PRACTICE PROBLEMS STD. XII Sci.

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2 MAHEMAICS PRACICE PROBEMS SD. XII Sci. irst Edition: November 2015 Salient eatures : Adequate Problems for Practice in each sub-topic opic and sub-topic wise classification of Problems at the beginning of every chapter. Printed at: Repro Knowledgecast td., 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. EID : 968 `

3 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. he 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. he 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 : mail@targetpublications.org Yours faithfully, Publisher APER Sr. No. Best of luck to all the aspirants! opic Name PAR - I Index Page No. Questions Answers 1 Mathematical ogic Matrices rigonometric unctions Pair of Straight ines Vectors hree Dimensional Geometry ine Plane inear Programming PAR - II 1 Continuity Differentiation Applications of Derivatives Integration Definite Integral Applications of Definite Integral Differential Equations Probability Distribution Binomial Distribution `

4 arget 01 Publications Pvt. td. Mathematical ogic Basic Physics (.Y.Dip.Sem.-1) Chapter MSBE 1: Mathematical ogic Chapter 01: Mathematical ogic ype of Problems Based on Exercise / Miscellaneous Q. Nos. Identify the statements and write down their ruth Value Express the statements in Symbolic orm/write the statement in Symbolic orm Write the ruth values of Statements Write the Negation of Statements/Using the Rules of Negation write the Negation of Statements Write the Verbal statement for the given Symbolic Statement Converse, Inverse and Contrapositive of the statement Using Quantifiers Convert Open sentences into rue statement Prepare the ruth able/ind Values of p and q for given cases ruth Examine the statement Patterns (autology, Contradiction, Contingency) Using ruth able, Verify ogical Equivalence 1.1 Q.1 Miscellaneous Q Q Q.1, 2 Miscellaneous Q Q Q.3, Q.1 Miscellaneous Q.2, 3, Q Q.1, 2, 4 Miscellaneous Q.4, Q.6, 7 Miscellaneous Q Q.4 Miscellaneous Q Q.2 Miscellaneous Q Q.1 Miscellaneous Q.12, Q.3 Miscellaneous Q.13, 14, Q.2 Miscellaneous Q.7, 18 Write Dual of the statement 1.7 Q.1, 2, 3, 4 Algebra of statements (without using ruth able verify the ogical Equivalence)/Rewrite the statement without using the conditional form 1.8 Q.3 Miscellaneous Q.8, 17 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

5 arget Publications Pvt. td. Based on Exercise 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. wo 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 aj Mahal? xi. 3 is root of equation x 2 5x + 6 = 0 xii. wo distinct points determine a unique line. Based on Exercise Express the following statements in symbolic form: i. wo 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 = 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. he 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 or Based on Exercise 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. Basic Physics (.Y.Dip.Sem.-1) Chapter MSBE 1: Mathematical ogic Std. XII Sci.: Practice Problems Based on Exercise Write the following statements in symbolic form. i. Paris is not in rance or ondon 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. ind the truth value of the following statements. i. If = 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 = 5 if and only if 2 > 3 vi. wo 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 ogic. 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 2

6 arget Publications Pvt. td. 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 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 If A = {1, 2, 3, 4,., 19, 20}, then determine the truth value of each of the following: i. x A, such that x 2 < 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. x 2 4 = 32 iv. x 2 + 2x + 5 = 13 Based on Exercise Write the duals of each of the following statements: i. p q r) ii. [(p q) r] iii. c (~p ~q) iv. (p c) (~r t) v. (p q) [Mar 14] Basic Physics (.Y.Dip.Sem.-1) Chapter MSBE 1: Mathematical ogic Chapter 01: Mathematical ogic 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. he 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 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 x 2 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. orm the negations of the following statements by giving justification i. p (p ~q) ii. (~p ~q) (p ~q) Based on Exercise Represent the following circuits symbolically and write the input-output or switching table. i. 3 3

