MATHEMATICS BASED ON STD XII (SCI) CURRICULUM (MAHARASHTRA BOARD) MULTIPLE CHOICE QUESTIONS docx

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 promotio

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:

9 Exhaustive coverage of MCQs subtopic wise

9 Each chapter contains three sections

9 Section 1 contains easy level questions

9 Section 2 contains competitive level questions

9 Section 3 contains questions from various competitive exams

9 Important formulae

9 Hints provided wherever relevant

9 Useful for MHT-CET and ISEET preparation

Tar g et 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

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Mumbai

Published by

Tar g et PUBLICATIONS PVT LTD.

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email: mail@targetpublications.in

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

Sr No Topic Name Page No

01 MATHEMATICAL LOGIC

1 Logical Connectives:

Connective Symbol Example

If … then (Conditional)

(Implication)

→ or ⇒ If p, then q: p → q

If and only if (Biconditional)

The truth table of above logical connectives are as given below:

p q p ∨ q p ∧ q p → q p ↔ q

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

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 Publications

Std XII: Triumph Maths

Mathematical Logic

2

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

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

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!

(C) What are you doing?

(D) Bombay is the capital of India

2 Which of the following is a statement?

(A) Open the door

(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!

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

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

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

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

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

TARGET Publications

Std XII: Triumph Maths

Mathematical Logic

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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

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

Mathematics is difficult

20 Which of the following is logically equivalent

to ~(p ∧ q)

21 ~(p → q) is equivalent to

22 Contrapositive of p → q is

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

(C) logical connective

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

(B) If I do not go to college, then it is not

(C) If I go to college then it is raining (D) Going to college depends on my mood

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

35 The connective in the statement “Earth

revolves round the Sun and Moon is a satellite

of earth”, is

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

40 The symbolic form of the statement ‘Since it

is raining the atmosphere is very cold’ is

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

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

45 Statement ~p ↔ ~q ≡ p ↔ q is (A) a tautology (B) a contradiction

46 Given that p is ‘false’ and q is ‘true’ then the statement which is ‘false’ is

TARGET Publications

Std XII: Triumph Maths

Mathematical Logic

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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

(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

(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

r