{"id":1730,"date":"2020-04-15T14:56:45","date_gmt":"2020-04-15T14:56:45","guid":{"rendered":"http:\/\/ibalmaths.com\/?post_type=knowledgebase&#038;p=1730"},"modified":"2020-07-01T07:30:34","modified_gmt":"2020-07-01T07:30:34","slug":"practice-reasoning-and-proof","status":"publish","type":"knowledgebase","link":"https:\/\/ibalmaths.com\/index.php\/ibdp-math-hl-2\/reasoning-and-proof\/practice-reasoning-and-proof\/","title":{"rendered":"Practice &#8211; Reasoning and Proof"},"content":{"rendered":"<p>1. Show that for all real numbers \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><\/math>\r\n and\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>b<\/mi><\/math>\r\n<\/p>\r\n<p>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfenced open=\"|\" close=\"|\"><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><\/mfenced><mo>&#8804;<\/mo><mfenced open=\"|\" close=\"|\"><mi>a<\/mi><\/mfenced><mo>+<\/mo><mfenced open=\"|\" close=\"|\"><mi>b<\/mi><\/mfenced><\/math>\r\n .\u00a0 \u00a0 \u00a0 \u00a0(Triangle Inequality)<\/p>\r\n<div id=\"link1-link-1730\" class=\"sh-link link1-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link1', 1730, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link1-toggle-1730\">Solution<\/span><\/a><\/div><div id=\"link1-content-1730\" class=\"sh-content link1-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><strong>Case1:\u00a0<\/strong>Let\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo>&#8805;<\/mo><mn>0<\/mn><\/math>\r\n . Then\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfenced open=\"|\" close=\"|\"><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><\/mfenced><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><\/math>\r\n &#8230;&#8230;&#8230;&#8230;..eq.(1)<\/p>\r\n<p>Also, we know that\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mo>&#8804;<\/mo><mfenced open=\"|\" close=\"|\"><mi>a<\/mi><\/mfenced><mo>&#160;<\/mo><mi>a<\/mi><mi>n<\/mi><mi>d<\/mi><mo>&#160;<\/mo><mi>b<\/mi><mo>&#8804;<\/mo><mfenced open=\"|\" close=\"|\"><mi>b<\/mi><\/mfenced><\/math>\r\n , so adding these two inequalities, we get\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo>&#8804;<\/mo><mfenced open=\"|\" close=\"|\"><mi>a<\/mi><\/mfenced><mo>+<\/mo><mfenced open=\"|\" close=\"|\"><mi>b<\/mi><\/mfenced><\/math>\r\n ,\u00a0 \u00a0 \u00a0 \u00a0then from eq.(1), we get<\/p>\r\n<p>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfenced open=\"|\" close=\"|\"><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><\/mfenced><mo>&#8804;<\/mo><mfenced open=\"|\" close=\"|\"><mi>a<\/mi><\/mfenced><mo>+<\/mo><mfenced open=\"|\" close=\"|\"><mi>b<\/mi><\/mfenced><\/math>\r\n<\/p>\r\n<p><strong>Case2:<\/strong> Let\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo>&#60;<\/mo><mn>0<\/mn><\/math>\r\n . Then \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfenced open=\"|\" close=\"|\"><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><\/mfenced><mo>=<\/mo><mo>&#8211;<\/mo><mi>a<\/mi><mo>&#8211;<\/mo><mi>b<\/mi><\/math>\r\n &#8230;&#8230;&#8230;&#8230;&#8230;eq.(2)<\/p>\r\n<p>Also, we know that \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo>&#8211;<\/mo><mi>a<\/mi><mo>&#8804;<\/mo><mfenced open=\"|\" close=\"|\"><mi>a<\/mi><\/mfenced><mo>&#160;<\/mo><mi>a<\/mi><mi>n<\/mi><mi>d<\/mi><mo>&#160;<\/mo><mo>&#8211;<\/mo><mi>b<\/mi><mo>&#8804;<\/mo><mfenced open=\"|\" close=\"|\"><mi>b<\/mi><\/mfenced><\/math>\r\n , so adding these two inequalities, we get\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo>&#8211;<\/mo><mi>a<\/mi><mo>&#8211;<\/mo><mi>b<\/mi><mo>&#8804;<\/mo><mfenced open=\"|\" close=\"|\"><mi>a<\/mi><\/mfenced><mo>+<\/mo><mfenced open=\"|\" close=\"|\"><mi>b<\/mi><\/mfenced><\/math>\r\n \u00a0 ,\u00a0 \u00a0 \u00a0 \u00a0then from eq.(2), we get<\/p>\r\n<p>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfenced open=\"|\" close=\"|\"><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><\/mfenced><mo>&#8804;<\/mo><mfenced open=\"|\" close=\"|\"><mi>a<\/mi><\/mfenced><mo>+<\/mo><mfenced open=\"|\" close=\"|\"><mi>b<\/mi><\/mfenced><\/math>\r\n<\/p>\r\n<p>\u00a0<\/div>\r\n<p>2. If\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><\/math>\r\n and\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><\/math>\r\n are not negative real numbers with\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><mo>&#8804;<\/mo><mi>y<\/mi><\/math>\r\n , then\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msqrt><mi>x<\/mi><\/msqrt><mo>&#8804;<\/mo><msqrt><mi>y<\/mi><\/msqrt><\/math>\r\n .