A book for Std. XII, MHT-CET, ISEET and other Competitive … · ... MHT-CET, ISEET and other Competitive Entrance Exams . Std. XII Sci. Triumph Maths Mr. Vinodkumar J. Pandey B.Sc

A book for Std. XII, MHT-CET, ISEET and other Competitive Entrance Exams

Std. XII Sci.

Triumph Maths

Mr. Vinodkumar J. Pandey B.Sc. (Mathematics)

G. N. Khalsa College, Mumbai

Mrs. Shama Mittal

M.Sc., (Mathematics), B.Ed. Punjabi University (Patiala)

Salient Features: Exhaustive coverage of MCQs subtopic wise. Each chapter contains three sections. Section 1 contains easy level questions. Section 2 contains competitive level questions. Section 3 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 : [email protected]

Written according to the New Text book (2012-2013) published by the Maharashtra State Board of Secondary and Higher Secondary Education, Pune.

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: [email protected]

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 1 consists of basic MCQs based on subtopics of Text Book.

Section 2 consists of MCQs of competitive level.

Section 3 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

Sr. No. Topic Name Page No.

1 Mathematical Logic 1

2 Matrices 16

3 Trigonometric Functions 40

4 Pair of Straight Lines 85

5 Circle 120

6 Conics 146

7 Vectors 174

8 Three Dimensional Geometry 198

9 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

TARGET Publications Std. XII: Triumph Maths

1Mathematical Logic

01 MATHEMATICAL LOGIC 1. Logical Connectives:

Connective Symbol Example And (Conjuction) ∧ p and q : p ∧ q Or (Disjunction) ∨ p or q : p ∨ q If … then (Conditional) (Implication)

→ or ⇒ If p, then q: p → q

If and only if (Biconditional) (iff) (Double implication)

↔ or ⇔ p iff q : p ↔ q

Not (Negation) ∼ p : ∼ p The truth table of above logical connectives are as given below:

p q p ∨ q p ∧ q p → q p ↔ q T T T T T T T F T F F F F T T F T F F F F F T T

2. 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.” 3. 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 q ~p ~q p→q q→p ~q→~p ~p→~q T T F F T T T T T F F T F T F T F T T F T F T F F F T T T T T T

From the above truth table, hypothesis and its contrapositive are logical equivalent. Also, the converse

and its inverse are equivalent. 4. 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. 5. 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)

p ~p T F F T

TARGET PublicationsStd. XII: Triumph Maths

Mathematical Logic2

6. Application of Logic to Switching Circuits: i. 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.

ii. 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. iii. 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 1. p ∨ q = q ∨ p p ∧ q = q ∧ p 2. (p ∨ q) ∨ r = p ∨ (q ∨ r) (p ∧ q) ∧ r = p ∧(q ∧ r) 3. p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r) p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r) 4. ~ (p ∨ q) = ~ p ∧ ~ q ~ (p ∧ q) ≡ ~ p ∨ ~ q 5. p → q ≡ ~ p ∨ q p ↔ q ≡ (p → q) ∧ (q → p) ≡ (~ p ∨ q) ∧ (~ q ∨ p) 6. p ∨ (p ∧ q) = p p ∧ (p ∨ q) = p 7. If T denotes the tautology and F denotes the contradiction, then for any statement ‘p’: i⋅ p ∨ T = T; p ∨ F = p ii. p ∧ T = p; p ∧ F = F 8. i. p ∨ ~ p = T ii. p ∧ ~ p = F iii. ∼(∼p) = p iv. ∼ T = F v. ∼ F = T 9. p ∨ p = p p ∧ p = p

Commutative property

Associative property

Distributive property

Demorgan’s law

Equivalent statements

Absorption laws

Identity laws

Complement laws

Idempotent laws

L

S1

S2

L

S1 S2


3Mathematical Logic

1.1 Statement, Logical Connectives, Compound

Statements and Truth Table 1. 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. 2. 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. 3. Which of the following is an open statement? (A) x + 5 = 11 (B) Good morning to all. (C) What is your problem? (D) Listen to me, Rahul! 4. Which of the following is not a proposition in

logic. (A) 3 is a prime

(B) 2 is an irrational number (C) Mathematics is interesting (D) 5 is an even integer 5. Which of the following is a statement in

Logic? (A) Go away (B) How beautiful! (C) x > 5 (D) 2 = 3 6. Using quantifiers ∀, ∃, convert the following

open statement into true statement. ‘x + 5 = 8, x ∈ N’ (A) ∀ x ∈ N, x + 5 = 8 (B) For every x ∈ N, x + 5 > 8 (C) ∃ x ∈ N, such that x + 5 = 8 (D) For every x ∈ N, x + 5 < 8 7. ~(p ∨ q) is (A) ~p ∨ q (B) p ∨ ~q (C) ~p ∨ ~q (D) ~p ∧ ~q 8. 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

9. 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. 10. 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.

