Unit 1

Unit	NUMBER SYSTEMS
01

Q.1 Define rational and irrational numbers. (Board 2014) 11901001

Q.2 Properties of Real Numbers 11901002

Example:2, 3 Î R Þ 2 + 3 = 5 Î R 11901003

Example: 2, 3, 4 Î R 11901004

Q.3 Write any two properties of equality of real numbers. 11901005

Q.4 Which of the following sets have closure property w.r.t addition and multiplication? 11901006 (i) {0} (ii) {1}

(iii) {0, -1} (iv) {1, -1}

Q.5 Define complex number and conjugate of a complex number. 11901007

Q.6 Simplify the following 11901008

(i)(-i)^-19 11301097 (ii)(-1)11901009

Q.7 Write in terms of i 11901010 (i)b 11301100 (ii) 11901011

(iii) 11301102 (iv) 11901012

Q.8 Simplify the following (7, 9)+(3, -5) 11901013

Q.9 Simplify (8, -5) - (-7, 4) 11901014

Q.10 Simplify (2, 6) . (3, 7) 11901015

Q.11 Simplify (5, -4) . (-3, -2) 11901016

Q.12 Simplify (0, 3) . (0, 5) 11901017

Q.13 Simplify (2, 6) ¸ (3, 7) 11901018

Q.14 (5, - 4) ¸( -3, - 8) 11901019

Q.15 Prove that the sum as well as the product of any complex number and its conjugate is a real number. 11901020

Q.16 Find the multiplicative inverse of each of the following numbers: 11901021

(i) (–4,7) 11901022 (ii) 11901023

(iii) (1,0) 11901024

Q.17 Factorize the following 11901025

(i) a² + 4b²11901026 (ii) 9a² + 16b²11901027

(iii) 3x² + 3y²11901028

Q.18 Separate into real and imaginary parts (write as a simple complex number): 11901029

(i) 11901030 (ii) 11901031

(iii) 11301132

Q.19 Define modulus of a complex number. 11901033

Q.20 Theorems: " z, z₁, z₂Î C. 11901034

(i) |-z| = |z| = || = |-| 11901035

(ii) = z 11301127 (iii)z = |z|²11901036

(iv) =₁ + ₂11901037

(v) = , z₂¹ 0 11901038

(vi) |z₁ . z₂| = |z₁| . |z₂| 11901039

Q.21 Show that

11901040

Q.22 Express the complex number in polar form. 11901041

Solution:

Q.23 Find the multiplicative inverse of the following numbers: 11901042

(i) - 3i 11901043

(ii) 1 - 2i 11901044

(iii) -3 -5i 11901045

(iv) (1, 2) 11901046

Q.24 Simplify

(i) i¹⁰¹ (Board 2008) 11901047

(ii) (-ai)⁴, a Î R 11901048

(iii) i^-³(Board 2014) 11901049

(iv) i^-¹⁰11901050

Q.25 Prove that = z iff z is real 11901051

Q.26 Simplify by expressing in the form a+bi 11901052

(i) 5 + 2 11901053

(ii) 11901054

(iii) 11901055

(iv) 11901056

Q.27 (i)Show that " z Î C,Z²+is a real number. 11901057

(ii) Show that (Z-)² is a real number for all z Î C. 11901058

number

Multiple Choice Questions

1. A number which cannot be written in the form , where p and q are relatively prime integers and q ¹ 0 is called the: 11901059

(a) rational number

(b) irrational number

(d) whole number

2. Division of a natural number by another natural number gives: 11901060

(a) always a natural number

(b) always an integer

(d) always an irrational number

3. Irrational numbers are: 11901061

(a) terminating decimals

(b) non- terminating decimals

(d) non- terminating , non-repeating

decimals

4. Rational numbers are: 11901062

(a) repeating decimals

(b) terminating decimals

(d) all of these

5. p, e are: 11901063 (a) integers (b)natural numbers

(d) irrational numbers

6. π is defined as: 11901064

(a) ratio of diameter of a circle to its

circumference

(b) ratio of the circumference of a circle to its

diameter

circumference

(d) ratio of the circumference of a circle to its

area

7. Zero is: 11901065

(a) a natural number

(b) a whole number

(d) anegativeinteger

8. Let x , y Î R, then x + iy is purely imaginary if: 11901066

(a) x¹ 0, y = 0 (b) x = 0 , y = 0

9. If x, y Î R and xy = 0, then: 11901067

(a) x = 0 (b) y = 0

(d) x = 0 or y = 0

10. If = –z , then: 11901068

(a) z is purely real

(b) z is any complex number

(d) real part of z=imaginary part of z

11. Real part of is: 11901069

(a) (b) 1

12. Imaginary part of is: 11901070

(a) (b) 1

(c)I (d) i

13. Which of the following is correct: 11901071

(a) 2 + 7i> 10 + i

(b) 1 + i> 1 – i

(d) None of these.

14. Product of a complex number and its conjugate is: 11901072

(a) a real number

(b) irrational number

(d) either real number or complex

number.

15. The ordered pairs (2, 5 ) and (5, 2) are:

(a) Not equal (b) Equal 11901073

16. Modulus of complex number Z =a+ib is the distance of a point from: 11901074

(a) x - axis (b) y - axis

17. Modulus of complex number z = a+ib is: 11901075

(a) (b)

(c)

(d) None of the above

18.Modulus of 15 i + 20 is: 11901076

(a) 20

(b) 15

(d) None of the above

19. Conjugate of complex number
(–a, –b) is: 11901077

(a) (–a, b) (b) (–a, –b)

20. Conjugate of a + i b is: 11901078

(a) – a + i b (b) a + i b

21. Conjugate of a – i b is: 11901079

(a) b + ia (b) –a + ib

22. Conjugate of –3 – 2 i is: 11901080

(a) 3 + 2i (b) – 3 + 2i

23. i+ 1 = 11901081

(a) -1 (b) 0

24. If z = (a, b), z = (c, d) are two complex numbers , then which expression defines the sum of z and z . 11901082

(a) (a + c, b + d) (b) (a + b, c + d)

25. If z = 4 i and z = 3–9 i ,then z + z =

11901083

(a) 3 – 5 i (b) 3 i- 5

26. belongs to the set of: 11901084

(a) real numbers

(b) complex numbers

(d) odd numbers

27. The real part of the complex number

a + bi is: 11901085

(a) b (b) –b

28. The imaginary part of the complex number a + bi is: 11901086

(a) b (b) bi

29. Every real number is also a/an: 11901087

(a) integer

(b) rational number

(d) complex number

30. Factors of 9a+ 25b in complex number system are: 11901088

(a) ( 3a – 5bi)( 3a + 5bi)

(b) ( 3a – 5b)( 3a + 5b)

(d) ( 3a – 5b)( 3a + 5bi)

31. If a, b, c and d Î R. Then a = b,c = d Þ

(a) a + c = b + d 11901089

(b) a + b = c + d

(d) None of these

32. If a, b, c Î R and a > b Þ ac <bc, then:

(a) c> 0 (b) c < 0 11901090

33. a> b Þ –a < –b Name of the property used in the above inequality is: 11901091

(a) Additive property

(b) Multiplicative property

(d) Transitive property

34. a> b Þ<, a ¹ 0 , b ¹ 0Name of the property used in the above in equation is:

(a) additive property 11901092

(b) multiplicative inverse property

(d) transitive property

35.For all x Î R, x = xWhat is above property called? 11901093

(a) Reflexive property

(b) Symmetric property

(d) Trichotomy property

36. The set of negative integers is closed with respect to: 11901094

(a) addition (b) multiplication

37. The identity element with respect to addition is: 11901095

(a) 0 (b) 1

38. The additive inverse of a real number a is: 11901096

(a) 0 (b) - a

39. The additive inverse of 3 is: 11901097

(a) 0 (b) 1

(c)– 3 (d)

40. The multiplicative inverse of a non-zero real number a is: 11901098

(a) 0 (b) - a

41. The multiplicative inverse of 3 is:

11901099

(a) 0 (b) 1

42. The multiplicative identity of real numbers is: 11901100

(a) 0 (b) 1

43. The additive identity of real numbers is: 11901101

(a) 0 (b) 1

44. For all x, y, z Î R z + x = z + y 11901102

x = y what is above property called?

(a) Cancellation property w.r.t.

Multiplication

(b) Cancellation property w.r.t. Addition

(d) Additive property

45. If x, y, z Î R, then name the property used in the equation given below? 11901103

x = z

(a) Closure property w.r.t.

Multiplication.

(b) Commutative property w.r.t.

Multiplication.

(d) Trichotomy property

46. If a, b Î R, where R is a set of real numbers, then the property used in the equation: a + b = b + a is called: 11901104

(a) Closure property

(b) Associative property

(d) Trichotomy property

47. If x, y Î R, where R is a set of real numbers, then the property used in the equation xy = yx is called: 11901105

(a) Closure property

(b) Trichotomy property

(d) Additive Inverse

48. Name the property used in the equation: 2 + 3 = 3 + 2? 11901106

(a) Closure property w.r.t.

Multiplication

(b) Commutative property w.r.t.

Multiplication

(d) Commutative Property w.r.t.

Addition.

49. If a, b, c Î R, where R is a set of real numbers, then the property used in the equation:

a + = + c is called: 11901107

(a) Closure property

(b) Associative property

(d) Additive inverse

50. If x, y, z Î R, where R is a set of real numbers, then the property used in the equation x (yz) = (xy) z is called: 11901108

(a) Closure property

(b) Associative property

(d) Additive Inverse

51. If a Î R, where R is a set of real numbers, then the property used in the equation a + 0 = 0 + a = a is called: 11901109

(a) Closure property

(b) Trichotomy property

(d) Additive Identity

52. For any x, y Î R, where R is a set of real numbers, then the property used in the equation x(y + z) = xy + xz is called: 11901110

(a) Closure property

(b) Associative property

(d) Distributive Property

53. For any x, y Î R, where R is a of real numbers. Then either x < y or x = y or x > y. The property used is called: 11901111

(a) Trichotomy Property

(b) Archmidean Property

(d) Multiplicative Property

54. For any x, y, z Î R, where R is a set of real numbers. x < y and y < z Þ x < z The property used is called: 11901112

(a) Trichotomy Property

(b) Archmidean Property

(d) Multiplicative Property

55. The set of all rational numbers between 2 , 3 is: 11901113

(a) an empty set (b) an infinite set

56. The reflexive property of equality of real numbers is: 11901114

(a) a = a "aÎ R (b) a¹a"aÎ R

57. The left distributive property of real numbers is: 11901115

(a) (b + c) a = a + b + c " a, b, c Î R

(b) (a + b) c = ac + bc" a, b, c Î R

(d) (a + b) c = ab + c " a, b, c Î R

58. The symmetric property of equality of real numbers is: 11901116

(a) a = b Þb = a "a, b Î R

(b) a = a Þ b = b " a Î R

(d) a = b Þ a – b = 0 " a, b Î R

59. The transitive property of equality of real numbers is: 11901117

(a) a = b Ùb = c Þ b = - c

"a, b, c Î R

(b) a = b Ùb = c Þ a = c

"a, b, c Î R

(d) a = b Ùb = c Þ a = b

"a, b, c Î R

60. The multiplicative property of equality of real number is: 11901118

(a) a = b Þ ac = bc" a, b, c Î R

(b) a = b Þ ac = b" a, b, c Î R

(d) a = b Þ a = bc" a, b, c Î R

61. The cancellation property with respect to addition of equality of the real numbers is: 11901119

(a) a+c= b + c Þ a ¹ b" a, b,cÎ R

(b) a + c = b + c Þ a = b"a,b,cÎ R

(d) a + c = b + c Þc = b"a,b,cÎ R

62. The cancellation property with respect to multiplication of equality of the real numbers is: 11901120

(a) ac = bcÞ a=c"a,b,cÎ R, c ¹ 0

(b) ac = bcÞ b=c"a,b,cÎ R, c ¹ 0

(d) ac = bcÞ a = b"a,b,cÎR,c¹ 0

63. The transitive property of order of the real numbers is: 11901121

(a) "a, b, c Î R, a>bÙb> c Þ a > c

(b) "a, b, c Î R, a>bÙb> c Þ a ³ c

(d) "a, b, c Î R, a>bÙb> c Þ a < c

64. The additive property of order of the real numbers is: 11901122

(a) "a,b,cÎ R, a<b Þa+c<b + c

(b) "a,b,cÎ R, a<bÞ a + c = b + c

(d) "a,b,cÎ R, a<b Þ a + c < b – c

65. The additive property of order of the real numbers is: 11901123

(a) "a,b,cÎ R, a>b Þ a + c = b + c

(b) "a,b,cÎ R, a >bÞ a + c < b + c

(d) "a,b,cÎ R, a>b Þ a + c > b – c

66. If z = x + i y = r , then modulus of z is: 11901124

(a) (b) cosq +sinq

67. If z = x + i y = r , then arg z is: 11901125

(a) tanq (b) cosq + sinq

68. = 11901126

(a) 2 (b) 2

69. Polar form of –3 i is: 11901127

(a) 3

(b) 3

(d) 3

70. cos+ is in in Cartesian form is:

(a) 0 (b) i 11901128

71. De Moivre’s theorem is: 11901129

(a)

(b)

(c)

(d)

72. is: 11901130

(a) integer (b) rational number

(d) natural number

73. If n is not a perfect square, then is:

(a) integer 11901131

(b) rational number

(d) natural number

74. Golden rule of fractions is that for
k≠ 0, = 11901132

(a) (b)

75. z = (a, b), then z^–1 = 11901133

(a) (b) (–a, –b)

(c)

(d)

76. If z₁ and z₂ are complex numbers, then 11901134

(a) (b)

77. If are complex numbers, then = __________,. 11901135

(a) (b)

78. = ____________. 11901136

(a) 0

(b)

(d) None of these

79. Multiplicative inverse of is:

(a) 11901137

(b)

(d) – 1

Unit 2

Unit	Sets, Functions and Groups
02

Q.1 What is set and order of a set? 11902001

Q.2 Explain the ways describing a set. 11902002

Illustrative example: 11902003

Q.3 What is the sub-set, proper sub-set and improper sub-set? 11902004

Q.4: What is difference between equal sets and equivalent sets? 11902005

Q.5 Is there any set which has no proper sub set? If so name that set. 11902007

Q.6 What is the difference between {a, b} and {{a, b}}? 11902008

Q.7 Which of the following sentences are true and which of them are false? 11902009

(i) {1, 2} = {2, 1}11302231 (ii) ÆÍ {{a}}11902010

(iii) {a} {{a}}11302233 (iv){a} Î {{a}}11902011

(v) aÎ {{a}} 11302235 (vi) ÆÎ {{a}} 11902012

Q.8 Define “union and intersection” of two sets and complement of a set? 11902013

Q.9 Under what conditions on A and B are the following statements true? 11902015

(i) A È B = A 11902016 (ii) A È B = B 11902017

(iii) A - B = A 11902018 (iv) A Ç B = B 11902019

(v) n (A È B) = n (A) + n(B) 11902020

(vi) n(A Ç B) = n(A) 11902021

(vii)A - B = A 11902022 (viii)n(AÇB)=0 11902023

(ix) A È B = U 11902024 (x) AÈB=BÈA 11902025

(xi) n(A Ç B) = n(B) 11902026

(xii) U - A = Æ 11902027

Q.10 Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {2, 4, 6, 8, 10}, B = {1, 2, 3, 4, 5}and

C = {1, 3, 5, 7, 9}

List the members of each of the following sets: 11902028

(i) A^c 11902029 (ii)B^c11902030 (iii)AÈB 11902031

(iv) A- B 11902032

(v) A Ç C 11902033 (vi) A^cÈC^c11902034

(vii) A^cÈC 11902035 (viii) U^c11902036

Q.11 Taking any set, say A = {1,2,3,4, 5} verify the following: 11902037

(i) AÈÆ = A 11902038 (ii) A È A = A 11902039

(iii) A Ç A = A 11902040

Q.12 If U = {1, 2, 3, 4, 5, …..20} and

A={1,3,5,…..,19} verify the following: 11902041

(i) AÈA¢ = U 11902042 (ii) AÇU=A 11902043

(iii) AÇA¢ = Æ 11902044

Q.13 From suitable properties of union and intersection deduce the following results. 11902045

(i) A Ç (A È B) = A È (A Ç B) 11902046

(ii) A È (A Ç B) = A Ç (A È B) 11902047

Q.14 Define sentence, statement, logic, inductive and deductive logic and compound statement. 11902048

Q.15 Construct truth tables of “Conjunction”, “Disjunction” and “Implication”. 11902049

Q.16 Define “Tautology”, “Absurdity” and “Contingency”. 11902050

Q.17: Write the converse, inverse and contra positive of the following conditionals: 11902051

(i) ~ p ® q 11902052 (ii) q ® p 11902053 (iii) ~p ® ~q 11902054 (iv)~ q ® ~ p 11902055

18. Construct truth tables for the following statements: 11902056

(i) (p ® ~p) Ú (p ® q) 11902057 (ii) (p Ù ~ p) ® q 11902058 (iii) ~(p ® q) « (p Ù ~q) 11902059

19. Show that each of the following statements is a tautology: 11902060

(i) (pÙq)®p 11902061 (ii) p®(pÚq) 11902062 (iii)~(p®q)®p 11902063 (iv) ~qÙ(p®q)®~p 11902064

20. Determine whether each of the following is a tautology, a contingency or an absurdity:

(i) pÙ ~p 11902065 (ii) p ® (q ® p) 11902066 (iii) q Ú (~q Ú p) 11902067

Q.21 What is “Relation”, “Domain” and “Range of a relation”? 11902068

B = . Examine which of the following are relations from A to B and from B to A. 11902069

(a) 11902070

(b) 11902071

(c) 11902072

Q.22 What is “Into function” and “On to function”? 11902073

Q.23 What is “Injective function” and “bijective function”? 11902074

Q.24 For A = {1,2,3,4}, find the following relations in A. State the domain and range of each relation. Also draw the graph of each. 11902075

(i){(x, y)} | y = x} 11902076

(ii){(x, y) | y+x=5} 11902077

(iii){(x, y) | x+y<5} 11902078

(iv){(x,y) | x+y>5} 11902079

Q.25 Which of the following diagrams represent functions and of which type? 11902080

11902081 Fig. (1)

11902082 Fig. (2)

11902083 Fig. (3)

11902084 Fig. (4)

Q.26 Find the inverse of each of the following relations. Tell whether each relation and its inverse is a function or not: 11902085

(i) {(2,1), (3,2), (4,3), (5,4), (6,5)} 11902086

(ii) {(1,3), (2,5), (3,7), (4,9), (5,11)} 11902087

(iii) {(x, y) | y = 2x + 3, x Î R} 11902088

(iv) {(x, y) | y² = 4ax, x ³ 0} 11902089

(v) {(x, y) | x² + y² = 9, | x| £ 3, | y | £ 3}

11902090

Q.27 What is “Unary operation” and “Binary operation”? 11902091

Q.28 Explain existence of “Binary operation”? 11902092

Q.29 Give the table for addition of elements of the set of residue classes modulo4. 11902093

Q.30 (i)In a set S having binary operation “a left identity and right identity” are the same? 11902094

(ii)In a set having associate binary operation “left inverse of an element in equal to its right inverse”. 11902095

Q.31 Show that the adjoining table is that of multiplication of the elements of the set of residue classes modulo 5. 11902096

	1	2	3	4
0	0	0	0	0
1	1	2	3	4
2	2	4	1	3
3	3	1	4	2
4	4	3	2	1

Q.32 Prepare a table of addition of the elements of the set of residue classes modulo 4. 11902097

Q.33 Define groupoid, semi group, monoid. 11902098

Q.34 What is “Group” and “Abelian Group”? (Board 2014) 11902099

Q.35Ifa,b G and G in a group, solve the equations: 11902100

(i) ax = b, 11902101 (ii) xa = b 11902102

Q.36 If a,b G and G in a group, then show that (ab)^-1 = b^-1 a^-1. 11902103

Q.37 Operation Å performed on the two member set G = {0, 1} is shown in the adjoining table. Answer the questions:

11902104

(i) Name the identity element if it exists?

11902105

(ii) What is the inverse of 1? 11902106

(iii) Is the set G, under the given operation a group? Abelian or non-Abelian? 11902107

Å	0	1
0	0	1
1	1	0

Q.38 Show that the set {1, w, w²}, when
w³ = 1 is an Abelian group w.r.t. ordinary multiplication. 11902108

Q.39 If G is a group under the operation ø and a, b Î G, find the solutions of the equations: 11902109

(i) a ø x = b, 11902110 (ii) x a = b 11902111

Multiple Choice Questions

1. A set is defined as: 11902112

(a) Collection of same objects.

(b) Well defined collection of same objects.

(d) None of these.

2. Distinct objects means: 11902113

(a) Identical objects(b) Not identical

3. The objects in a set are called: 11902114

(a) Elements (b) Sub-sets

(d) Overlapping sets

4. A set can be described by: 11902115

(a) One way (b) Two ways

5. If a set is described in words, the method is called: 11902116

(a) Tabular form

(b) Descriptive form

(d) Non-tabular method

6. If a set is described by listing its elements within brackets is called:

11902117

(a) Set builder notation

(b) Tabular form

(d) None of these

7. If a set is described as

{ x | x N Ù x < 100} is the: 11902118

(a) Set builder notation

(b) Tabular form

(d) Non-set builder method

8. aA means: 11902119

(a) a is an element of set A

(b) a is subset of A

(d) a contains A

9. Two sets A and B are said to be equivalent if: 11902120

(a) n(A) = n(B)

(b) n(A) ¹ n(B)

(d) None of these

10. If set A = {1,2,3} and B = {2,1,3} then sets A and B are: 11902121

(a) Not equal (b) Equal

11. The well defined collection of disjoint object is a: 11902122

(a) Complex number

(b) Rational number

(d) Set

12. A ÍB (i.e., A Ì B and A = B) then:

11902123

(a) A is improper subset of B

(b) A is proper subset of B

(d) B is proper subset of A

13. If A Í B and B Í A then: 11902122

(a) A = Æ (b) A = B

14. A Ê B means: 11902125

(a) A is super set of B

(b) B is supper set of A

(d) A is equivalent to B

15. If n(S) = 3 then n {P(S)} = 11902126

(a) 2 (b) 4

16.The number of subsets of a set having three elements is: 11902127

(a) 2 (b) 3

17. If A = {1, 2, 3, ...., 99}, B = {x | x Î N,

0 < x < 100} then: 11902128

(a) AÌ B (b) A ¹ B

18. The number of elements of the set

{x : xÎ N, x= 1}, where N is the set of all natural numbers, is: 11902129

(a) 0 (b) 1

19. A set having no element is called: 11902130

(a) Null set (b) Subset

20. The proper subset E of a set F is denoted by: 11902131

(a) F Ì E (b) E Ì F

21. An Improper subset of a set F is: 11902132

(a) U (b) X

22. If A È B = Æ, and A = Æ then: 11902133

(a) B = Æ (b) B = {Æ}

23. If sets A and B are equal then: 11902134

(a) A É B (b) B É A

24. If A and B are two sets such that

A Ç B = A È B, then: 11902135

(a) A and B are power sets

(b) A and B are disjoint sets

(d) A and B are equal sets.