<\/p>\r\n<div id=\"link2-link-1730\" class=\"sh-link link2-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link2', 1730, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link2-toggle-1730\">Solution<\/span><\/a><\/div><div id=\"link2-content-1730\" class=\"sh-content link2-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><em><strong>[Proof by Contradiction]<\/strong><\/em><\/p>\r\n<p>Let\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mo>=<\/mo><msqrt><mi>x<\/mi><\/msqrt><\/math>\r\n and\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>b<\/mi><mo>=<\/mo><msqrt><mi>y<\/mi><\/msqrt><\/math>\r\n such that\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0<\/mn><mo>&#8804;<\/mo><mi>x<\/mi><mo>&#8804;<\/mo><mi>y<\/mi><\/math>\r\n . Since both\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><\/math>\r\n and\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>b<\/mi><\/math>\r\n are non negative and\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><mo>&#8804;<\/mo><mi>y<\/mi><\/math>\r\n , we get\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>&#8804;<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><\/math>\r\n .<\/p>\r\n<p>Let us assume\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>b<\/mi><mo>&#60;<\/mo><mi>a<\/mi><\/math>\r\n . Since \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>b<\/mi><mo>&#8805;<\/mo><mn>0<\/mn><\/math>\r\n , we can multiply\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>b<\/mi><\/math>\r\n on both sides of\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>b<\/mi><mo>&#60;<\/mo><mi>a<\/mi><\/math>\r\n to get\u00a0 \u00a0 \u00a0\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>&#8804;<\/mo><mi>a<\/mi><mi>b<\/mi><\/math>\r\n .<\/p>\r\n<p>Multiplying \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><\/math>\r\n on both sides of\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>b<\/mi><mo>&#60;<\/mo><mi>a<\/mi><\/math>\r\n , we get\u00a0\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mi>b<\/mi><mo>&#60;<\/mo><msup><mi>a<\/mi><mn>2<\/mn><\/msup><\/math>\r\n<\/p>\r\n<p>Putting the above two inequalities together, we get\u00a0<\/p>\r\n<p>\r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>&#60;<\/mo><msup><mi>a<\/mi><mn>2<\/mn><\/msup><\/math>\r\n . Which is a contradiction to our hypothesis\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>&#8804;<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><\/math>\r\n . So, out assumption that \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>b<\/mi><mo>&#60;<\/mo><mi>a<\/mi><\/math>\r\n is wrong. Hence\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mo>&#8804;<\/mo><mi>b<\/mi><\/math>\r\n \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo>&#8658;<\/mo><\/math>\r\n \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msqrt><mi>x<\/mi><\/msqrt><mo>&#60;<\/mo><msqrt><mi>y<\/mi><\/msqrt><\/math>\r\n .<\/p>\r\n<p>\u00a0<\/div>\r\n<p>3. For any real numbers\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><\/math>\r\n and\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><\/math>\r\n , show that\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msqrt><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>y<\/mi><mn>2<\/mn><\/msup><\/msqrt><mo>&#8804;<\/mo><mfenced open=\"|\" close=\"|\"><mi>x<\/mi><\/mfenced><mo>+<\/mo><mfenced open=\"|\" close=\"|\"><mi>y<\/mi><\/mfenced><\/math>\r\n .<\/p>\r\n<div id=\"link3-link-1730\" class=\"sh-link link3-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link3', 1730, 'Solution', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link3-toggle-1730\">Solution<\/span><\/a><\/div><div id=\"link3-content-1730\" class=\"sh-content link3-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p>Let\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><\/math>\r\n and\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><\/math>\r\n be real numbers. Now, we know that \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0<\/mn><mo>&#8804;<\/mo><mfenced open=\"|\" close=\"|\"><mi>x<\/mi><\/mfenced><\/math>\r\n and\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0<\/mn><mo>&#8804;<\/mo><mfenced open=\"|\" close=\"|\"><mi>y<\/mi><\/mfenced><\/math>\r\n , so we can say