{"id":16554,"date":"2019-12-31T09:45:44","date_gmt":"2019-12-31T04:15:44","guid":{"rendered":"https:\/\/kseebsolutions.guru\/?p=16554"},"modified":"2021-07-02T15:33:25","modified_gmt":"2021-07-02T10:03:25","slug":"1st-puc-maths-question-bank-chapter-14","status":"publish","type":"post","link":"https:\/\/kseebsolutions.guru\/1st-puc-maths-question-bank-chapter-14\/","title":{"rendered":"1st PUC Maths Question Bank Chapter 14 Mathematical Reasoning"},"content":{"rendered":"<p>Students can Download Maths Chapter 14 Mathematical Reasoning Questions and Answers, Notes Pdf, <a href=\"https:\/\/kseebsolutions.guru\/1st-puc-maths-question-bank\/\">1st PUC Maths Question Bank with Answers<\/a> helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.<\/p>\n<h2>Karnataka 1st PUC Maths Question Bank Chapter 14 Mathematical Reasoning<\/h2>\n<p>Question 1.<br \/>\nWhat is the basic unit involved in mathematical reasoning?<br \/>\nAnswer:<br \/>\nThe basic unit involved in mathematical statement.<\/p>\n<p>Question 2.<br \/>\nDefine mathematical statement.<br \/>\nAnswer:<br \/>\nMathematical statement is a sentence which, in a given context is either true or false but not both.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 3.<br \/>\nDefine mathematically acceptable statements.<br \/>\nAnswer:<br \/>\nA sentence is called mathematically acceptable statement if it either true or false but not both.<\/p>\n<ul>\n<li>Note: In this chapter, statement means it is mathematically acceptable statement.<\/li>\n<li>Note: If a sentence, which is vague or ambiguous then that sentence is not a statement.<\/li>\n<li>Note: Open sentences are not statements.<\/li>\n<li>Note: A sentence which contains where, why, what, he, she, other pronouns is not a statement.<\/li>\n<\/ul>\n<p>Question 4.<br \/>\nWhich of the following sentences are statements? Give reasons for your answers.<br \/>\n(i) Two plus two equals four.<br \/>\n(ii) The sum of two equals four<br \/>\n(iii) All prime numbers are add numbers<br \/>\n(iv) The sum of x and y is greater than zero<br \/>\n(v) How beautiful?<br \/>\n(vi) Open the door.<br \/>\n(vii) Where are you going?<br \/>\n(viii) Tomorrow is Friday<br \/>\n(ix) She is mathematic graduate.<br \/>\n(x) There are 40 days in a month.<br \/>\n(xi) 8 is less than 6.<br \/>\n(xii) Every set is a finite set.<br \/>\n(xiii) The sun is a star<br \/>\n(xiv) Mathematics is fun.<br \/>\n(xv) There is no rain without clouds.<br \/>\n(xvi) There are 35 days in a month.<br \/>\n(xvii) Mathematics is difficult.<br \/>\n(xviii) The sum of 5 and 7 is greater than 10<br \/>\n(xix) The square of a number is an even number.<br \/>\n(xx) The sides of a quadrilateral have equal length.<br \/>\n(xxi) Answer this question<br \/>\n(xxii) The product of (-1) and 8 is 8<br \/>\n(xxiii) Today is windy day.<br \/>\n(xxiv) All real numbers are complex numbers<br \/>\n(xxv) The sum of all interior angles of a triangle is 180\u00b0<br \/>\nAnswer:<br \/>\n(i) It is a statement because it is always true.<br \/>\n(ii) It is a statement because it is always true.<br \/>\n(iii) It is a statement because it is false.<br \/>\n(vi) It is not a statement because it is open sentence.<br \/>\n(v) It is not a statement because it is an exclamation.<br \/>\n(vi) It is not a statement \u2235 It is an order.<br \/>\n(vii) It is not a statement \u2235 It is a question.<br \/>\n(viii) Not a statement, \u2235 It is true on Thursday but not on other days.<br \/>\n(ix) Not a statement \u2235 Sentence with variable pronoun like\u2018she\u2019.<br \/>\n(x) A statement \u2235 It is false (statement) sentence.<br \/>\n(xi) A statement \u2235 It is a false sentence.<br \/>\n(xii) Not a statement \u2235 It may be true or false<br \/>\n(xiii) A statement \u2235 It is true<br \/>\n(xiv) Not a statement \u2235 This sentence is not always true<br \/>\n(xv) A statement \u2235 This sentence is always true, as is natural phenomenon.<br \/>\n(xvi) Not a statement \u2235 It is false sentence maximum number of days in a month can never exceed.