Std. 12 - Mathematics Practice Problems · Preface In the case of good books, the point is not how many of them you can get through, but rather how many can get through to you. “Std

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MATHEMATICS PRACTICE PROBLEMS STD. XII Sci.

TEID : 968

Printed at: Repro Knowledgecast Ltd., Mumbai

No part of this book may be reproduced or transmitted in any form or by any means, C.D. ROM/Audio Video Cassettes or electronic, mechanical including photocopying; recording or by any information storage and retrieval system without permission in writing from the Publisher.

First Edition: November 2015

Salient Features :

• Adequate Problems for Practice in each sub-topic • Topic and sub-topic wise classification of Problems at the beginning of every chapter.

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Preface

In the case of good books, the point is not how many of them you can get through, but rather how many can get

through to you. “Std. XII Sci. - Mathematics: Practice Problems” is a complete and thorough guide, which is critically

analysed and drafted to serve as a supplementary problem solving book for all the HSC aspiring students. The book is

designed as per the Maharashtra State Board Syllabus. At the beginning of every chapter, sub-topic wise classification

of all problems has been provided for simpler understanding of the different types of questions. Coverage of different

variety of problems based on each sub-topic has been assured. Board Problems similar to text book exercises with the final answer have been provided to help the student get

accustomed to the different standards of board problems. The journey to create a complete book is strewn with triumphs, failures and near misses. If you think we’ve

nearly missed something or want to applaud us for our triumphs, we’d love to hear from you. Please write to us on : [email protected]

Best of luck to all the aspirants!

Yours faithfully, Publisher APER

Index

Sr. No. Topic Name

Page No. Questions Answers

PART - I

1 Mathematical Logic 1 67 2 Matrices 6 70 3 Trigonometric Functions 11 72 4 Pair of Straight Lines 14 74 5 Vectors 17 75 6 Three Dimensional Geometry 21 76 7 Line 23 77 8 Plane 26 78 9 Linear Programming 29 79

PART - II 1 Continuity 32 81 2 Differentiation 36 83 3 Applications of Derivatives 42 87 4 Integration 45 89 5 Definite Integral 52 95 6 Applications of Definite Integral 55 97 7 Differential Equations 56 98 8 Probability Distribution 60 100 9 Binomial Distribution 64 102

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Basic Physics (F.Y.Dip.Sem.-1) MSBTEChapter 1: Mathematical LogicChapter 01: Mathematical Logic

Publications Pvt. Ltd. Target

Type of Problems Based on Exercise / Miscellaneous Q. Nos.

Identify the statements and write down their Truth Value

1.1 Q.1

Miscellaneous Q.1

Express the statements in Symbolic Form/Write the statement in Symbolic Form

1.2 Q.1

1.4 Q.1, 2

Miscellaneous Q.5

Write the Truth values of Statements

1.2 Q.2

1.4 Q.3, 5

1.6 Q.1

Miscellaneous Q.2, 3, 9

Write the Negation of Statements/Using the Rules of Negation write the Negation of Statements

1.3 Q.1

1.8 Q.1, 2, 4

Miscellaneous Q.4, 20

Write the Verbal statement for the given Symbolic Statement

1.4 Q.6, 7

Miscellaneous Q.6

Converse, Inverse and Contrapositive of the statement

1.4 Q.4

Miscellaneous Q.19

Using Quantifiers Convert Open sentences into True statement

1.6 Q.2

Miscellaneous Q.11

Prepare the Truth Table/Find Truth Values of p and q for given cases

1.5 Q.1


Examine the statement Patterns (Tautology, Contradiction, Contingency)

1.5 Q.3

Miscellaneous Q.13, 14, 16

Using Truth Table, Verify Logical Equivalence

1.5 Q.2


Write Dual of the statement 1.7 Q.1, 2, 3, 4

Algebra of statements (without using Truth Table verify the Logical Equivalence)/Rewrite the statement without using the conditional form

1.8 Q.3


Change the statements in the form if then Miscellaneous Q.10

Applications of logic to switching circuits

1.9 Q.1 to 6

Miscellaneous Q.21 to 23

Mathematical Logic 01

2

Basic Physics (F.Y.Dip.Sem.-1) MSBTEChapter 1: Mathematical LogicStd. XII Sci.: Practice Problems

Publications Pvt. Ltd. Target Based on Exercise 1.1 1. State which of the following sentences are

statements. Justify your answer. In case of statements, write down the truth value.