25. For any two sets A = B if and only if

A È B = 11902136

(a) A¢ (b) B¢

26. Which is the commutative law? 11902137

(a) A Ç B¢ = B Ç A¢

(b) A Ç B = B Ç A

(d) A Ç B = B Ç A¢

27. If A Í B and B Í A, then: 11902138

(a) A and B are power sets

(b) A and B are disjoint sets

(d) A and B are equal sets.

28. If A Ì B, then A Ç B is equal to: 11902139

(a) A (b) B

29. If A Ì B, then A È B is equal to: 11902140

(a) A (b)

30. A – B is a subset of: 11902141

(a) A (b) B

31. B – A is a subset of: 11902142

(a) A (b) B

32. A ÇÆ = 11902143

(a) A (b) Æ

33. A ÈÆ = 11902144

(a) A (b) Æ

34. A ÇA= 11902145

(a) U (b) {0}

35. A È A= 11902146

(a) U (b) {0}

36. A È (B Ç C) = 11902147

(a) È

(b) ( AÈ B) Ç ( A È C )

(d) ( AÇ B) Ç ( A È C )

37. If A and B are two sets, then

A È (A Ç B) is equal to: 11902148

(a) B (b) A

38. If A and B are two sets, then

A ÇA is equal to: 11902149

(a) A (b) B

39. The intersection of two sets A and B is represented by: 11902150

(a) A È B (b) A ´ B

40. The difference of two sets A and B is represented by: 11902151

(a) A È B (b) A Ç B

41. If A and B are both subsets of the same universal set X and AÈB =X ,AÇB =Æ then A and B are called: 11902152

(a) disjoint sets

(b) equal sets

(d) overlapping sets

42. A Ç (B – A) = 11902153

(a) A (b) {Æ}

43. If A ËB , B Ë A and A and B have at least one element common, they are called: 11902154

(a) equal sets (b) null sets

44. A set containing finite numbers of elements is called: 11902155

(a) null set (b) super set

45. If A = {1, 2, 3, 4} and B = {5, 6, 7} and

A Ç B is: 11902156

(a) {1, 2, 3} (b) {5, 6, 7}

46. If W= {0, 1, 2, 3, 4,….}, N={1, 2, 3, 4,....} then N – W = ? 11902157

(a) W (b) {O}

47. If A ={ 1, 2, 7, 9 } , B = { 1, 4, 7, 11 } then A and B are called: 11902158

(a) disjoint sets

(b) equal sets

(d) complementary sets

48. The ordered pairs (4,5) and (5 , 4) are:

(a) same (b) different 11902159

49. { 0, ± 1, ± 2, ± 3, ± 4, ....... } is known as the set of: 11902160

(a) numbers (b) positive numbers

50. { 2, 4, 6, 8,......} represents the set of:

11902161

(a) Positive odd numbers

(b) Natural numbers

(d) Positive even numbers

51. If two sets have no element common, they are called: 11902162

(a) disjoint (b) over lapping

52. If P = {1, 3} and Q = , then:

(a) P É Q (b) Q É P 11902163

53. If A = and B = , then:

(a) A Í B (b) B Í A 11902164

54. If X = { 1 , 2 , 3 }, then P( X ) is: 11902165

(a) { Æ , {3} }

(b) { Æ , {1}, {2}, {1,2,3} }

(d) None of these.

55. If A = {{5}}, then P(A) is equal to:

11902166

(a) {{Æ , {5}}} (b) {Æ , {5}}

56. {0} È {1} is equal to: 11902167

(a) {{0} , {1}} (b) {0 , 1}

57. If a relation is given by R= then Range

R is: 11902168

(a) { 0, 1, 3 } (b) { 1, 2, 3 }

58. If A= { 1 , –1 } then number of elements in A ´A are: 11902169

(a) 2 (b) 6

59. S = {1, –1, 2, –2} is a group under:

11902170

(a) multiplication (b) subtraction

60. S = {1 ,w , w} where w is a cube root of unity form an abelian group with respect to: 11902171

(a) multiplication (b) division

61. S = {1, – 1, i, – i} where i = form an abelian group with respect to: 11902172

(a) multiplication (b) division

62. The set S = {0 , 1} has closure property w.r.t. 11902173

(a) + (b) –

63. An element e Î S is said to be an identity element of S w.r.t if
a e = e a = 11902174

(a) 1 (b) 0

64. An element b Î S is said to be an inverse of a Î S w.r.t if a b = ba = 11902175

(a) 1 (b) e

65. The identity element in a group is:

11902176

(a) unique (b) infinite

66. In a group G, if b ø b=b, then b = 11902177

(a) 1 (b) e

67. Inverse of an element in a group is:

11902178

(a) infinite (b) finite

68. To draw general conclusions from a limited number of observations is called: 11902179

(a) logic (b) proposition

69. To draw general conclusions from well-known facts is called: 11902180

(a) logic (b) proposition

70. A declarative statement which is either true or false but not both is called:

11902180

(a) logic (b) proposition

71. A biconditional is written in symbols as:

(a) p« q (b) p Ú q 11902181

72. (p® q) Ù (q ® p) is logically equivalent to: 11902182

(a) p« q (b) q ® p

73. Which is the converse of the sentence ~p ® q? 11902183

(a) q® p (b) p ®~q

74. If ~p ® q be a given conditional, then its inverse is: 11902184

(a) ~p®~q (b) q ® p

75. If q® p be a given conditional, then its inverse is: 11902185

(a) ~p®~q (b) q ® p

76. If p ® q be a given conditional, then its contrapositive is: 11902186

(a) ~p®~q (b) q ® p

77. If ~p ®~ q be a given conditional, then its contrapositive is: 11902187

(a) ~p®~q (b) q ® p

78. The conjunction of two statements p and q is denoted by: 11902188

(a) p~ q (b) p ® q

79. The sentence pÙq is true if and only if:

(a) p is false and q is true 11902189

(b) both p and q are false

(d) both p and q are true

80. The sentence p Ú q is false if and only if:

(a) p is false and q is true 11902190

(b) both p and q are false

(d) both p and q are true

81. The disjunction of two statements p and q is denoted by: 11902191

(a) p~ q (b) p ® q

82. Which sentence is always false? 11902192

(a) pÚ~p (b) q Ù~q

83. If p « q is true, which sentence is also true? 11902193

(a) p® q (b) p Ù q

84. The proposition (p® q) Ù (q ® p) is shortly written as: 11902194

(a) p = q (b) p ¹ q

85. Given the true statement: If the polygon is a rectangle, then it has four sides. Which statement must also be true? 11902195

(a) If the polygon has four sides, then it is not a rectangle.

(b) If the polygon does not have four sides, then it is not a rectangle.

(d) If the polygon has four sides, then itis a rectangle.

Hint: A conditional and its contrapositive always have the same truth-values.

86. Which of the following sentences is equivalent to ~(p Ú q)? 11902196

(a) ~pÚ~q (b) ~p Ù~q

87. Additive inverse of is: 11902197

(a) – (b)

88. If R = A ´ B then R is an onto function if: 11902198

(a) Dom R =B , Range of R = A

(b) Dom R =A , Range of R = B

(d) Dom R =B , Range of R = B

89. R is a relation from A to B if and only if R Í 11902199

(a) B´ A (b) A´ A

90. The phrase, “For all x in S”, is abbreviated as: 11902200

(a) $xÎ S (b) x Î S

91. The phrase, “There exist an x in S”, is abbreviated as: 11902201

(a) $xÎ S (b) x Î S

92. If two sets P and Q are equivalent, they are denoted by: 11902202

(a) P Î Q (b) P « Q

93.If A Ì B, then A – B = 11902203

(a) A (b) Æ

94. Set A is proper subset of B is denoted by: 11902204

(a) BÌ A (b)A Ì B

95. If (x – 2, 2) = (3, 2) , then: 11902205

(a) x= 5 (b) x = 2

96. An element b Î S is said to be an inverse of a ÎS w. r. t * if: 11902206
(a) a * b = b * a = e

(b) a * b = b * a = 0

(d) a * b = b * a = a

97. In a binary relation, the set consisting of all the first elements of the ordered pairs is called: 11902207

(a) function (b) range

98. In a binary relation, the set consisting of all the second elements of the ordered pairs is called: 11902208

(a) function (b) range

99. In the conditional p ® q, p is called:

11902209

(a) antecedent (b) consequent

100. In the conditional p ® q, q is called:

11902210

(a) antecedent (b) consequent

101. A statement which is true for all possible values of the variables involved in it, is called a: 11902211

(a) tautology (b) conditional

102. A compound statement of the form “if p then q” is called an: 11902212

(a) tautology (b) conditional

103. A groupoid (S) is called ------------ if it is associative in S. 11902213

(a) group (b) abelian-group

(d) associative -group

104. The inverse of the linear function {(x, y): y = mx + c} is: 11902214

(a) {(x, y): x = my + c}

(b) {(x, y): y = mx + c}

(d) {(x, y): y = mx + d}

105. The graph of the quadratic function is a: 11902215

(a) straight line (b) line segment

106. Q = {x | x = where p, q Z Ù q ¹ 0} is a set of: 11902216

(a) Rational numbers

(b) Irrational numbers

(d) Set of integers

107. is called converse of: 11902217

(a)

(b)

(c)

(d)

108. If then complement of A in B is:

11902218

(a) A – B

(b) B – A

(c)

(d)

Unit 3

Unit	MATRICES AND DETERMINANTS
03

Q. 1 Define a matrix 11903001

Examples: 11903002

(i)

(ii)

Q. 2. What do you mean by order of a matrix? (Board 2014) 11903003

Q. 3. What is row matrix? 11903004

Examples: 11903005

, ,

Q. 4. What is Column Matrix? 11903006

Examples: , , 11903007

Q.5 What is a diagonal Matrix? 11903008

(Board 2014)

Q.6 What is a scalar matrix? 11903009

Examples: 11903010

Q. 7 What is a unit matrix? 11903011

Examples: I = , 11903012

Q. 8 What is a null matrix? 11903013

, , 11903014

Q.9.What is determinant of a matrix? 11903015

Q. 10. What is a singular matrix? 11903016

Q. 11. What is non singular matrix? 11903017

Q. 12. What is inverse of 2 x 2 matrix?

11903018

Q. 13. If A = , show that A=I.

11903019

Q. 14. Find x and y if 11903020

+ 2 =

Q. 15. If A = and A = , find the values of a and b. (Board 2014) 11903021

Q. 16. If A = and A2 = , find the values of a and b. 11903022

Q. 17. Find the matrix X if; 11903023

(i) X = 11903024

(ii) X = 11903025

(ii) X = 11903026

Q. 18. Show that 11903027

= rI₃

Q.19. Find the inverse of the following matrix 11903028

Q. 20. 11903029

Q.21. Solve the following system of linear equations. 11903030

Q.22. If A and B are square matrices of the same order, then explain why in general;

11903031

(i) (A + B)¹ A+ 2AB + B 11903032

(ii) (A – B) ¹ A– 2AB + B 11903033

(iii) (A + B) (A – B) ¹ A– B 11903034

(A – B) ¹ A– 2AB + B

(iii) (A + B) (A – B) ¹ A– B

Q. 23. Solve the following matrix equations for A: 11903035

(i) A – =

11903036

(ii) A – = 11903037

(ii) A – = 11903038

Q. 24. What is the minor of element of a matrix 11903039

Q. 25. What is co-factor of an element of a matrix? 11903040

Q. 26. If any row (or column) of a determinant is multiplied by a non-zero number k, the value of the new determinant is equal to k time the value of the original determinant. 11903041

