060010105-Mathematics for Computer Applicationsutu.ac.in/dcst/download/2015-16/INTMSCIT/Sem1/QB/QBMScIT... · 2015. 7. 24. · Convert the following binary numbers to hexadecimal

060010105-Mathematics for Computer Applications

2015

Mr. Nikhil Choksi Page 1

Unit-1: Data Representation & Binary Mathematics Short Questions

1. How many unique strings are there with 6 bits in each string? 2. How many bits per character does ASCII code and ISCII use? 3. If the capital English letters, digits and 16 special characters are to be coded as

strings of bits, how many bits will be required for each string? 4. Define a byte. 5. How many types of representations are there for representing the negative

numbers? 6. Which are codes defined by ASCII coding System? 7. What are the ASCII and ISCII codes? 8. Why non-printable characters are assigned code in ASCII? 9. Convert the following binary numbers into equivalent decimal numbers:

(a) 1011010 (b) 0101111 (c) 111111 (d) 100001 (e) 110110 (f) 111000111 (g) 001100 (h) 000011 (i) 011100 (j) 111100 (k) 101010

10. What is the decimal equivalent of the fraction 0.647? 11. What are the binary equivalents of the following decimal fractions?

(a) 0.7625 (b) 0.8 (c) 0.245 (d) 0.390625 (e) 24.625 (f) 0.8 (g) 0.3 (h) 34.75 (i) 25.25 (j) 27.1875

10. Convert the following decimal numbers into equivalent binary numbers: (a) 7 (b) 20 (c) 31 (d) 64 (e) 285(f) 473 (g) 694(h) 64 (i) 100 (j) 111 (k)145 (l) 255.

12. What are the decimal equivalents of the following binary fractions? (a)0.01101 (b) 0.11010 (c) 0.00011 (d) 110.11, (d) 1010.10101 (e)1111011.101 (f) 11000.0011 (g) 0.1010101 (h) 1010101.1010101 (i)11100.011 (j) 110011.10011 (k) 1010101010.1

13. Convert the following binary numbers into its hexadecimal values: (a)100101010101 (b) 0101110111101 (c) 1011000001 (d) 1010101000 (e)111110001010 (f) 1010101010100 (g) 1110001010 (h) 10100001010

14. Convert the following binary numbers to hexadecimal and then to decimal: (a) 1000100101 (b) 1101011 (c) 101001011111.0010001 (d) 100101010101 (e)0101110111101 (f) 1011000001 (g) 101010101010 (h) 11100101010

15. Convert the following decimal numbers to hexadecimal equivalents: (a)285.48 (b) 3452.645 (c) 678920.45 (d) 204.125 (e) 255.875 (f) 631.25 (g)10000.00390625

16. Give the place value of each underlined bit: (a)111000 (b) 11001100 (c) 111.000111 (d) 11.00110011

17. Rewrite in expanded notation the following: (a)11001100 (b) 111.000111.

18. Find the Binary Sums: (a)1101+111 (b) 110011+11101 (c) 11100111+11000011 (d) 110.1101+1011.101 (e)1110.1101 + 110101.01101 (f) 1011011.111 + 1010110.1010


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(g)110111.11 + 11011101.0101 (h) 11001 +11100 + 1011 + 110011 19. Find the Binary products:

(a)11100111 x 11 (b) 111011 x 1011 (c) 11.101 x 11.01 (d)10011 x 1101 (e) 110.101 x 1011.001 (f) 1101.101 x 110101.11

20. Find the Binary differences: (a)1100011-110111 (b) 10101010-110011 (c) 110.001-11.111 (d) 10101.1010 – 10001.0011 (e) 11011.011 - 11110.101 (f) 1001000.001 – 1000011.011

21. Find the quotients:

(a) 1011011 111 (b) 100.0001 10.1 (c) 1011 11 (d)1011/011 (e) 110111/1011 (f) 110001.110/111.110

22. Find the twos complements of the binary numbers (a)110011 (b) 10001000 (c) 10111011101 (d) 111000000111

23. Convert the following hexadecimal numbers to their decimal equivalents: (a) C (b) 9F (c) D52 (d) 67E (e) ABCD

23. Convert the following hexadecimal numbers to their decimal equivalents: (a) F.4 (b) D3.E (c) 1111.1 (d) 888.8 (e) EBA.C

24. Convert the following decimal numbers to their hexadecimal equivalents: (a) 16 (b) 80 (c) 2560 (d) 3000 (d) 62.500

25. Convert the following hexadecimal numbers to their binary equivalents: (a)E (b) 1C (c) A64 (d) IF.C (e) 239.4

26. Convert the following binary numbers to their decimal equivalents: (a)1001.1111 (b) 110101.011001 (c) 10100111.111011

Long Questions 1. What is Unicode? What is the advantage of using Unicode? 2. Under what circumstances are decimal digits coded using ASCII? 3. Under what circumstances are decimal numbers converted into binary numbers?

4. What is the difference between the bits used in a code such as ASCII and the bits used in binary numbers?

5. What is the advantage of using hexadecimal numbers?

6. Define the base of a number system and state the radix of the Decimal, Binary and Hexadecimal number systems.

7. Explain the terms “External representation” and the “Internal representation” of data in a Computer?

8. What are the considerations that govern the selection of a representation for storing and processing data in a computer?

9. Why are binary digits used to code data to be stored in a computer? 10. How many average binary digits are required to represent a decimal digit?

Explain in detail. 11. Explain in detail the types of the representations for the negative numbers. 12. Explain the normalized floating point mode of representing and storing real

numbers?

13. Subtract the following binary numbers using 2‟s complement representation of


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negative numbers: (a) 10101-10001 (b) 11011-1110 (c) 100100-100011

14. Convert the following binary numbers to their decimal equivalents: (a)1001.1111 (b) 110101.011001 (c) 10100111.111011

15. Subtract the following binary numbers using 2‟s complement representation of negative numbers: (a) 101101.0011-100101.0001 (b) 11011.110-101.001 (c) 10111.1001-11000.1101

16. Add the following binary numbers using 16-bit floating point representation: (a) 1011011.110101 + 110101.0101 (b) 11011.1101 + 1011.10110 (c) 0.000111 + 111.00111

17. Subtract the following binary numbers using a 16-bit floating point representation: (a) 1011011.1101 – 01110.1101 (b) 101110.110 – 1110.0011 (c)100110.101 –

011.110011 18. Multiply the following numbers using a 16-bit floating point representation:

(a) 1011.110 x 1010.110 (b) 0.11011 x 11011.1101 (b) 1110111.111 x 11011.1101

19. Divide the following numbers using a 16-bit floating point representation: (a) 1011.110 / 1010.110 (b) 0.11011 / 0.111011 (c) 110111.111 / 11011.1101

Multiple Choice Questions 1. What does a decimal number represents?

a. Quality b. Quantity c. Position d. None of the above

2. Why the decimal number system is also called as positional number system? a. Since the values of the numbers are decided by multiplying the values. b. Since the values of the numbers are decided by the weight of the values. c. Since the values of the numbers are decided by adding the values. d. Since the values of the numbers are decided by the position of the values.