i. 18 is less than 16. ii. Two plus three is five. iii. Every square is a rectangle. iv. May you live long! v. Switch on the light. vi. New Delhi is in Nepal. vii. Bring some fruits from the fruit shop. viii. Please do me a favour. ix. Is every set finite? x. Have your ever seen Taj Mahal? xi. 3 is root of equation x2 5x + 6 = 0 xii. Two distinct points determine a unique

line. Based on Exercise 1.2 1. Express the following statements in symbolic

form: i. Two lines intersect at a point or they are

parallel. ii. Sun rises or moon sets. iii. 10 is multiple of both 2 and 5. iv. Delhi is in England and 2 + 2 = 4. v. Mumbai is the capital of Gujarat or

Maharashtra. vi. A triangle is equilateral if and only if it is

equiangular. [Mar 13] vii. Price increases and demand falls.

[Mar 13] 2. Write the truth values of following statements. i. 100 is a multiple of 4 and 5 ii. Square of an integer is positive or negative. iii. The earth is round or the sun is cold. iv. Sion station is a part of central railway

route or 1 is a prime number. v. 4 + 5 9 or 3 5 10 Based on Exercise 1.3 1. Write the negation of each of the following

statements. i. 1 is greater than 5. ii. e is an irrational number. iii. 5 2 = 10 iv. It is not true that students are smart. v. Both the diagonals of a rectangle have

the same length.

Based on Exercise 1.4 1. Write the following statements in symbolic

form. i. Paris is not in France or London is not in

England. ii. If you access the website, then you will

have to pay the subscription fee. iii. It does not rains and I shall go to school. iv. Sita does not get promotion if and only

if sita is transferred to pune. v. It is not true that 5 is complex number. 2. If p: Sunday is a holiday and q: Ram does not study on holiday, express the

following statements in symbolic form. i. Sunday is not a holiday or Ram studies

on holiday. ii. If Sunday is not holiday then Ram

studies on holiday. iii. Sunday is holiday and Ram studies on

holiday. 3. Find the truth value of the following statements. i. If 5 + 4 = 9, then 9 3 = 12 ii. Square of any even number is even and

square of any negative number is negative. iii. 3 is an integer and 4 divides 19. iv. 23 is a prime number or 32 is a perfect

square. v. 2 + 3 = 5 if and only if 2 > 3 vi. Two parallel lines meet at a point. 4. Write the converse, contrapositive and inverse

of the following conditional statements: i. If you are good in Mathematics then you

are good in Logic. ii. If a triangle is equilateral, then it is

equiangular. 5. If p and q are true and r and s are false

statements, find the truth value of each of the following statements.

i. (~p q) ~(p q) ii. ~p (p ~q) iii. (~p q) (~q p) iv. (p q) (p r) v. ~(p q) (r s) 6. If p : Stock prices are high, q : Stocks are rising Give the compound statements in verbal form

denoted by i. p q ii. p q iii. p q iv. ~q p

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Publications Pvt. Ltd. Target 7. If p : It is a day time, q : It is warm, write the

compound statements in verbal form denoted by ~p q [Oct 14] Based on Exercise 1.5 1. Prepare the truth tables for the following

statement patterns: i. p (p q) ii. p [q (p q)] iii. (p q) (q p) iv. p (q r) 2. Using truth tables, prove the following logical

equivalances: i. (p q) (p q) p ii. p (p r) (p q) (p r) iii. ~p q (p q) ~ p

[Mar 14, Oct 13] 3. Using truth tables examine whether the

following statement patterns are tautology, contradiction or contingency.

i. (p q) r ii. p (p q ) q iii. [(q) p] [p (p)] iv (p q) (r q) v. ~(~p ~q) q [Mar 15] Based on Exercise 1.6 1. If A = {1, 2, 3, 4, …., 19, 20}, then determine

the truth value of each of the following: i. x A, such that x2 < 20 ii. x A, x 3 < 15 iii. x A, such that x is a even prime

number iv. x A, x 1 W 2. Use quantifiers to convert each of the

following open sentences defined on W, into a true statement:

i. x 5 < 1 ii. 3x + 7 < 16 iii. x2 4 = 32 iv. x2 + 2x + 5 = 13 Based on Exercise 1.7 1. Write the duals of each of the following

statements: i. p q r) ii. [(p q) r] F iii. c (~p ~q) iv. (p c) (~r t) v. (p q) T [Mar 14]