Q. 27. If any row (or column) of a determinant is multiplied by a non-zero number k and the result is added to the corresponding entries of another row (or column), the value of the determinant does not change. 11903042

Q. 28. What do you mean by Adjoint and Inverse of a Square Matrix of Order 3: 11903043

Inverse of square matrix of order 3:

Q. 29.Inverse of a Square Matrix of Order 3:

11903044

Q. 30. Evaluate the following determinants. , 11903045

Q. 31 Show that = 4 abc

11903046

Q. 32. Show that = 9

(Board 2008) 11903047

Q.33. Show that 11903048

Q. 35. Show that=r 11903050

Q. 36. If A = then find

A , A , A and 11903051

Q. 37. Without expansion verify that 11903052

(i) = 0 (Bord 2014) 11903053

(ii) = 0 11903054

(iii) = 0 11903055

(iv) = 0 11903056

(v) = 0 11903057

(vi)=

11903058

Q. 38. Find values of x if 11903060

(i) = –30 11903061

(ii) = 0 (Board 2014) 11903062

Q. 39. If A is a square matrix of order 3, then show that |kA| = k³ |A|. 11903053

Q. 40. Find the values of l if A is singular.

A = 11903074

Q. 41. Find whether the given matrix is singular or non-singular 11903075

Q. 42. Verify that (AB)^-1 = B^-1 A^-1 if

A = , B = 11903076

Q. 43. Verify that (AB)^t = B^tA^tand if

A = and B = 11903077

Q.44. If A = verify that

(A^-¹)^t = (A^t)^-¹11903078

Q. 45. What is upper triangular matrix?

11903079

Q. 46. What is lower triangular matrix?

11903080

Q. 47. What is triangular matrix? 11903081

Q. 48. What is Symmetric matrix? 11903082

Q. 49. What is Skew Symmetric matrix?

11903083

Q. 51. What is Conjugate of matrix? 11903084

Q.52. What is Hermitian matrix? 11903085

Q.53.What is Skew Hermitian matrix?

11903086

Q.54. What is echelon form of a matrix?11903087

Q. 56. What is the rank of a matrix? 11903088

Q.57. If the matrices A and B are symmetric and AB = BA, show that AB is symmetric. (Board 2008) 11903089

Q.58 Show that AA^t and A^tA are symmetric for any matrix of order 2´3. 11903090

(ii) A – () is skew Hermitian. 11903093

Q.60 If A is symmetric or skew symmetric show that A² is symmetric. 11903094

Q.61 If A = , find A () 11903095

Q. 62. What are homogeneous linear equations? 11903096

Q. 63. What are non-homogeneous linear equations? 11903097

Q.64. What is trivial solution of homogeneous linear equations? 11903098

Q. 65. What is consistent and in consistent system of linear equations? 11903099

MULTIPLE CHOICE QUESTIONS

q Each question has four possible answers. Select the correct answer and encircle it.

Q.1 The order of a matrix is shown by:

11903100

(a) number of columns ´ noumber of rows

(b) number of columns + number of rows

(d) number of columns – number of rows

Q.2 The order of the matrix is:

11903101

(a) 3 ´ 3 (b) 3 ´ 2

Q.3 A matrix of order m´1 is called? 11903102

(a) Row matrix (b) Column matrix

Q.4 is a: 11903103

(a) Row matrix (b)Column matrix

Q.5 If A and B are two square matrices of same order, (A + B)= 11903104

(a) A+ 2AB + B

(b) A+ 2BA + B

(d) A+ B

Q.6 If A and B are two square matrices of same order and (A+B)=A+2AB+B, then: 11903105

(a) A = B (b) AB = BA

Q.7 If A is a matrix of order m ´ n such that m ¹ n, then A is called: 11903106

(a) A rectangular matrix

(b) A square matrix

(d) An identity matrix

Q.8 If A is a matrix of order m ´ n such that m = n, then what is A called?

11903107

(a) A rectangular matrix

(b) A square matrix

(d) An identity matrix

Q.9 If A is a matrix of order m ´ n, then the number of elements in each row of A is: (Board 2009) 11903108

(a) m (b) n

Q.10 If A and B are two matrices, then:

(a) A B = O (b) AB = BA

(d) AB may not be defined

Q.11 A matrix in which each element is 0 is called: 11903110

(a) Square matrix (b) Null matrix

(d) Rectangular matrix

Q.12 If A= , then matrix A is singular if: 11903111

(a) ab – cd = 0 (b) ac – bd = 0

Q.13 If A = , then: 11903112

(a) A= A (b) A= – Adj. A

Q.14 If A = , B = we can find:

(a) A + B (b) 11903113

Q.15 If A = , B = then BA is:

11903114

(a) null matrix (b) rectangular matrix

Q.16 The matrix is……… 11903115

(a) null matrix (b) diagonal matrix

Q.17 is ….. 11903116

(a) scalar matrix (b) null matrix

Q.18 The matrix is: 11903117

(a) singular (b) non-singular

Q.19 The matrix is: 11903118

(a) scalar matrix (b) diagonal matrix

(d) upper triangular matrix

Q.20 The matrix is: 11903119

(a) scalar matrix

(b) diagonal matrix

(d) upper triangular matrix

Q.21 The matrix is: 11903120

(a) scalar matrix

(b) diagonal matrix

(d) none of these

Q.22 An element a of a square matrix

A = is said to be a diagonal element if: 11903121

(a) i = j (b) i < j

Q.23 An element a of a square matrix A = is said to be above the diagonal if : 11903122

(a) i = j (b) i < j

Q.24 If Iis the identity matrix of order n, then rank of Iis: 11903123

(a) equal to n (b) less than n

Q.25 If A = and B = then A + B = 11903124

(a) (b)

Q.26 If AB=BA=I, then A and B are: 11903125

(a) equal to each other.

(b) multiplicative inverse of each other.

(d) both singular.

Q.27 If two rows (or two columns) in a square matrix are identical
(i.e. corresponding elements are equal), the value of the determinant is:

(a) 0 (b) 1 11903126

Q.28 If each element in any row or each element in any column of a square matrix is zero, then value of the determinant is: 11903127

(a) 0 (b) 1

Q.29 If any two rows of a square matrix are interchanged, the determinant of the resulting matrix: 11903128

(a) is zero.

(b) is multiplicative inverse of the determinant of the original matrix.

(d) none of these.

Q.30 If a matrix A is symmetric as well as skew symmetric, then: 11903129

(a) A is null matrix

(b) A is unit matrix

(d) A is diagonal matrix

Q.31 If for a matrix A, |A| ¹ 0 then we say matrix A is: 11903130

(a) zero (b) non-singular

Q.32 If A and B are non-singular matrices, then = 11903131

(a) BA (b) AB

Q.33 If P = and Q = are two matrices of same order p ´ q, then order of P + Q is: 11903132

(a) p – q (b) p ´ q

Q.34 If A = and B = are two matrices of same order r ´ s, then order of A – B is: 11903133

(a) r – s (b) r ´ s

Q.35 The matrix A is Hermitian if =

(a) A (b) – A 11903134

Q.36 The matrix A is skew Hermitian if = 11903135

(a) A (b) – A

Q.37 In a diagonal matrix, all elements except those of the diagonal are ………… . 11903136

(a) equal (b) not equal

Q.38 Let be a square matrix. Then the cofactor of a denoted by A is defined as: 11903137

(a) M (b) M

Q.39 Minors and co-factors of the elements in a determinant are equal in magnitude but they may differ in :

11903138

(a) order (b) position

Q.40 Two matrices X and Y are equal if and only if: 11903139

(a) X and Y are of same order

(b) Their corresponding

elements are equal

(d) none of these.

Q.41 If A = , then A is: 11903140

(a) scalar matrix (b) diagonal matrix

(d) skew symmetric matrix

Q.42 If A is a non-singular matrix, then = 11903141

(a) A (b) A

Q.43 If A is a square matrix, then: 11903142

(a) = A (b) = –A

Q.44 For a square matrix A, equals: 11903143

(a) A (b)

Q.45 If A is a square matrix, then A+A is:

(a) null matrix 11903144

(b) unit null matrix

(d) skew symmetric matrix

Q.46 If A is a square matrix, then A-A is:

(a) null matrix 11903145

(b) unit null matrix

(d) skew symmetric matrix

Q.47 If matrix A is non-singular, then:

11903146

(a) A=

(b) A =

(d) A =

Q.48 If each element of a 3 ´ 3 matrix A is multiplied by 3, then the determinant of the resulting matrix is: 11903147

(a) (b) 27

Q.49 If A is a square matrix of order 3 ´ 3, then |kA| equals: 11903148

(a) k |A| (b) k² |A|

Q.50 The solution set of a first degree equation in two variables has: 11903149

(a) one element (b) two elements

(d) infinite number of elements

Q.51 = ………… 11903150

(a) – tany (b) tany

Q.52 = ………… 11903151

(a) (b)

Q.53 If A = , then |A| = ? 11903152

(a) 1 (b) – A

Q.54 = ………. 11903153

(a) 0 (b) 1

Q.55 The value of is: 11903154

(a) 0 (b) 1

Q.56 Rank of the matrix is:

(a) 1 (b) 2 11903155

Q.57 The value of is: 11903156

(a) ah + bg + cm (b) ab + c d + fgh

Q.58 If A = , then:

a A + a A + a A = 11903157

(a) 0 (b)

Q.59 If D =then: 11903158

(a) D = 0 (b) D = 10

Q.60 If = 5, then = ……… 11903159

(a) 25 (b) 20

Q.61 If = 5, then = ……… 11903160

(a) 10 (b) 5

Q.62 If = 5, then =……… 11903161

(a) 10 (b) – 5

Q.63 Matrix form of the equations 11903161

ax + by + c = 0

ax + by + c = 0 is:

(a) =

(b) + =

(d) – =

Q.64 In the homogeneous system of linear equations 11903161

a x+ a x+ a x=0

if = 0

then the system has:

(a) no solution

(b) infinitely many solutions

(d) one trivial and one non-trivial solution

Q.65 In the homogeneous system of linear equations 11903162

a x+ a x+ a x=0

If ¹ 0

then the system has:

(a) no solution

(b) infinitely many solutions

(d) one trivial and one non-trivial solution

Q.66 If A is square matrix of order 2 then equals: (Board 2014) 11903163

(a)

(b)

(c)

(d)

Q.67 If then order of is:

11903164

(a) (b)

Q.68 If , then is equal to:

(Board 2014) 11903165

(a) 5 (b) 20

Unit 4

Unit	QUADRATIC EQUATIONS
04

Q.1. What is a quadratic equation? 11904001

Example: (i) x² + 5x + 6 = 0 11904002

(ii) ax² + bx + c = 0, a 0 11904003

Q. 2. What is the standard form of the quadratic equation? 11904004

Q. 3. What is quadratic formula? 11904005

Q. 4. Solve x² - x = 2 11904006

Hence solution set is {2, -1}.

Q. 5. Solve x(x + 7) = (2x – 1) (x + 4) 11904007

(Board 2014)

Q. 6. Solve + = ;

x ¹ – 1,–2,– 5 11904008

Q. 7. Solve + = a + b ;

x ¹ , 11904009

Q.8. Solve the following equation by completing the square: x – 2x – 899 = 0

11904010

Q. 9. Solve by completing square

2x² + 12x - 110 = 0 11904011

Q.10. Solve by quadratic formula?

15x + 2ax – a = 0 11904012

Q.11. Solve the equation

11904013

Q.12. What is an exponential equation?