SECTION - 1


Mathematical Logic4

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. The inverse of logical statement p → q is (A) ~p → ~q (B) p ↔ q (C) q → p (D) q ↔ p 19. 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. 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) ~p ∨ q (C) p ∨ ~q (D) ~p ∧ ~q 22. Contrapositive of p → q is (A) q → p (B) ~q → p (C) ~q → ~p (D) q → ~p 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.

24. 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 do 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.


5Mathematical Logic

31. 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 32. 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.

33. 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 “2 + 7 > 9 or 2 + 7 < 9” is

(A) and (B) or (C) > (D) < 35. 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 36. 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 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

38. 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


Mathematical Logic6

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. 1.3 Duality 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. 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 + 5 > 4’. (A) ∀ x ∈ N, x + 5 < 4 (B) ∀ x ∈ N, x − 5 < 4 (C) For every integer x, x + 5 < 4 (D) There exists a natural number x, for

which x + 5 ≤ 4 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’.

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. 55. 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 3 and 4 is 9’, is (A) The product of 3 and 4 is not 12. (B) The product of 3 and 4 is 12. (C) It is false that the product of 3 and 4 is

not 9. (D) It is false that the product of 3 and 4 is

9. 57. The contrapositive of the statement ‘If 7 is

greater than 5, then 8 is greater than 6’, is (A) If 8 is greater than 6, then 7 is greater

than 5. (B) If 8 is not greater than 6, then 7 is

greater than 5. (C) If 8 is not greater than 6, then 7 is not

greater than 5. (D) If 8 is greater than 6, then 7 is not

greater than 5. 1.5 Switching circuit 58. Consider the circuit, Then, the current flow in the circuit is (A) (p ∧ q) ∨ r (B) (p ∧ q) (C) (p ∨ q) (D) None of these

p q

r


7Mathematical Logic

1.1 Statement, Logical Connectives, Compound

Statements and Truth Table 1. If p and q have truth value ‘F’ then

(~p ∨ q) ↔ ~(p ∧ q) and ~p ↔ (p → ~q) respectively are

(A) T, T (B) F, F (C) T, F (D) F, T 2. 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 3. 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 4. Truth value of the statement ‘It is false that 3 + 3 = 33 or 1 + 2 = 12’ is (A) T (B) F (C) both T and F (D) 54 5. 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) 6. 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 7. 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 8. 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. 9. Which of the following is true? (A) p ∧ ∼p ≡ T (B) p ∨ ∼p ≡ F (C) p → q ≡ q → p (D) p → q ≡ (~q) → (∼p)

10. 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)

SECTION - 2


Mathematical Logic8

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 → ~(p ∧ ~q) is (A) contradiction. (B) a tautology. (C) contingency. (D) none of these 18. The proposition (p → q) ↔ ( ∼p → ∼q) is a (A) tautology (B) contradiction (C) contingency (D) None of these 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

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


9Mathematical Logic

1.5 Switching circuit 30. The switching circuit for the statement

[p ∧ (q ∨ r)] ∨ (~p ∨ s) is (A) (B) (C) (D) 31. The simplified circuit for the following circuit

is (A) (B) (C) (D)

32. If the symbolic form is (p ∧ r) ∨ (~q ∧ ~r) ∨ (~p ∧ ~r), then switching circuit is:

(A) (B) (C) (D) HF011 33. The symbolic form of logic for the following

circuit is: (A) (p ∨ q) ∧ (~p ∧ r ∨ ~q) ∨ ~r (B) (p ∧ q) ∧ (~p ∨ r ∧ ~q) ∨ ~r (C) (p ∧ q) ∨ [~p ∧ (r ∨ ~q)] ∨ ~r (D) (p ∨ q) ∧ [~p ∨ (r ∧ ~q)] ∨ ~rHF012

p q

S′1

S1

S′2 S′3

S2 S3

S′1 S3

S′1

S2 S′3

S3

S′3

S′1S′2

S1 S2

S3

1S′ S3

S′1

2S′3S′

S2

S1

2S′

S3

1S′ 3S′

3S′

p p′ s

r

q

p p′

s

q

r

r

q

s

p

p′

q

p′r

p

s′

p q

q p′

p

q

q′ p


Mathematical Logic10

34. The simplified circuit for the following circuit is

(A)

(B) (C) (D) 35. For the symbolic form (p ∨ q) ∧ [~p ∨ (r ∧ ~q)] the switching circuit

is: (A) (B) (C) (D) 36. The switching circuit 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