<br \/>\n(xvii) Not a statement \u2235 It may be true or false for some people mathematics can be easy and some people mathematics can be difficult.<br \/>\n(xviii) A statement\u2235 It is true sentence. Since 5 + 7 = 12 &gt;10<br \/>\n(xix) Not a statement, \u2235 It is sometimes true and sometimes false. Since (2)<sup>2<\/sup> = 4, even and (3)<sup>2<\/sup> = 9, odd.<br \/>\n(xx) Not a statement \u2235 It may be true or false.<br \/>\nSince square has equal length sides, rectangle has unequal length sides.<br \/>\n(xxi) Not a statement, \u2235 It is an order.<br \/>\n(xxii) A statement \u2235 It is false sentence.<br \/>\n(xxiii) Not a statement \u2235 which day is not mentioned<br \/>\n(xxiv) A statement \u2235 It is true sentence.<br \/>\n(xxv) A statement \u2235 It is always true.<\/p>\n<p>Question 5.<br \/>\nDefine negation of a statement.<br \/>\nAnswer:<br \/>\nIf p is a statement, then the negation of p is also a statement and is denoted ~p and read as \u2018not p&#8217;.<br \/>\nNote: While forming the negation of a statement, phrases like \u2018It is not the case\u2019 or \u2018It is false that\u2019 are also used.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 6.<br \/>\nDefine compound statement.<br \/>\nAnswer:<br \/>\nCompound statement is a statement which is made up of two or more statements using some connecting words like \u2018and\u2019 \u2018or\u2019, \u2018if, then\u2019, \u2018if and only if etc.<br \/>\nNote: Connecting words like \u2018And\u2019 \u2018Or\u2019, \u2018If, then\u2019, \u2018If and only if are called connectives and each statement is called a component statement.<\/p>\n<p>Question 7.<br \/>\nWrite the negation of the following statements:<br \/>\n(i) New Delhi is a city<br \/>\n(ii) Everyone in Germany speaks German<br \/>\n(iii) \\(\\sqrt{7}\\) is rational<br \/>\n(iv) Chennai is the capital of Tamil Nadu<br \/>\n(v) \\(\\sqrt{2}\\) is not a complex number<br \/>\n(vi) All triangles are not equilateral triangle<br \/>\n(vii) The number 2 is greater than 7.<br \/>\n(viii) Every natural number is an integer.<br \/>\nAnswer:<br \/>\n(i) New Delhi is not a city.<br \/>\n(ii) It is false that everyone in Germany speaks German.<br \/>\n(iii) \\(\\sqrt{7}\\) is not rational<br \/>\n(iv) Chennai is not the capital of Tamil Nadu<br \/>\n(v) \\(\\sqrt{2}\\) is a complex number<br \/>\n(vi) The number 2 is not greater than 7<br \/>\n(vii) Every natural number is not an integer.<\/p>\n<p>Question 8.<br \/>\nAre the following pairs of statements negations of each other,<br \/>\n(i) The number x is not a rational number The number x is not an irrational number,<br \/>\n(ii) The number x is a rational number. The number x is an irrational number<br \/>\nAnswer:<br \/>\n(i) Let p : the number x is not a rational number<br \/>\nq : the number x is not an irrational number<br \/>\nNow ~ p : The number x is a rational number<br \/>\nq : The number x is a rational number (\u2235 when a number is not irrational, it is a rational)<br \/>\n~p = q and ~q = p<br \/>\n\u21d2 given pairs of statements are negations of each other.<\/p>\n<p>(ii) Let p : the number x is a rational number<br \/>\nq : the number x is an irrational number. Now, ~p: the number x is an irrational number (\u2235When a number is not rational, it is irrational)<br \/>\n\u2234 ~p = q and ~q=p<br \/>\n\u21d2The pairs are negations of each other.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 9.<br \/>\nFind the component statements of the following compound statements and check whether they are true or false.<br \/>\n(i) Numbers 3 is prime or it is odd<br \/>\n(ii) All integers are positive or negative<br \/>\n(iii) 100 is divisible by 3,11 and 5<br \/>\n(iv) A square is a quadrilateral and its four sides equal<br \/>\n(v) All prime numbers are either even or odd<br \/>\n(vi) Chandigarh is the capital of Haryana and P.