11904014

Q.13. Solve the equation

11904015

Q.14. What is a reciprocal equation? 11904016

Q.15. Solve the equation x – 10 = 3x

11904017

Q.16. Solve the equation x + 8 = 6 x

11904018

Q.17. Solve the equation = 24 11904019

Q. 18. Solve the equation 4×2–9×2+1=0

11904020

Q. 19. Solve the equation 11904021

– 3 – 4 = 0

Q.20. What is a radical equation? 11904022

Q.22. Find the three cube roots of unity

11904023

Q.23. Prove that the sum of three cube roots of unity is zero i.e. 1+ w+ w²= 0

11904024

Q.24. Prove that: 11904025

Q.25. Prove that: 11904026

Q.26. Find four fourth roots of unity:

11904027

Q. 27. Evaluate: 11904028

(ii) Evaluate: w²⁸ + w²⁹ + 1 11904029

(iii) Evaluate: 11904030

Q.28. Show that: 11904031

x- y=

(ii) Show that: ….. 2n factors = 1 11904032

Q.29. If w is a cube root of unity, form an equation whose roots are 2w and 2w.

11904033

Q. 30. Solve the given equations: 11904034

x+x+x+1 = 0

Q.31.What is a polynomial function? 11904035

Q. 32. Sate and prove Remainder Theorem

(Board 2014) 11904036

Q. 33. State and prove Factor Theorem

11904037

Q. 34. Use factor theorem to determine if the first polynomial is a factor of the second polynomial.

a) w+ 2 , 2w³ + w² - 4w + 7 11904038

b) x - a, xⁿ - aⁿ where n is a positive integer. 11904039

Q.35. When x⁴ + 2x³ + kx² + 3 is divided by x-2, the remainder is 1. Find the value of k.

11904040

Q. 36. When the polynomial x³+2x²+kx+4 is divided by x - 2, the remainder is 14. Find the value of k. 11904041

Q.37. Find the Relations between the roots and the coefficients of a quadratic equation. 11904042

Q.38. Form an equation whose roots are a and b 11904043

Q.39. If a, b are the roots of

x – px – p – c = 0, 11904044

Prove that (1 + a)(1 + b) = 1 – c

Q. 40. If the roots of the equation x– px +q=0 differ by unity, prove that p=4q+ 1. 11904045

Q.41. If the roots of px+ qx + q = 0 are a and b then prove that 11904046

+ + = 0

Q. 42. If a , b are the roots of

5x– x – 2 = 0, form the equation whose roots are and . 11904047

Q. 43. If a and b are the roots of

x– 3x + 5 = 0, form the equation whose roots are and . 11904048

Q.44. Discuss the nature of the roots of a quadratic equation 11904049

Q.45. For what values of m will the equation (m+1)x2 + 2(m+3)x + 2m+3 = 0 have equal roots? 11904050

Q.46. Show that the roots of the given equation will be real: 11904051

x – 2 x + 3 = 0 ; m ¹ 0

Q.47. Show that the roots of the given equations will be rational: 11904052

(p + q)x – px – q = 0 (Board 2008)

Q. 48. For what values of m will the roots of the given equations be equal? 11904053

(m + 1)x + 2(m + 3)x + m + 8 = 0

Q. 49. Show that the roots of x + (mx + c) = a will be equal, if c = a (1 + m). 11904054

Q. 50. Show that the roots of

= 4ax will be equal, if c = ; m ¹ 0 11904055

Q.51. What is system of simultaneous equations? 11904056

MULTIPLE CHOICE QUESTIONS

q Each question has four possible answers. Select the correct answer and encircle it.

Q.1 The polynomial ax+ bx + c = 0 is quadratic if: 11904057

(a) a ¹ 0 (b) a ¹ 0, b ¹ 0

Q.2 If one root of the equation
a x+ b x + c = 0 be reciprocal of other, then: 11904058

(a) a = 0 , c ¹ 0 (b) b = c

Q.3 Only one of the roots of
a x²+ b x + c = 0, a ¹ 0, is zero if :

11904059

(a) c = 0 (b) b = 0 , c = 0

Q.4 Both the roots of the equation

a x²+ b x + c = 0, are zero if : 11904060

(a) a = 0 and b = 0

(b) a = 0 and c = 0

(d) a = b = c = 0

Q.5 If P(x) is a polynomial of degree m and Q(x) is a polynomial of degree n, the product P(x) × Q(x) will be a polynomial of degree: 11904061

(a) m × n (b) m - n

Q.6 If P(x) is a polynomial of degree m and Q(x) is a polynomial of degree n, the quotient P(x) ¸ Q(x) will produce a polynomial of degree: 11904062

(a) m × n, plus a quotient

(b) m - n, plus a remainder

(d) m + n, plus a remainder

Q.7 Zeros of a function ¦(x) means the values of x which make: 11904063

(a) ¦(x) < 0 (b) ¦(x) > 0

Q.8 (ax - b) is a factor of the polynomial ¦(x), if and only if: 11904064

(a) ¦ = 0 (b) ¦ = 0

Q.9 The roots of the quadratic equation a x+ b x + c = 0 are: 11904065

(a)

(b)

(c)

(d)

Q.10 Synthetic division is a process of:

(a) subtraction (b) addition 11904066

Q.11 The roots of the equation

x+ x + 1 = 0 are: 11904067

(a) complex (b) irrational

Q.12 If the Discriminant of a quadratic equation is a perfect square, then roots are: 11904068

(a) real and equal (b) complex

Q.13 A quadratic equation with 1 and 2 as roots is: 11904069

(a) x²+ 3x – 2 = 0 (b) x² – 3x + 2 = 0

Q.14 Complex roots of real quadratic equation always occur in: 11904070

(a) conjugate pair (b) ordered pair

Q.15 If w is one of complex cube roots of unity, then conjugate of w is: 11904071

(a) – w (b) – w

Q.16 Solution set of the equation: 11904072
x– 3x + 2 = 0 is

(a) (b)

Q.17 Equations having a common solution are called: 11904073

(a) linear (b) quadratic

(d) simultaneous

Q.18 Solution set of the simultaneous equations: x + y = 1, x - y = 1 is : 11904074

(a) (b)

Q.19 A quadratic equation with 1 and –5 as roots is: 11904075

(a) x+ 4x – 5 = 0

(b) x– 4x – 5 = 0

(d) 4x+ x – 5 = 0

Q.20 The real quadratic equation whose one root is 2 – is: 11904076

(a) x+ 4x – 1 = 0 (b) x– 4x–1 = 0

Q.21 If w is the cube root of unity, then a quadratic equation whose roots are 2w and 2wis: 11904077

(a) x– 2x + 4 = 0 (b) x+ 2x + 4 = 0

(c)x+ x + 4 = 0 (d) x+ 2x – 4 = 0

Q.22 How many complex cube roots of unity are there: 11904078

(a) 0 (b)1

Q.23 One of the roots of the equation 3x²+ 2x + k = 0 is the reciprocal of the other, then k = --------- 11904079

(a) 1 (b) 2 (c) 3 (d) 4

Q.24 If the sum of the roots of the equation kx–2x + 2k = 0 is equal to their product, then the value of k is:

(a) 1 (b) 2 (c) 3 (d) 4 11904080

Q.25 For what value of k, the roots of the equation x+ x + 2 = 0 are equal:

(a) 1 (b) 2 (c) 4 (d) 8 11904081

Q.26 For what value of k, the sum of the roots of the equation x+ kx + 4 = 0 is equal to the product of its roots:

(a) ± 1 (b) 4 (c) ± 4 (d) –4 11904082

Q.27 If a , b be the roots of a x²+b x+c= 0 where a ¹ 0 , c ¹ 0 then roots of cx+ bx + a = 0 are: 11904083

(a) , (b) – a , – b

Q.28 If the roots of x – bx + c = 0 are two consecutive integers, then:
b– 4c = --------- 11904084

(a) 0 (b) – 1 (c) 2 (d) 1

Q.29 If the sum of the roots of
ax– x + = 0 is 2, then the product of the roots is: 11904085

(a) 1 (b) 2 (c) 3 (d) 4

Q.30 If –1 is a root of x³– 5x+7x+13 = 0 the depressed equation is: 11904086

(a) – 5x+ 7x + 13 = 0

(b) x– 6x + 12 = 0

(d) 5x+ 6x + 13 = 0

Q.31 If a , b are non-real cube roots of unity, then which one of the following is incorrect: 11904087

(a) a = b (b) b = a

Q.32 If a , b are complex cube roots of unity, then 1 + a + b = ------ where n is a positive integer divisible by 3. 11904088

(a) 1 (b) 2 (c) 3 (d) 4

Q.33 If a , b are the roots of

3x² - 2x + 4 = 0, then the equation whose roots are 2a, 2b is: (Board 2009) 11904089

(a) 2x² + 6x + 8 = 0

(b) 4x² - 2x + 3 = 0

(d) 3x² + 16x - 4 = 0

Q.34 The cube roots of unity are 1, w, w where w = --------- 11904090

(a) (b)

Q.35 ax+by+c = 0 where a , b ¹ 0 is a:

(a) Linear equation 11904091

(b) Quadratic equation

(d) Radical equation

Q.36 If 4^x = 2, then x equals: (Board 2009)

(a) 2 (b) - 11904093

Q.37 Sum of all three cube roots of unity is: (Board 2014) 11904094

(a) 1 (b) – 1

Q.38 If and perfect square the roots are: (Board 2014) 11904095

(a) Rational (b) Irrational

Q.39 If a polynomial P(x) = x³ + 4x² – 2x + 5 is divided by x – 1, then the reminder is:

(Board 2014) 11904096

(a) 4 (b) – 2

Unit 5

MULTIPLE CHOICE QUESTIONS

q Each question has four possible answers. Select the correct answer and encircle it.

Q.1 The quotient of two polynomials where Q(x) ¹ 0, with no common factors, is called a: 11905015

(a) Irrational fraction

(b) Polynomial

(d) None of these.

Q.2 When a rational fraction is separated into partial fractions, the result is: 11905016

(a) an equation

(b) an identity

(d) none of these.

Q.3 The conditional equation = 3 holds if x = ------ 11905017

(a) (b)

Q.4 If degree of P(x) = 3 and degree of Q(x) = 4, then will be: 11905018

(a) Proper rational fraction

(b) Improper rational fraction

(d) Conditional equation

Q.5 If degree of P(x) = 4 and degree of Q(x) = 3, then will be: 11905019

(a) Proper rational fraction

(b) Improper rational fraction

(d) Conditional equation

Q.6 The rational fraction where Q(x) ¹ 0 is proper rational fraction if:11905020

(a) Degree of P(x) = Degree of Q(x)

(b) Degree of P(x) < Degree of Q(x)

(d) None of these.

Q.7 will be improper rational fraction if: 11905021

(a) Degree of Q(x) = 2

(b) Degree of Q(x) = 3

(d) None of these.

Q.8 will be proper rational fraction if: 11905022

(a) Degree of Q(x) = 1

(b) Degree of Q(x) = 2

(d) None of these.

Q.9 The rational fraction is:

(a) Proper (b) Improper 11905023

Q.10 The rational fraction is: 11905024

(a) Proper (b) Improper

Q.11 The rational fraction is: 11905025

(a) Proper (b) Improper

Q.12 The rational fraction is: 11905026

(a) Proper (b) Improper

Q.13 The rational fraction is: 11905027

(a) Proper (b) Improper

Q.14 Partial fractions of will be of the from: 11905028

(a) +

(b) +

(d) +

Q.15 Partial fractions of will be of the from: 11905029

(a) + +

(b) + +

(d) + +

Q.16 If = + + , then A is: 11905030

(a) (b)

Q.17 If = + , then B is:

(a) (b) – 11905031

Q.18 Partial fractions of are of the form: (Board 2014) 11905032

(a) (b)

Q.19 Partial fractions of are of the form: (Board 2014) 11905033

(a)

(b)

(c)

(d)

Unit	PARTIAL FRACTIONS
05

Q.1 What is an open sentence? 11905001

Q.2 What is conditional equation and identity? 11905002

Q.3 What is the rational fraction? 11905003

Note:Any improper rational fraction can be reduced by division to a mixed form, consisting of the sum of a polynomial and a proper rational fraction.

