[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-657320-105":59,"doc-detail-657320-en":130},{"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":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","argand-diagrams-question-paper","Argand Diagrams - Question Paper","","A mark-scheme style set of mathematics questions and answers focused on Argand diagrams and complex numbers. It includes problems requiring shading loci on the complex plane, finding ranges of values, computing moduli and arguments in specified intervals, and working with equations of polynomials whose roots are given. Tasks also cover interpreting geometric regions defined by inequalities, determining exact areas of shaded regions, and using the relationships between roots of complex polynomials with real coefficients. Solutions emphasize exact values and coordinate-based reasoning.",{"@graph":69,"@context":122},[70,84,105],{"@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/argand-diagrams-question-paper/657320/",{"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/argand-diagrams-question-paper/657320.png","ImageObject",300,407,{"name":92,"@type":93},"SALUTone","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-26",true,{"@type":102,"interactionType":103,"userInteractionCount":19},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How do you shade the required set of points on an Argand diagram?","Question",{"text":112,"@type":113},"Use the given modulus/argument or distance inequalities to identify the locus (e.g., circles, arcs, or intersection regions) and shade the corresponding area on the complex plane.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What is the difference between finding |w|^2 and finding arg(w)?",{"text":117,"@type":113},"|w|^2 is determined from the modulus condition of a complex number, while arg(w) is the angular component measured in radians and restricted to the stated interval.",{"name":119,"@type":110,"acceptedAnswer":120},"Why can’t all listed complex numbers be roots of a quartic with real coefficients?",{"text":121,"@type":113},"For real-coefficient polynomials, non-real roots occur in conjugate pairs. If the given numbers violate the required conjugate-pair structure, they cannot all be roots.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},657320,1793101528,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":19,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"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":145},5909897206756,"https://ap-avatar.wpscdn.com/avatar/5600152dbe4a7e99502?x-image-process=image/resize,m_fixed,w_180,h_180&k=1792654152478566863","# Argand Diagrams\n\n   \nQuestions  \n   \nQ1.  \n(a)Shade on an Argand diagram the set of points  \n(5)  \nThe complex number w satisfies  \n(b)Find,in simplest form,the exact value of |w|²  \n(4)  \n(Total for question =9marks)  \nQ2.  \nGiven that there are two distinct complex numbers z that satisfy  \ndetermine the exact range of values for the real constant r.  \n(Total for question =7 marks)  \nQ3.  \n(a)Shade on an Argand diagram the set of points  \n(6)  \nThe complex number w satisfies  \nw-4i|=3  \n(b)Find the maximum value of arg win the interval(-π,π).  \nGive your answer in radians correct to 2 decimal places.  \n(2)  \n(Total for question=8marks)  \nQ4.  \nf(z)=Z3+Z2+pz+q  \nwhere pand qare real constants.  \nThe equation f(z)=0 has roots Z1,Z2 and Z₃  \nWhen plotted on an Argand diagram,the points representing Z1,Z2 and z₃form the verticesof a triangle of area 35  \nGiven that z₁=3,find the values of pand q.  \n(7)  \n(Total for question =7marks)  \nQ5.  \nf(z)=z³+az+52where ais a real constant  \nGiven that 2-3i is a root of the equation f(z)=0  \n(a)write down the other complex root.  \n(1)  \n(b)Hence  \n(i)solve completely f(z)=0  \nii)determine the value of a  \n(4)  \n(c)Show all the roots of the equation f(z)=0 on a single Argand diagram.  \n(1)  \n(Total for question =6marks)  \nFigure 1  \nThe complex numbers Z₁=-2,Z2=-1+2i and z₃=1+iare plotted in Figure 1,on anArgand diagram for the complex plane with z=x+iy  \n(a)Explain why Z1,Z2 and z₃,cannot all be roots of a quartic polynomial equation with realcoefficients.  \n(2)  \n(b)Show that arg  \n(3)  \n()Hence show ta arctan(2)-=acan  \n(2)  \nDiagram 1  \n(d)Shade,on Diagram 1,the set of points of the complex plane that satisfy the inequality  \nz+2|≤z-1-i|  \n(2)  \n(Total for question =9marks)  \nQ7.  \n(a)Express the complex number w=4√3-4i in the form r(cosθθ+isine)where r>0 and  \n-π\u003Cθ≤π  \n(4)  \n(b)Show,on a single Argand diagram,  \n(3)  \n(c)Hence determine te minimum distance of w trom he locus arg(z+10)=3  \n(3)  \n(Total for question =10 marks)  \nFigure 1  \nFigure 1 shows an Argand diagram.  \nThe set P,of points that lie within the shaded region including its boundaries,is defined by  \nP={z∈C:a≤|z+b+ci|≤d}  \nwhere a,b,cand dare integers.  \n(a)Write down the values of a,b,cand d.  \n(3)  \nThe set Qis defined by  \nQ={z∈C:a≤|z+b+ci|≤d}n{z∈C:|z-i|≤|z-3i|}  \n(b)Determine the exact area of the region defined by Q,giving your answer in simplestform.  \n(7)  \n(Total for question =10 marks)  \nQ9.  \nf(z)=z⁴+az³+bz²+CZ+d  \nwhere a,b,cand dare real constants.  \nGiven that-1+2i and 3-iare two roots of the equation f (z)=0  \n(a)show all the roots of f(z)=0 on a single Argand diagram,  \n(4)  \n(b)find the values of a,b,cand d.  \n(5)  \n(Total for question =9marks)  \nQ10.  \nf(z)=⁴+az³+bz²+cZ+d  \nwhere a,b,cand dare real constants.  \nThe equation f(z)=0 has complex roots Z1,Z2,Z3 and z₄  \nWhen plotted on an Argand diagram,the points representing Z1,Z2,Z and zq form thevertices of a square,with one vertex in each quadrant.Given that z1=2+3i,determine the values of a,b,cand d.  \n(Total for question=6marks)  \nQ11.  \nGiven that  \nz₁=2+3i  \n2₁z₂|=39√2  \nwhere z and z2 are complex numbers,  \n(a)write z₁in the form r(cosθ+isin θ)Give the exact value of rand give the value of θ in radians to 4 significant figures.  \n(2)  \n(b)Find z giving your answer in the form a+ib where a and bare integers.  \n(6)  \n(Total for question =8marks)  \n# Mark Scheme-Argand Diagrams\n\n   \nQ1.  \n\n| Question   | Scheme   | Marks   | AOs   |\n| --- | --- | --- | --- |\n| (a)   |  | M1   | 1.1b   |\n|  |  | M1   | 1.1b   |\n|  |  | A1   | 2.2a   |\n|  |  | M1   | 3.1a   |\n|  |  | A1   | 1.1b   |\n|  |  | (5)   |  |\n| (b)   | (x-1)²+(y-1)²=9,y=x-2=x=….,or y=…   | M1   | 3.1a   |\n|  |  | A1   | 1.1b   |\n|  |  | M1   | 1.1b   |\n|  | =11+2√14   | A1   | 1.1b   |\n|  |  | (4)   |  |\n| (9 marks)   |  |  |  |\n| Notes   |  |  |  |\n| (a)  \u003Cbr>M1:Circle or arc of a circle with centre in first quadrant and with the circle in all 4 quadrants or  \u003Cbr>arc of circle in quadrants 1","cbCaijJlA5awrpzi","https://ap.wps.com/l/cbCaijJlA5awrpzi","pdf",3027899,28,"English","# Questions\n## Q1-Q5\n## Q7-Q11\n# Mark Scheme","[{\"question\":\"How do you shade the required set of points on an Argand diagram?\",\"answer\":\"Use the given modulus/argument or distance inequalities to identify the locus (e.g., circles, arcs, or intersection regions) and shade the corresponding area on the complex plane.\"},{\"question\":\"What is the difference between finding |w|^2 and finding arg(w)?\",\"answer\":\"|w|^2 is determined from the modulus condition of a complex number, while arg(w) is the angular component measured in radians and restricted to the stated interval.\"},{\"question\":\"Why can’t all listed complex numbers be roots of a quartic with real coefficients?\",\"answer\":\"For real-coefficient polynomials, non-real roots occur in conjugate pairs. If the given numbers violate the required conjugate-pair structure, they cannot all be roots.\"}]","Argand Diagrams - Question Paper | PDF",1792977384,71]