2. Write the dual statement of each of the following compound statements.

i. Sachin is a lawyer and he is honest. ii. Rahul plays hockey or cricket. iii. The film receives an award for its story

or for its direction. iv. India is in Asia or Rome is in Europe. 3. Write the duals of the following statements. i. p (q r) (p q) r ii. (p q) (p r) p (q r) 4. Write duals of each of the following statements

where t is a tautology and c is a contradiction. i. p q t ii. (q t) p iii. (p t) (c q) Based on Exercise 1.8 1. Write the negations of following statements: i. All pictures are colourful. ii. Some integers are not natural numbers. iii. Some dolls are attractive. iv. Every user has paid the bills. v. x I, x 2 < 2 vi. x R, such that x2 x 2 < 0 vii. 7 is a positive integer if and only if

Nasik is in Maharashtra. viii. If a quadrilateral is a rectangle then it is

a parallelogram. 2. Using the rules of negation. Write the

negations of the following: i. (p q) (p q) ii. (p q) (q r). iii. (p q) r 3. Without using truth table, show that p [(~p q) ~q] p 4. Form the negations of the following

statements by giving justification i. p (p ~q) ii. (~p ~q) (p ~q) Based on Exercise 1.9 1. Represent the following circuits symbolically

and write the input-output or switching table. i.

S1

S1 S2

S1

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Basic Physics (F.Y.Dip.Sem.-1) MSBTEChapter 1: Mathematical LogicStd. XII Sci.: Practice Problems

Publications Pvt. Ltd. Target ii. iii. 2. Construct the switching circuits of the

following statements. i. [p (p q)] (q p) ii. (p q r) [p (q r)] 3. Give an alternative arrangement for the following

circuit, so that the new circuit has three switches only. Also write the switching table.

4. Find the symbolic form of the following

switching circuit, construct its switching table and interpret your result.

5. Construct the new switching circuit for the

following circuit with only one switch by simplifying the given circuit: [Oct 13]

OR Construct the simplified circuit for the

following circuit: [Oct 15]

6. Find the symbolic form of the following switching circuit, construct its switching table and interpret it. [Mar 14]

Based on Miscellaneous Exercise – 1 1. State which of the following sentences are

statements. Justify your answer. In case of statements, write down the truth value.

i. Washington D.C. is in America. ii. Every relation is a function. iii. Every rectangle is a square. iv. 6 has three prime factors. v. The real number x is less than 2. 2. Write the truth value of each of the following

statements. i. 10 – 2 6 or 5 8 > 14 ii. A rectangle is quadrilateral or a 5-sided

polygon. iii. n N, such that n + 7 > 12 iv. n N, n + 5 > 7 3. If B = {7, 8, 9, 11, 13}, determine the truth

value of each of the following quantified statements.

i. x B, x + 7 54 ii. x B, such that x2 1 is divisible by 5 4. Write the negation of each of the following

statements. i. Australia is a continent. ii. There does not exist a quadrilateral

which has all its sides equal. iii. Bangalore is the capital of Karnataka. 5. Express the following statements in symbolic

form. i. If the school is closed, then there is a

holiday. ii. All rational numbers are real and all real

numbers are complex. iii. If it is raining, then the game is

cancelled. iv. Either ram is not in class X or ram is not

in class XII. v. He is not fat and he is not hard working.

LS3

S1 S2 S3

S1

S3

S2 S1

L

S1

S2 S3

S2 S1

S2

L

S1

S1

S2

S1

L

S2

S1

S2

S1

3S

L2S S1

S3

S2

1S S2

S1

2S

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Publications Pvt. Ltd. Target 6. If p : she is beautiful q : she is clever Give the verbal statements for the following

symbolic statements. i. ~p q ii. p ~q iii. ~p ~q iv. q p v. ~p q 7. If p : Girls are playing, q : Girls are happy,

which of the following statements are logically equivalent? Justify?

i. Girls are happy only if they are playing. ii. If girls are not playing then they are not

happy iii. Girls are playing it and only it they are

happy. iv. Girls are playing but they are not happy 8. Rewrite the following statements without

using the conditional form: i. If the problem is difficult, we take more

time to solve. ii. I can score good marks if I study hard. 9. If p and q are true and r and s are false

statements, find the truth value of each of the following.