Q.4 Resolve into Partial Fractions. [L.B. 2008 (G-II) Short] 11905004

Q.5 Resolve in to partial fractions 11905005

Q.6 Resolve into partial fractions 11905006

Q.7 Resolve into partial fractions. 11905007

Q.8 Define irreducible quadratic factor.

11305051

Q.9. Resolve into partial fractions. 11905008

Q.10 Resolve into partial fraction. 11905009

Q.11 Write identity equation of 11905010

Q.12 Write identity equation of

11905011

Q.13 Convert into mixed form.

11905012

Q.14 Write the identity equation of 11905013

Q.15 Write the identity equation of 11905014

Unit 6

MULTIPLE CHOICE QUESTIONS

q Each question has four possible answers. Select the correct answer and encircle it.

Q.1 Sequences are also called: 11906074

(a) series (b) progressions

Q.2 A function whose domain is the set of natural numbers is called the: 11906075

(a) series (b) sequence

Q.3 A sequence is denoted by: 11906076

(a) (b)

Q.4 What is called the arrangement of numbers formed according to some definite rule? 11906077

(a) arithmetic sequence

(b) geometric sequence

(d) none of these.

Q.5 In the standard form of an A.P. a, a+d, a + 2d, a + 3d, ……., where a is the first term and d is the common difference, its general term “a_n” is written as: 11906078

(a) a_n = a₁ + (n – 1)d

(b) a_n = a₁ + nd

(d) a_n = a₁ + (n – 2)d

Q.6 If are in A.P., then 11906079

(a) (b)

Q.7 The sum of n A.Ms between a and b is:

(a) (b) 11906080

Q.8 Which number cannot be a term of a geometric sequence? 11906081

(a) 1 (b) -1

Q.9 What is called the difference between two consecutive terms of an arithmetic sequence? 11906082

(a) common ratio

(b) common difference

(d) none of these.

Q.10 If “a₁” is the first term and “d” is the common difference of an A.P., then sum upto n terms of the series is:

11906083

(a)

(b)

(c)

(d)

Q.11 The n^th term of the sequence is , , , … is : 11906084

(a) (b)

Q.12 Which is not an example of arithmetic sequence? 11906085

(a) 1, 2, 3, 4, ……

(b) 1 + 1, 2 + 2, 3 + 3, 4 + 4, ………

(d) 1², 2², 3², 4², ……

Q.13 A sequence of numbers whose reciprocal form an arithmetic sequence, is known as: 11906086

(a) arithmetic sequence

(b) geometric sequence

(d) none of these.

Q.14 If a₁ is the first term and d is the common difference of an A.P. then its (2m + 1)^th term is: 11906087

(a) a₁ + 2md (b) a₁ + m

Q.15 If A, G, H are the A.M., G.M. and H.M. between two distinct positive real numbers, then: 11906088

(a) A < G > H (b) A > G > H

Q.16 If A, G and H are the arithmetic, geometric and harmonic means between a and b respectively, then

G²= 11906089

(a) G (b) HG

Q.17 Three numbers , a, ar are in:

(a) A.P. (b) G.P. 11906090

Q.18 If a = (n + 1) a , a = 1, second term of the sequence is: 11906091

(a) 1 (b) 2

Q.19 If the standard form of an A.P. a₁, a₁ + d, a₁ + 2d, a₁ + 3d, ……., where a₁ is the first term and d is the common difference, then its general term a_n is written as: 11906092

(a) a = a₁ – (n + 1)d

(b) a = a₁+ (n + 1) d

(d) a = a₁ + (n – 1)d

Q.20 , , , …… is a/an: 11906093

(a) A.P. (b) G.P.

Q.21 Reciprocals of the terms of the geometric sequence form: 11906094

(a) A.P. (b) G.P.

Q.22 What is the general term of the sequence 2, 4, 6, 8, ……….? 11906095

(a) 2n (b) n + 1

Q.23 What is the general term of the geometric sequence –1, 1, –1, 1, ……..? 11906096

(a) (–1) (b) (1)

Q.24 What is the next term in the sequence 10, 7, 4, 1 ……..? 11906097

(a) 2 (b) –2

Q.25 Fifth term of the sequence 2, 6, 11, 17, …… is: 11906098

(a) 24 (b) 41

(d) none of the foregoing numbers.

Q.26 What is the common difference of the sequence 11, 5, – 1 , ……? 11906099

(a) 6 (b) –6

(c)

(d) none of the foregoing numbers.

Q.27 The series 3 + 33 + 333 + ..… is: 11906100

(a) A.P. (b) G.P.

Q.28 The series r + (1 + k) r+ (1 + k + k) r+ …… is 11906101

(a) A.P. (b) G.P.

Q.29 The series 2 + (1 – i) + + …… is:

(a) A.P. (b) G.P. 11906102

Q.30 If a = (n + 1) a , a = 1, third term of the sequence is: 11906103

(a) 3 (b) 60

(d) none of the foregoing numbers.

Q.31 If a₁ = 3, r = 2, then the general term of G.P. is: 11906104

(a) (b)

Q.32 The sum of first n natural numbers is:

(a) n (b) (n + 1) 11906105

Q.33 The sum of n terms of a series is , the series is in: 11906106

(a) A.P. (b) G.P.

Q.34 If a , b , c are in A.P., then , , are in:

(a) A.P. (b) G.P. 11906107

Q.35 The series y = 1 + + + ××××is convergent in the interval: 11906108

(a) – 2 < x < 2

(b) – 3 < x < 3

(d) none of these.

Q.36 If x = 1 + y + y+ y + ......... ¥, then y: = ------------- 11906109

(a) (b)

Q.37 The nth term of the series 1+ + + ........ is: 11906110

(a) n (n + 1) (b) (n + 1)

Q.38 The sum up to n terms of the series + + + + ....... is: 11906111

(a) n (n + 1) (b)

Hint: + + + + .... + a= =

Q.39 For any two numbers a and b is:

11906112

(a) A.M (b) G.M

Q.40 5, is: (Board 2014) 11906113

(a) Series (b) A.P

Q.41The sum of an infinite geometric series exists if: (Board 2014) 11906114

(a) (b)

Unit	SEQUENCES AND SERIES
06

Q.1 What is Sequence? 11906001

Q.2 What is real sequence? 11906002

Q.3 What is complex sequence? 11906003

Q.4 What is finite sequence? 11906004

Q.5 What is infinite sequence? 11906005

Q.6 Write the first four terms of the sequence a = (–1)n. 11906006

Q.7 Write the first four terms of the sequence a = na , a = 1. 11906007

Q.8 Write the first four terms of the sequence a = 3n – 5 11906008

Q.9 Find the indicated terms of the sequence 2, 6, 11, 17, ….,a₇11906009

Q.10 Find the indicated term of the sequence 1, 3, 12, 60, …….,a₆11906010

Q.11 Find the indicated term of the sequence 1,1, -3, 5, -7, ……, a₈11906011

Q.12 Find the indicated terms of the sequence 1, -3, 5, -7, 9, -11, …..a₈(Board 2014)11906012

Q.13 Find the next two terms of the sequence.7, 9, 12, 16, …… 11906013

Q.14 Find the next two terms of the sequence -1, 2, 12, 40, ……. 11906014

Q.15 Define A.P and common difference(d). 11906015

Q.16 Find rule for the nth term of an A.P.

11906016

Q.17 If a_n_-₂ = 3n - 11, find the nth term of the sequence. 11906017

Solution:

Given

a_n_-₂ = 3n - 11 …..(1)

Q.18 If a = 2n – 5, find the nth term of the sequence. 11906018

Q.19 Find the 13^th term of the sequence x, 1, 2 – x, 3 – 2x, ... 11906019

Q.20 Which term of the A.P. 5,2,–1,…is– 85? (Board 2014) 11906020

Q.21 Which term of the A.P.–2,4,10,…is 148? 11906021

Q.22 How many terms are there in A.P. in which a = 11, a = 68, d = 3? 11906022

Q.23 If the nth term of an A.P is 3n – 1, find the A.P. 11906023

Q.24 Determine whether 2 is the term of A.P. 17, 13, 9, … or not 11906024

Q.25 Find the nth term of the sequence, , , , … 11906025

Q.26 If , and are in A.P. , Show that b = 11906026

Q.27. If , and are in A.P. Show that the common difference is 11906027

Q.28 Define arithmetic mean (A.M)

11906028

Q.29. Derive the formula to find A.M between a and b. 11906029

Q.30. Write the general formula to find A.M between a_n_-₁ and a_n₊₁ 11906030

Q.31. If there are n A.Ms between a and b. Then find common difference. 11906031

Q.32 Find A.M between 3 and 5

11906032

Q.33 Find A.M between x – 3 and x + 5

11906033

Q.34 Find A.M between 1–x + x and

1+x + x 11906034

Q.35 If 5 , 8 are two A.Ms between a and b, find a and b. 11906035

Q.36 What is series? 11906036

Q.37 What is finite series? 11906037

Q.38 What is infinite series? 11906038

Q.39 Write the formulae to find sum of first n terms of an arithmetic series. 11906039

Q.40 Find the sum of all the integral multiples of 3 between 4 and 97. 11906040

Q.41 Sum the series–3+(–1)+1+3+5+ ×××+ a

11906041

Q.42 How many terms of the series

– 7 + (–5) + (–3) + … amount to 65? 11906042

Q.43 How many terms of the series

–7 + (–4) + (–1) + … amount to 114?

11906043

Q.44 A man deposits in a bank Rs.10 in the first month; Rs.15 in the second month; Rs.20 in the third month and so on. Find how much he will have deposited in the bank by the 9^th month. 11906044

Q.45 A clock strikes once when its hour hand is at one, twice when it is at two and so on. How many times does the clock strike in twelve hours? 11906045

= 6 (13) = 78

Q.46 The sum of interior angles of polygons having sides 3, 4, 5, … etc. form an A.P. Find the sum of the interior angles for a 16 sides polygon. 11906046

Q.47 Define geometric progression (G.P) and common ratio (r). 11906047

Q.48 Derive the formula for nth term of a G.P. 11906048

Q.49 Find the 5th term of the G.P., 3, 6, 12, ¼ 11906049

Q.50 If , and are in G.P. Show that the common ratio is ± 11906050

Q.51 Define geometric means between two numbers a and b. 11906051

Q.52 Derive the formula to find geometric means between two numbers a and b.

11906052

Q.53 Find the geometric mean between 4 and 16. 11906053

Q.54 Find G.M. between –2i and 8i 11906054

Q.55 Insert two G.Ms. between 1 and 8 11906055

Q.56 Insert two G.Ms. between 2 and 16

11906056

Q.57. Find sum of n terms of G. Series.

11906057

Q.58 Find the sum of the infinite G.P.

