A book for Std XII, MHT-CET, ISEET and other Competitive Entrance Exams Written according to the New Text book (2012-2013) published by the Maharashtra State Board of Secondary and Higher Secondary Education, Pune Std XII Sci Triumph Maths Mr Vinodkumar J Pandey Mrs Shama Mittal B.Sc (Mathematics) G N Khalsa College, Mumbai M.Sc., (Mathematics), B.Ed Punjabi University (Patiala) Salient Features: Exhaustive coverage of MCQs subtopic wise Each chapter contains three sections Section contains easy level questions Section contains competitive level questions Section contains questions from various competitive exams Important formulae Hints provided wherever relevant Useful for MHT-CET and ISEET preparation Target PUBLICATIONS PVT LTD Mumbai, Maharashtra Tel: 022 – 6551 6551 Website : www.targetpublications.in www.targetpublications.org email : mail@targetpublications.in Std XII Triumph Maths © Target Publications Pvt Ltd First Edition : October 2012 Price : ` 330/- Printed at: Vijaya Enterprises Sion, Mumbai Published by Target PUBLICATIONS PVT LTD Shiv Mandir Sabhagriha, Mhatre Nagar, Near LIC Colony, Mithagar Road, Mulund (E), Mumbai - 400 081 Off.Tel: 022 – 6551 6551 email: mail@targetpublications.in PREFACE With the change in educational curriculum it’s now time for a change in Competitive Examinations NEET and ISEET are all poised to take over the decade old MHT-CET The change is obvious not merely in the names but also at the competitive levels The state level entrance examination is ushered aside and the battleground is ready for a National level platform However, keeping up with the tradition, Target Publications is ready for this challenge To be at pace with the changing scenario and equip students for a fierce competition, Target Publications has launched the Triumph series Triumph Maths is entirely based on Std XII (Science) curriculum of the Maharashtra Board This book will not only assist students with MCQs of Std XII but will also help them prepare for MHT-CET / NEET and ISEET and various other competitive examinations The content of this book has evolved from the State Board prescribed Text Book and we’ve made every effort to include most precise and updated information in it Multiple Choice Questions form the crux of this book We have framed them on every sub topic included in the curriculum Each chapter is divided into three sections: Section consists of basic MCQs based on subtopics of Text Book Section consists of MCQs of competitive level Section consists of MCQs compiled from various competitive examinations To end on a candid note, we make a humble request for students: Preserve this book as a Holy Grail This book would prove as an absolute weapon in your arsenal for your combat against Medical and Engineering entrance examinations Best of luck to all the aspirants! Yours faithfully Publisher Contents Topic Name Sr No Page No Mathematical Logic Matrices 16 Trigonometric Functions 40 Pair of Straight Lines 85 Circle 120 Conics 146 Vectors 174 Three Dimensional Geometry 198 Line 217 10 Plane 239 11 Linear Programming 269 12 Continuity 303 13 Differentiation 330 14 Applications of Derivatives 376 15 Integration 434 16 Definite Integral 494 17 Applications of Definite Integral 531 18 Differential Equations 554 19 Bivariate Frequency Distribution 589 20 Probability Distribution 604 21 Binomial Distribution 619 Std XII: Triumph Maths TARGET Publications MATHEMATICAL LOGIC 01 Logical Connectives: Connective And (Conjuction) Or (Disjunction) If … then (Conditional) (Implication) If and only if (Biconditional) (iff) (Double implication) Not (Negation) Symbol ∧ ∨ → or ⇒ Example p and q : p ∧ q p or q : p ∨ q If p, then q: p → q ↔ or ⇔ p iff q : p ↔ q ∼ p:∼p The truth table of above logical connectives are as given below: p T T F F q T F T F p ∨ q T T T F p ∧ q T F F F p → q T F T T p ↔ q T F F T p T F ~p F T Types of Statements: i If a statement is always true, then the statement is called “tautology.” ii If a statement is always false, then the statement is called “contradiction.” iii If a statement is neither tautology nor a contradiction, then it is called “contingency.” Converse, Contrapositive, Inverse of a Statement: If p → q is a hypothesis, then i Converse: q → p ii Contrapositive: ~q → ~ p iii Inverse: ~p → ~q Consider the truth table for each of the above: p T T F F q T F T F ~p F F T T ~q F T F T p→q T F T T q→p T T F T ~q→~p T F T T ~p→~q T T F T From the above truth table, hypothesis and its contrapositive are logical equivalent Also, the converse and its inverse are equivalent Principles