<br \/>\n(vii) 24 is a multiple of 2, 4 and 8<br \/>\nAnswer:<br \/>\n(i) The component statements are<br \/>\np : number 3 is prime<br \/>\nq : number 3 is odd<br \/>\nBoth p and q are true<\/p>\n<p>(ii) The component statements are<br \/>\np : all integers are positive<br \/>\nq : all integers are negative<br \/>\nBoth p and q are false<\/p>\n<p>(iii) The component statements are<br \/>\np : 100 is divisible by 3 (false)<br \/>\nq : 100 is divisible by 11 (false)<br \/>\nr : 100 is divisible by 5 (true)<\/p>\n<p>(iv) The component statements are<br \/>\np : a square is a quadrilateral<br \/>\nq : a square has all its sides equal<br \/>\nBoth p and q are true<\/p>\n<p>(v) The component statements are<br \/>\np : all prime numbers are even<br \/>\nq : all prime numbers are odd<br \/>\nBoth p and q are false<\/p>\n<p>(vi) The component statements are<br \/>\np : Chandigarh is the capital of Haryana (True)<br \/>\nq : Chandigarh is the capital of U.P. (False)<\/p>\n<p>(vii) The component statements are<br \/>\np : 24 is a multiple of 2 (True)<br \/>\nq : 24 is a multiple of 4 (True)<br \/>\nr : 24 is a multiple of 8 (True)<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 10.<br \/>\nWrite the rules for<br \/>\n(i) \u2018And\u2019 (conjunction) (\u2227)<br \/>\n(ii) \u2018Or\u2019 (disjunction) (\u2228)<br \/>\n(iii) \u2018If-Then\u2019 (implication) (\u21d2)<br \/>\n(iv) \u2018If and only if (double implication) (\u21d4)<br \/>\nAnswer:<br \/>\nLet p and q be the statements of the compound statement. Then<\/p>\n<p>(i) The compound statement p and q is the conjunction of p and q and is denoted by<br \/>\np \u2227 q . (read: p and q)<br \/>\nRule for compound statement with \u2018And\u2019 (conjunction)<br \/>\nThis can be shown by the truth table<\/p>\n<p>\u2234 Truth table for p \u2227 q as follows<\/p>\n<table style=\"height: 122px;\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\" width=\"31\"><strong>p<\/strong><\/td>\n<td style=\"text-align: center;\" width=\"31\"><strong>q<\/strong><\/td>\n<td style=\"text-align: center;\" width=\"53\"><strong>p \u2227 q<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"31\">T<\/td>\n<td style=\"text-align: center;\" width=\"31\">T<\/td>\n<td style=\"text-align: center;\" width=\"53\">T<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"31\">T<\/td>\n<td style=\"text-align: center;\" width=\"31\">F<\/td>\n<td style=\"text-align: center;\" width=\"53\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"31\">F<\/td>\n<td style=\"text-align: center;\" width=\"31\">T<\/td>\n<td style=\"text-align: center;\" width=\"53\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"31\">F<\/td>\n<td style=\"text-align: center;\" width=\"31\">F<\/td>\n<td style=\"text-align: center;\" width=\"53\">F<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>(ii) The compound statement p or q is the disjunction of p and q, and is denoted by p v q (read: p \u2228 q)<br \/>\nRule: The disjunction p \u2228 q is false if both p and q are false, otherwise it is true.<\/p>\n<p>This can be shown by truth table<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\" width=\"31\"><strong>P<\/strong><\/td>\n<td style=\"text-align: center;\" width=\"31\"><strong>q<\/strong><\/td>\n<td style=\"text-align: center;\" width=\"53\"><strong>p \u2227 q<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"31\">T<\/td>\n<td style=\"text-align: center;\" width=\"31\">T<\/td>\n<td style=\"text-align: center;\" width=\"53\">T<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"31\">T<\/td>\n<td style=\"text-align: center;\" width=\"31\">F<\/td>\n<td style=\"text-align: center;\" width=\"53\">T<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"31\">F<\/td>\n<td style=\"text-align: center;\" width=\"31\">T<\/td>\n<td style=\"text-align: center;\" width=\"53\">T<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"31\">F<\/td>\n<td style=\"text-align: center;\" width=\"31\">F<\/td>\n<td style=\"text-align: center;\" width=\"53\">F<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>(iii) The compound statement \u2018if p then q\u2019 is implication of p and It is denoted by p \u2192 q or p \u21d2 q. (read : p implication q)<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Rule:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\" width=\"31\"><strong>P<\/strong><\/td>\n<td