2, , 1, ¼ 11906058

Q.59 Find the sum of the infinite geometric series + + + ××× 11906059

Q.60 Find the sum of the infinite geometric series + + + …… 11906060

Q.61 Find the sum of the infinite geometric series + + 1 + + …. 11906061

Q.62 Find the sum of the infinite geometric series 2 + 1 + 0.5 + …. 11906062

Q.63 Find a vulgar fraction equivalent to the recurring decimal 0. 11906063

Q.64 If y = 1 + 2x + 4x + 8x + ×××× Show that x = 11906064

Q.65 If y = 1 + 2x + 4x + 8x + ×××× Find the interval in which the series is convergent. 11906065

Q.66 A man deposits in a bank Rs. 8 in the first year, Rs.24 in the second year; Rs.72 in the third year and so on. Find the amount he will have deposited in the bank by the fifth year. 11906066

Q.67 The population of a certain village is 62500. What will be its population after 3 years if it increases geometrically at the rate of 4% annually? 11906067

Q.68 A singular cholera bacteria produces two complete bacteria in hours. If we start with a colony of A bacteria, how many bacteria will we have in n hours? 11906068

Q.69. Define Harmonic progression (H.P).

11906060

Q.70 Write the general form of harmonic sequence. 11906061

Q.71 Define harmonic mean. 11906062

Q.72 Derive the formula to find harmonic mean between two number a and b. 11906063

Q.73 Find the 9th term of the harmonic sequence , , , … (L.B. 2014) 11906064

Q.74 Find the 9th term of the harmonic sequence-, -, -1,…… 11906065

Q.75 Find the 12^th term of the following harmonic sequence ,,, … 11906066

Q.76 Find the 12^th term of the following harmonic sequences ,, , … 11906067

Q.77 If 5 is the harmonic mean between 2 and b, find b. (L.B. 2014) 11906068

Q.78 If the numbers , and are in harmonic sequence, find k. 11906069

Q.79 If A, G and H are the arithmetic, geometric and harmonic means between a and b respectively, show that G = A H.

11906070

Q.80 Find A, G, H and verify that

A > G > H , if a = 2, b = 8

11906071

Q.81 Write three formulas for the sums of series. 11906072

Q.82 Sum the series upto n terms. 1´1 + 2´4 + 3´7 + ……… 11906073

Unit 7

MULTIPLE CHOICE QUESTIONS

q Each question has four possible answers. Select the correct answer and encircle it.

Q.1 n! stands for: 11907065

(a) product of first n natural numbers

(b) sum of first n natural numbers

(d) None of these.

Q.2 = ------------- 11907066

(a) (b)

Q.3 If n is a positive integer, then n! = 11907067

(a) n(n + 1)(n + 2)(n + 3)…3.2.1

(b) n(n – 1)(n – 2)(n – 3)…3.2.1

(d) n(n – 1)(n – 2)(n – 3)…

Q.4 For a positive integer n: 11907068

(a) (n + 1)! = (n + 1)n!

(b) (n + 1)! = (n + 1)(n – 1)!

(d) (n + 1)! = (n)(n - 1)!

Q.5 Number of permutations of n different things taken r at a time is denoted by:

11907069

(a) C (b) P

Q.6 How many arrangements of the letters of the word pakistan can be made, taken all together? 11907070

(a) 21160 (b) 20160

Q.7 How many arrangements of the letters of the word pakpattan can be made, taken all together? 11907071

(a) 15130 (b) 1512

Q.8 How many necklaces can be made from 6 beads of different colours? 11907072

(a) 120 (b) 60

Q.9 The number of handshakes that can be exchanged among a party of 10 students if every student shakes hands once with every student is: 11907073

(a) P (b) P

Q.10 How many diagonals can be formed by joining the vertices of the polygon having 5 sides? 11907074

(a) 10 (b) 15

Q.11 How many diagonals can be formed by joining the vertices of the polygon having 12 sides? 11907075

(a) 70 (b) 54

Q.12 How many triangles can be formed by joining the vertices of the polygon having 5 sides? 11907076

(a) 20 (b) 15

Q.13 How many triangles can be formed by joining the vertices of the polygon having 12 sides? 11907077

(a) 202 (b) 220

Q.14 The number of diagonals of a polygon with n sides are: 11907078

(a) (b)

Q.15 How many different numbers can be formed by taking 4 out of the six digits 1,2,3,4,5,6: 11907079

(a) 360 (b) 120

Q.16 Numbers are formed by using all the digits 1 , 2 , 3 , 4 , 5 , 6 no digit being repeated, then the numbers which are divisible by 5 are: 11907080

(a) 110 (b) 120

Q.17 How may numbers of six digits can be formed from the digits 4 , 5 , 6 , 7 , 8 , 9, no digit being repeated? 11907081

(a) 750 (b) 720

Q.18 If we denote the set of all outcomes, favourable to an event by E, the set of all equally possible outcomes by S and the probability of an event happening by P(E), then P(E) is: 11907082

(a) (b)

Q.19 If A and B are overlapping two events, then: 11907083

(a) P( A È B ) = P( A ) + P( B )

(b) P( A È B ) = P( A ) + P( B ) + P( A Ç B )

(d) P( A È B ) = P( A ) + P( B ) + P( A È B )

Q.20 For two mutually exclusive events A and B: 11907084

(a) P( A È B ) = P( A ) + P( B ) + P( A Ç B )

(b) P( A È B ) = P( A ) + P( B ) – P( A È B )

(d) P( A È B ) = P( A ) + P( B )

Q.21 Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn bears a number which is a multiple of 3? 11907085

(a) (b)

Q.22 A dice is thrown. What is the probability to get an odd number?

(a) 1 (b) 11907086

Q.23 A dice is thrown. What is the probability to get an even number?

(a) 1 (b) 11907087

Q.24 The probability that an egg will be broken during delivery from a farm to a supermarket is . How many broken eggs would there be in 2400 eggs? 11907088

(a) 4 (b) 6

Q.25 How many arrangements can be made of 4 letters a , b , c , d taken 2 at a time? 11907089

(a) 8 (b) 10

Q.26 The number of ways in which five persons can sit at a round table is:

(a) 4! (b) 5! 11907090

Q.27 Probability of an impossible event is:

(a) 0 (b) 1 11907091

Q.28 In a simultaneous throw of two dice, what is the probability of getting a total of 10 or 11? 11907092

(a) (b)

Q.29 A dice is rolled, the probability that it does not show an even number is: 11907093

(a) 1 , 3 , 5 (b) 2 , 4 , 6

Q.30 What is the probability that a number selected from the numbers 1, 2, 3, 4, 5, ….., 16 is a prime number is? 11907094

(a) (b)

Q.31 In a simultaneous throw of two coins, the probability of getting at least one head is: 11907095

(a) (b)

Q.32 One card is drawn at random from a pack of 52 cards. What is the probability that the card drawn is a king? 11907096

(a) (b)

Q.33 In a simultaneous throw of two dice, what is the probability of getting a total of 7? 11907097

(a) (b)

Q.34 A dice is rolled, the probability of getting a number which is even or greater than 4 is: 11907098

(a) (b)

Q.35 One card is drawn at random from a pack of 52 cards. What is the probability that the card drawn is either a diamond card or a king? 11907099

(a) (b)

Q.36 Number of ways of arranging 5 keys in a circular ring is: (Board 2014) 11907100

(a) 24 (b) 12

Q.37 The value of ⁵C₂ is: (Board 2014) 11907101

(a) 1 (b) 10

Q.38 If A and B are independent events and and then is: (Board 2014) 11907102

(a) (b)

Q.39 is equal to: (Board 2014) 11907103

(a) (b)

Unit

PERMUTATION, COMBINATION

AND PROBABILITY

Q. 1 What is Permutation? 11907001

Q. 3 How many signals can be made with 4-different flags when any number of them are to be used at a time? 11907003

Q.4 In how many ways can be letters of the word MISSISSIPPI be arranged when all the letters are to be used? 11907004

Q. 5 In how many ways can a necklace of 8 beads of different colours be made? 11907005

Q. 6 What is Combination. 11907006

Q. 7 Prove That ⁿC_r = 11907007

Q.8 ⁿC_r = ⁿC_n-r(Board 2014)11907008

Q. 9 Find the number of the diagonals of a 6-sided figure. 11907009

Q.10 What is “Sample space” and an event? 11907010

Q. 11 What is Probability? 11907011

Q.12 What are mutually exclusive events?

11907012

Example: 11907013

Q. 13 What are equally Likely Events?

11907014

Q.14 A die is rolled. What is the probability that the dots on the top are greater than 4?

Q. 15 A die is thrown. Find the probability that the dots on the top are prime numbers or odd numbers. 11907016

Q. 16 The probabilities that a man and his wife will be alive in the next 20 years are 0.8 and 0.75 respectively. Find the probability that both of them will be alive in the next 20 years. 11907017

Q. 17 Write n(n – 1)(n – 2) …. (n – r + 1) in factorial form. 11907018

Q. 18 Evaluate P 11907019

Q. 19 Find the value of n when: 11907020

(i) nP2 = 30 11907021 (ii) = 11.10.9 11907022

Q.20 How many signals can be given by 5 flags of different colours, using 3 flags at a time? 11907023

Q. 21 How many signals can be given by 6 flags of different colours, when any number of them are used at a time? (Board 2008) 11907024

Q. 22 How many words can be formed from the letters of the following words using all letters when no letter is to be repeated?

11907025

(i) PLANE 11907026

(ii) OBJECT 11907027

(iii) FASTING 11907028

Q. 23 Find the numbers greater than 23000 that can be formed from the digits 1, 2, 3, 5, 6 without repeating the digit. 11907029

Q.24 How many arrangements of the letters of the following words, taken all together, can be made? 11907030

pakpattan

Q.25 How many arrangements of the letters of the following words, taken all together, can be made? pakistan 11907031

Q. 26 How many arrangements of the letters of the following words, taken all together, can be made? mathematics 11907032

Q.27 How many arrangements of the letters of the following words, taken all together, can be made? 11907033

assassination

Q. 28 In how many ways can 4 keys be arranged on a circular key ring? 11907034

Q. 29 How many necklaces can be made from 6 beads of different colours? 11907035

Q. 30 Find the value of n, when 11907036

(i) ⁿC₅ = ⁿC₄ 11907037

(ii) ⁿC₁₀ = 11907038

Q. 31 Find the values of n and r when

11907039

(i) nCr = 35 , nPr = 210

Q. 32 In how many ways can a hockey team of 11 players be selected out of 15 players? How many of them will include a particular player? 11907040

Q. 33 Pakistan and India play a cricket match. Find probability of events happening: 11907041

Q. 34 A fair coin is tossed three times. Find probability when events are.

11907042

One tail 11907043

At least one head 11907044

Q. 35 A die is rolled. Find probability when events are. 11907045

3 or 4 dots 11907046

dots less than 5 11907047

Q. 36 A coin is tossed four times. Find probability when top shows 11907048

(i) All heads 11907049

(ii) 2 heads and 2 tails 11907050

Q. 37 A box contains 10 red, 30 white and 20 black marbles. A marble is drawn at random. Find the probability that it is either red or white. 11907051

Q. 38 A natural number is chosen out of the first fifty natural numbers. What is the probability that the chosen numbers is a multiple of 3 or of 5? 11907052

Þ n(A Ç B) = 3

Q. 39 A card is drawn from a deck of 52 playing cards. What is the probability that it is a diamond card or an ace? 11907053

Q. 40 A die is thrown twice. What is the probability that the sum of the number of dots shown is 3 or 11? 11907054

Q. 41 Two dice are thrown. What is the probability that the sum of the number of dots appearing on them is 4 or 6 11907055

Q. 42 A die is rolled twice: Event E is the appearance of even number of dots and event E is the appearance of more than 4 dots. Prove that: 11907056

P (EÇ E) = P (E) . P (E)

Q. 43 Determine the probability of getting 2 heads in two successive tosses of a balanced coin. 11907057

Q. 44 Two coins are tossed twice each. Find the probability that the head appears on the first toss and the same faces appear in the two tosses. 11907058

Q. 45 When cards are drawn from a deck of 52 playing cards. If one card is drawn and replaced before drawing the second card, find the probability that both the cards are aces. 11907059

Q. 46 Two cards from a deck of 52 playing cards are drawn in such a way that the card is replaced after the first draw. Find the probabilities in the following cases: 11907060

(i) First card is king and second is queen.