of Duality: Two compound statements are said to be dual of each other, if one can be obtained from other by replacing “∧” by “∨” and vice versa The connectives “∧” and “∨” are duals of each other Negation of a Statement: i ~ (p ∨ q) ≡ ~ p ∧ ~ q ii ~ (p ∧ q) ≡ ~ p ∨ ~ q iii ~ (p → q) ≡ p ∧ ~ q iv ~ (p ↔ q) ≡ (p ∧ ~ q) ∨ (q ∧ ~ p) Mathematical Logic Std XII: Triumph Maths TARGET Publications Application of Logic to Switching Circuits: i ∴ ii ∴ iii AND : [∧] Let p : S1 switch is ON q : S2 switch is ON then for the lamp L to be ‘ON’ both S1 and S2 must be put ON Which logically indicates truth table of AND the adjacent circuit resembles p ∧ q OR : [∨] Let p : S1 switch is ON q : S2 switch is ON for lamp L to be put ON either of S1 or S2 must be put ON even both can be put ON Which resembles truth table of OR the adjacent circuit resembles p ∨ q S2 S1 L S1 S2 L If two or more switch open or close simultaneously then the switches are denoted by the same letter If p : switch S is closed ~ p : switch S is open If S1 and S2 are two switches such that if S1 is open; S2 is closed and vice versa then S1 ≡ ~ S2 or S2 ≡ ~ S1 Shortcuts p∨q=q∨p p∧q=q∧p Commutative property (p ∨ q) ∨ r = p ∨ (q ∨ r) (p ∧ q) ∧ r = p ∧(q ∧ r) Associative property p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r) p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r) Distributive property ~ (p ∨ q) = ~ p ∧ ~ q ~ (p ∧ q) ≡ ~ p ∨ ~ q Demorgan’s law p→q≡~p∨q p ↔ q ≡ (p → q) ∧ (q → p) ≡ (~ p ∨ q) ∧ (~ q ∨ p) Equivalent statements p ∨ (p ∧ q) = p p ∧ (p ∨ q) = p Absorption laws If T denotes the tautology and F denotes the contradiction, then for any statement ‘p’: i⋅ p ∨ T = T; p ∨ F = p Identity laws ii p ∧ T = p; p ∧ F = F i ii iii iv v p∨~p=T p∧~p=F ∼(∼p) = p ∼T=F ∼F=T p∨p=p p∧p=p Complement laws Idempotent laws Mathematical Logic Std XII: Triumph Maths TARGET Publications SECTION - 1.1 Which of the following is a statement? (A) Open the door (B) Do your homework (C) Switch on the fan (D) Two plus two is four Which of the following is an open statement? (A) x + = 11 (B) Good morning to all (C) What is your problem? (D) Listen to me, Rahul! Which of the following is not a proposition in logic p = There are clouds in the sky and q = it is not raining The symbolic form is (A) p → q (B) p → ~q (C) p ∧ ~q (D) ~p ∧ q 11 Write in verbal form: p: he is fat, w: he is hard working, then (~p) ∨ (~w) is (A) If he is fat or he is hard working (B) He is not fat and he is not hard working (C) He is not fat or he is not hard working (D) He is fat or hard working 12 If p: Rohit is tall, q: Rohit is handsome, then the statement ‘Rohit is tall or he is short and handsome’ can be written symbolically as (A) p ∨ (~p ∧ q) (B) p ∧ (~p ∨ q) (C) p ∨ (p ∧ ~q) (D) ~p ∧ (~p ∧ ~q) 13 p: Sunday is a holiday, q: Ram does not study on holiday The symbolic form of the statement ‘Sunday is a holiday and Ram studies on holiday’ is (A) p ∧ ~q (B) p ∧ q (C) ~p ∧ ~q (D) p ∨ ~q 14 The converse of the statement ‘If I work hard then I get the grade’ is (A) If I get the grade then I work hard (B) If I don’t work hard then I don’t get the grade (C) If I don’t get the grade then I don’t work hard (D) If I work hard then I don’t get the grade 15 The converse of ‘If x is zero then we cannot divide by x’ is (A) If we cannot divide by x then x is zero (B) If we divide by x then x is non-zero (C) If x is non-zero then we can divide by x (D) If we cannot divide by x then x is non-zero Which of the following is a statement in logic? (A) What a wonderful day! (B) Shut up! (C) What are you doing? (D) Bombay is the capital of India 10 Statement, Logical Connectives, Compound Statements and Truth Table If p: Sita gets promotion, q: Sita is transferred to Pune The verbal form of ~p ↔ q is written as (A) Sita gets promotion and Sita gets transferred to Pune (B) Sita does not get promotion then Sita will be transferred to Pune (C) Sita gets promotion if Sita is transferred to Pune (D) Sita does not get promotion if and only if Sita is transferred to Pune (A) (B) (C) (D) is a prime is an irrational number Mathematics is interesting is an even integer Which of the following is a statement in Logic? (A) Go away (B) How beautiful! (C) x > (D) = Using quantifiers ∀, ∃, convert the following open statement into true statement ‘x + = 8, x ∈ N’ (A) ∀ x ∈ N, x + = (B) For every x ∈ N, x + > (C) ∃ x ∈ N, such that x + = (D) For every x ∈ N, x + < 8 ~(p ∨ q) is (A) ~p ∨ q (C) ~p ∨ ~q (B) (D) p ∨ ~q ~p ∧ ~q If p: The sun has set, q: The moon has risen, then symbolically the statement ‘The sun has