является убывающей на всей своей области определения $[-1, 1]$. Поэтому при переходе от арккосинусов к их аргументам знаки неравенства меняются на противоположные: <br> $\cos(0) \geq x^2 - \frac{5}{2}x + 2 > \cos(\frac{\pi}{3})$ <br> Подставим значения косинусов: $\cos(0) = 1$ и $\cos(\frac{\pi}{3}) = \frac{1}{2}$. <br> $1 \geq x^2 - \frac{5}{2}x + 2 > \frac{1}{2}$ <br> Это система из двух неравенств: <br> 1) $x^2 - \frac{5}{2}x + 2 > \frac{1}{2}$ <br> 2) $x^2 - \frac{5}{2}x + 2 \leq 1$ <br> Решим первое неравенство: <br> $x^2 - \frac{5}{2}x + \frac{3}{2} > 0$ <br> $2x^2 - 5x + 3 > 0$ <br> Корни соответствующего уравнения $2x^2 - 5x + 3 = 0$ мы нашли в пункте а): $x_1 = 1, x_2 = \frac{3}{2}$. <br> Так как ветви параболы $y = 2x^2 - 5x + 3$ направлены вверх, неравенство выполняется вне интервала между корнями: $x \in (-\infty, 1) \cup (\frac{3}{2}, +\infty)$. <br> Решим второе неравенство: <br> $x^2 - \frac{5}{2}x + 1 \leq 0$ <br> $2x^2 - 5x + 2 \leq 0$ <br> Найдем корни уравнения $2x^2 - 5x + 2 = 0$: <br> $D = (-5)^2 - 4 \cdot 2 \cdot 2 = 25 - 16 = 9$ <br> $x_{1,2} = \frac{5 \pm 3}{4}$, откуда $x_1 = \frac{1}{2}, x_2 = 2$. <br> Ветви параболы $y = 2x^2 - 5x + 2$ направлены вверх, поэтому неравенство выполняется между корнями: $x \in [\frac{1}{2}, 2]$. <br> Теперь найдем пересечение решений обоих неравенств: <br> $(-\infty, 1) \cup (\frac{3}{2}, +\infty) \cap [\frac{1}{2}, 2]$ <br> Пересечение дает нам два интервала: $[\frac{1}{2}, 1)$ и $(\frac{3}{2}, 2]$. <br> Ответ: $x \in [\frac{1}{2}, 1) \cup (\frac{3}{2}, 2]$.</p><p><strong>в) (3)</strong> Решите неравенство $\arccos(x^2 - \frac{5}{2}x + 2) > \frac{\pi}{3}$</p><p> Область значений функции арккосинус - это отрезок $[0, \pi]$. Таким образом, данное неравенство равносильно двойному неравенству: <br> $\frac{\pi}{3} < \arccos(x^2 - \frac{5}{2}x + 2) \leq \pi$ <br> Функция $y = \arccos(t)$ является убывающей. Применяя функцию косинус к частям неравенства, мы меняем знаки на противоположные: <br> $\cos(\frac{\pi}{3}) > x^2 - \frac{5}{2}x + 2 \geq \cos(\pi)$ <br> Подставим значения косинусов: $\cos(\frac{\pi}{3}) = \frac{1}{2}$ и $\cos(\pi) = -1$. <br> $\frac{1}{2} > x^2 - \frac{5}{2}x + 2 \geq -1$ <br> Это система из двух неравенств: <br> 1) $x^2 - \frac{5}{2}x + 2 < \frac{1}{2}$ <br> 2) $x^2 - \frac{5}{2}x + 2 \geq -1$ <br> Решим первое неравенство: <br> $x^2 - \frac{5}{2}x + \frac{3}{2} < 0$ <br> $2x^2 - 5x + 3 < 0$ <br> Корни уравнения $2x^2 - 5x + 3 = 0$ равны $x_1 = 1, x_2 = \frac{3}{2}$. <br> Ветви параболы направлены вверх, поэтому неравенство выполняется между корнями: $x \in (1, \frac{3}{2})$. <br> Решим второе неравенство: <br> $x^2 - \frac{5}{2}x + 3 \geq 0$ <br> $2x^2 - 5x + 6 \geq 0$ <br> Найдем дискриминант: $D = (-5)^2 - 4 \cdot 2 \cdot 6 = 25 - 48 = -23$. <br> Так как $D < 0$ и старший коэффициент $2 > 0$, парабола $y = 2x^2 - 5x + 6$ целиком лежит выше оси Ох, и неравенство выполняется для всех действительных чисел $x$. <br> Пересечением решений $x \in (1, \frac{3}{2})$ и $x \in (-\infty, +\infty)$ является интервал $(1, \frac{3}{2})$. <br> Ответ: $x \in (1, \frac{3}{2})$.