[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-316196-105":59,"doc-detail-316196-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","2007-vcaa-mathematical-methods-exam-2-solutions-free-download-and-print","2007 VCAA Mathematical Methods Exam 2 Solutions - Free download and print","","Answer key and worked solutions for VCAA Mathematical Methods Exam 2, Section 1 multiple-choice questions and Section 2 multi-part problems. Includes final options (A–E) for each question, step-by-step algebraic, calculus, and inverse-function reasoning, definite integral results, and domain/range determinations. Probability and trigonometry questions use transformations to standard forms (e.g., Z-scores, angle solutions). Later parts include application problems involving geometry, optimization, graphing calculator guidance, and checking continuity and intersections.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/2007-vcaa-mathematical-methods-exam-2-solutions-free-download-and-print/316196/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/2007-vcaa-mathematical-methods-exam-2-solutions-free-download-and-print/316196.png","ImageObject",300,407,{"name":92,"@type":93},"Bulrr","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-05","2026-09-20",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",17,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What does Section 1 provide for the exam?","Question",{"text":113,"@type":114},"Section 1 lists multiple-choice questions and gives the selected answer options (A–E) for each question from Q1 to Q22.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How are solutions presented in Section 1 and early parts of Section 2?",{"text":118,"@type":114},"Solutions show the key substitution or transformation steps and then compute the required result, such as solving for x, evaluating integrals, or determining domain and range.",{"name":120,"@type":111,"acceptedAnswer":121},"What kinds of problems are included in Section 2?",{"text":122,"@type":114},"Section 2 includes multi-part tasks covering geometry and calculus applications, continuity checks, finding intersections using a horizontal line, and using graphing calculator steps to sketch and verify results.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},316196,1790024601,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":81,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":39,"preview_page_count":4},5909892115043,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","2007 VCAA Mathematical Methods Exam 2 Solutions Free download & print [from www.itute.com](from www.itute.com)  \nDo not photocopy ©[Copyright 2007 itute.com](Copyright 2007 itute.com)  \nSECTION 1: Multiple-choice questions  \n\n| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| A | D | A | C | D | B | A | B | E | E | E |\n\n\n| 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| C | D | A | B | E | C | D | A | A | A | E |\n\nQ1 f (x ) = 6 − 2x .  \n12 = 6 − 2x −4 = 6 − 2x  \n,  \n,  \nx = −3 .  \nx = 5 .  \n∴ D is [− 3,5]  \nQ2 f (g (x )) = e 2g (x )+3 = e 2x2 +4x−3  \nQ3 ∫ 􀂧􀂩􀂨 x12~~ − ~~co1s2~~ ~~x~~ ~~ 􀂷􀂹􀂸dx = ∫ 􀂨􀂧􀂩 ~~ ~~x12~~ ~~ − sec 2 x􀂷􀂹􀂸dx 1  \n= −  − tan x  \nx  \nQ4 f (x ) = x 3 −  x +1 , f (0) = 03 −  0 +1 = −1 , f (3) = 33 −  3 +1 = 25 . Average rate = ~~ ~~253−−(−0~~ ~~1) = ~~ ~~263 .  \nQ5 ∫ (sin 2x + 24x3 )dx = − ~~ ~~12~~ ~~ cos 2x + 6x4 + c Q6 By Graphics calculator.  \n2  \n0 ~~ ~~π3~~ ~~ x  \nThe range off is [2,5] .  \nQ7 Pr (X \u003C 10 .5) = Pr (X \u003C 11− 2× 0.25) = Pr (X \u003C µ − 2σ)  \n= Pr (Z \u003C −2) = Pr (Z > 2)  \nQ8 f ′(x ) is undefined at 2x + 4 = 0 , i.e. x = −2 .  \n∴ f ′(x ) is discontinuous at x = −2 .  \nQ9 k = 1xdx = −  ~~ ~~1xdx = − [log e x]21 = −loge 2 = loge 􀂨􀂧􀂩~~ ~~12􀂷􀂹􀂸 .  \nk 1  \n∴e = .  \n2  \nQ10 f (x ) is an increasing function. f (0) = −2 , f (1) = e 2 − 3 . The range of f (x ) is [0, e 2 − 3) .  \n∴ the domain of f −1 (x ) is [0, e 2 − 3) . Let y = e 2 x − 3 be the equation of f (x ) . y + 3 = e 2x , ∴ x = ~~ ~~12~~ ~~ loge (y + 3)  \n∴ the equation of f −1 (x ) is y = ~~ ~~12~~ ~~ loge (x + 3) .  \nQ11 (e 2x )2 − 5(e 2x )+ 4 = 0 ,(e 2x − 4)(e 2x −1) = 0 .  \ne 2x = 1 , x = 0 ,  \nor e 2x = 4 , 2x = loge 4 , x = loge 2 .  \nSolution set is {0, loge 2} .  \nQ12 f ′(x )  \n0 3 x  \nf (x )  \n0 3 x  \nQ13 By graphics calculator.  \nLocal maximum at x = −5 , local minimum at x = ~~ ~~12 . Gradient is negative for x ∈ 􀂧􀂨 − 5,  1  􀂷􀂸 .  \n􀂩 2 􀂹  \nQ14 f (x ) = log e x − 3 + 6 is defined for x ≠ 3 .  \nMaximal domain is R \\ {3} .  \nQ15  \ny = 3x52 → y = −3x52 → y = −3(x − 3)52 → y = −3(x − 3)52 − 4 .  \nQ16 f (x ) = (x − a)2 g (x ) ,  \nf ′(x ) = 2(x − a)g (x ) + (x − a)2 g ′(x )  \nf ′(x ) = (x − a)(2g (x) + (x − a)g ′(x )) .  \n© Copyright [itute.com 2007](itute.com 2007) Not to be photocopied or reproduced by other means Free download and print [from www.itute.com](from www.itute.com) 1  \nQ17 E(X ) =  x􀂨􀂧􀂩 ~~ ~~x2~~ ~~ 􀂷􀂹􀂸dx =  ~~ ~~x22~~ ~~dx = 􀂫􀂪􀂬 ~~ ~~x63~~ ~~􀂺􀂼􀂻 20 = ~~ ~~43 .  \nQ18 Pr (X \u003C x) = 0.35 , x = invNorm(0.35,130,2.7) = 129 . Q19 a + b = 1− (0.2 + 0 .2 + 0 .3) = 0.3 .  \nE(X ) = 0a + 2× 0.2 + 4× 0.2 + 6× 0.3 + 8b = 5 ,  \n∴ 8b = 2 , b = 0.25 and a = 0.05 . Q20 tan 2 ~~ ~~θ3 = 1 and θ ∈ [0,2π] .  \n∴ tan ~~ ~~θ3 = 1 , θ3 = ~~ ~~π4~~ ~~ , θ = ~~ ~~3π4~~ ~~ .  \nNote: When ∴ tan ~~ ~~θ3 = −1 , θ3 = ~~ ~~3π4~~ ~~ , θ = ~~ ~~9π4~~ ~~ ∉ [0,2π] . Q21 cos 2 x + 2cos x = 0 , cos x(cos x + 2) = 0 .  \nSince cos x + 2 ≠ 0 , ∴ cos x = 0 .  \nQ22 Let f (x ) = x(x −1) and g (x ) = − x .  \ny = f (g (x )) = g (x )(g (x ) −1) = − x(− x −1) = x 2 + x Graphics calculator:  \n0 x  \nSECTION 2:  \n2 2 1000 Q1a V = πr h , 1000 = πr h , h =  πr2  .  \nQ1b Area of top plus bottom = 2 × πr2 .  \nArea of curved surface = 2πrh = 2πr􀂧􀂨 1000  􀂷􀂸 =  2000 .  \n􀂩 πr2 􀂹 r  \n∴ A =  2000 + 2πr 2 cm2.  \nr  \nQ1c dAdr = − ~~ ~~20r020 + 4πr . Let dAdr = 0 .  \n1  \n− ~~ ~~20020 + 4πr = 0 , r 3 =  500 , r = 􀂧􀂨 500  􀂷􀂸  3 . r π 􀂩 π 􀂹  \n2  \nQ1d Amin = 2000 + 2π􀂧􀂨 500  􀂷􀂸  3 = 553.58 cm2.  \n􀂧 500 􀂷 13 􀂩 π 􀂹  \n􀂨 ~~ ~~ 􀂸  \n􀂩 π 􀂹  \nQ2a When t = 8 , C = 1000(cos0 + 2)2 −1000 = 8000 . When t = 16 , C = 1000(cos 4π + 2)2 −1000 = 8000 .  \n∴ m = 8000 for C(t ) to be continuous.  \nQ2b Use graphics calculator to sketch the middle section.  \nQ2c Cmin = 0 when t = 10 or 14.  \nQ2d Use graphics calculator to find the first intersection of the middle section and the horizontal line C = 1250 .  \nt = ~~ ~~283 hours after midnight, i.e. 9.20 am.  \nQ2e  \nt  \n283~~ ~~ 10","cbCaivAnUCU4Z8SY","https://ap.wps.com/l/cbCaivAnUCU4Z8SY","pdf",143342,"English","# SECTION 1: Multiple-choice questions\n## Q1-Q22\n# SECTION 2\n## Q1 (a-d)\n## Q2 (a-g)\n## Q3 (a-c)\n## Q4 (a-d)\n## Q5 (a-b)","[{\"question\":\"What does Section 1 provide for the exam?\",\"answer\":\"Section 1 lists multiple-choice questions and gives the selected answer options (A–E) for each question from Q1 to Q22.\"},{\"question\":\"How are solutions presented in Section 1 and early parts of Section 2?\",\"answer\":\"Solutions show the key substitution or transformation steps and then compute the required result, such as solving for x, evaluating integrals, or determining domain and range.\"},{\"question\":\"What kinds of problems are included in Section 2?\",\"answer\":\"Section 2 includes multi-part tasks covering geometry and calculus applications, continuity checks, finding intersections using a horizontal line, and using graphing calculator steps to sketch and verify results.\"}]","2007 VCAA Mathematical Methods Exam 2 Solutions - Free download and print | PDF",1789911022]