3. In binary number system a digit ranges from _. a. 0 to 9 b. 0 to 1 c. 0 to 15 d. 0 to 7

4. How can you represent a decimal point? a. By a series of coefficients. b. By weight decided by its position. c. By location as well as base d. None of the above

5. A digit in base R will have a range from .

a. 1 to R-1 b. 0 to R-1 c. 1 to R+1


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d. 0 to R+1 6. Conversion from any base to decimal base is done by each digit by its

corresponding weight and then all the individual products to get the equivalent decimal value. a. Multiplying, Adding b. Adding, Multiplying c. Dividing, Adding d. Adding, Subtracting

7. Which method is used to convert a number from an octal base to decimal base? a. Direct conversion method b. Decimal equivalent method c. Octal equivalent method d. Positional notation method

8. In which conversion the product of number 16 raised by the location and then adds all the products to get the final decimal value? a. Octal to decimal b. Binary to Decimal c. Hexadecimal to decimal d. None of the above

9. Binary numbers can be converted into equivalent octal numbers by making groups of three bits _. a. Starting from the MSB b. Starting from the LSB c. Ending at the MSB d. Ending at the LSB

10. What is the octal equivalent of 58, 3-bit binary number? a. 0102

b. 1102

c. 0002

d. 1012

11. In direct conversion from binary to hexadecimal, if the last group does not have 4- bits, then it is padded with to make it four bits. a. Zeros b. Ones c. Two zeros and two ones d. One zero and three ones

12. What is the hex equivalent of 916, a 4-bit binary number? a. 11112

b. 10012

c. 01102

d. 11002

13. The sign information has to be encoded along with the to represent the integers completely. a. No. of bits


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b. Position c. Magnitude d. Weight

14. Which one is the possible technique for representing signed integers? a. Signed Magnitude Representation b. Diminished Radix-Complement Representation c. Radix-Complement Representation d. All of the above

15. What is used to represent the signed magnitude? a. MSB b. LSB c. Both d. None of the above

16. What is the corresponding hex number of the signed magnitude -127? a. (7F)16

b. (FF)16

c. (00)16

d. (80)16

17. What are the two ways of representing the 0 with signed magnitude representation? a. -0 and -0 b. +0 and +0 c. -0 and +0 d. None of the above

18. (FA)16 is the one’s complement representation of -5. a. 4-bit b. 8-bit c. 16-bit d. 2-bit

19. 2’s complement is used to represent signed integers, especially integers. a. Negative b. Positive c. Both A and B d. None of the above

20. In _, to encode a negative number, first the binary representation of its magnitude is taken, complement each bit and then add 1.

a. Signed integer representation b. 1’s complement representation c. 2’s complement representation d. Radix complement representation

Unit-2: Mathematical Logic Short Questions

1. What do you mean by logic? 2. Which of the following sentences are propositions (or statements)? What are the

truth values of those that are propositions?


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a. What a beautiful solar eclipse? b. Every triangle is an equilateral triangle. c. Please, stand up. d. Every quadrilateral is a square. e. 4 + x = 5. f. There is no pollution in Surat. g. 2 +1=3. h. The summer in India is hot and sunny.

3. Define the following terms and give the truth tables for each one by considering the propositions p and q.

a. Conjunction and Disjunction Connectives. b. Conditional and Bi-conditional propositions. c. Exclusive Or

4. Define the connectives conjunction and disjunction and give the truth tables for p ʌ q and p v q.

5. Give the equation of commutative law. 6. Define conditional and biconditional propositions and also give the truth table for

p → q and p ↔ q. 7. Define tautology and contradiction with simple examples. 8. When do you say that two compound propositions are equivalent? 9. Define the dual of compound propositions with an example. 10. What is the law of duality? 11. Give the primal and dual forms of the distributive law. 12. Give the primal and dual forms of the absorption law. 13. Give the primal and dual forms of the De Morgan’s law. 14. Define tautological implication with an example. 15. When is a statement said to be satisfiable? 16. Define disjunctive and conjunctive normal forms of a statement. 17. Define PDNF and PCNF of a statement. 18. What do you mean by declarative sentence? 19. What is universe of discourse? 20. Which symbol is used to define universal quantifier? 21. Give the equation of idempotent law. 22. What is DNF?

Long Questions 1. State and prove the following by using Truth Table.

a. Laws of Tautology and Contradiction. b. Commutative Laws. c. Associative Laws. d. De-Morgan’s Laws. e. Distributive Laws. f. Law for Negation of a negation (or involution law)

2. Laws for absorption. 3. Construct the truth table for each of the following compound propositions: 4. (p ʌ q) → (p v q)


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5. (p → q) ↔ ( ˥q → ˥p) 6. (p →q) v (˥p → q) 7. (p →q) ʌ (˥p → q)

8. (p ↔q) v (˥p → q) 9. Determine which of the following statements are tautologies or contradictions. 10. (p → ˥p) → ˥p 11. P → (p v q) 12. (˥q → p) ʌ q 13. (q → p) ʌ (˥p ʌ q) 14. Prove the following equivalences: 15. ˥ (p → q) ≡ p ʌ ˥q 16. (p ʌ q) v (p ʌ ˥q) ≡ p 17. (p v q) ʌ (p v ˥q) ≡ p

18. ˥ (p ↔ q) ≡ (p ʌ ˥q) v (˥p ʌ q) 19. Write down the duals of the following statements: 20. ˥ (p v q) v [(˥p) ʌ q ] v p 21. ˥p → (p → q) 22. (p ʌ q) → (p → q) 23. (p → q) → (˥q → ˥p) 24. Prove the following implications, using truth tables:

25. (p ʌ q) ⇒ (p → q)

26. (q →p) ⇒ (˥p → ˥q) 27. (p → q) ⇒p → (p ʌ q)

28. (p → q) → q ⇒ p v q 29. Find a DNF or a CNF of the following: 30. (p ʌ (p → q)) 31. ˥(p → q)

32. ˥(p↔q) 33. q ʌ (p v ˥q) 34. Find the PDNF of the following statements using truth tables: 35. p ʌ (p → q)

36. ˥ (p v q) ↔ p ʌ q

37. q ʌ (p v ˥q) 38. (q → p) ʌ (˥p ʌ q) 39. Find the PCNF of the following statements using truth tables:

40. p ↔ q 41. (p v q) → (p ʌ q) 42. (˥(p v q)) v (p ʌ q) 43. (q → p) ʌ (˥p ʌ q) 44. p → (p ʌ (q → p)) 45. Without using truth tables, prove the following equivalences: 46. (˥p →˥q) → (q → p) ≡ T

47. (p v q) ʌ (˥p ʌ (˥p ʌ q)) ≡ (˥p ʌ q) 48. Find the CNF of the following statements:

49. ˥(p v q) ↔(p ʌ q)


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50. ˥((p v ˥q) ʌ ˥ r)

Fill in the blanks 1. A(n) sentence which is true or false, but not both, is called a

proposition. 2. Sentences which are , or in

nature are not propositions. 3. If a proposition is true, then the truth value of that proposition is true, denoted by

or _. 4. Mathematical statements which can be constructed by combining one or more

atomic statements using connectives are called propositions. 5. Propositions which do not contain any of the logical operators or connectives are

called propositions. 6. The area of logic that deals with propositions is called _. 7. The compound proposition “p if and only if q” is denoted by and is

called a proposition. 8. is the primal form of an Identity law. 9. is the dual form of an Absorption law. 10. A formula which consists of a product of elementary sums and which is equivalent to

a given formula is called a of the given formula.