not set or the moon has not risen’ is written as (A) p ∧ ~q (B) ~q ∨ p (C) ~p ∧ q (D) ~p ∨ ~q Mathematical Logic Std XII: Triumph Maths TARGET Publications 16 Write verbally ~p ∨ q where p: She is beautiful; q: She is clever (A) She is beautiful but not clever (B) She is not beautiful or she is clever (C) She is not beautiful or she is not clever (D) She is beautiful and clever 17 If p: Ram is lazy, q: Ram fails in the examination, then the verbal form of ~p ∨ ~q is (A) Ram is not lazy and he fails in the examination (B) Ram is not lazy or he does not fail in the examination (C) Ram is lazy or he does not fail in the examination (D) Ram is not lazy and he does not fail in the examination 18 Let p: Mathematics is interesting, q: Mathematics is difficult, then the symbol p → q means (A) Mathematics is interesting implies that Mathematics is difficult (B) Mathematics is interesting is implied by Mathematics is difficult (C) Mathematics is interesting and Mathematics is difficult (D) Mathematics is interesting or Mathematics is difficult When two statements are connected by the connective ‘if’ then the compound statement is called (A) conjunctive statement (B) disjunctive statement (C) biconditional statement (D) conditional statement 25 For the statements ‘p’ and ‘q’ ‘p → q’ is read as if p then q Here, the statement ‘q’ is called (A) antecedent (B) consequent (C) logical connective (D) prime component 26 The contrapositive of the statement: “If a child concentrates then he learns” is (A) If a child does not concentrate he can not learn (B) If a child does not learn then he does not concentrate (C) If a child practises then he learns (D) If a child concentrates, he can’t forget 27 A compound statement p or q is false only when (A) p is false (B) q is false (C) both p and q are false (D) depends on p and q 28 A compound statement p and q is true only when (A) p is true (B) q is true (C) both p and q are true (D) none of p and q is true 29 A compound statement p → q is false only when (A) p is true and q is false (B) p is false but q is true (C) atleast one of p or q is false (D) both p and q are false 30 The statement, ‘if it is raining then I will go to college’ is equivalent to (A) If it is not raining then I will not go to college (B) If I not go to college, then it is not raining (C) If I go to college then it is raining (D) Going to college depends on my mood The inverse of logical statement p → q is (A) ~p → ~q (B) p ↔ q (C) q → p (D) q ↔ p 19 24 20 Which of the following is logically equivalent to ~(p ∧ q) (A) p ∧ q (B) ~p ∨ ~q (C) ~(p ∨ q) (D) ~p ∧ ~q 21 ~(p → q) is equivalent to (A) p ∧ ∼q (B) (C) p ∨ ~q (D) ~p ∨ q ~p ∧ ~q Contrapositive of p → q is (A) q → p (B) (C) ~q → ~p (D) ~q → p q → ~p 22 23 When two statements are connected by logical connective ‘and’, then the compound statement is called (A) conjunctive statement (B) disjunctive statement (C) negation statement (D) conditional statement Mathematical Logic Std XII: Triumph Maths TARGET Publications 31 32 33 The converse of the statement “If Sun is not shining, then sky is filled with clouds” is (A) If sky is filled with clouds, then the Sun is not shining (B) If Sun is shining, then sky is filled with clouds (C) If sky is clear, then Sun is shining (D) If Sun is not shining, then sky is not filled with clouds Which of the following is the converse of the statement ‘If Billu secures good marks, then he will get a bicycle’? (A) If Billu will not get bicycle, then he will secure good marks (B) If Billu will get a bicycle, then he will secure good marks (C) If Billu will get a bicycle, then he will not secure good marks (D) If Billu will not get a bicycle, then he will not secure good marks The contrapositive of the statement ‘If Chandigarh is capital of Punjab, then Chandigarh is in India’, is (A) If Chandigarh is not in India, then Chandigarh is not a capital of Punjab (B) If Chandigarh is in India, then Chandigarh is capital of Punjab (C) If Chandigarh is not capital of Punjab, then Chandigarh is not capital of India (D) If Chandigarh is capital of Punjab, then Chandigarh is not in India 34 The connective in the statement “Earth revolves round the Sun and Moon is a satellite of earth”, is (A) or (B) Earth (C) Sun (D) and The converse of the statement “If x > y, then x + a > y + a”, is (A) If x < y, then x + a < y + a (B) If x + a > y + a, then x > y (C) If x < y, then x + a > y + a (D) If x > y, then x + a < y + a Every conditional statement is equivalent to (A) its contrapositive (B) its inverse (C) its converse (D) only itself 39 If p : Pappu