7 arget Publications Pvt. td. Basic Physics (.Y.Dip.Sem.-1) Chapter MSBE 1: Mathematical ogic Std. XII Sci.: Practice Problems ii. 6. ind the symbolic form of the following switching circuit, construct its switching table and interpret it. [Mar 14] 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. ind 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] S 3 S 2 S 2 S 2 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. he real number x is less than Write the truth value of each of the following statements. i 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 ii. x B, such that x 2 1 is divisible by 5 4. Write the negation of each of the following statements. i. Australia is a continent. ii. here 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. 4

8 arget Publications Pvt. td. 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. x 3 = 216 ii. 4x + 5 > 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. ind truth value of p and q in the following cases: i. (p q) is and (p q) is ii. (p q) is and (p q) q is Basic Physics (.Y.Dip.Sem.-1) Chapter MSBE 1: Mathematical ogic Chapter 01: Mathematical ogic 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 ii. [(p q ) ( p r)] (p q ) (p r) 18. Using truth table, verify i. p (q ~p p 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. 5 5

9 arget Publications Pvt. td. 01: Mathematical ogic Basic Physics (.Y.Dip.Sem.-1) Chapter MSBE 1: Mathematical ogic Chapter 01: Mathematical ogic Based on Exercise rue statement are (ii), (iii), (xi), (xii). they have truth value alse statements are (i), (vi) they have truth value Remaining are not the statements. Based on Exercise i. p q ii. p q iii. p q iv. p q v. p q vi. p q vii. p q 2. i. ii. iii. iv. v. Based on Exercise i. 1 is not greater than 5. ii. e is not an irrational number. iii iv. It is true that students are smart. v. Both the diagonals of rectangle does not have same length. Based on Exercise 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. ii. iii. iv. v. vi. 4. i. Converse : If you are good in ogic then you are good in Mathematics. Contrapositive : If you are not good in ogic then you are not good in Mathematics. Inverse : If you are not good in Mathematics then you are not good in ogic. 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. ii. iii. iv. v. 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 i. p q ~ p ~ p q p (~ p q) ii. p q p q q (p q) p [q (p q)] iii. p q ~p ~q p ~q q ~p ~(q ~p) (p ~ q) ~ (q ~ p) 67 67

10 arget Publications Pvt. td. iv. p q r q r p (q r) 3. i. contingency ii. contradiction iii. tautology iv. contingency v. contingency Based on Exercise i. ii. iii. iv. 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 x 2 4 = 32. It is true statement since x = 6 W satisfies x 2 4 = 32. iv. x W, such that x 2 + 2x + 5 = 13. It is true statement since x = 2 W satiesfies x 2 + 2x + 5 = 13. Based on Exercise i. p ( q r) ii. [(p q) r] iii. t (~p ~q) iv. (p t) (~r c) v. (p q) 2. i. Sachin is a lawyer or he is honest. ii. Rahul plays hockey and cricket. iii. he 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 i. Some pictures are not colourful. Basic Physics (.Y.Dip.Sem.-1) Chapter MSBE 1: Mathematical ogic Std. XII Sci.: Answers to Practice Problems 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, x 2 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 i. [( q) p] [p ( p)] 2. i. 3. ii. [( p) q] [( q) p] iii. ii [(p q) r] [( r) p] [r (q p)]

11 arget Publications Pvt. td 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 (p q) (~p q) Based on Miscellaneous Exercise 1 1. rue statement are (i), (ii) they have truth value alse statements are (iii), (iv), (v) they have truth value i. ii. iii. iv. 3. i. ii. 4. i. Australia is not a continent. ii. here 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. he problem is not difficult or we take more time to solve. ii. I do not study hard or I pass examination. 9. i. ii. 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 x 3 = 216. It is a true statement. Since, x = 6 N satisfies x 3 = 216 ii. x N, such that 4x + 5 > 7. It is true statement. Since, all x N satisfies 4x + 5 > i. Basic Physics (.Y.Dip.Sem.-1) Chapter MSBE 1: Mathematical ogic Chapter 01: Mathematical ogic p q r p r q p (p r) (q p) ii. p q ~p ~q p q ~p ~q (p q) (~p ~q) 15. i. p is and q is ii. p is and q is 16. i. Contradicition ii. Contradicition iii. Contingency 19. Converse: If your credentials are good Inverse: 69 then you get a job. 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