i. (p q) (~q ~p) ii. ~[s (p q)] [(r s) (~r ~p)] 10. Change each of the following statement in the

form if …. then …. i. x = 0 only if x + n = n ii. Paying the electric bill is necessary

condition for me to get electric supply. 11. Use quantifiers to convert each of the

following open sentences defined on N, into true statement.

i. x3 = 216 ii. 4x + 5 > 7 12. Prepare the truth tables of the following

statements patterns. i. (p r) (q p) ii. (p q) (~p ~q) 13. Prove that the following statement pattern is a

tautology. (p q) p 14. Prove that the following statement pattern is a

contradiction. [~(p q)] (p q) 15. Find truth value of p and q in the following cases: i. (p q) is F and (p q) is T ii. (p ↔ q) is T and (p ↔ q) q is F

16. Examine, whether each of the following statement patterns is a tautology or a contradicition or a contingency.

i. ~[p (p q)] ii. (p ~p) (~p p) iii. (p q) (p r) 17. Using the rules of logic, prove the following

logical equivalances. i. (p q) p ≡ T ii. [(p q) (p r)] ≡ (p q) (p r) 18. Using truth table, verify i. p (q ~pp q ii. (p q) (~p q) [(p ~p) (q ~p)] q q 19. Write the converse, inverse and contrapositive of the following statement. If you get a job, then your credentials are

good. 20. With proper justification, state the negation of

the following. i. (~p q) (p r) ii. p (r s) 21. Represent the following circuit in symbolic form. 22. Construct the switching circuits of the

following statements: i. [(p q) r] [r (p q)] ii. (p q) (p q) (p q) (p q) 23. Give alternative arrangement of the following

circuit, so that the new circuit has minimum switches only.

L

S1 S3

S2 S3

S1 S3

S2 S3

S3 S2 S1

S1S2

S3 L


6767

Publications Pvt. Ltd. Target 01: Mathematical Logic

Based on Exercise 1.1 1. True statement are (ii), (iii), (xi), (xii). they have truth value T False statements are (i), (vi) they have truth value F Remaining are not the statements. Based on Exercise 1.2 1. i. p q ii. p q iii. p q iv. p q v. p q vi. p q vii. p q 2. i. T T T ii. T F T iii. T F T iv. T F T v. F F F Based on Exercise 1.3 1. i. 1 is not greater than 5. ii. e is not an irrational number. iii. 5 2 10 iv. It is true that students are smart. v. Both the diagonals of rectangle does not

have same length. Based on Exercise 1.4 1. i. ~p ~q ii. p q iii. ~p q iv. ~p q v. ~p 2. i. p q ii. p q iii. p q 3. i. F ii. F iii. F iv. T v. F vi. F 4. i. Converse : If you are good in

Logic then you are good in Mathematics.

Contrapositive : If you are not good in Logic then you are not good in Mathematics.

Inverse : If you are not good in Mathematics then you are not good in Logic.

ii. Converse : If a triangle is equiangular, then it is equilateral.

Contrapositive : If a triangle is not equiangular, then it is not equilateral.

Inverse : If a triangle is not equilateral, then it is not equiangular.

5. i. F ii. T iii. T iv. F v. F 6. i. Stock prices are high or stocks are rising. ii. If stock prices are high then stocks are

rising. iii. Stock prices are high and stocks are not

rising. iv. Stock are not rising if and only if stock

prices are high. 7. If it is not day time then it is warm. Based on Exercise 1.5 1. i.

p q ~ p ~ p q p (~ p q)

T T F T T

T F F F F

F T T T T

F F T T T ii.

p q p q q (p q) p [q (p q)]T T T T T T F F T T F T F F T F F F T T

iii.

p q ~p ~q p ~q q ~p ~(q ~p)(p ~ q)

~ (q ~ p)

T T F F F F T F T F F T T F T T

F T T F F T F F

F F T T F F T F

Basic Physics (F.Y.Dip.Sem.-1) MSBTEChapter 1: Mathematical LogicStd. XII Sci.: Answers to Practice Problems

68

Publications Pvt. Ltd. Target iv.

p q r q r p (q r) T T T T T

T T F F T

T F T F T T F F F T F T T T T F T F F F F F T F F F F F F F

3. i. contingency ii. contradiction iii. tautology iv. contingency v. contingency Based on Exercise 1.6 1. i. T ii. F iii. T iv. T 2. i. x W, such that x 5 < 1. It is true

statement since x = 5 W satiesfies x 5 < 1.

ii. x W, such that 3x + 7 < 16. It is true statement since x = 0 W satiesfies 3x + 7 < 16.

iii. x W, such that x2 4 = 32. It is true statement since x = 6 W satisfies x2 4 = 32.

iv. x W, such that x2 + 2x + 5 = 13. It is true statement since x = 2 W satiesfies x2 + 2x + 5 = 13.