that\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0<\/mn><mo>&#8804;<\/mo><mn>2<\/mn><mfenced open=\"|\" close=\"|\"><mi>x<\/mi><\/mfenced><mfenced open=\"|\" close=\"|\"><mi>y<\/mi><\/mfenced><\/math>\r\n<\/p>\r\n<p>Adding \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>y<\/mi><mn>2<\/mn><\/msup><\/math>\r\n on both sides , we get<\/p>\r\n<p>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>y<\/mi><mn>2<\/mn><\/msup><mo>&#8804;<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>y<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mfenced open=\"|\" close=\"|\"><mi>x<\/mi><\/mfenced><mfenced open=\"|\" close=\"|\"><mi>y<\/mi><\/mfenced><\/math>\r\n<\/p>\r\n<p>Also, we know that \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mfenced open=\"|\" close=\"|\"><mi>x<\/mi><\/mfenced><mn>2<\/mn><\/msup><\/math>\r\n \u00a0 and\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>y<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mfenced open=\"|\" close=\"|\"><mi>y<\/mi><\/mfenced><mn>2<\/mn><\/msup><\/math>\r\n \u00a0<\/p>\r\n<p>\r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo>&#8658;<\/mo><mo>&#160;<\/mo><mo>&#160;<\/mo><mo>&#160;<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>y<\/mi><mn>2<\/mn><\/msup><mo>&#8804;<\/mo><msup><mfenced open=\"|\" close=\"|\"><mi>x<\/mi><\/mfenced><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mfenced open=\"|\" close=\"|\"><mi>y<\/mi><\/mfenced><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mfenced open=\"|\" close=\"|\"><mi>x<\/mi><\/mfenced><mfenced open=\"|\" close=\"|\"><mi>y<\/mi><\/mfenced><\/math>\r\n<\/p>\r\n<p>\r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo>&#8658;<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>y<\/mi><mn>2<\/mn><\/msup><mo>&#8804;<\/mo><msup><mfenced><mrow><mfenced open=\"|\" close=\"|\"><mi>x<\/mi><\/mfenced><mo>+<\/mo><mfenced open=\"|\" close=\"|\"><mi>y<\/mi><\/mfenced><\/mrow><\/mfenced><mn>2<\/mn><\/msup><\/math>\r\n<\/p>\r\n<p>\r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo>&#8658;<\/mo><msqrt><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>y<\/mi><mn>2<\/mn><\/msup><\/msqrt><mo>&#8804;<\/mo><mfenced open=\"|\" close=\"|\"><mi>x<\/mi><\/mfenced><mo>+<\/mo><mfenced open=\"|\" close=\"|\"><mi>y<\/mi><\/mfenced><\/math>\r\n<\/p>\r\n<p>\u00a0<\/div>\r\n<p>4. Prove that if a, b,\u00a0 c are positive integers with a<sup>2<\/sup> + b<sup>2<\/sup> = c<sup>2<\/sup> then either a or b is even.<\/p>\r\n<p>5. Prove that\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msqrt><mn>3<\/mn><\/msqrt><mo>&#8211;<\/mo><msqrt><mn>2<\/mn><\/msqrt><\/math>\r\n is irrational. (prove by contradiction)<\/p>\r\n<p>6. Prove that there does not exist a smallest positive real number.<\/p>\r\n<p>7. If <em>x , y , z<\/em> are positive real numbers. Prove that <em>x &gt; z<\/em> and<em> y<sup>2<\/sup> = xz<\/em> implies that <em>x &gt; y &gt; z.<\/em><\/p>\r\n<p>8. Prove that if \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><\/math>\r\n and\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>g<\/mi><\/math>\r\n are differentiable functions with \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><mo>=<\/mo><mi>f<\/mi><mfenced><mi>x<\/mi><\/mfenced><mi>g<\/mi><mfenced><mi>x<\/mi><\/mfenced><\/math>\r\n , then either \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>f<\/mi><mfenced><mn>0<\/mn><\/mfenced><mo>&#8800;<\/mo><mn>0<\/mn><\/math>\r\n or \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>g<\/mi><mfenced><mn>0<\/mn><\/mfenced><mo>&#8800;<\/mo><mn>0<\/mn><\/math>\r\n .<\/p>\r\n<p>9. Prove by contradiction that \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mroot><mn>2<\/mn><mn>3<\/mn><\/mroot><\/math>\r\n \u00a0is an irrational number.<\/p>\r\n<div id=\"link9-link-1730\" class=\"sh-link link9-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link9', 1730, 'Solution9', 'Hide Solution9'); return false;\" aria-expanded=\"false\"><span id=\"link9-toggle-1730\">Solution9<\/span><\/a><\/div><div id=\"link9-content-1730\" class=\"sh-content link9-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2084 size-medium\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-51-174x300.jpg\" alt=\"\" width=\"174\" height=\"300\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-51-174x300.jpg 174w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-51.jpg 427w\" sizes=\"auto, (max-width: 174px) 100vw, 174px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p>10. Prove that<br \/>\r\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0n is a prime number greater than 5 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo>&#8658;<\/mo><msup><mi>n<\/mi><mn>4<\/mn><\/msup><\/math>\r\n has final digit 1.