37. Simplified form of the switching circuit (A) (B) (C) (D) 1.1 Statement, Logical Connectives, Compound

Statements and Truth Table 1. 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 2. 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 3. 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) 4. 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′1

S1 S3

S′2

S2

S3

S1

S2

S′1

S2

S1 S′1 S3

S′2

S2

S1 S′1

S3 S′2

S2

S′2

S1

S′1

S1

1S′

S2

S1

S1

1S′ 2S′

S′1 S′2

S1 S2

S2 S′1

SECTION - 3

1S′ 2S′

S2

S1 S′1S2 S2

S′2S3

S1 S′1

S3 S′2S3


11Mathematical Logic

1.2 Statement Pattern and Logical Equivalence: Tautology, Contradiction, Contingency

5. 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) 6. 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 7. (p ∧ ∼q) ∧ (∼p ∧ q) is a [Karn. 2003] (A) Tautology (B) Contradiction (C) Tautology and a contradiction (D) Contingency 8. 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 1.4 Negation of compound statements 9. 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

10. 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) 1.5 Switching circuit 11. When does the current flow through the

following circuit. [Karn. CET 2002] (A) p, q should be closed and r is open (B) p, q, r should be open (C) p, q, r should be closed (D) none of these 12. The following circuit represent symbolically

in logic when the current flow in the circuit. [Karn. CET 1999] (A) (∼p ∨ q) ∨ (p ∨ ∼q) (B) (∼p ∧ p) ∧ (∼q ∧ q) (C) (∼p ∧ ∼q) ∧ (q ∧ p) (D) (∼p ∧ q) ∨ (p ∧ ∼q)

Answers Key to Multiple Choice Questions

Section 1 1. (D) 2. (D) 3. (A) 4. (C) 5. (D) 6. (C) 7. (D) 8. (D) 9. (D) 10. (C) 11. (C) 12. (A) 13. (A) 14. (A) 15. (A) 16. (B) 17. (B) 18. (A) 19. (A) 20. (B) 21. (A) 22. (C) 23. (A) 24. (D) 25. (B) 26. (B) 27. (C) 28. (C) 29. (A) 30. (B) 31. (A) 32. (B) 33. (A) 34. (B) 35. (D) 36. (B) 37. (B) 38. (A) 39. (A) 40. (A) 41. (D) 42. (A) 43. (C) 44. (B) 45. (A) 46. (A) 47. (B) 48. (C) 49. (A) 50. (D) 51. (B) 52. (D) 53. (B) 54. (A) 55. (C) 56. (D) 57. (C) 58. (A)

Section 2 1. (A) 2. (B) 3. (C) 4. (A) 5. (B) 6. (B) 7. (C) 8. (D) 9. (D) 10. (D) 11. (C) 12. (D) 13. (B) 14. (C) 15. (D) 16. (A) 17. (C) 18. (C) 19. (B) 20. (A) 21. (D) 22. (C) 23. (B) 24. (D) 25. (A) 26. (A) 27. (C) 28. (B) 29. (C) 30. (C) 31. (B) 32. (B) 33. (C) 34. (D) 35. (A) 36. (A) 37. (B)

Section 3 1. (D) 2. (A) 3. (A) 4. (C) 5. (C) 6. (D) 7. (B) 8. (D) 9. (D) 10. (B) 11. (C) 12. (D)

q′

q

rp

q~ p

~ q p



Hints to Multiple Choice Questions

Section 1 1. ‘Bombay is the capital of India’ is a statement. 2. ‘Two plus two is four’ is a statement. 3. As value of ‘x’ is not defined. 4. It may be interesting for some person and may

not be interesting for other. 5. Even though 2 = 3, is false, it is a statement in logic with value F. 6. It is a true statement, since x = 3 ∈ N satisfies

x + 5 = 8. 7. ~(p ∨ q) ≡ ~p ∧ ~q 8. ~p: The sun has not set, ~q: The moon has not

risen, ‘or’ is expressed by ‘∨’ symbol. ∴ ~p ∨ ~q 9. ~p: Sita does not get promotion and ‘↔’

symbol indicates if and only if. 10. p: There are clouds in the sky, q: It is raining,

‘and’ is expressed by ‘∧’ symbol. ∴ p ∧ ~q 11. ~p: He is not fat, ~w: He is not hard working,

‘∨’ symbol indicates ‘or’. 12. ~p: Rohit is short, ‘or’ is expressed by ‘∨’

symbol and ‘and’ is expressed by ‘∧’ symbol. 13. Symbolic form is p ∧ ~q 14. Converse of p → q is q → p. 15. Converse of p → q is q → p. 16. ~p: She is not beautiful, ‘∨’ indicates ‘or’. 17. ~p: Ram is not lazy, ~q: Ram does not fail in

the examination, ‘∨’ indicates ‘or’. 18. It is a property. 19. p → q means Mathematics is interesting

implies Mathematics is difficult. 20. ~(p ∧ q) ≡ ~p ∨ ~q 21. p → q ≡ ~p ∨ q ∴ ~(p → q) ≡ ~(~p ∨ q) ≡ ~(~p) ∧ ~q ≡ p ∧ ∼q 22. It is a property. 26. p → q ≡ ~q → ~p 27. It is a property.