style=\"text-align: center;\" width=\"30\"><strong>q<\/strong><\/td>\n<td style=\"text-align: center;\" width=\"53\"><strong>p \u2192 q<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"31\">T<\/td>\n<td style=\"text-align: center;\" width=\"30\">T<\/td>\n<td style=\"text-align: center;\" width=\"53\">T<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"31\">T<\/td>\n<td style=\"text-align: center;\" width=\"30\">F<\/td>\n<td style=\"text-align: center;\" width=\"53\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"31\">F<\/td>\n<td style=\"text-align: center;\" width=\"30\">T<\/td>\n<td style=\"text-align: center;\" width=\"53\">T<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"31\">F<\/td>\n<td style=\"text-align: center;\" width=\"30\">F<\/td>\n<td style=\"text-align: center;\" width=\"53\">T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Note:<br \/>\nIf &#8216;p&#8217; and then &#8216;q&#8217; is small following:<\/p>\n<ul>\n<li>p \u21d2 q (i.e., p implies q)<\/li>\n<li>p is sufficient condition for q<\/li>\n<li>p only if q<\/li>\n<li>q is necessary condition for p<\/li>\n<li>~q implies ~p (i.e., ~q \u21d2 ~p)<\/li>\n<li>The compound statement &#8216;p if and only if q&#8217; is double implication of p and It is denoted by p \u21d4 q (read: p double implication q)<\/li>\n<\/ul>\n<p>Rule:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\" width=\"31\"><strong>p<\/strong><\/td>\n<td style=\"text-align: center;\" width=\"31\"><strong>q<\/strong><\/td>\n<td style=\"text-align: center;\" width=\"61\"><strong>P \u21d4 q<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"31\">T<\/td>\n<td style=\"text-align: center;\" width=\"31\">T<\/td>\n<td style=\"text-align: center;\" width=\"61\">T<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"31\">T<\/td>\n<td style=\"text-align: center;\" width=\"31\">F<\/td>\n<td style=\"text-align: center;\" width=\"61\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"31\">F<\/td>\n<td style=\"text-align: center;\" width=\"31\">T<\/td>\n<td style=\"text-align: center;\" width=\"61\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"31\">F<\/td>\n<td style=\"text-align: center;\" width=\"31\">F<\/td>\n<td style=\"text-align: center;\" width=\"61\">T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Note: \u2018p if and only if q\u2019 is same as the following.<\/p>\n<ul>\n<li>p \u21d4 q<\/li>\n<li>p if and only if q<\/li>\n<li>q if and only if p<\/li>\n<li>p is necessary and sufficient condition for q and vice-versa<\/li>\n<\/ul>\n<p>Question 11.<br \/>\nDefine quantifiers (quantifiers)<br \/>\nAnswer:<br \/>\n\u2018Quantifiers\u2019 are the phrases like \u2018there exists\u2019 and \u2018for all\u2019 (or for all).<\/p>\n<p>Question 12.<br \/>\nDefine converse and contrapositive.<br \/>\nAnswer:<br \/>\nConverse and contra positive are certain other statements which can be formed from a given statement with \u2018if-then\u2019.<\/p>\n<ul>\n<li>Converse: The converse of the statement if p, then q is defined as \u2018if q then p\u2019.<br \/>\ni.e., the converse of a implication p \u21d2 q is q \u21d2 q<\/li>\n<li>Contrapositive: The contrapositive of the statement if p, then q is defined as \u2018if ~q, then ~p\u2019 i.e., the contrapositive of a implication p \u21d2 q is ~q \u21d2 ~p<\/li>\n<\/ul>\n<p>Note:<br \/>\nInverse of a implication p \u21d2 q is ~p \u21d2 ~q<\/p>\n<p>Question 13.<br \/>\nFor each of the following compounds statements first identify the connecting words and then break it into component statements.<br \/>\n(i) All rational numbers are real and all real numbers are not complex.<br \/>\n(ii) Square of an integer is positive or negative<br \/>\n(iii) The sand heats up quickly in the sun and doesnot cool down fast at night.<br \/>\n(iv) x = 2 and x = 3 are the roots of the equation 3x<sup>2<\/sup> &#8211; x -10 = 0<br \/>\nAnswer:<br \/>\n(i) \u2018and\u2019 is the connecting word.<br \/>\n\u2234 component statements are<br \/>\np : all rational numbers are real<br \/>\nq : all real numbers are not complex.