Multiple Choice Questions 1. Which of the following is a statement in logic?

a. What a wonderful day! b. Shut up! c. What are you doing? d. New Delhi 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. 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 4. ~(p ∨ q) is

a. ~p ∨ q b. p ∨ ~q c. ~p ∨ ~q

d. ~p ∧ ~q 5. 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


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a. p → q b. p ↔ q

c. p ∧ q

d. p ∨ q 6. The connective in the statement “2 + 7 > 9 or 2 + 7 < 9” is

a. and b. or c. > d. <

7. 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

8. 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 is sweet fruits.

9. ~[p ∨ (~q)] is equal to

a. ~p ∨ q

b. (~p) ∧ q

c. ~p ∨ ~p

d. ~p ∧ ~q 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. 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

12. 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

13. The symbolic form of the statement ‘It is not true that intelligent persons are neither polite nor helpful’ is

a. (~(p ∨ q)


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Mr. Nikhil Choksi & Ms. Chetna Tailor Page 10

b. ~(~p ∧ ~q) c. ~(~p ∨ ~q) d. ~(p ∧ q)

14. 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]

15. 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

16. If p → (q ∨ r) is false then the truth values of p, q, r are respectively. a. T, F, F b. F, F, F c. F, T, F d. T, T, F

17. (p ∧ ~q) ∧ (~p ∧ q) is a a. Tautology b. Contradiction c. Tautology and a contradiction d. Contingency

18. The proposition (p → ∼p) ∧ (∼p → p) is a a. Neither tautology nor contradiction b. Tautology c. Tautology and contradiction d. Contradiction

19. Negation of the statement: “If Dhoni loses 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. 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

True-False 1. Sentences which are exclamatory, interrogative or imperative in nature are

propositions. 2. “Take a cup of coffee” is propositions.

3. “New Delhi is the capital city of India” is propositions. 4. Propositions which do not contain any of the logical operators or connectives are

called atomic propositions.


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5. The propositions “p or q” is denoted by p ʌ q. 6. The conditional and biconditional operators have higher precedence than other

operators. 7. The propositions “p and q” is denoted by p v q. 8. A product of the variables and their negations is called an elementary product. 9. A compound proposition which consists of a sum of elementary products and which

is equivalent to a given proposition is called a conjunctive normal form of the given formula.

10. The products in which each variable or its negation, but not both, occurs only once are called the minterms.

11. The disjunction operator has precedence over the conjunction operator. 12. A formula consisting of conjunctions of maxterms in the variables in the variables

only and equivalent to given formula is known as its principal conjunctive normal form.

13. The negation operator has precedence over all other logical operators. 14. The compound proposition p → q is true when p is true and q is false and false

otherwise. 15. In conditional proposition, p is called the consequence and q is called the premise. 16. The value of p → q is F (false) then value of p and q are F (false) and T (true)

respectively. 17. The compound proposition “p if and only if q”, which is true when p and q have the

same truth values and is false otherwise.

18. The symbol ∃ is called the existential quantifier. 19. “How beautiful is rose?” is called a proposition. 20. If P(x) ≡ {(-x)2 = x}, where the universe consists of all integers, then the truth value of

∀x({(-x)2 = x2) is T. Unit-3: Set Theory Short Questions

1. Define: set 2. Define subset and proper subset. 3. Explain the roster notation and set builder notation of sets with examples. 4. Define null set and singleton set. 5. What is cardinality of a set? 6. State the relation between the cardinalities of a finite set and its power set. 7. When are two sets said to be equal? 8. What is a power set? 9. Define the Cartesian product of two sets and give an example. 10. Define complement and relative complement of a set. Give examples. 11. When are two sets said to be disjoint? 12. Define union and intersection of two sets. Give their Venn diagram representation. 13. Define the symmetric difference of two sets. 14. Define the following terms with an example.

a. Identity law b. Domination law c. Idempotent law


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d. Inverse law e. Commutative law f. Associative law g. Distributive law h. De Morgan’s law i. The principle of duality

15. State the following properties of a relation with an example: a. Reflexive

b. Symmetric

c. Antisymmetric

d. Transitive

16. Give an example of a relation that is both symmetric and antisymmetric. 17. Give an example of a relation that is neither symmetric nor antisymmetric. 18. Give an example of a relation that is reflexive and transitive but not symmetric. 19. Give an example of a relation that is symmetric and transitive but not reflexive. 20. Define a lattice and give an example of a lattice. 21. State the principle of duality with respect to lattices 22. State the four basic properties of a lattice.

23. Draw the Hasse diagram of all lattices with 5 elements. 24. State the isotonic property of a lattice. 25. Write down the distributive inequalities of a lattice. 26. State the modular inequality of a lattice. 27. Define a lattice as an algebraic system. 28. Define sublattice. 29. Define a function, domain, codomain and range of the function. 30. Define the following terms of a function:

a. Domain b. Co domain c. Range d. Injective function e. Bijective function

31. Define binary and n-ary operations. 32. State the closure property of a binary operation with an example. 33. When is an element � ∈ � said to be (a) idempotent and (b) identity with respect to

a binary operation?

34. Define the inverse of an element � ∈ � with respect to a binary operation.

Long Questions 1. Given that U = {1,2,3,….,9,10}, A = {1,2,3,4,5}, B = {1,2,4,8}, C = {1,2,3,5,7} and

D = {2,4,6,8}, find each of the following:

a. (� ∪ �) ∩ �

b. A ∪ (B ∩ C)

c. C′ ∪ D′

d. (C ∪ D)′

e. (A ∪ B) − C


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f. A ∪ (B − C)

g. (B − C) − D

h. B − (C − D)

i. (A ∪ B) − (C ∩ D)

j. (A − B) ∪ (C − D)

k. A (B ∩ C)

l. A ∪ (B C) 2. If R is the relation from A = {1,2,3,4,5,6} to B = {3,4,5,6}, list the element in R Defined

by ��, if � and � are both odd. Write also the domain and range of R. 3. If R is a relation from A = {1,2,3} to B = {4,5} given by R = {(1,4), (2,4), (1,5), (3,5)},

Find R-1 (the inverse of R) and R'(the complement of R). 4. If R = {(1,1), (2,2), (3,3)} and S = {(1,1), (1,2), (1,3), (1,4)} find R S . 5. Write the dual of each of the following statements in a lattice:

a. (a ˄ b) ˅ c = (b ˅ c) ˄ (c ˅ a)

b. (� ˄ �) ˅ � = � ˄ (� ˅ �) 6. Give a function f ∶ A → B, where |A|, |B| ≥ 4, give an example of f as a set of ordered pairs such that f is

a. Neither one-to-one nor onto. b. One-to-one but not onto. c. Onto but not one-to-one. d. One-to-one and onto.

