{"id":1859,"date":"2020-05-27T05:56:44","date_gmt":"2020-05-27T05:56:44","guid":{"rendered":"http:\/\/ibalmaths.com\/?post_type=knowledgebase&#038;p=1859"},"modified":"2020-05-28T06:05:50","modified_gmt":"2020-05-28T06:05:50","slug":"practice-questions-complex-numbers","status":"publish","type":"knowledgebase","link":"https:\/\/ibalmaths.com\/index.php\/ibdp-math-hl-2\/complex-numbers\/practice-questions-complex-numbers\/","title":{"rendered":"Practice Questions &#8211; Complex Numbers"},"content":{"rendered":"<p>Q1. The complex numbers\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>z<\/mi><mn>1<\/mn><\/msub><\/math>\r\n and\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>z<\/mi><mn>2<\/mn><\/msub><\/math>\r\n are given by\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>z<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><msqrt><mn>2<\/mn><\/msqrt><mo>+<\/mo><msqrt><mn>2<\/mn><\/msqrt><mi>i<\/mi><\/math>\r\n and\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>z<\/mi><mn>2<\/mn><\/msub><mo>=<\/mo><msqrt><mn>3<\/mn><\/msqrt><mo>&#8211;<\/mo><mi>i<\/mi><\/math>\r\n<\/p>\r\n<p>(i) Find the modulus and argument of each of\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>z<\/mi><mn>1<\/mn><\/msub><\/math>\r\n and\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>z<\/mi><mn>2<\/mn><\/msub><\/math>\r\n .\u00a0 [6 marks]<br \/>\r\n(ii) Plot the points representing each of \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>z<\/mi><mn>1<\/mn><\/msub><\/math>\r\n ,\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>z<\/mi><mn>2<\/mn><\/msub><\/math>\r\n and\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>z<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>z<\/mi><mn>2<\/mn><\/msub><\/math>\r\n on an Argand diagram. [3 marks]<br \/>\r\n(iii) Hence find the exact value of\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>tan<\/mi><mfrac><mi mathvariant=\"normal\">&#960;<\/mi><mn>24<\/mn><\/mfrac><\/math>\r\n . [5 marks]<\/p>\r\n<div id=\"link1-link-1859\" class=\"sh-link link1-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link1', 1859, 'Solution1', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link1-toggle-1859\">Solution1<\/span><\/a><\/div><div id=\"link1-content-1859\" class=\"sh-content link1-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1862\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-3.jpg\" alt=\"\" width=\"631\" height=\"1116\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-3.jpg 631w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-3-170x300.jpg 170w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-3-579x1024.jpg 579w\" sizes=\"auto, (max-width: 631px) 100vw, 631px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p>Q2.\u00a0<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1865\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-1.png\" alt=\"\" width=\"1143\" height=\"654\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-1.png 1143w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-1-300x172.png 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-1-1024x586.png 1024w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-1-768x439.png 768w\" sizes=\"auto, (max-width: 1143px) 100vw, 1143px\" \/><\/p>\r\n<div id=\"link2-link-1859\" class=\"sh-link link2-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link2', 1859, 'Solution2', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link2-toggle-1859\">Solution2<\/span><\/a><\/div><div id=\"link2-content-1859\" class=\"sh-content link2-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1866\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-4.jpg\" alt=\"\" width=\"878\" height=\"986\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-4.jpg 878w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-4-267x300.jpg 267w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-4-768x862.jpg 768w\" sizes=\"auto, (max-width: 878px) 100vw, 878px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p>Q3.\u00a0<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1868\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-5.jpg\" alt=\"\" width=\"969\" height=\"784\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-5.jpg 969w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-5-300x243.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-5-768x621.jpg 768w\" sizes=\"auto, (max-width: 969px) 100vw, 969px\" \/><\/p>\r\n<div id=\"link3-link-1859\" class=\"sh-link link3-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link3', 1859, 'Solution3', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link3-toggle-1859\">Solution3<\/span><\/a><\/div><div id=\"link3-content-1859\" class=\"sh-content link3-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1869\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-6.jpg\" alt=\"\" width=\"838\" height=\"2244\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-6.jpg 838w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-6-112x300.jpg 112w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-6-382x1024.jpg 382w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-6-768x2057.jpg 768w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-6-574x1536.jpg 574w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-6-765x2048.jpg 765w\" sizes=\"auto, (max-width: 838px) 100vw, 838px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p>Q4.\u00a0<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1872\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-7.jpg\" alt=\"\" width=\"976\" height=\"510\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-7.jpg 976w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-7-300x157.