passes the exam, q : Papa will give him a bicycle Then the statement ‘Pappu passing the exam, implies that his papa will give him a bicycle’ can be symbolically written as (A) p → q (B) p ↔ q (C) p ∧ q (D) p ∨ q 40 The symbolic form of the statement ‘Since it is raining the atmosphere is very cold’ is (A) p → q (B) p ↔ q (C) p ∧ q (D) p ∨ q 41 Assuming the first part of each statement as p, second as q and the third as r, the statement ‘Candidates are present, and voters are ready to vote but no ballot papers’ in symbolic form is (A) (p ∨ q) ∧ ∼r (B) (p ∧ ~q) ∧ r (C) (~p ∧ q) ∧ ∼r (D) (p ∧ q) ∧ ∼r 42 Assuming the first part of each statement as p, second as q and the third as r, the statement ‘A monotonic increasing sequence which is bounded above is convergent’ in symbolic form is (A) (p ∧ q) → r (B) (p ∨ q) → r (C) (p ∧ q) ↔ r (D) (p ∨ q) ↔ r 43 Assuming the first part of each statement as p, second as q and the third as r, the statement ‘If A, B, C are three distinct points, then either they are collinear or they form a triangle’ in symbolic form is (A) p ↔ (q ∨ r) (B) (p ∧ q) → r (C) p → (q ∨ r) (D) p → (q ∧ r) 44 If d: Drunk, a: accident, translate the statement ‘If the Driver is not drunk, then he cannot meet with an accident’ into symbols (A) ∼a → ∼d (B) ∼d → ∼a (C) ~d ∧ a (D) a ∧ ~d 1.2 Statement Pattern and Logical Equivalence: Tautology, Contradiction, Contingency 45 Statement ~p ↔ ~q ≡ p ↔ q is (A) a tautology (B) a contradiction (C) contingency (D) proposition 46 Given that p is ‘false’ and q is ‘true’ then the statement which is ‘false’ is (A) ~p → ~q (B) p → (q ∧ p) (C) p → ~q (D) q → ~p The connective in the statement “2 + > or + < 9” is (A) and (B) or (C) > (D) < 35 38 36 37 The statement “If x2 is not even then x is not even”, is the converse of the statement (A) If x2 is odd, then x is even (B) If x is not even, then x2 is not even (C) If x is even, then x2 is even (D) If x is odd, then x2 is even Mathematical Logic Std XII: Triumph Maths TARGET Publications 47 When the compound statement is true for all its components then the statement is called (A) negation statement (B) tautology statement (C) contradiction statement (D) contingency statement 54 One of the negations of the statement ‘I will have tea or coffee’ is wrong Point it out (A) I will not have both tea and coffee (B) I will neither have tea nor coffee (C) I won’t have any of tea or coffee (D) I will have none of tea and coffee 1.3 Duality 55 48 Dual of the statement (p ∧ q) ∨ ~q ≡ p ∨ ~q is (A) (p ∨ q) ∨ ~q ≡ p ∨ ~q (B) (p ∧ q) ∧ ~q ≡ p ∧ ~q (C) (p ∨ q) ∧ ~q ≡ p ∧ ~q (D) (~p ∨ ~q) ∧ q ≡ ~p ∧ q 49 The dual of the statement “Manoj has the job but he is not happy” is (A) Manoj has the job or he is not happy (B) Manoj has the job and he is not happy (C) Manoj has the job and he is happy (D) Manoj does not have the job and he is happy The negation of ‘If it is Sunday then it is a holiday’ is (A) It is a holiday but not a Sunday (B) No Sunday then no holiday (C) Even though it is Sunday, it is not a holiday, (D) No holiday therefore no Sunday 56 The negation of the statement ‘The product of and is 9’, is (A) The product of and is not 12 (B) The product of and is 12 (C) It is false that the product of and is not (D) It is false that the product of and is 57 The contrapositive of the statement ‘If is greater than 5, then is greater than 6’, is (A) If is greater than 6, then is greater than (B) If is not greater than 6, then is greater than (C) If is not greater than 6, then is not greater than (D) If is greater than 6, then is not greater than 50 The dual of the statement ‘Mango and Apple are sweet fruits’ is (A) Mango and Apple are not sweet fruits (B) Mango is sweet fruit but not apple (C) Apple is sweet fruit but not mango (D) Mango or Apple are sweet fruits 1.4 Negation of compound statements 51 ~[p ∨ (~q)] is equal to (A) ~p ∨ q (B) (~p) ∧ q (C) ~p ∨ ~p (D) ~p ∧ ~q 52 Write Negation of ‘For every natural number x, x + > 4’ (A) ∀ x ∈ N, x + < (B) ∀ x ∈ N, x − < (C) For every integer x, x + < (D) There exists a natural number x, for which x + ≤ 53 One of the negations of the statement ‘Some people are honest’ given below is incorrect Point it out (A) All are dishonest (B) All are not honest (C) None is honest (D) It is not true that, ‘Some people are honest’ 1.5 Switching circuit 58 Consider the circuit, p q r Then, the current flow in the circuit is (A) (p ∧ q) ∨ r (B) (p ∧ q) (C) (p ∨ q) (D) None of these Mathematical Logic Std XII: Triumph Maths TARGET Publications 10 SECTION - 1.1 Statement, Logical Connectives, Compound