<\/p>\n<p>(ii) Connecting word is \u2018or\u2019 component statements are<br \/>\np : square of an integer is positive<br \/>\nq : square of an integer is negative<\/p>\n<p>(iii) Connecting word is \u2018and\u2019 component statements are<br \/>\np : sand heats up quickly in the sun<br \/>\nq : sand does not cool down fast at night<\/p>\n<p>(iv) Connecting word is \u2018and\u2019Component statements are<br \/>\np : x &#8211; 2 is a root of 3x<sup>2<\/sup> &#8211; x &#8211; 10 = 0<br \/>\nq : x = 3 is a root of 3x<sup>2<\/sup> &#8211; x &#8211; 10 = 0<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 14.<br \/>\nIdentify the quantifier in the following statements and write the negation of the statement.<br \/>\n(i) There exists a number which is equal to its square<br \/>\n(ii) For every real number x, x is less than x + 1<br \/>\n(iii) There exists a capital for every state in India<br \/>\nAnswer:<br \/>\n(i) Quantifier is\u2018there exists\u2019<br \/>\nThe negation is,<br \/>\nThere doesn\u2019t exist a number which is equal to its square.<\/p>\n<p>(ii) Quantifier is \u2018For every\u2019<br \/>\nThe negation is, there exist a real number x such that x is not less than x + 1<\/p>\n<p>(iii) Quantifier is \u2018there exists\u2019<br \/>\nThe negation state in India which doesn\u2019t have a capital<\/p>\n<p>Question 15.<br \/>\nCheck whether the following pair of statements are negation of each other. Give reasons for your answer.<br \/>\n(i) x + y = y + x is true for every real numbers x and y<br \/>\n(ii) There exists real numbers x and y for which x + y = y + x<br \/>\nAnswer:<br \/>\nNo<br \/>\nNegation of (i) is \u2018there exists real number x and y for which x + y &amp; y + x\u2019, which is not same as (ii).<\/p>\n<p>Question 16.<br \/>\nState whether \u2018or\u2019 used in the following statements is exclusive or inclusive. Give reason for your answer.<br \/>\n(i) Sun rises or moon sets<br \/>\n(ii) To apply for a driving license, you should have a ration card or a pass port<br \/>\n(iii) All integers are positive or negative<br \/>\nAnswer:<br \/>\n(i) Here \u2018or\u2019 is exclusive<br \/>\n\u2235 \u2018sun rises\u2019 and \u2018moon sets\u2019 cannot be true simultaneously<\/p>\n<p>(ii) Here \u2018or\u2019 is inclusive<br \/>\n\u2235 For license, one can have both ration card and a passport.<\/p>\n<p>(iii) Here \u2018or\u2019 is exclusive<br \/>\n\u2235 Integer cannot be both positive as well as negative.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 17.<br \/>\nRewrite the following statements with \u2018if-then\u2019 in five different ways conveying the same meaning.<br \/>\n(i) \u2018If a natural number is odd, then its square is also odd\u2019<br \/>\n(ii) If a number is a multiple of 9, then it is a multiple of 3<br \/>\nAnswer:<br \/>\n(i) Given: \u2018If a natural number is odd, then its square is also odd.<br \/>\nLet p : a natural number is odd<br \/>\nq : a natural number square is also odd. Then, if p then q is same as<\/p>\n<ul>\n<li>p \u21d2q (p implies q)<br \/>\ni.e., A natural number is odd implies that its square is also odd.<\/li>\n<li>p is sufficient condition for q e., for the square of a natural number to be odd it is sufficient that the number itself is odd.<\/li>\n<li>p only if q e., A natural number is odd only if its square is odd.<\/li>\n<li>q is a necessary condition for p e., for a natural number to be odd it is necessary that its square must be odd.<\/li>\n<li>~q implies ~p<\/li>\n<li>If the square of a natural number is not odd, then the number itself is also not odd.<\/li>\n<\/ul>\n<p>(ii) Try yourself<\/p>\n<p>Question 18.<br \/>\nWrite the contrapositive and converse of the following statements.<br \/>\n(i) If a number is divisible by 9, then it is divisible by 3<br \/>\n(ii) If you are born in India, then you are a citizen of India.<br \/>\n(iii) If a triangle is equilateral, then it is isosceles<br \/>\n(iv) If a number n is even, then n<sup>2<\/sup> is even<br \/>\n(v) If you do all the exercises in the book you get an A grade in the class.