7. Show that the function f ∶ N × N → N givumen by f(m, n) = m + n is onto but not one-to-one.

8. Show that the function f ∶ N × N → N given by f(m, n) = mn is onto but not one-to- one.

9. If � ∶ � → � is given by (a) �(�) = 3� − 7 and �(�) = x3 – 2, find inverse of � in each case.

10. If �, � ∶ � → � are defined by �(�) = 2� + 5 and �(�) = 1

(� − 5), show that � 2

and � are inverses of each other.

11. Show that x ∗ y = x − y is not a binary operation on the set of natural numbers, but it is a binary operation on the set of integers.

12. Show that the binary operation on the set of nature numbers given by a ∗ b = a is not commutative, but is associative.

13. Determine whether the operation ∗ on the set of natural numbers given by a ∗ b = a+b is a binary operation. ab 14. Prove the following analytically or graphically:

a. A – B = A ∩ B′

b. A – (A ∩ B)=A – B

c. A – B = B′ − A′

15. Find the quotient set of {1, 2, 3} under the relation {(1,1),(1,2),(2,1),(2,2),(3,3)}. 16. Draw the directed graph representing the relation on {1,2,3,4} given by the ordered

pairs {(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)}. 17. For each of the following statements in which A, B, C and C are arbitrary sets, either

prove that it is true or give a counter example to show that it is false.


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a. A ∩ C = B ∩ C → A = B b. (A – C) = (B - C) → A = B

c. A x (B ∪ C) = (A x B) * (A x C) Fill in the blanks

1. A set is a well-defined collection of objects, called the or of the set.

2. The set which contains no elements at all is called the set. 3. The set which contains all the objects under consideration is called the

set and denoted as . 4. The Null set or Empty set is denoted by the symbol or _. 5. A set which contains only one element is called a(n) set. 6. A pair of objects whose components occur in a specific order is called a(n)

_. 7. A set is represented in two ways, namely and . 8. A set which contains element is called a singleton sets. 9. A set which contains a finite number of element is called a(n) set and a

set with infinite number of elements is called a(n) set. 10. The cardinality or size of set A is denoted by _. 11. The union of two sets A and B is denoted by _.

12. The of two sets A and B, denoted by A∩ B. 13. The difference of two sets A and B is denoted by _. 14. A relation R on a set A is called a(n) _, if R is reflexive, symmetric

and transitive. 15. A relation R on a set A is called a(n) relation, if R is reflexive,

antisymmetric and transitive.

Multiple Choice Questions

1. Let A = {2, {4, 5}, 4}. Which statement is correct?

a. 5 is an element of A. b. {5} is an element of A. c. {4, 5} is an element of A. d. {5} is a subset of A.

2. Which of these sets is finite?

a. {x | x is even} b. {x | x < 5} c. {1, 2, 3,...} d. {1, 2, 3,...,999,1000}

3. Which of these sets is not a null set?


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a. A = {x | 6x = 24 and 3x = 1} b. B = {x | x + 10 = 10} c. C = {x | x is a man older than 200 years} d. D = {x | x < x}

4. Let D E. Suppose a D and b E. Which of the following statements must be true?

a. c D b. b D c. a E d. a D

5. Let A = {x | x is even}, B = {1, 2, 3,..., 99, 100}, C = {3, 5, 7, 9}, D = {101, 102} and E = {101, 103, 105}. Which of these sets can equal S if S A and S and B are disjoint?

a. A b. B c. C d. D e. E

6. Which statement best describes the Venn diagram below?

a. A = B b. A and B are not comparable c. A B d. A B

7. Let A = {x, y, z}, B = {v, w, x}. Which of the following statements is correct?

a. A B = {v, w, x, y, z} b. A B = {v, w, y, z} c. A B = {v, w, x, y} d. A B = {x, w, x, y, z}

8. Let A = {1, 2, 3, ..., 8, 9} and B = {3, 5, 7, 9}. Which of the following statements is correct?.

a. A B = {2, 4, 6}

b. A B = {1, 2, 3, 4, 5, 6, 7, 8, 9} c. A B = {1, 2, 4, 6, 8} d. A B = {2, 4, 6, 8}

9. Let A = {2, 3, 4}, B = {3} and C = {x | x is even}. Which statement is correct?

a. C A = B b. C B = A


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c. A C d. C / A = B

10. Let A B, B C and D A = C. Which statement is always false?

a. B D b. A C c. A = B

d. B D = and B A 51. What is shaded in the Venn diagram below?.

a. A B b. A B c. A d. B

52. What is shaded in the Venn diagram below?.

a. A B b. A' c. A - B d. B - A

13. Let U = {1, 2, 3, ..., 8, 9} and A = {1, 3, 5, 7}. Find A'.

a. A' = {2, 4, 6, 8} b. A' = {2, 4, 6, 8, 9} c. A' = {2, 4, 6}


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d. A' = {9}

14. Which of the diagrams defines a function of A = {a, b, c, d} into B = {1, 2, 3}

a.

b.

c.

d.

15. Let f 1, f 2, f 3, f 4, f 5 be functions of R into R and let f 1(x) = x2 + 3x - 4. Which of these functions are equal to f 1?

a. f2(x) = x2

b. f3(y) = y2 - 4 c. f4(z) = z2 + 3z - 4 d. f5(x) = x2 + 3x

16. Which of the following functions is one-one?

a. To each show assign its first performance. b. To each student assign his mentor. c. To each pair of shoes assign its price. d. To each school assign the number of computers it has.


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17. Let A = {1, 2, 3, 4}. Let f, g and h be functions of A into R. Which one of them is one-one?

a. f(1) = 3, f(2) = 4, f(3) = 5, f(4) = 3 b. g(1) = 2, g(2) = 4, g(3) = 5, g(4) = 3 c. h(1) = 2, h(2) =4, h(3) = 3, h(4) = 2

18. Let A = [-2,-3], B = [-3,5] and C = [2,-2]. Which of the following functions is not one-one?

a. f : A R b. g : B R c. h : C R

19. Find the ordered pairs corresponding to the points A and B, which appear in the coordinate diagram {1, 2, 3} × {1, 2, 3} below.

a. A = (2, 1), B = (1, 3) b. A = (1, 2), B = (3, 1) c. A = (3, 1), B = (1, 2) d. A = (2, 2), B = (2, 3)

20. Let S = {a, b, c, d}. Which of the following sets of ordered pairs is a function of S into S?

a. {(a, b), (c, a), (b, d), (d, c), (c, a)}

b. {(a, c), (b, c), (d, a), (c, b), (b, d)} c. {(a, c), (b, d), (d, b)} d. {(d, b), (c, a), (b, e), a, c)}

True-False 1. N= {0, 1, 2, 3, ...], is the set of integers. 2. Z+= {1,2,3, ...}, is the set of natural numbers. 3. A set which contains more than one element is called a Singleton set. 4. A set with infinite number of elements is called an infinite set. 5. The set A is said to be a subset of B, if and only if every element of A is also an

element of B. 6. If A is a subset of B, then A is called the superset of B. 7. The union of two sets A and B is the set of elements that belong to A and to B and to

both.