Statements and Truth Table If p and q have truth (~p ∨ q) ↔ ~(p ∧ q) and respectively are (A) T, T (B) (C) T, F (D) If p is false and q is true, then (A) p ∧ q is true (B) p ∨ ∼q is true (C) q → p is true (D) p → q is true 11 Assuming the first part of the sentence as p and the second as q, write the following statement symbolically: ‘Irrespective of one being lucky or not, one should not stop working’ (A) (p ∧ ~p) ∨ q (B) (p ∨ ~p) ∧ q (C) (p ∨ ~p) ∧ ~q (D) (p ∧ ~p) ∨ ~q 12 If first part of the sentence is p and the second is q, the symbolic form of the statement ‘It is not true that Mathematics is not interesting or difficult’ (A) ∼(∼p ∧ q) (B) (∼p ∨ q) (C) (∼p ∨ ~q) (D) ∼(∼p ∨ q) 13 The symbolic form of the statement ‘It is not true that intelligent persons are neither polite nor helpful’ is (A) ~(p ∨ q) (B) ∼(∼p ∧ ∼q) (C) ~(~p ∨ ~q) (D) ~(p ∧ q) 14 Find out which of the following statements have the same meaning: i If Seema solves a problem then she is happy ii If Seema does not solve a problem then she is not happy iii If Seema is not happy then she hasn’t solved the problem iv If Seema is happy then she has solved the problem (A) (i, ii) and (iii, iv) (B) i, ii, iii (C) (i, iii) and (ii, iv) (D) ii, iii, iv 15 Find out which of the following statements have the same meaning: i If Humpty sit on a wall then he will fall ii If Humpty falls then he was sitting on a wall iii If Humpty does not fall then he was not sitting on the wall iv If Humpty does not sit on a wall then he does not fall (A) (i, iv) and (ii, iii) (B) (i, ii) and (iii, iv) (C) i, ii, iii (D) (i, iii) and (ii, iv) value ‘F’ then ~p ↔ (p → ~q) F, F F, T Given ‘p’ and ‘q’ as true and ‘r’ as false, the truth values of ~p ∧ (q ∨ ~r) and (p → q) ∧ r respectively are (A) T, F (B) F, F (C) T, T (D) F, T If p is true and q is false then (p → q) ↔ (~q → ~p) and (~p ∨ q) ∧ (~q ∨ p) respectively are (A) F, F (B) F, T (C) T, F (D) T, T Truth value of the statement ‘It is false that + = 33 or + = 12’ is (A) T (B) F (C) both T and F (D) 54 Which of the following is logically equivalent to ~[p → (p ∨ ~q)]? (A) p ∨ (~p ∧ q ) (B) p ∧ (~p ∧ q) (C) p ∧ (p ∨ ~q) (D) p ∨ (p ∧ ~q) If ∼q ∨ p is F then which of the following is correct? (A) p ↔ q is T (B) p → q is T (C) q → p is T (D) p → q is F If p, q are true and r is false statement then which of the following is true statement? (A) (p ∧ q) ∨ r is F (B) (p ∧ q) → r is T (C) (p ∨ q) ∧ (p ∨ r) is T (D) (p → q) ↔ (p → r) is T If p is the statement ‘Sun rises in the West’, and q is any statement, state which one of the following is incorrect (A) (p and q), is always false (B) (p → q), is always true (C) (∼p or q), is always true (D) depends on what q is Which of the following is true? (A) p ∧ ∼p ≡ T (B) p ∨ ∼p ≡ F (C) p → q ≡ q → p (D) p → q ≡ (~q) → (∼p) Mathematical Logic Std XII: Triumph Maths 16 Find which of the following statements convey the same meanings? i If it is the bride’s dress then it has to be red ii If it is not bride’s dress then it cannot be red iii If it is a red dress then it must be the bride’s dress iv If it is not a red dress then it can’t be the bride’s dress (A) (i, iv) and (ii, iii) (B) (i, ii) and (iii, iv) (C) (i), (ii), (iii) (D) (i, iii) and (ii, iv) 1.2 Statement Pattern and Logical Equivalence: Tautology, Contradiction, Contingency 17 The proposition (p → q) ↔ ( ∼p → ∼q) is a (A) tautology (B) contradiction (C) contingency (D) None of these 1.3 Duality 24 Duals of the following statements are given which one is not correct? (A) (p ∨ q) ∧ (r ∨ s), (p ∧ q) ∨ (r ∧ s) (B) [p ∨ (~q)] ∧ (~p), [p ∧ (~q)] ∨ (~p) (C) (p ∧ q) ∨ r, (p ∨ q) ∧ r (D) (p ∨ q) ∨ s, (p ∧ q) ∨ s 25 Which of the following statements is dual of the statement (p ∨ q) ∨ r? (A) (p ∧ q) ∧ r (B) (p ∨ q) ∧ r (C) (p ∧ q) ∨ r (D) ~[(p ∨ q) ∨ r] 26 The dual of ‘(p ∧ t) ∨ (c ∧ ~q)’ where t is a tautology and c is a contradiction, is (A) (p ∨ c) ∧ (t ∨ ~q) (B) (~p ∧ c) ∧ (t ∨ q) (C) (~p ∨ c) ∧ (t ∨ q) (D) (~p ∨ t) ∧ (c ∨ ~q) 1.4 Negation of compound statements 27 The negation of the statement “If Saral Mart does not reduce the prices, I will not shop there any more” is (A) Saral Mart reduces the prices and still I will shop there (B) Saral Mart reduces the prices and I will not shop there (C) Saral Mart does not reduce the prices and still I will shop there (D) Saral Mart does not reduce the prices or I will shop there 28 Negation of the statement: “If Dhoni looses the toss then the team wins”, is (A) Dhoni does not lose the toss and the team does not win (B) Dhoni loses the toss but the team does not win (C) Either Dhoni loses the