</p>" ] #original: array:6 [ "id" => 1551507 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1122} "task" => array:2 [ "refs" => "1119786" "type" => "task" ] "text" => "<p><strong>а) (1)</strong> Решите уравнение $\arccos(x^2 - \frac{5}{2}x + 2) = \frac{\pi}{3}$</p><p> По определению арккосинуса, если $\arccos(y) = a$, то $y = \cos(a)$. <br> Применим это к нашему уравнению: <br> $x^2 - \frac{5}{2}x + 2 = \cos(\frac{\pi}{3})$ <br> Так как $\cos(\frac{\pi}{3}) = \frac{1}{2}$, получаем: <br> $x^2 - \frac{5}{2}x + 2 = \frac{1}{2}$ <br> Перенесем все в левую часть и приведем подобные слагаемые: <br> $x^2 - \frac{5}{2}x + 2 - \frac{1}{2} = 0$ <br> $x^2 - \frac{5}{2}x + \frac{3}{2} = 0$ <br> Умножим обе части уравнения на 2, чтобы избавиться от дробей: <br> $2x^2 - 5x + 3 = 0$ <br> Это квадратное уравнение. Найдем его корни через дискриминант: <br> $D = b^2 - 4ac = (-5)^2 - 4 \cdot 2 \cdot 3 = 25 - 24 = 1$ <br> $x_{1,2} = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{1}}{2 \cdot 2} = \frac{5 \pm 1}{4}$ <br> $x_1 = \frac{5 - 1}{4} = \frac{4}{4} = 1$ <br> $x_2 = \frac{5 + 1}{4} = \frac{6}{4} = \frac{3}{2}$ <br> При решении уравнения с арккосинусом необходимо, чтобы его аргумент принадлежал отрезку $[-1, 1]$. В нашем случае аргумент равен $\cos(\frac{\pi}{3}) = \frac{1}{2}$, что удовлетворяет этому условию. Следовательно, найденные действительные корни являются решениями. <br> Ответ: $x=1, x=\frac{3}{2}$.</p><p><strong>б) (2)</strong> Решите неравенство $\arccos(x^2 - \frac{5}{2}x + 2) < \frac{\pi}{3}$</p><p> Область значений функции арккосинус - это отрезок $[0, \pi]$. Таким образом, данное неравенство равносильно двойному неравенству: <br> $0 \leq \arccos(x^2 - \frac{5}{2}x + 2) < \frac{\pi}{3}$ <br> Функция $y = \arccos(t)$ является убывающей на всей своей области определения $[-1, 1]$. Поэтому при переходе от арккосинусов к их аргументам знаки неравенства меняются на противоположные: <br> $\cos(0) \geq x^2 - \frac{5}{2}x + 2 > \cos(\frac{\pi}{3})$ <br> Подставим значения косинусов: $\cos(0) = 1$ и $\cos(\frac{\pi}{3}) = \frac{1}{2}$. <br> $1 \geq x^2 - \frac{5}{2}x + 2 > \frac{1}{2}$ <br> Это система из двух неравенств: <br> 1) $x^2 - \frac{5}{2}x + 2 > \frac{1}{2}$ <br> 2) $x^2 - \frac{5}{2}x + 2 \leq 1$ <br> Решим первое неравенство: <br> $x^2 - \frac{5}{2}x + \frac{3}{2} > 0$ <br> $2x^2 - 5x + 3 > 0$ <br> Корни соответствующего уравнения $2x^2 - 5x + 3 = 0$ мы нашли в пункте а): $x_1 = 1, x_2 = \frac{3}{2}$. <br> Так как ветви параболы $y = 2x^2 - 5x + 3$ направлены вверх, неравенство выполняется вне интервала между корнями: $x \in (-\infty, 1) \cup (\frac{3}{2}, +\infty)$. <br> Решим второе неравенство: <br> $x^2 - \frac{5}{2}x + 1 \leq 0$ <br> $2x^2 - 5x + 2 \leq 0$ <br> Найдем корни уравнения $2x^2 - 5x + 2 = 0$: <br> $D = (-5)^2 - 4 \cdot 2 \cdot 2 = 25 - 16 = 9$ <br> $x_{1,2} = \frac{5 \pm 3}{4}$, откуда $x_1 = \frac{1}{2}, x_2 = 2$. <br> Ветви параболы $y = 2x^2 - 5x + 2$ направлены вверх, поэтому неравенство выполняется между корнями: $x \in [\frac{1}{2}, 2]$. <br> Теперь найдем пересечение решений обоих неравенств: <br> $(-\infty, 1) \cup (\frac{3}{2}, +\infty) \cap [\frac{1}{2}, 2]$ <br> Пересечение дает нам два интервала: $[\frac{1}{2}, 1)$ и $(\frac{3}{2}, 2]$. <br> Ответ: $x \in [\frac{1}{2}, 1) \cup (\frac{3}{2}, 2]$.