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8. If A={1,2,3}, B={2,3,4} then A U B= {1,2,3,4}. 9. If two sets A and B do not have any element in common, then the sets A and B are

said to be disjoint. 10. If A and B are any two sets, then the set of elements that belong to A but do not

belong to B is called the relative component of B with respect to A and is denoted by A\B.

11. A relation can be thought of as a structure that represents the relationship of elements of a set to the elements of another set. 12. A relation R on a set A is called a universal relation, if R=A+A.

13. A relation R on set A is said to be symmetric, if a R a for every a ∈ A, if (a, a) ∈ R for every a ∈ A. 14. A relation R on a set A is said to be antisymmetric, if whenever a R b then b R a, if

whenever (a, b) ∈ R then (b, a) also ∈ R. 15. A relation R on a set A is said to be transitive, if whenever a R b and b R c then a R c. 16. A relation R on a set A is called an equivalence relation, if R is reflexive,

antisymmetric and transitive. 17. A relation R on a set A is called a partial ordering or partial order relation, if R is

reflexive, antisymmetric and transitive. 18. The lattices {L, ≤} and {L, ≥} are called the duals of each other. 19. A function which is not algebraic is called a transcendental function.

20. The set {a ∈ A|a R b, for some b ∈ B} is called the domain of R and denoted by D(R).

Unit-4: Elementary Combinatorics Short Questions

1. State the sum rule in Counting Principle. 2. State the product rule in Counting Principle. 3. A new company with just two employees, A and B, rents a floor of a building with 12

offices. How many ways are there to assign different offices to these two employees?

4. The chairs of an auditorium are to be labelled with a letter and a positive integer not exceeding 100. What is the largest number of chairs that can be labelled differently?

5. What are the different circular arrangements of n objects? 6. There are 32 microcomputers in a computer centre. Each microcomputer has 24

ports. How many different ports to a microcomputer in the center are there? 7. How many different bit strings of length seven are there? 8. How many bit strings of length eight either start with a 1 bit or end with the two bits

00? 9. State the Pigeonhole Principle. 10. Among any group of 367 people, how many of them be at-least with the same birth-

date? 11. In any group of 27 English words, how many must be at-least begin with same letter? 12. List all the permutations of {a, b, c}. 13. How many different permutations are there of the set {a, b, c, d, e, f, g}? 14. How many different permutations of {a, b, c, d, e, f, g} end with a? 15. Find the values of each of these quantities.

(a) P(6,3); (b) P(6,5); (c) P(8,1); (d) P(8,5); (e) P(8,8); (f) P(10,9)


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16. Find the values of each of these quantities. (a) C(5,1); (b) C(5,3); (c) C(8,4); (d) C(8,8); (e) C(8,0); (f) C(12,6)

Long Questions 1. If there are 22 mathematics majors and 335 computer science majors at a college,

then find (a) How many ways are there to pick two representatives so that one is mathematics major and the other is a computer science major? (b) How many ways are there to pick one representative who is either a mathematics major or a computer science major?

2. A multiple choice test contains 10 questions. There are five possible answers for each question. (a) How many ways can a student answer the questions on the test if the student answers every question? (b) How many ways can a student answer the questions on the test if the student can leave answers blank?

3. Assuming that repetitions are not permitted, (a) How many four digit numbers can be formed from the six digits 1,2,3,5,7, 8? (b) How many of these numbers are less than 4000? (c) How many of the numbers in part (a) are even? (d) How many of the numbers in part (a) are odd? (e) How many of the numbers in part (a) are multiples of 5? (f) How many of the numbers in part (a) contain both the digits 3 and 5.

4. (a) In how many ways can 6 boys and 4 girls sit in a row? (b) In how many ways can they sit in a row if the boys are to sit together and the girls are to sit together? (c) In how many ways can they sit in a row if the girls are to sit together? (d) In how many ways can they sit in a row if just the girls are to sit together?

5. Suppose that either a member of the mathematics faculty or a student who is a mathematics major is chosen as a representative to a university committee. How many different choices are there for this representative if there are 37 members of the mathematics faculty and 83 mathematics majors and no one is both a faculty member and a student?

6. A student can choose a computer project from one of three lists. The three lists contain 23, 15, and 19 possible projects, respectively. No project is on more than one list. How many possible projects are there to choose from?

7. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building?

8. In how many ways can we select three students from a group of five students to stand in line for a picture? In how many ways can we arrange all five of these students in a line for a picture?

9. How many ways are there to select a first prize winner, a second prize winner, and a third prize winner from 100 different people who have entered a contest?

10. Suppose that there are eight runners in a race. The winner receives a gold medal, the


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second place finisher receives a silver medal, and the third place finisher receives a bronze medal. How many different ways are there to award these medals, if all possible outcomes of the race can occur and there are no ties?

11. Suppose that a saleswoman has to visit eight different cities. She must begin her trip in a specified city, but she can visit the other seven cities in any order she wishes. How many possible orders can the saleswoman use when visiting these cities?

12. How many poker hands of five cards can be dealt from a standard deck of 52 cards? Also, how many ways are there to select 47 cards from a standard deck of 52 cards?

13. A group of 30 people have been trained as astronauts to go on the first mission to Mars. How many ways are there to select a crew of six people to go on this mission (assuming that all crew members have the same job)?

14. Suppose that there are 9 faculty members in the mathematics department and 11 in the Computer science department. How many ways are there to select a committee to develop a discrete mathematics course at a school if the committee is to consist of three faculty members from the mathematics department and four from the computer science department?

15. How many positive integers n can be formed using the digits 3,4,4,5,5,6,7 if n has to exceed 50,00,000?

16. How many bit strings of length 10 contain (a) exactly four 1’s, (b) at-most four ‘1s (c) at least four 1’s (d) an equal number of 0’s and 1’s?

17. How many permutations of letters A B C D E F G contain (a) the string BCD , (b) the string CFGA, (c) the strings BA and GF, (d) the strings ABC and DE, (e) the strings ABC and CDE (f) the strings CBA and BED?

18. If 6 people A, B, C, D, E, F are seated about a round table, how many different circular arrangements are possible, if arrangements are considered the same when one can be obtained from the other by rotation? If A, B ,C are females and the others are males, in how many arrangements do the genders alternate?

19. From a club consisting of a men and 7 women, in how many ways can we select a committee of:

(a) 3 men and 4 women? (b) 4 persons which has at-least one woman? (c) 4 persons that has at-most one man? (d) 4 persons that has persons of both genders? (e) 4 persons that so that two specific members are not included?

20. In how many ways can 20 students out of 30 be selected for an extra-curricular activity, if (a) Rama refuses to be selected? (b) Raja insists on being selected? (c) Gopal and Govind insist on being selected? (d) Either Gopal or Govind or both get selected? (e) Just one of Gopal and Govind gets selected? (f) Rama and Raja refuse to be selected together?