Based on Exercise 1.7 1. i. p (q r) ii. [(p q) r] iii. t (~p ~q) iv. (p t) (~r c) v. (p q) F 2. i. Sachin is a lawyer or he is honest. ii. Rahul plays hockey and cricket. iii. The film receives an award for its story

and for its direction. iv. India is in Asia and Rome is in Europe. 3. i. p (q r) (p q) r ii. (p q) (p r) p (q r) 4. i. p q c ii. (q c) p iii. (p c) (t q) Based on Exercise 1.8 1. i. Some pictures are not colourful.

ii. All integers are natural numbers. iii. All dolls are not attractive. iv. Some users have not paid the bills. v. x I, such that x 2 2 vi. x R, x2 x 2 0 vii. 7 is a positive integer and Nasik is not in

Maharashtra or Nasik is in Maharashtra and 7 is not a positive integer.

viii. A quadrilateral is a rectangle and it is not a parallelogram.

2. i. (p q) (p q) ii. (p q) (q r) iii. (p q) (r) 4. i. p (~p q) ii. (p q) (~p q) Based on Exercise 1.9 1. i. [(q) p] [p (p)] ii. [(p) q] [(q) p] iii. [(p q) r] [(r) p] [r (q p)] 2. i. ii. 3.

1 1 1 1

0 0 0 0

S1

S2 S1

S2 S1 L

S3S2 S1

S1S2

S3 L

S1

LS2

S3

1 0 1 0 0 0 0 0


6969

Publications Pvt. Ltd. Target 4. is the switching table. Given circuit will always be off irrespective of

the status of the switches i.e. irrespective of the status of the switches, the lamp will never be on.

5. 6. (p q) (~p q) Based on Miscellaneous Exercise‐1 1. True statement are (i), (ii) they have truth value T False statements are (iii), (iv), (v) they have truth value F 2. i. T T ≡ T ii. T F ≡ T iii. T iv. F 3. i. T ii. T 4. i. Australia is not a continent. ii. There exists a quadrilateral which has

all its sides equal. iii. Bangalore is not the capital of Karnataka. 5. i. p q ii. p q iii. p q iv. ~p ~q v. ~p ~q 6. i. If she is not beautiful then she is clever. ii. she is beautiful or she is not clever. iii. she is neither beautiful nor clever. iv. she is clever if and only if she is beautiful v. she is not beautiful or she is clever 7. Statements (i) and (ii) are logically equivalent 8. i. The problem is not difficult or we take

more time to solve. ii. I do not study hard or I pass examination. 9. i. T ii. T 10. i. If x = 0 then x + n = n ii. If I get electric supply then I pay the

electric bill. 11. i. x N such that x3 = 216. It is a true

statement. Since, x = 6 N satisfies x3 = 216 ii. x N, such that 4x + 5 > 7. It is

true statement. Since, all x N satisfies 4x + 5 > 7

12. i.

p q r p r q p (p r) (q p)

TTTTFFFF

T T F F T T F F

T F T F T F T F

T F T F F T F T

T T F F F F T T

T F F F F F F T

ii.

p q ~p ~q p q ~p ~q (p q) (~p ~q)

T T F F T F T T F F T F F F F T T F F F F F F T T F T T

15. i. p is F and q is F ii. p is F and q is F 16. i. Contradicition ii. Contradicition iii. Contingency 19. Converse: If your credentials are good

then you get a job. Inverse: If you do not get a job then

your credentials are not good. Contrapositive: If your credentials are not

good then you do not get a job.

20. i. (~p q) (~p r) ii. ~p (r s)(s ~r)] 21. [(p q) r] [p (q r)] 22. i. ii.

23.

0 0 0 0

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1 1 1 1

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Documents

Std. 12 - Mathematics Practice Problems · Preface In the case of good books, the point is not how many of them you can get through, but rather how many can get through to you. “Std