</p><p><strong>в) (3)</strong> Решите неравенство $\arccos(x^2 - \frac{5}{2}x + 2) > \frac{\pi}{3}$</p><p> Область значений функции арккосинус - это отрезок $[0, \pi]$. Таким образом, данное неравенство равносильно двойному неравенству: <br> $\frac{\pi}{3} < \arccos(x^2 - \frac{5}{2}x + 2) \leq \pi$ <br> Функция $y = \arccos(t)$ является убывающей. Применяя функцию косинус к частям неравенства, мы меняем знаки на противоположные: <br> $\cos(\frac{\pi}{3}) > x^2 - \frac{5}{2}x + 2 \geq \cos(\pi)$ <br> Подставим значения косинусов: $\cos(\frac{\pi}{3}) = \frac{1}{2}$ и $\cos(\pi) = -1$. <br> $\frac{1}{2} > x^2 - \frac{5}{2}x + 2 \geq -1$ <br> Это система из двух неравенств: <br> 1) $x^2 - \frac{5}{2}x + 2 < \frac{1}{2}$ <br> 2) $x^2 - \frac{5}{2}x + 2 \geq -1$ <br> Решим первое неравенство: <br> $x^2 - \frac{5}{2}x + \frac{3}{2} < 0$ <br> $2x^2 - 5x + 3 < 0$ <br> Корни уравнения $2x^2 - 5x + 3 = 0$ равны $x_1 = 1, x_2 = \frac{3}{2}$. <br> Ветви параболы направлены вверх, поэтому неравенство выполняется между корнями: $x \in (1, \frac{3}{2})$. <br> Решим второе неравенство: <br> $x^2 - \frac{5}{2}x + 3 \geq 0$ <br> $2x^2 - 5x + 6 \geq 0$ <br> Найдем дискриминант: $D = (-5)^2 - 4 \cdot 2 \cdot 6 = 25 - 48 = -23$. <br> Так как $D < 0$ и старший коэффициент $2 > 0$, парабола $y = 2x^2 - 5x + 6$ целиком лежит выше оси Ох, и неравенство 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№3 (с. 118)
Условие. №3 (с. 118)
скриншот условия
3. a)(1) Решите уравнение $arccos\\left(x^2 - \\frac{5}{2}x + 2\\right) = \\frac{\\pi}{3}$;
б)(2) Решите неравенство $arccos\\left(x^2 - \\frac{5}{2}x + 2\\right) < \\frac{\\pi}{3}$;
в)(3) Решите неравенство $arccos\\left(x^2 - \\frac{5}{2}x + 2\\right) > \\frac{\\pi}{3}$.
Решение 2 (rus). №3 (с. 118)
а) (1) Решите уравнение $\arccos(x^2 - \frac{5}{2}x + 2) = \frac{\pi}{3}$
По определению арккосинуса, если $\arccos(y) = a$, то $y = \cos(a)$.
Применим это к нашему уравнению:
$x^2 - \frac{5}{2}x + 2 = \cos(\frac{\pi}{3})$
Так как $\cos(\frac{\pi}{3}) = \frac{1}{2}$, получаем:
$x^2 - \frac{5}{2}x + 2 = \frac{1}{2}$
Перенесем все в левую часть и приведем подобные слагаемые:
$x^2 - \frac{5}{2}x + 2 - \frac{1}{2} = 0$
$x^2 - \frac{5}{2}x + \frac{3}{2} = 0$
Умножим обе части уравнения на 2, чтобы избавиться от дробей:
$2x^2 - 5x + 3 = 0$
Это квадратное уравнение. Найдем его корни через дискриминант:
$D = b^2 - 4ac = (-5)^2 - 4 \cdot 2 \cdot 3 = 25 - 24 = 1$
$x_{1,2} = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{1}}{2 \cdot 2} = \frac{5 \pm 1}{4}$
$x_1 = \frac{5 - 1}{4} = \frac{4}{4} = 1$
$x_2 = \frac{5 + 1}{4} = \frac{6}{4} = \frac{3}{2}$
При решении уравнения с арккосинусом необходимо, чтобы его аргумент принадлежал отрезку $[-1, 1]$. В нашем случае аргумент равен $\cos(\frac{\pi}{3}) = \frac{1}{2}$, что удовлетворяет этому условию. Следовательно, найденные действительные корни являются решениями.
Ответ: $x=1, x=\frac{3}{2}$.