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. 32. Converse of p → q is q → p. 33. Contrapositive of p → q is ∼q → ∼p. 34. The given statement is a disjunction. 35. The given statement is a conjunction. 36. Converse of p → q is q → p. 37. Converse of p → q is q → p. 38. It is a property. 39. Implies is indicated by → sign. 40. p: It is raining q: The atmosphere is very cold. 41. 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) 44. (~d: Driver is not drunk) implies (~a: He cannot meet with an accident). 45. Plain logic, both are equivalent. 46. Consider (A), ~p → ~q i.e., ~F → ~T i.e., T → F which is false. 47. It is a property. 48. It is a property. 49. 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’. 51. ~[p ∨ (~q)] ≡ ~p ∧ ~(~q) ≡ (~p) ∧ q 52. Negation of Existential quantifier. 53. All are not honest means some can still be

honest. Therefore, it is a wrong negation. 54. 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.



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 3 and 4 is 9’.

57. Contrapositive of p → q is ~q → ~p 58. 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. Section 2

1. (~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 2. ~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 3. (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 4. p: 3 + 3 = 33, q: 1 + 2 = 12 Truth values of both p and q is F. ∴ ~(F ∨ F) ≡ ~F ≡ T 5. ~[p → (p ∨ (~q))] ≡ ~[~p ∨ (p ∨ (~q))] ≡ p ∧ ~[p ∨ (~q)] ≡ p ∧ (~p ∧ q) 6.

p q ∼q ∼q ∨ p p ↔ q p → q T T F T T T T F T T F F F T F F F T F F T T T 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

7. (p ∨ q) ∧ (p ∨ r) ≡ (T ∨ T) ∧ (T ∨ F) ≡ T ∧ T ≡ T 8. ‘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.

9. (∼q) → (∼p) is contrapositive of p → q and both convey the same meaning.

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. 17. p q ~q p ∧ ~q ~(p ∧ ~q) p → ~(p ∧ ~q)T T F F T T T F T T F F F T F F T T F F T F T T



18. p q p→q ∼ 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

T F F T

19. (p ∧ q) → p ≡ ∼ (p ∧ q) ∨ p ≡ (∼p ∨ ∼q) ∨ p ≡ (∼p ∨ p) ∨ ∼q ≡ T ∨ ∼q ≡ T 20. 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) 21. p q p ∧ q p ∨ q (p ∧ q) → (p ∨ q) T T F F

T F T F

T F F F

T T T F

T T T T

22.

p q ∼q p→q ∼(p→q) p∧∼q ∼(p→q)↔(p∧∼q)T T F T F F T T F T F T T T F T F T F F T F F T T F F T

23. p q ∼p ∼q p∧q ∼p→ ∼ q (p∧q)↔(∼p→∼q)T T F F T T T T F F T F T F F T T F F F T F F T T F T F

(p ∧ q) ↔ (∼p → ∼q) is contingency. ∴ (B) is correct answer. 24. 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’.

27. Let p: Saral Mart does not reduce the prices. q: I will not shop there any more. (~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. 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. 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. 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

Section 3 1. 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. 2. 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.



3. Contrapositive of (p ∨ q) → r is ∼r → ∼(p ∨ q) i.e. ∼r → ∼p ∧ ∼q

4. Given p → q Its contrapositive is ∼q → ∼p and its converse is ∼p → ∼q ∴ (C) is correct answer. 5. We know, p ↔ q ≡ (p → q) ∧ (q → p) 6. p ∼ p p → ∼ p ∼ p → p (p→∼p)∧(∼p→ p)T F

F T

F T

T F

F F

7. Contradiction. 8. p → q is logically equivalent to ∼q → ∼p ∴ (p → q) ↔ (∼q → ∼p) is tautology But, it is given contradiction. Hence, it is false statement. 9. 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)

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A book for Std. XII, MHT-CET, ISEET and other Competitive … · ... MHT-CET, ISEET and other Competitive Entrance Exams . Std. XII Sci. Triumph Maths Mr. Vinodkumar J. Pandey B.Sc