{"id":5192,"date":"2021-03-04T12:18:35","date_gmt":"2021-03-04T06:48:35","guid":{"rendered":"https:\/\/apboardsolutions.guru\/?p=5192"},"modified":"2021-09-15T11:39:30","modified_gmt":"2021-09-15T06:09:30","slug":"ap-board-9th-class-maths-solutions-chapter-15-ex-15-4","status":"publish","type":"post","link":"https:\/\/apboardsolutions.guru\/ap-board-9th-class-maths-solutions-chapter-15-ex-15-4\/","title":{"rendered":"AP Board 9th Class Maths Solutions Chapter 15 Proofs in Mathematics Ex 15.4"},"content":{"rendered":"

AP State Syllabus\u00a0AP Board 9th Class Maths Solutions<\/a> Chapter 15 Proofs in Mathematics Ex 15.4 Textbook Questions and Answers.<\/p>\n

AP State Syllabus 9th Class Maths Solutions 15th Lesson Proofs in Mathematics Exercise 15.4<\/h2>\n

\"AP<\/p>\n

Question 1.
\nState which of the following are mathematical statements and which are not ? Give reason.
\ni) She has blue eyes.
\nSolution:
\nThis is not a mathematical statement. no mathematics is involved in it.<\/p>\n

ii) x + 7 = 18
\nSolution:
\nThis is not a statement, as its truthness cant be determined.<\/p>\n

iii) Today is not Sunday.
\nSolution:
\nThis is not a statement. This is an am biguous open sentence.<\/p>\n

iv) For each counting number x, x + 0 = x.
\nSolution:
\nThis is a mathematical statement.<\/p>\n

\"AP<\/p>\n

v) What tune is it?
\nSolution:
\nThis is not a riathematical statement.<\/p>\n

Question 2.
\nFind counter examples to disprove the following statements.
\ni) Every rectangle is a square.
\nSolution:
\n\"AP
\nA rectangle and square are equiangular i.e., all the four angles are right angles. This doesn’t mean that they have equal sides.<\/p>\n

ii) For any integers x and y,
\n\\(\\sqrt{x^{2}+y^{2}}\\) = x + y
\nSolution:
\nLet x = 3; y = 8
\n\\(\\sqrt{x^{2}+y^{2}}=\\sqrt{3^{2}+8^{2}}\\)
\n= \\(\\sqrt{9+64}=\\sqrt{73}\\)<\/p>\n

x + y = 3 + 8 = 11
\nHere, \u221a73 \u2260 11
\ni.e., \\(\\sqrt{x^{2}+y^{2}}\\) \u2260 x + y<\/p>\n

iii) If n is a whole number then 2n2<\/sup> +11 is a prime.
\nSolution:
\nIf n = 11 then 2n2<\/sup>+ 11 = 2 (11)2<\/sup> + 11
\n= 11 (2 \u00d7 11 + 1) = 11 \u00d7 (22 + 1)
\n= 11 \u00d7 23 is not a prime.<\/p>\n

\"AP<\/p>\n

iv) Two triangles are congruent if all their corresponding angles are equal.
\nSolution:
\n\"AP
\nIf the corresponding angles are equal then the triangles are only similar.<\/p>\n

v) A quadrilateral with all sides are equal is a square.
\nSolution:
\nA rhombus is not a square, but all its sides are equal.<\/p>\n

Question 3.
\nProve that the sum of two odd numbers is even.
\nSolution:<\/p>\n\n\n\n\n\n
Steps<\/td>\nReasons<\/td>\n<\/tr>\n
1) (2m + 1); (2n + 1) be the two odd numbers<\/td>\nGeneral form of an odd number.<\/td>\n<\/tr>\n
2) (2m + 1) + (2n + 1) = (2m + 2n + 2)
\n= 2 (m + n + 1) = 2K
\nHence proved.<\/td>\n		Adding the two numbers
\nGeneral form of an even number.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\"AP<\/p>\n

Question 4.
\nProve that the product of two even numbers is an even number.
\nSolution:<\/p>\n\n\n\n\n\n\n\n
Steps<\/td>\nReasons<\/td>\n<\/tr>\n
1) Let 2m and 2n be two even numbers.<\/td>\nGeneral form of an even number.<\/td>\n<\/tr>\n
2) 2m.2n = 4mn = 2(2mn) = 2K<\/td>\nTaking the product Rearranging the numbers.<\/td>\n<\/tr>\n
3) 2K where K = 2mn<\/td>\nK=2mn<\/td>\n<\/tr>\n
4) Even number
\nHence proved.<\/td>\n		General form of an even number.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Question 5.
\nProve that if x is odd, then x2<\/sup> is also odd.
\nSolution:
\nLet x be an odd number.
\nThen x = 2m + 1
\n(general form of ah odd number) Squaring on both sides,
\nx2<\/sup> = (2m + 1)2<\/sup>
\n= 4m2<\/sup> + 4m +1
\n= 2 (2m2<\/sup> + 2m) + 1
\n= 2K + 1 where K = 2m2<\/sup>+ 2
\nHence x2<\/sup> is also odd.<\/p>\n

\"AP<\/p>\n

Question 6.
\nExamine why they work ?
\nChoose a number. Double it. Add nine. Add your original number. Divide by three. Add four. Subtract your original number. Your result is seven.
\nSolution:
\nChoose a number = x say
\nDouble it = 2x
\nAdd nine = 2x + 9
\nAdd your original number
\n= 2x + 9 + x = 3x + 9
\nDivide by 3 = (3x + 9) \u00f7 3
\n= \\(\\frac{3 x}{3}+\\frac{9}{3}\\) = x + 3
\nAdd 4 \u21d2 x + 3 + 4 = x + 7
\nSubtract your original number =
\nx + 7 – x = 7
\nYour result is 7 – True.<\/p>\n

\"AP<\/p>\n

ii) Write down any three digit number (for example, 425). Make a six digit number by repeating these digits in the same order 425425. Your new number is divisible by 7, 11 and 13.
\nSolution:
\nLet a three digit number be xyz.
\nRepeat the digit = xyzxyz
\n= xyz \u00d7 (1001)
\n= xyz \u00d7 (7 \u00d7 11 \u00d7 13)
\nHence the given conjecture is true.<\/p>\n","protected":false},"excerpt":{"rendered":"

AP State Syllabus\u00a0AP Board 9th Class Maths Solutions Chapter 15 Proofs in Mathematics Ex 15.4 Textbook Questions and Answers. AP State Syllabus 9th Class Maths Solutions 15th Lesson Proofs in Mathematics Exercise 15.4 Question 1. State which of the following are mathematical statements and which are not ? Give reason. i) She has blue eyes. …<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[5],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts\/5192"}],"collection":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/comments?post=5192"}],"version-history":[{"count":0,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts\/5192\/revisions"}],"wp:attachment":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/media?parent=5192"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/categories?post=5192"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/tags?post=5192"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}