<br \/>\n(vi) If x is a prime number, then x is odd<br \/>\n(vii) If the two lines are parallel, then they donot intersect in the same plane<br \/>\n(viii) Something is cold implies that it has low temperature<br \/>\n(ix) x is even number implies that x is divisible by 4<br \/>\nAnswer:<br \/>\n(i) Let p : A number is divisible by 9<br \/>\nq : A number is divisible by 3<br \/>\nIts converse is q \u21d2 p and its contrapositive is ~q \u21d2~P<br \/>\n\u2234 contrapositive: \u2018If a number is not divisible by 3, then it is not divisible by 9\u2019.<br \/>\nConverse: \u2018If a number is divisible by 3 then it is divisible by 9.<\/p>\n<p>(ii) Contrapositive: \u2018If you are not a citizen of India then you were not born in India\u2019. Converse: \u2018If you are a citizen of India then you are born in India\u2019.<\/p>\n<p>(iii) Contrapositive: \u2018If a triangle is not isosceles, then it is not equilateral\u2019.<br \/>\nConverse: \u2018If a triangle is isosceles, then it is equilateral\u2019.<\/p>\n<p>(iv) Try<\/p>\n<p>(v) Try<\/p>\n<p>(vi) Contrapositive: \u2018If x is not odd, then x is not a prime number\u2019.<br \/>\nConverse: If x is odd then x is a prime number<\/p>\n<p>(vii) Try<\/p>\n<p>(viii) and (ix) yourself<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/kseebsolutions.guru\/wp-content\/uploads\/2019\/11\/KSEEB-Solutions-300x28.png\" alt=\"KSEEB Solutions\" width=\"172\" height=\"16\" \/><\/p>\n<p>Question 19.<br \/>\nWrite each of the following statements in the form \u2018if-then\u2019<br \/>\n(i) You get a job implies that your credentials are good.<br \/>\n(ii) The Banana trees will bloom if it stays warm for a month<br \/>\n(iii) A quadrilateral is a parallelogram if its diagonals bisect each other<br \/>\n(iv) To get an A<sup>+<\/sup> in the class, it is necessary that you do all the exercise of the book<br \/>\nAnswer:<br \/>\n(i) \u2018If you get a job, then your credentials are good\u2019.<br \/>\n(ii) \u2018If the banana tree stays warm for a month, then it will bloom\u2019<br \/>\n(iii) \u2018If the diagonals of a quadrilateral bisect each other, then it is a parallelogram\u2019.<br \/>\n(iv) \u2018If you get an A<sup>+<\/sup> in the class then you have done all exercises of the book\u2019.<\/p>\n<p>Question 20.<br \/>\nVerify by the method of contradiction that \\(\\sqrt{7}\\) is irrational<br \/>\nAnswer :<br \/>\nLet\u00a0 p : \\(\\sqrt{7}\\) is irrational<br \/>\nLet us assume p is not true i.e.., \\(\\sqrt{7}\\) is rational .<br \/>\n\u21d2 \\(\\sqrt{7}=\\frac{a}{b}\\),where a and b are integers having no common<br \/>\nfactor.<br \/>\n\u21d2 \\(7=\\frac{a^{2}}{b^{2}}\\)<br \/>\n\u21d2 a<sup>2<\/sup> =7b<sup>2<br \/>\n<\/sup>\u21d2 7 divides a<sup>2<br \/>\n<\/sup>\u21d2 7 divides a<br \/>\n\u21d2 a = 7c, for some integer c.<br \/>\n\u21d2 a<sup>2<\/sup> = 49c<sup>2<br \/>\n<\/sup>\u21d2 7b<sup>2<\/sup> = 49c<sup>2<br \/>\n<\/sup>\u21d2 b<sup>2<\/sup> = 7c<sup>2<br \/>\n<\/sup>\u21d2 7 divides b<sup>2<\/sup><br \/>\n\u21d2 7 divides b<br \/>\nThus, 7 is common factor of both a and b. This contradicts that a and b have no common factor. So, our assumption is wrong. Hence, \\(\\sqrt{7}\\) is irrational is true.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Students can Download Maths Chapter 14 Mathematical Reasoning Questions and Answers, Notes Pdf, 1st PUC Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations. Karnataka 1st PUC Maths Question Bank Chapter 14 Mathematical Reasoning Question 1. What is the basic unit involved &hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[81],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/16554"}],"collection":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/comments?post=16554"}],"version-history":[{"count":0,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/16554\/revisions"}],"wp:attachment":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/media?parent=16554"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/categories?post=16554"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/tags?post=16554"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}