21. 5 balls are to be placed in 3 boxes. Each can hold all the 5 balls. In how many different ways can we place the balls so that no box is left empty, if (a) Balls and boxes are different?


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(b) Balls are identical and boxes are different? (c) Balls are different and boxes are identical? (d) Balls as well as boxes are identical?

22. Prove that 12 + 32 + 52 + … … . +(2� + 1)2 = (𝑛+1)(2𝑛+1)(2𝑛+3)

whenever n is a

3 nonnegative integer.

23. Prove that 1.1! + 2.2! + ............n.n! = (n+1)! – 1 whenever n is a positive integer 𝑛+1

24. Prove that 3 + 3.5 + 3.52 + … … . +3. 5𝑛 = 3(5 −1)

whenever n is a nonnegative

4 integer.

25. Use mathematical induction to prove that �3 - n is divisible by 3 whenever n is a positive integer.

Multiple Choice Questions 1. Sarah goes to her local pizza parlor and orders a pizza. She can choose either a large or a

medium pizza, has a choice of seven different toppings, and can have three different choices of crust. How many different pizzas could Sarah order?

(a) 12 (b) 20 (c) 27 (d) 42 2. Ben can take any one of three routes from school to the town center , and can then take five

possible routes from the town center to his home . He doesn't retrace his steps. How many different possible ways can Ben walk home from school?

(a) 7 (b) 8 (c) 15 (d) 35

3. Derek must choose a four-digit PIN number. Each digit can be chosen from 0 to 9. How many different possible PIN numbers can Derek choose?

(a) 5,040 (b) 6,561 (c) 9,000 (d) 10,000 4. For her literature course, Rachel has to choose one novel to study from a list of four, one

poem from a list of six and one short story from a list of five. How many different choices does Rachel have?

(a) 15 (b) 120 (c) 225 (d) 240

5. James has to choose his options in his final year at school. He must choose one subject from

each of five different option groups. In the first group, there are six subjects to choose from. In the second group, there are four subjects to choose from. In the third group, there are five subjects to choose from. In the fourth group, there are two subjects to choose from. In the fifth group, there are four subjects to choose from. How many possible choices of subjects can James make?

(a) 480 (b) 960 (c) 1200 (d) 4800 6. Jenny has nine different skirts, seven different tops, ten different pairs of shoes, two

different necklaces and five different bracelets. In how many ways can Jenny dress up?

(a) 6300 (b) 7560 (c) 12,600 (d) 63,000 7. Simon visits his local fast-food restaurant and orders a burger, a side, a drink and an ice

cream. He can choose either a hamburger, a cheeseburger or an egg burger; he can choose either fries or salad as a side; he can choose one drink from coke, lemonade, orange or raspberry; and for his ice cream he can choose either a cone or a sundae. How many different possible meals could Simon choose?


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(a) 11 (b) 24 (c) 40 (d) 48 8. How many even five digit whole numbers are there?

(a) 5000 (b) 45,000 (c) 50,000 (d)100,000 9. How many 'words' with five letters are there that start with a vowel and end with an S?

Note: They don't have to be real words, results like AGXFS or UFQWS will do.

(a) 3,380 (b) 87,880 (c) 105,456 (d) 2,284,880 10. How many odd numbers are there greater than 1,000 but less than 100,000?

(a) 49,500 (b) 49,950 (c) 49,995 (d) 50,000 11. From a group of 7 men and 6 women, five persons are to be selected to form a committee

so that at least 3 men are there on the committee. In how many ways can it be done?

(a) 564 (b) 645 (c) 735 (d) 756 12. In how many different ways can the letters of the word 'LEADING' be arranged in such a way

that the vowels always come together?

(a) 360 (b) 480 (c) 720 (d) 5040 13. In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that

the vowels always come together?

(a) 10080 (b) 4989600 (c) 120960 (d) None of these

14. There are 20 people who work in an office together. Four of these people are selected to go to the same conference together. How many such selections are possible?

(a) 80 (b) 4845 (c) 116280 (d) None of these 15. In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways

can they be selected such that at least one boy should be there?

(a) 159 (b) 194 (c) 205 (d) 209 16. A box contains 2 white balls, 3 black balls and 4 red balls. In how many ways can 3 balls be

drawn from the box, if at least one black ball is to be included in the draw? (a) 32 (b) 48 (c) 64 (d) 96

1

17. A sequence is defined by the recurrence relation �𝑛+1= 4 �𝑛+8 with �0 = 32. Evaluate �2. (a) 10 (b) 12 (c) 16 (d) 32

2

18. A sequence is defined by the recurrence relation �𝑛+1= 5 �𝑛+6 with �0 = -10. What

is the limit of sequence? (a) 10 (b) 2/5 (c) -2/25 (d) -30

19. Consider the recurrence relation �𝑘 = −8�𝑘−1 − 15�𝑘−2 �𝑖�ℎ 𝑖�𝑖�𝑖�� 𝑖�𝑖�� 0 = 0 �� 1 = 2. Which of the following is an explicit solution to this recurrence relation?

(a) �𝑘 = (−3)𝑘 − (−5)𝑘 (b) �𝑘 = �(−3)𝑘 − �(−5)𝑘

(c) �𝑘 = �(−3)𝑘 − (−5)𝑘 (d) �𝑘=(−5)𝑘 − (−3)𝑘

20. Consider the recurrence relation �𝑘 = −6�𝑘−1 − 9�𝑘−2 �𝑖�ℎ 𝑖�𝑖�𝑖�� 𝑖�𝑖�� 0 = 0 �� 1 = 2. Which of the following is an explicit solution to this recurrence relation, provided the constants A and B are chosen correctly?

(a) �𝑛 = �3𝑛 + �3𝑛 (b) �𝑛 = �3𝑛 + �(−3)𝑛

(c) �𝑛 = �3𝑛 + ��3𝑛 (d)�𝑘=(−5)𝑘 − (−3)𝑘

Unit-5: Analytical Geometry


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Short Questions

1. If P(�1,�1) and Q(�2,�2) are two points in �2, find the distance PQ. 2. Find the distance between the following pairs of points:

(a) (7,2), (3,1) (b) (-2,11) ,(5,0) (c) (4,-8), (-1,-2) (d) (5,0), (0,4) 3. Find the co-ordinate of the centroid of a triangle whose vertices are (1,5),(-6,0) and

(5,1) 4. The centroid of a triangle is (1,-2), Two of the vertices of the triangle are (-3,-4) and

(5,-2) find the third vertex. 5. Find the slope of the lines joining the following pairs of points:

(a) (7,11),(5,2) (b) (-3,5),(6,-7) (c) (-2,-9),(0,-6) (d) (-2,-3),(-4,-11) 6. Find the equations of lines joining the following pairs of points:

(a) (6,4),(2,1) (b) (3,-5),(4,7) (c) (-3,1),(-4,-5) (d) (0,3),(-7,2) 7. Find the equation of a line with slope 1/3 and passing through (-2,7). 8. Find the equation of a line with slope 3 and passing through (2,5). 9. Find the area of triangle whose vertices are (-3,0),(2,8) and (5,1).