toss or the team wins (D) Dhoni loses the toss iff the team wins 29 Negation of the proposition (p ∨ q) ∧ (∼q ∧ r) is (A) (p ∧ q) ∨ (q ∨ ∼r) (B) (∼p ∨ ∼q) ∧ (∼q ∧ r) (C) (∼p ∧ ∼q) ∨ (q ∨ ∼r) (D) None of these The proposition p → ~(p ∧ ~q) is (A) contradiction (B) a tautology (C) contingency (D) none of these 18 TARGET Publications 19 The statement (p ∧ q) → p is (A) a contradiction (B) a tautology (C) either (A) or (B) (D) a contingency 20 ∼(p ↔ q) is equivalent to (A) (p ∧ ∼q) ∨ (q ∧ ∼p) (B) (p ∨ ∼q) ∧ (q ∨ ∼p) (C) (p → q) ∧ (q → p) (D) None of these 21 The proposition (p ∧ q) → (p ∨ q) is a (A) tautology and contradiction (B) neither tautology nor contradiction (C) contradiction (D) tautology 22 Which of the following is a tautology? (A) p → (p ∧ q) (B) q ∧ (p → q) (C) ∼(p → q) ↔ p ∧ ∼q (D) (p ∧ q) ↔ ∼q 23 Which of the following statement is a contingency (A) (p ∧ ∼q) ∨ ∼(p ∧ ∼q) (B) (p ∧ q) ↔ (∼p → ∼q) (C) [p ∧ (p → ∼q)] → q (D) None of these Mathematical Logic Std XII: Triumph Maths TARGET Publications 1.5 30 32 Switching circuit The switching circuit for the statement [p ∧ (q ∨ r)] ∨ (~p ∨ s) is If the symbolic form is (p ∧ r) ∨ (~q ∧ ~r) ∨ (~p ∧ ~r), then switching circuit is: q (A) p p′ S′1 (A) r q p′ r (B) s S2 ′ S1 s S3 S′ S′ p S1 S′ S′ (B) p r (C) S′ ′ S1 q S3 p′ s (C) q r (D) S2 S3 S′2 S′3 S1 p S′1 p′ s′ 31 S′1 S2 (D) q p HF011 q p′ 33 (A) p S′3 S′1 The simplified circuit for the following circuit is S3 S3 The symbolic form of logic for the following circuit is: S1 (B) q S2 S3 S′1 S′2 (C) (D) p q q′ p Mathematical Logic (A) (B) (C) (D) S′3 (p ∨ q) ∧ (~p ∧ r ∨ ~q) ∨ ~r (p ∧ q) ∧ (~p ∨ r ∧ ~q) ∨ ~r (p ∧ q) ∨ [~p ∧ (r ∨ ~q)] ∨ ~r (p ∨ q) ∧ [~p ∨ (r ∧ ~q)] ∨ ~rHF012 Std XII: Triumph Maths 34 TARGET Publications The simplified circuit for the following circuit is S′1 37 Simplified form of the switching circuit S′1 S′2 S1 S1 S3 S2 (A) (B) (C) S3 For the symbolic form (p ∨ q) ∧ [~p ∨ (r ∧ ~q)] the switching circuit is: (A) S′ S2 S′1 (D) 35 ′ S1 (A) S2 (C) S2 S2 S′1 S1 (B) S′2 S1 ′ S1 S2 S1 S1 (D) S′1 ′ S1 S′ SECTION - S3 S2 S′2 S2 S1 The converse of the contrapositive of p → q is [Karn CET 2005] (A) ∼p → q (B) p → ∼q (C) ∼p → ∼q (D) ∼q → p S′2 S1 S′1 S3 The switching circuit S1 The contrapositive of (p ∨ q) → r is [Karn 1990] (A) ∼r → ∼p ∧ ∼q (B) ∼r → (p ∨ q) (C) r → (p ∨ q) (D) p → (q ∨ r) S3 S′1 S3 36 S′2 S2 (D) If p → (q ∨ r) is false then the truth values of p, q, r are respectively [Karn CET 1997] (A) T, F, F (B) F, F, F (C) F, T, F (D) T, T, F S′1 S3 (C) If p ⇒ (∼p ∨ q) is false, the truth values of p and q respectively, are [Karn 02] (A) F, T (B) F, F (C) T, T (D) T, F S2 Statement, Logical Connectives, Compound Statements and Truth Table S1 (B) 1.1 S′2 S2 S′1 S1 S′2 in symbolic form of logic, is: (A) (p ∧ q) ∨ (~p) ∨ (p ∧ ~q) (B) (p ∨ q) ∨ (~p) ∨ (p ∧ ~q) (C) (p ∧ q) ∧ (~p) ∨ (p ∧ ~q) (D) (p ∨ q) ∧ (~p) ∨ (p ∧ ~q)019 10 Mathematical Logic Std XII: Triumph Maths TARGET Publications 1.2 Statement Pattern and Logical Equivalence: Tautology, Contradiction, Contingency The logically equivalent statement of p ↔ q is [Karn 2000] (A) (p ∧ q) ∨ (q → p) (B) (p ∧ q) → ( p ∨ q) (C) (p → q) ∧ (q →p) (D) (p ∧ q) ∨ (p ∧ q) 10 1.5 11 The proposition (p → ∼p) ∧ (∼p → p) is a [MHT Asso 2006], [Karn 1997] (A) Neither tautology nor contradiction (B) Tautology (C) Tautology and contradiction (D) Contradiction 1.4 When does the current flow through the following circuit [Karn CET 2002] p (A) (B) (C) (D) 12 The negation of the statement given by “He is rich and happy” is [MH-CET 2006] (A) He is not rich and not happy (B) He is rich but not happy (C) He is not rich but happy (D) Either he is not rich or he is not happy r q′ p, q should be closed and r is open p, q, r should be open p, q, r should be closed none of these The following circuit represent symbolically in logic when the current flow in the circuit [Karn CET 1999] q ~p Negation of compound statements Switching circuit q (p ∧ ∼q) ∧ (∼p ∧ q) is a [Karn 2003] (A) Tautology (B) Contradiction (C) Tautology and a contradiction (D) Contingency The false statement in the following is [Karn CET 2002] (A) p ∧ (∼p) is a contradiction (B) p ∨ (∼p) is a tautology (C) ∼ (∼p) ↔ p is tautology (D) (p → q) ↔ (∼q ⇒ ∼p) is a contradiction The negation of q ∨ ∼(p ∧ r) is [Karn CET 1997] (A) ∼q ∧ ∼(p ∨ r) (B) ∼q ∧ (p ∧ r) (C) ∼q ∨ (p ∧ r) (D) ∼q ∨ (p ∧ r) p (A) (B) (C) (D) ~q (∼p ∨ q) ∨ (p ∨ ∼q) (∼p ∧ p) ∧ (∼q ∧ q) (∼p ∧ ∼q) ∧ (q ∧ p) (∼p ∧ q) ∨ (p ∧ ∼q) 11 21 31 41 51 (D) (C) (A) (A) (D) (B) 12 22 32 42 52 (D) (A) (C) (B) (A) (D) 13 23 33 43 53 Answers Key to Multiple Choice