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-7-768x401.jpg 768w\" sizes=\"auto, (max-width: 976px) 100vw, 976px\" \/><\/p>\r\n<div id=\"link4-link-1859\" class=\"sh-link link4-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link4', 1859, 'Solution4', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link4-toggle-1859\">Solution4<\/span><\/a><\/div><div id=\"link4-content-1859\" class=\"sh-content link4-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1871\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-2.png\" alt=\"\" width=\"732\" height=\"1201\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-2.png 732w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-2-183x300.png 183w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-2-624x1024.png 624w\" sizes=\"auto, (max-width: 732px) 100vw, 732px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p>Q5.<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1874\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-8.jpg\" alt=\"\" width=\"981\" height=\"653\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-8.jpg 981w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-8-300x200.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-8-768x511.jpg 768w\" sizes=\"auto, (max-width: 981px) 100vw, 981px\" \/><\/p>\r\n<div id=\"link5-link-1859\" class=\"sh-link link5-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link5', 1859, 'Solution5', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link5-toggle-1859\">Solution5<\/span><\/a><\/div><div id=\"link5-content-1859\" class=\"sh-content link5-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1876\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-9.jpg\" alt=\"\" width=\"607\" height=\"1533\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-9.jpg 607w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-9-119x300.jpg 119w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-9-405x1024.jpg 405w\" sizes=\"auto, (max-width: 607px) 100vw, 607px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p>Q6. (i) The complex number\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>z<\/mi><\/math>\r\n is such that \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>z<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>1<\/mn><mo>+<\/mo><mi>i<\/mi><msqrt><mn>3<\/mn><\/msqrt><\/math>\r\n . Find the two possible values of\u00a0 in the forma \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mo>+<\/mo><mi>i<\/mi><mi>b<\/mi><\/math>\r\n , where\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><\/math>\r\n and \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>b<\/mi><\/math>\r\n are exact real numbers. [5 marks]<br \/>\r\n(ii) With the value of\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>z<\/mi><\/math>\r\n from part (i)such that the real part of \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>z<\/mi><\/math>\r\n is positive, show on an Argand diagram the points A and B representing\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>z<\/mi><\/math>\r\n and \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>z<\/mi><mn>2<\/mn><\/msup><\/math>\r\n respectively. [2 marks]<br \/>\r\n(iii) Specify two transformations which together map the line segment OA to the line segment OB, where O is the origin. [4 marks]<\/p>\r\n<p>Q7.<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1894\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-12.jpg\" alt=\"\" width=\"930\" height=\"1054\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-12.jpg 930w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-12-265x300.jpg 265w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-12-904x1024.jpg 904w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-12-768x870.jpg 768w\" sizes=\"auto, (max-width: 930px) 100vw, 930px\" \/><\/p>\r\n<div id=\"link7-link-1859\" class=\"sh-link link7-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link7', 1859, 'Solution7', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link7-toggle-1859\">Solution7<\/span><\/a><\/div><div id=\"link7-content-1859\" class=\"sh-content link7-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1895\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-13.jpg\" alt=\"\" width=\"385\" height=\"908\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-13.jpg 385w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/05\/1-13-127x300.jpg 127w\" sizes=\"auto, (max-width: 385px) 100vw, 385px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p>&nbsp;<\/p>\r\n<p>&nbsp;<\/p><!-- AddThis Advanced Settings generic via filter on the_content --><!-- AddThis Share Buttons generic via filter on the_content -->","protected":false},"excerpt":{"rendered":"Q1. The complex numbers\u00a0 z1 and\u00a0 z2 are given by\u00a0 z1=2+2i and\u00a0 z2=3&#8211;i (i) Find the modulus and argument of each of\u00a0 z1 and\u00a0 z2 .\u00a0 [6 marks] (ii) Plot the points representing each of z1 ,\u00a0 z2 and\u00a0 z1+z2 on an Argand diagram. [3 marks] (iii) Hence find the exact value of\u00a0 tan&#960;24 . [&hellip;]<!-- AddThis Advanced Settings generic via filter on get_the_excerpt --><!-- AddThis Share Buttons generic via filter on get_the_excerpt -->","protected":false},"author":4,"featured_media":0,"comment_status":"closed","ping_status":"closed","template":"","class_list":["post-1859","knowledgebase","type-knowledgebase","status-publish","hentry","knowledgebase_cat-complex-numbers","no-wpautop"],"_links":{"self":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1859","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase"}],"about":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/types\/knowledgebase"}],"author":[{"embeddable":true,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/comments?post=1859"}],"version-history":[{"count":14,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1859\/revisions"}],"predecessor-version":[{"id":1898,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1859\/revisions\/1898"}],"wp:attachment":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/media?parent=1859"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}