10. If the distance between (a,-5) and (2,a) is 13 units, find the value of a 11. Find the slope and intercept on Y-axis of the following lines:

a) 2x+5y = 11 b) x/2 + y/5 = 0 c) x = 7y -11 d) y - 11 = 0 e) 7x = 11y f) lx + my – p = 0

Long Questions 1. Find a point which divides line joining A(x1, y1) and B(x2, y2) in the ratio m : n.

2. If Q(x, y) divides the line joining A(x1, y1) and B(x2, y2) externally in the ratio m : n,

write down the co-ordinates of Q.

3. Find the co-ordinates of the centered of a triangle whose vertices are A(x1, y1), B(x2,

y2) and C(x3, y3). 4. Find the area of a triangle formed by the points (x1, y1), (x2, y2) and (x3, y3). 5. Write down the condition that the points (x1, y1), (x2, y2) and (x3, y3) are collinear. 6. Prove that (3, 0), (6, 4) and (-1, 3) are the vertices of a right angled triangle.

7. Show that (10, -18), (3, 6) and (-5, 2) form an isosceles triangle.

8. Prove that (0, -1), (2, 1), (0, 3) and (-2, 1) are the vertices of a square.

9. Show that (2, -2), (8, 4), (5, 7) and (-1, 1) form a rectangle.

10. Show that (-2, -1), (1, 0), (4, 3) and (1, 2) are the vertices of a parallelogram.

11. Prove that (0, 2), (2, 10) and (5, 22) are collinear points.

12. Find the circumcentre of the triangle formed by the points (2, 3), (4, 7) and (-2, 5).

Also find circum radius t.

13. Find the circum centre of a triangle formed by the points (1, 4), (5, 8) and (9, 3).

14. Prove that the points (7, 0), (6, -2), (3, 4) and (4, 6) form a parallelogram.

15. Show that the points (6, 6), (2,3), and (4, 7) are the vertices of a right angled

triangle.

16. Prove that the points (4, 3), (7, -1) and (9, 3) are the vertices of an isosceles triangle.

17. Prove that the points (-4, 0), (0, -3), (4, 0) and (0, 3) are the vertices of a rhombus.

18. Find the co-ordinates of point which divides the line joining the points (1, -2) and


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(4, 7) in the ratio 2:5. 19. Find the ratio in which the point (3, 16) divides the line joining the points (1, 6) and

(4, 21).

20. Show that the line joining (-1, 5) and (7, 7) bisects the line joining (3, 9) and (3, 3). 21. Find the ratio in which Y axis divides the line joining (-4,1) and (8, 5).

22. Find the ratio in which the line joining (2, 2) and (7, 8) is divided by X axis.

23. The points (0, 0), (0,1) and (x, y) form an equilateral triangle, find (x, y)

24. If (4, 3), (6, 5) and (2, 4) are the mid-points of sides of a triangle, find the vertices of

the triangle.

25. Find the lengths of median of a triangle whose vertices are (-2, 2),(8, 4) and(2, 8).

26. The area of a triangle formed by the points (-1, 6) (0, 1) and (x, 2) is 3 square units

find the values of x.

27. Find the area of a quadrilateral formed by the points (-3, 0), (0, 6), (4, 3) and (1, -2).

28. Prove that (1, 1), (3, 9), (4, 13) are collinear points.

29. If (0, 3), (-1, 1), (a, 7) are collinear points find a.

30. Find the area of a triangle formed by the lines. 7x + y – 11 = 0, x + 3y + 7 = 0 and

3x – y +1 = 0.

31. Find the equations of sides of a triangle formed by the points (1, 4), (2, -3), (-1, -2).

32. A line passes through (2, -3) and its slope is equal to the slope of a line joining (1, 3)

and (2, -1), find its equation.

33. Find the equation of a line joining (1, -3) and the point of intersection of the lines

x+y+1=0 and 3x+y-5=0.

34. Find an equation of a line passing through the point of intersection of x-4y+18 = 0

and x+y-12 = 0 and having slope 2.

35. Prove that the line 3x+4y+2 = 0 and 12x+16y-7 = 0 are parallel.

36. A line passes through the point of intersection of x-2y+3 = 0 and 2x-3y+4 = 0 and it

is parallel to the line joining the points (1, 1) and (0, -1). Find its equation.

37. The line joining the points (k, 3) and (-2, 1) is parallel to the line joining the points

(-3, 2) and (1, 0). Find the value of k.

38. Prove that the line 4x+3y+2 = 0 and 6x-8y+11 = 0 are perpendiculars to each other.

39. Find the equation of the line perpendicular to the line joining (3, 2) and (4, 0) and

passing through (5, 7).

40. Find the equation of a line passing through the point of intersection of

4x+y-13 = 0 and 5x+2y-20 = 0 and perpendicular to 3x = 2y

Fill in the blanks

1. The point (0, -3) lies on axis. 2. A point whose both of coordinates are negative lies in the quadrant.

3. The length of the line joining points A(8, 5) and B(5, 5+3√3) is . 4. If distance between the points (8, 7) and (2, a) is 10 units, then the value


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of a is 5. point which divides the line joining A(4,12) and B(2,4) in the ratio

4:5.

6. A line cut off intercepts 3 and 4 on X and Y axes respectively, then its equation is .

7. x = - 4 is a parallel to Y-axis and at a distance of units from it.

Multiple Choice Questions 1. The point (-2, 3) lies in the

a) 1st quadrant

b) 2nd quadrant

c) 3rd quadrant

d) 4th quadrant

2. The sign of x coordinate of a point lying in 3rd quadrant is

a) +

b) –

c) ±

d) None of the above

3. The distance of the point (4, -3) from the x axis is

a) -3 units

b) 4 units

c) 3 units

d) 5 units

4. The origin lies on

a) x-axis only

b) y-axis only

c) Both axes

d) None of the axes 5. Find the distance between the points (3, -2) and (6,4).

a) √85 b) √79 c) 5√3 d) 3√5

6. What is the slope of the line passing through the points (4,6) and (-1,-2)? a) 4/3

b) 3/4

c) 8/5

d) 5/8

7. Find the slope of a line perpendicular to the line whose equation is

2y + 6x = 24.

a) -3 b) 6 c) 1/3

d) -1/6


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8. Find the midpoint of the segment connecting the points (a, b) and (5a, -7b).

a) (3a, -3b)

b) (2a, -3b)

c) (3a, -4b)

d) (-2a, 4b)

9. Find the equation of the line parallel to the line whose equation is y = 6x + 7 and

whose y-intercept is 8.

a) y = -6x + 8

b) y = (-1/6)x + 8

c) y = (1/6)x + 8 d) y = 6x + 8

10. Find the area of triangle whose vertices are (2, 3). (8, 5) and (4, 7).

a) 12 sq. units

b) 10 sq. units

c) 13 sq. units

d) 9 sq. units

11. In what ratio the line joining A(6, 10) and B(3, 6) is divided by a point P(5,11).

a) 2:1

b) 3:2

c) 1:2

d) None of these

True-False

1. The point (-5, 6) lies in the 2nd quadrant.

2. The coordinates of origin is (0, 0).

3. The points (-2, 6) and (6, -2) lies in the different quadrant.

4. The distance between A(2, 5) and B(0, 1) is 20.

5. If (1, 2), (2, 7) and (3, b) are collinear points, then the value of b = 12.

6. The area of a triangle whose vertices are (4,5), (1, 3) and (2,1) is 11 sq. units.

7. The slope of the line joining the pair of points (0, 6) and (-3, -1) is -1/3.

8. A line makes an intercept of 5/2 units on Y-axis and its slope is ¾, then its equation is

3x – 4y = -10.