Questions Section (A) (C) (D) (C) (D) (A) 14 (A) 15 (A) 16 (B) 17 (B) (A) 24 (D) 25 (B) 26 (B) 27 (C) (A) 34 (B) 35 (D) 36 (B) 37 (B) (C) 44 (B) 45 (A) 46 (A) 47 (B) (B) 54 (A) 55 (C) 56 (D) 57 (C) 11 21 31 (A) (C) (D) (B) 12 22 32 (B) (D) (C) (B) 13 23 33 (C) (B) (B) (C) 14 24 34 (A) (C) (D) (D) (A) (C) (D) 11 (C) (A) 12 (D) Mathematical Logic Section (B) 15 (D) 16 25 (A) 26 35 (A) 36 Section (C) 18 28 38 48 58 (D) (A) (C) (A) (C) (A) 19 29 39 49 (D) (A) (A) (A) (A) 10 20 30 40 50 (C) (B) (B) (A) (D) (B) (A) (A) (A) 17 27 37 (C) (C) (C) (B) (D) 18 (C) 28 (B) (D) 19 (B) 29 (C) 10 (D) 20 (A) 30 (C) (D) (B) 10 (B) (D) (D) 11 Std XII: Triumph Maths TARGET Publications Hints to Multiple Choice Questions Section 28 It is a property 30 r: It is raining, c: I will go to college The given statement is r → c ≡ ∼c → ∼r 31 Converse of p → q is q → p ‘Bombay is the capital of India’ is a statement ‘Two plus two is four’ is a statement As value of ‘x’ is not defined It may be interesting for some person and may not be interesting for other 32 Converse of p → q is q → p 33 Contrapositive of p → q is ∼q → ∼p Even though = 3, is false, it is a statement in logic with value F 34 The given statement is a disjunction The given statement is a conjunction It is a true statement, since x = ∈ N satisfies x + = 35 36 Converse of p → q is q → p ~(p ∨ q) ≡ ~p ∧ ~q 37 Converse of p → q is q → p ~p: The sun has not set, ~q: The moon has not risen, ‘or’ is expressed by ‘∨’ symbol ~p ∨ ~q 38 It is a property 39 Implies is indicated by → sign 40 ~p: Sita does not get promotion and ‘↔’ symbol indicates if and only if p: It is raining q: The atmosphere is very cold 41 p: There are clouds in the sky, q: It is raining, ‘and’ is expressed by ‘∧’ symbol p ∧ ~q p: Candidates are present, q: Voters are ready to vote r: Ballot papers 42 (p and q) → r 43 p statement implies (q or r) ∴ 10 ∴ 11 ~p: He is not fat, ~w: He is not hard working, ‘∨’ symbol indicates ‘or’ 44 12 ~p: Rohit is short, ‘or’ is expressed by ‘∨’ symbol and ‘and’ is expressed by ‘∧’ symbol (~d: Driver is not drunk) implies (~a: He cannot meet with an accident) 45 Plain logic, both are equivalent 13 Symbolic form is p ∧ ~q 46 14 Converse of p → q is q → p 15 Converse of p → q is q → p Consider (A), ~p → ~q i.e., ~F → ~T i.e., T → F which is false 16 ~p: She is not beautiful, ‘∨’ indicates ‘or’ 47 It is a property 17 ~p: Ram is not lazy, ~q: Ram does not fail in the examination, ‘∨’ indicates ‘or’ 48 It is a property 49 18 It is a property 19 p → q means Mathematics is interesting implies Mathematics is difficult ∴ p: Manoj has the job, q: he is not happy Symbolic form is p ∧ q Its dual is p ∨ q Manoj has the job or he is not happy 50 Dual of ‘and’ is ‘or’ 20 ~(p ∧ q) ≡ ~p ∨ ~q 51 ~[p ∨ (~q)] ≡ ~p ∧ ~(~q) ≡ (~p) ∧ q 21 ∴ p → q ≡ ~p ∨ q ~(p → q) ≡ ~(~p ∨ q) ≡ ~(~p) ∧ ~q ≡ p ∧ ∼q 52 Negation of Existential quantifier 53 22 It is a property All are not honest means some can still be honest Therefore, it is a wrong negation 26 p → q ≡ ~q → ~p 54 27 It is a property I will not have both tea and coffee, means that, I can’t have both but I can still have one of tea or coffee Therefore, it is a wrong negation 12 Mathematical Logic Std XII: Triumph Maths TARGET Publications 55 The given statement is ‘Sunday has to be a holiday’ Therefore its negation is ‘Even though it is a Sunday, it is Not a holiday’ 56 The negation of the given statement is ‘It is false that the product of and is 9’ 57 Current in the upper part will flow only if both the switches p and q are closed It is represented by p ∧ q Current will flow in the circuit if switch p and q are closed or switch r is closed It is represented by (p ∧ q) ∨ r (A) is correct answer (p ∨ q) ∧ (p ∨ r) ≡ (T ∨ T) ∧ (T ∨ F) ≡ T ∧ T ≡T ‘depends on what q is’ is incorrect Since p is false (A) (p and q), is always false for all q (B) (p → q), is always true for all q, (C) p is false, thus, ∼p is true, therefore, (∼p or q) is true for all q Hence, the options (A), (B), (C) are valid irrespective of what q is (∼q) → (∼p) is contrapositive of p → q and both convey the same meaning Contrapositive of p → q is ~q → ~p 58 ∴ 10 When p is false and q is true, then p ∧ q is false, p ∨ ∼q is false, (∵ both p and ∼q are false) and q → p is also false, only p → q is true 11 p: One being lucky, q: One should stop working 12 p: Mathematics is interesting q: Mathematics is difficult 13 p: Intelligent persons are polite q: Intelligent persons are helpful 14 p: Seema solves a problem q: She is happy i p→q ii ∼p → ∼q iii ∼q → ∼p iv q → p (i) and (iii) have the same