б) (2) Решите неравенство $\arccos(x^2 - \frac{5}{2}x + 2) < \frac{\pi}{3}$
Область значений функции арккосинус - это отрезок $[0, \pi]$. Таким образом, данное неравенство равносильно двойному неравенству:
$0 \leq \arccos(x^2 - \frac{5}{2}x + 2) < \frac{\pi}{3}$
Функция $y = \arccos(t)$ является убывающей на всей своей области определения $[-1, 1]$. Поэтому при переходе от арккосинусов к их аргументам знаки неравенства меняются на противоположные:
$\cos(0) \geq x^2 - \frac{5}{2}x + 2 > \cos(\frac{\pi}{3})$
Подставим значения косинусов: $\cos(0) = 1$ и $\cos(\frac{\pi}{3}) = \frac{1}{2}$.
$1 \geq x^2 - \frac{5}{2}x + 2 > \frac{1}{2}$
Это система из двух неравенств:
1) $x^2 - \frac{5}{2}x + 2 > \frac{1}{2}$
2) $x^2 - \frac{5}{2}x + 2 \leq 1$
Решим первое неравенство:
$x^2 - \frac{5}{2}x + \frac{3}{2} > 0$
$2x^2 - 5x + 3 > 0$
Корни соответствующего уравнения $2x^2 - 5x + 3 = 0$ мы нашли в пункте а): $x_1 = 1, x_2 = \frac{3}{2}$.
Так как ветви параболы $y = 2x^2 - 5x + 3$ направлены вверх, неравенство выполняется вне интервала между корнями: $x \in (-\infty, 1) \cup (\frac{3}{2}, +\infty)$.
Решим второе неравенство:
$x^2 - \frac{5}{2}x + 1 \leq 0$
$2x^2 - 5x + 2 \leq 0$
Найдем корни уравнения $2x^2 - 5x + 2 = 0$:
$D = (-5)^2 - 4 \cdot 2 \cdot 2 = 25 - 16 = 9$
$x_{1,2} = \frac{5 \pm 3}{4}$, откуда $x_1 = \frac{1}{2}, x_2 = 2$.
Ветви параболы $y = 2x^2 - 5x + 2$ направлены вверх, поэтому неравенство выполняется между корнями: $x \in [\frac{1}{2}, 2]$.
Теперь найдем пересечение решений обоих неравенств:
$(-\infty, 1) \cup (\frac{3}{2}, +\infty) \cap [\frac{1}{2}, 2]$
Пересечение дает нам два интервала: $[\frac{1}{2}, 1)$ и $(\frac{3}{2}, 2]$.
Ответ: $x \in [\frac{1}{2}, 1) \cup (\frac{3}{2}, 2]$.
в) (3) Решите неравенство $\arccos(x^2 - \frac{5}{2}x + 2) > \frac{\pi}{3}$
Область значений функции арккосинус - это отрезок $[0, \pi]$. Таким образом, данное неравенство равносильно двойному неравенству:
$\frac{\pi}{3} < \arccos(x^2 - \frac{5}{2}x + 2) \leq \pi$
Функция $y = \arccos(t)$ является убывающей. Применяя функцию косинус к частям неравенства, мы меняем знаки на противоположные:
$\cos(\frac{\pi}{3}) > x^2 - \frac{5}{2}x + 2 \geq \cos(\pi)$
Подставим значения косинусов: $\cos(\frac{\pi}{3}) = \frac{1}{2}$ и $\cos(\pi) = -1$.
$\frac{1}{2} > x^2 - \frac{5}{2}x + 2 \geq -1$
Это система из двух неравенств:
1) $x^2 - \frac{5}{2}x + 2 < \frac{1}{2}$
2) $x^2 - \frac{5}{2}x + 2 \geq -1$
Решим первое неравенство:
$x^2 - \frac{5}{2}x + \frac{3}{2} < 0$
$2x^2 - 5x + 3 < 0$
Корни уравнения $2x^2 - 5x + 3 = 0$ равны $x_1 = 1, x_2 = \frac{3}{2}$.
Ветви параболы направлены вверх, поэтому неравенство выполняется между корнями: $x \in (1, \frac{3}{2})$.
Решим второе неравенство:
$x^2 - \frac{5}{2}x + 3 \geq 0$
$2x^2 - 5x + 6 \geq 0$
Найдем дискриминант: $D = (-5)^2 - 4 \cdot 2 \cdot 6 = 25 - 48 = -23$.
Так как $D < 0$ и старший коэффициент $2 > 0$, парабола $y = 2x^2 - 5x + 6$ целиком лежит выше оси Ох, и неравенство выполняется для всех действительных чисел $x$.
Пересечением решений $x \in (1, \frac{3}{2})$ и $x \in (-\infty, +\infty)$ является интервал $(1, \frac{3}{2})$.
Ответ: $x \in (1, \frac{3}{2})$.
Другие задания:
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