9. x = 3 is a parallel to x-axis and at a distance of 3units from it.

Unit-6: Determinants Short Questions 1. Evaluate the following determinants:

a) 5 3|

| 2 4

b) −6 2|

|

−3 −4


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c)

7 28| | 3 12 � + � �

d) | � � − �| e) � + � � − �| | � − � � + �

f) 2 + � 2 | |

3 3 + � g) � �

| |� � − � � + � �

h) | � � − �|

Long Questions 1. Solve the following equations using Cramer’s method.

a) 2x + 3y = 13

x + y = 5

b) y = 1 – 3x

2y = 4 – 5x

c) 8x + 3y – 13 = 0

7x + 5y – 9 = 0

d) 2x + 3y + 5 = 0

3x + 4y + 7 = 0

e) 2x – 3y = 3

5x – 7y = 2

f) 2y = 3 – x

5y = 7 – 4x g)

� +

7� = 9,

3� − �

− 8 = 0 3 3 2 3

h) 1

− 2

= 6, 3

+ 5

= 7 � � � �

i) 3� − 4� = 24 3y + 2x + 1 = 0

j) 4(x – 1) + 3(y – 1) = 15

3(x – 1) + 4(y + 1) = 21

k) 2x + 5y = 2xy

4x + 5y = 3xy

l) 2x – 6y = 5xy

6x – 5y = 2xy m) �� + �� − �� = 0, �� + �� − �� = 0 n) |� + 2 3 � − 1 � − 1

� + 1 5| = 8, | 1 6

| = 4 o) � + 2 2�

| = 25, | 3 −4

= 23

| 1 3 � − 2 �

|

p) 3x + 5y + 6z = 4, x + 2y + 3z = 2, 2x + 4y + 5z = 3 q) x + y = -1,y + z = 1, z + x = 0

r) 2x + 3y – z = 5, 3x + 2y +z = 10, x – 5y + 3z = 0

s) 2x – 3y + z = 3, x + y – 2z = -1, 3x – 2y + 2z = 8

t) x + 2y = 3, y – 3z = 4, 3x – 2z = 5

2. Find the values of the following determinants:


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2 3 4 2 −3 −3 � 1 2 5 −4 −12

a) |2 7 −7|, b) |5 −7 −2|, c) |2 2 −�| , d) |0 3 −5 | 4 9 1 7 −8 17 3 � 4 0 2 1 11 40 28 3. If | 3 12 8 | = 0 , Find the value of A.

� 2 2 16 8 26 1 2 5

4. If | 6 3 9 | = |2 𝐾 0|, Find the value of K. 2 1 4 7 14 9

4 5 −7 5. Find the value of K if | −2 𝐾 6| = 43

1 𝐾 1 6. Solve the following equations:

1 2 −3 3 � −8 a) |� 2 −5| = 0 , b) |5 −2 −3| = 0 ,

4 −3 −1 1 −3 2 � � 1 � � 3

7. Solve: |2 3 4| = 0 and |3 4 5| = 0.

4 5 6 5 6 8 8. Prove the following:

� + � � + � � + � � � � a) |� + � � + � � + �| = 2 |� � �| � + � � + � � + � � � � 1 � �² b) � = |1 � �²| = (� − �)(� − �)(� − �)

1 � �²

Fill in the blanks 1. The solution of x + y = 5 and x – y = 0 is . 2. If

3 −⋋ 1 | = 0, then the value of ⋋ is or .

| 2 2 −⋋

3. The solution of x = 3 and y = 2 is . 4. The value of

−7 0| is . |

2 −3 � + � � �

5. The value of | � � + � � | is .

� � � + � 6. The solution of x = 2 and y = 3 is .

Multiple Choice Questions 1. Which of these determinants has the value 6?

a) 0 2

| | 3 5 b) 6 6

| |

1 1


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|

1 2 3 c) |−1 1 1| 2 4 8

0 0 1 d) |0 2 4|

3 2 1 2. Suppose

� �| = 4 ,Which of the following is false?

� � a)

2� 2�| = 8

2� 2� b) |

� �

� − � � − �

c) | � �

| = 12 3� 3�

d) | �

�

| = 4

| = 4

� + 2� � + 2� 3. Which of the following determinants is not 0?

2

4. If 2� + 1 � + 2

2� + 3 � − 3 a) 2 b) -2

c) 1

d) None of these

| = 0 ,then the value of x is

5. The value of � −�

| is � � − �

a) �² − �² b) �² c) �² − �²

d) �² � � �

6. The value of |−� � �| is

−� −� � a) 4xyz b) xyz

c) -4xy

|

|

|

1 0 3 a) |−1 0 1| 2

1 2 0 8

b) |2 2 1| 5 8 7 1 0 0

c) |0 1 0| 0 0 1 1 2 3

d) |3 6 9|

3 2 1


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d) None of these

7. Which of the following determinants is not 0?

1 2 3

a) |3 6 9|

3 2 1

1 0 3 b) |−1 0 1|

2 0 8

1 2 2 c) |2 2 1|

5 8 7

0 1 0 d) |0 0 1|

1 0 0

� � 2 8. Find all the values of t for which the following determinant (4 −2 8) is equal to

1 1 � 0?

a) The determinant is never 0

b) t = -1 and t = 3

c) t = 3

d) t = -1

9. If A and B are 3X3 matrices with |A| = 2 and |B| = 3 , which of the following is

generally false?

a) |2A| = 16

b) |A + B| = 5

c) 3|B| = 9

d) |AB| = 6 True-False 1. If |

8 4 = 0, then the value of A is 4.

� |

2


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2.

3.

4.

5.

The value of | � � + �

| is

−�². If |

1 −⋋ 2 � − �

�

3 2 −⋋ | = 0, then the value of ⋋ is -4 or 1.

The solution of the given equations 2x + 3y – 8 = 0 and 5x – 4y +3 = 0 is (1, 2).

7. The value of |

�² + �² �² 1

�² + �² �² 1

�² + �² �² 1 | is 0.

1 0 0 6. The value of |−4 3 0| is 0. 3 1 7

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060010105-Mathematics for Computer Applicationsutu.ac.in/dcst/download/2015-16/INTMSCIT/Sem1/QB/QBMScIT... · 2015. 7. 24. · Convert the following binary numbers to hexadecimal