meaning, (ii) and (iv) have the same meaning 15 w: Humpty sit on a wall f: Humpty will fall i w→f ii f→w iii ∼f → ∼w iv ∼w → ∼f (i) and (iii) have the same meaning, (ii) and (iv) have the same meaning 16 i b→r ii ∼b → ∼r iii r → b v ∼r → ∼b (i) and (iv) are the same and (ii) and (iii) are the same Section ∴ ∴ ∴ (~p ∨ q) ↔ ~(p ∧ q) and ~p ↔ (p → ~q) (~F ∨ F) ↔ ~(F ∧ F) and ~F ↔ (F → ~F) (T ∨ F) ↔ ~F and T ↔ (F → T) T ↔ T and T ↔ T T and T ∴ ∴ ~p ∧ (q ∨ ~r) and (p → q) ∧ r ~T ∧ (T ∨ ~F) and (T → T) ∧ F F ∧ (T ∨ T) and (T ∧ F) = F ∧ T and T ∧ F = F, F ∴ ∴ ∴ (p → q) ↔ (~q → ~p) and (~p ∨ q) ∧ (~q ∨ p) (T → F) ↔ (~F → ~T) and (~T ∨ F) ∧ (~F ∨ T) F ↔ (T → F) and (F ∨ F) ∧ (T ∨ T) F ↔ F and F ∧ T ⇒ T and F p: + = 33, q: + = 12 Truth values of both p and q is F ~(F ∨ F) ≡ ~F ≡ T ∴ ~[p → (p ∨ (~q))] ≡ ~[~p ∨ (p ∨ (~q))] ≡ p ∧ ~[p ∨ (~q)] ≡ p ∧ (~p ∧ q) p T T F F ∴ ∴ q T F T F ∼q F T F T ∼q ∨ p T T F T Alternate Method: ~ q ∨ p: F ~ q is F, p is F i.e q is T, p is F p→q≡F→T≡T Mathematical Logic p↔q T F F T p→q T F T T 17 p T T F F q T F T F ~q F T F T p ∧ ~q F T F F ~(p ∧ ~q) T F T T p → ~(p ∧ ~q) T F T T 13 Std XII: Triumph Maths TARGET Publications 27 18 p q p→q ∼p ∼q ∼p→∼q T T F F T F T F T F T T F F T T F T F T T T F T 19 20 ∴ (p→q)↔ (∼p→∼q) T F F T (p ∧ q) → p ≡ ∼ (p ∧ q) ∨ p ≡ (∼p ∨ ∼q) ∨ p ≡ (∼p ∨ p) ∨ ∼q ≡ T ∨ ∼q ≡T (~p ∨ q): Either Saral Mart reduces the prices or I will not shop there any more The negation of the given statement is (p ∧ ~q), given by Saral Mart does not reduce the prices and still I will shop there 28 ∴ We know that, p ↔ q = (p → q) ∧ (q → p) ∼(p ↔ q) = ∼[(p → q) ∧ (q → p)] = ∼ (p → q) ∨ ∼(q → p) (By Demorgan’s Law) = (p ∧ ∼q) ∨ (q ∧ ∼p) (∵ ∼(p → q) = p ∧ ∼q) q T F T F p∧q T F F F p∨q T T T F Negation of (p ∨ q) ∧ (∼q ∧ r) is ∼[(p ∨ q) ∧ (∼q ∧ r)] = ∼(p ∨ q) ∨ ∼(∼q ∧ r) = (∼p ∧ ∼q) ∨ [∼(∼q) ∨ ∼r] = (∼p ∧ ∼q) ∨ (q ∨ ∼r) 31 (p ∧ q) ∨ (~p ∧ q) ≡ (p ∨ ~p) ∧ q ≡T∧q ≡q 34 (p ∧ q) → (p ∨ q) T T T T The Symbolic form is ≡ [(~p ∧ ~q) ∨ p ∨ q ] ∧ r ≡ [~(p ∨ q) ∨ (p ∨ q)] ∧ r ≡T∧r ≡r 37 (~p ∧ ~q) ∨ (p ∧ q) ∨ (~p ∧ q) ≡ ~p ∧ (~q ∨ q) ∨ (p ∧ q) ≡ (~p ∧ T) ∨ (p ∧ q) ≡ ~p ∨ (p ∧ q) ≡ (~p ∨ p) ∧ (~p ∨ q) ≡ T ∧ (~p ∨ q) ≡ ~p ∨ q 22 p q ∼q T T F F T F T F F T F T p→q ∼(p→q) p∧∼q ∼(p→q)↔(p∧∼q) T F T T F T F F F T F F T T T T 23 p q ∼p ∼q p∧q ∼p→ ∼ q T T F F T T T F F T F T F T T F F F F F T T F T (p∧q)↔(∼p→∼q) T F T F ∴ Dual of (p ∨ q) ∨ s is (p ∧ q) ∧ s 25 Dual of (p ∨ q) ∨ r is (p ∧ q) ∧ r 26 Dual of ‘∨’ is ‘∧’ and of ‘t’ is ‘c’ Section p ⇒ (∼p ∨ q) is false mean p is true and ∼p ∨ q is false ⇒ p is true and both ∼p and q are false ⇒ p is true and q is false Since p → q is false, when p is true and q is false p → (q ∨ r) is false when p is true and q ∨ r is false, when both q and r are false (p ∧ q) ↔ (∼p → ∼q) is contingency (B) is correct answer 24 14 p: Dhoni looses the toss, q: The team wins ∼(p → q) ≡ p ∧ ∼q Dhoni loses the toss but (and) the team does not win 29 21 p T T F F Let p: Saral Mart does not reduce the prices q: I will not shop there any more ∴ Mathematical Logic Std XII: Triumph Maths TARGET Publications Contrapositive of (p ∨ q) → r is ∼r → ∼(p ∨ q) i.e ∼r → ∼p ∧ ∼q ∴ Given p → q Its contrapositive is ∼q → ∼p and its converse is ∼p → ∼q (C) is correct answer We know, p ↔ q ≡ (p → q) ∧ (q → p) p T F ∼p p→∼p ∼p→p F F T T T F (p→∼p)∧(∼p→ p) F F Contradiction ∴ p → q is logically equivalent to ∼q → ∼p (p → q) ↔ (∼q → ∼p) is tautology But, it is given contradiction Hence, it is false statement Either he is not rich or he is not happy 10 Negation of q ∨ ∼(p ∧ r) is ∼(q ∨ ∼(p ∧ r)) = ∼q ∧ ∼(∼(p ∧ r)) = ∼q ∧ (p ∧ r) 11 The current will be flow to the circuit if p, q, r should be closed or p, q′, r should be closed (C) is correct answer ∴ 12 Let p : s1 is closed q: switch s2 is closed ∼q : switch s2 is open ~p : switch s1 is open or switch s1′ is closed The current can flow in the circuit iff either s1′ and s2 are closed or s1 and s2′ are closed It is represented by (∼p ∧ q)∨ (p ∧ ∼q) Mathematical Logic 15 ... MCQs subtopic wise Each chapter contains three sections Section contains easy level questions Section contains competitive level questions Section contains questions from various competitive exams... Distribution 604 21 Binomial Distribution 619 Std XII: Triumph Maths TARGET Publications MATHEMATICAL LOGIC 01 Logical Connectives: Connective And (Conjuction) Or (Disjunction) If … then (Conditional)... have framed them on every sub topic included in the curriculum Each chapter is divided into three sections: Section consists of basic MCQs based on subtopics of Text Book Section consists of MCQs