<\/p>\r\n<div id=\"link10-link-1730\" class=\"sh-link link10-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link10', 1730, 'Solution10', 'Hide Solution10'); return false;\" aria-expanded=\"false\"><span id=\"link10-toggle-1730\">Solution10<\/span><\/a><\/div><div id=\"link10-content-1730\" class=\"sh-content link10-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2086 size-full\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-52.jpg\" alt=\"\" width=\"428\" height=\"566\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-52.jpg 428w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-52-227x300.jpg 227w\" sizes=\"auto, (max-width: 428px) 100vw, 428px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p>11. A student argues that when a rational number is multiplied by an irrational number the result will always be an irrational number.<br \/>\r\n(a) Identify the rational number for which the student\u2019s argument is not true.<\/p>\r\n<p>(b) Prove that the student is right for all rational numbers other than the one you have identified in part <strong>(a)<\/strong>.<\/p>\r\n<div id=\"link11-link-1730\" class=\"sh-link link11-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link11', 1730, 'Solution11', 'Hide Solution11'); return false;\" aria-expanded=\"false\"><span id=\"link11-toggle-1730\">Solution11<\/span><\/a><\/div><div id=\"link11-content-1730\" class=\"sh-content link11-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p>(a) 0<\/p>\r\n<p>(b)<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2099 size-full\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-53.jpg\" alt=\"\" width=\"397\" height=\"580\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-53.jpg 397w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-53-205x300.jpg 205w\" sizes=\"auto, (max-width: 397px) 100vw, 397px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p>12<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-large wp-image-2102\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-54-1024x993.jpg\" alt=\"\" width=\"580\" height=\"562\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-54-1024x993.jpg 1024w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-54-300x291.jpg 300w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-54-768x745.jpg 768w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-54.jpg 1098w\" sizes=\"auto, (max-width: 580px) 100vw, 580px\" \/><\/p>\r\n<div id=\"link12-link-1730\" class=\"sh-link link12-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link12', 1730, 'Solution12', 'Hide Solution12'); return false;\" aria-expanded=\"false\"><span id=\"link12-toggle-1730\">Solution12<\/span><\/a><\/div><div id=\"link12-content-1730\" class=\"sh-content link12-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-2103\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-1.png\" alt=\"\" width=\"470\" height=\"714\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-1.png 470w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/04\/1-1-197x300.png 197w\" sizes=\"auto, (max-width: 470px) 100vw, 470px\" \/><\/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":"1. Show that for all real numbers a and\u00a0 b \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 a+b&#8804;a+b .\u00a0 \u00a0 \u00a0 \u00a0(Triangle Inequality) 2. If\u00a0 x and\u00a0 y are not negative real numbers with\u00a0 x&#8804;y , then\u00a0 x&#8804;y . 3. For any real numbers\u00a0 [&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-1730","knowledgebase","type-knowledgebase","status-publish","hentry","knowledgebase_cat-reasoning-and-proof","no-wpautop"],"_links":{"self":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1730","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=1730"}],"version-history":[{"count":18,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1730\/revisions"}],"predecessor-version":[{"id":2104,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1730\/revisions\/2104"}],"wp:attachment":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/media?parent=1730"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}