(ii) both the cards are faced cards i.e. king, queen, jack. 11907062

Q. 47 Two dice are thrown twice what is probability that sum of the dots shown in the first throw is 7 and that of the second throw is 11? 11907063

Q. 48 Find the probability that the sum of dots appearing in two successive throws of two dice is every time 7. 11907064

Unit 8

MULTIPLE CHOICE QUESTIONS

q Each question has four possible answers. Select the correct answer and encircle it.

Q.1 If a statement P(n) is true for n = 1 and the truth of P(n) for n = k implies the truth of P(n) for n = k + 1, then P(n) is true for all: 11908036

(a) integers n

(b) real numbers n

(d) positive integers n

Q.2 If n is any positive integer, then
a₁ + (a₁ + d) + (a₁ + 2d) + … + = -----------------11908037

(a)

(b)

(c)

(d)

Q.3 If n is any positive integer, then
r + r+ r + … + r = -----------11908038

(a) ; (r ¹ 1) (b) ; (r ¹ 1)

Q.4 If n is a positive integer, then
+ + + … + =

11908039

(a) (b)

Q.5 If n is a positive integer, then
+ + +..…+ = 11908040

(a) (b)

Q.6 The expansion of where n is a positive integer, is: 11908031

(a) a + a b + a b + ....

+ C a b + ...+ a b + b

(b) a + C a + C a + ....

+ C a + ...+ C a + Cb

+ C a b + ...+ n a b + b

(d) None of these.

Q.7 Binomial coefficients in the expansion of are: 11908042

(a)	–1	–5	–10	–10	–5	–1
(b)	1	–5	10	–10	5	–1
(c)	–1	5	–10	10	–5	1
(d)	1	5	10	10	5	1

Q.8 The T in the binomial expansion of is: 11908043

(a) C x (b) C x

Q.9 The middle term in the expansion of is: 11908044

(a) 10^th term (b) 11^th term

Q.10 If the middle term in the expansion of (a+b)is th term, then n is:

11908045

(a) even (b) odd

Q.11 In the expansion of (x – y) , the terms are alternatively positive and ----------

11908046

(a) Negative (b) Undefined

Q.12 The expansion of is valid if: 11908047

(a) < 1 (b) > 1

Q.13 The middle term of (x– y) is : 11908048

(a) 9th (b) 10th

Q.14 The sum of the series 1+ + + + … ¥ is: 11908049

(a) (b)

Q.15 Number of terms in the expansion of (a + b) is: 11908050

(a) n (b) n + 1

Q.16 The middle terms of (x + y)are:

11908051

(a) 10 and 11 (b) 11 and 12

Q.17 In the expansion of (1 + x), the sum of the coefficients of odd powers of x is:

(a) 0 (b) 2⁴⁹11908052

Q.18 Number of terms in the expansion of (x + y) is: 11908053

(a) 6 (b) 2

Q.19 If n is a positive integer, then the binomial co-efficients equidistant from the beginning and the end in the expansion of are: 11908054

(a) equal

(b) not equal

(d) multiplicative inverse of each other.

Q.20 If –1 < x < 1, then

1 – x + x² – x³ + … = _________. 11908055

(a) (b)

Q.21 The expansion of is valid if:

(Board 2014) 11908056

(a) (b)

Q.22 If n is any positive integer then + n³ equals: 11908057

(Board 2014)

(a) (b)

Q.23 In the expansion of middle term is: (Board 2014) 11908058

(a) (b)

Q.24 The number of terms in the expansion of is: (Board 2014) 11908059

(a) (b)

Unit	MATHEMATICAL INDUCTION AND BINOMIAL THEOREM
08

Q.1 What is principle of mathematical induction? 11908001

Q.2 What is principle of extended mathematical induction? 11908002

Q.3 Use mathematical induction to prove the following formula for every positive integer n 1 + 3 + 5 + … + (2n – 1) = n

11908003

Q.4 Use mathematical induction to prove the following formula for every positive integer a = a + d when a , a+ d , a + 2 d, … form an A.P. 11908004

Q.5 Use mathematical induction prove the given formula for every positive integer n a = ar when a , a r , ar, … form a G.P. 11908005

Q.6 Prove the following for n = 1, 2 11908006

i. n+ n is divisible by 2 11908007

ii. 5 – 2 is divisible by 3 11908008

iii. 5 – 1 is divisible by 4. 11908009

Q.7 Prove that x – y is a factor of

x– y ; for n = 1, 2 11908010

Q.8 Prove that x + y is a factor of 11908011

x + y; for n = 1, 2

Q.9 What is binomial theorem? 11908012

Q.10 Evaluate (9.9)⁵. 11908013

Q.11 Using binomial theorem to expand the following: 11908014

Q.12 Calculate by means of binomial theorem. 11908015

(i) (0.97) 11908016

(ii) (2.02) 11908017

Q.13 Expand and simplify the following:

– 11908018

Q.14 Find 6^th term in the expansion of 11908019

Q.15 Expand (1-2x)^1/3 to four terms and apply it to evaluate (.8)^1/3 correct to three places of decimal. 11908020

Q.16 Evaluate correct to three places of decimal. 11908021

Q.17 Expand the following up to 4 terms, taking the values of x such that the expansion in each case is valid. 11908022

i. (4 – 3x), 11908023 ii. (8 – 2x) 11908024

Q.18 Using Binomial theorem to find the value of the following upto three places of decimals. 11908025

i. 11908026 ii. 11908027

iii. 11908028 iv. 11908029

Q.19 If x is so small that its square and higher powers can be neglected, then show that 11908030

i. »1 – x 11908031

ii.»1+ x 11908032

Q.20 If x is very small nearly equal to 1 then prove that px^p – qx^q»(p – q)x^p+q

11908033

Q.21 Use binomial theorem to show that

1 + + + + …¥ = 11908034

Q.22 If 2y = + + + ×××, prove that 4y+ 4y – 1 = 0 11908035

Unit 9

MULTIPLE CHOICE QUESTIONS

q Each question has four possible answers. Select the correct answer and encircle it.

Q.1 The distance between the points and is: 11909034

(a)

(b)

(c)

(d)

Q.2 If the initial side of an angle is the positive x-axis and the vertex is at the origin, the angle is said to be in the
----------------- 11909035

(a) initial position

(b) final position

(d) standard position

Q.3 The area of a sector of a circular region of radius r with length of the arc of the sector equal to s is ---- 11909036

(a) r s (b) r q

Q.4 The system of measurement in which the angle is measured in degrees, and its sub-units, minutes and seconds is called the: 11909037

(a) circular system

(b) sexagesimal system

(d) degree system

Q.5 In a circle of radius r, an arc of length k r will subtend an angle of -------- radians at the center. 11909038

(a) s (b) k

Q.6 In circular system the angle is measured in: 11909039

(a) radians (b) degrees

Q.7 To convert any angle in degrees into radians, we multiply the measure by:

(a) (b) 11909040

Q.8 To convert any angle in radians into degrees, we multiply the measure by:

(a) (b) 11909041

Q.9 1 radian is equal to: 11909042

(a) (b)

Q.10 1° is equal to: 11909043

(a) radian (b) radian

Q.11 180° = ------------- 11909044

(a) radian (b) radian

Q.12 The direction of an angle q is determined by its: 11909045

(a) value (b) magnitude

Q.13 The quadrant of an angle q is determined by its: 11909046

(a) sign (b) value

Q.14 If q = 60° Then: 11909047

(a) q = (b) sec q = 2

Q.15 p can represent: 11909048

(a) the ratio of the circumference to the radius of a circle

(b) an angle whose cosine is zero

(d) half a revolution

Q.16 If sine of an angle is times its cosine, then angle q is: 11909049

(a) 30° (b) 45°

Hint: By given condition

sin q = cos q = 1

Þ tan q = Þ q = 60°

Q.17 If sin²q = then tan²q = ……….11909050

(a) (b)

Q.18 The expression can be written in the simplified form as:

11909051

(a) sin a (b) cos a

Q.19 tan²a is equal to: 11909052

(a) coseca – 1 (b) cosa – 1

Q.20 cosec²a – 1 is equal to: 11909053

(a) cosa (b) cota

Q.21 is equal to: 11909054

(a) sin a (b) cos a

Q.22 is equal to: 11909055

(a) sin a (b) cos a

Q.23 + is equal to: 11909056

(a) 0 (b) 1

Q.24 If tan q > 0 and sin q < 0 then terminal arm of the angle lies in quadrant:

11909057

(a) I (b) II

Q.25 If cosec q > 0 and cot q < 0, then terminal arm of the angle lies in:

(a) First quadrant 11909058

(b) Second quadrant

(d) Fourth quadrant

Q.26 If sin a < 0 and cos a > 0, then a lies in: 11909059

(a) First quadrant

(b) Second quadrant

(d) Fourth quadrant

Q.27 In a triangle, the side opposite to 90 is called: 11909060

(a) Base (b) Perpendicular

Q.28 In a right angled triangle, the side adjacent to angle is called: 11909061

(a) Base

(b) Perpendicular

(d) None of these.

Q.29 If q lies in third quadrant, then Sinq + cos q is: 11909062

(a) Negative

(b) Positive

(d) Negative or positive

Q.30 An angle q is such that tan q = 1 and cos q is negative, then: 11909063

(a) cosec q is positive

(b) cot q = -1

(d) sec q = -

Q.31 If tan q = x and q is acute, cosec q =

(a) (b) 11909064

Q.32 If sinq = x and q is acute, tanq = 11909065

(a) (b)

Q.33 If cos q = x and q is acute, cot q =11909066

(a) (b)

Q.34 If sec q = x and q is acute, sin q =11909067

(a) (b)

Q.35 If tan q = , then sin q = ----------where 0 < q < : 11909068

(a) (b)

Q.36 cosec q = , for all q Î R except:

11909069

(a) q = np , n Î z (b) q ¹ np , n Î z

(d) q ¹ (2n + 1) , n Î z

Q.37 cot q = , for all q Î R but: 11909070

(a) q = np , n Î z (b) q ¹ np , n Î z

(d) q ¹ (2n + 1) , n Î z

Q.38 sec q = , for all q Î R but: 11909071

(a) q ¹ np , n Î z

(b) q ¹ (2n + 1) , n Î z

(d) none of these.

Q.39 tan q = , for all q Î R but: 11909072

(a) q ¹ np , n Î z

(b) q ¹ (2n + 1) , n Î z

(d) none of these.

Q.40 1 + tanq = secq, for all q Î R but:

(a) q ¹ np , n Î z 11909073

(b) q ¹ (2n + 1) , n Î z

(d) none of these.

Q.41 1 + cotq = cscq , for all q Î R but:

(a) q = np , n Î z 11909074

(b) q ¹ np , n Î z

(d) q ¹ (2n + 1) , n Î z

Q.42 sinq + cosq = 1 , for all q Î R but:

(a) q = np , n Î z 11909075

(b) q ¹ np , n Î z

(d) none of these.

Q.43 Which one is a quadrantal angle?

(a) 60° (b) 30° 11909076

Q.44 Which one is not a quadrantal angle?

(a) 0° (b) 90° 11909077

Q.45 (1 – tan q) + (1 + tan q) is equal to:

(a) 2 tanq (b) 2 tan q 11909078

Q.46 is equal to: 11909079

(a) cot A cot B (b) tan A tan B

Q.47 = 11909080

(a) sin 2q (b) cos 2q

Q.48 cosq – sinq = 11909081

(a) sin 2q (b) cos 2q

Q.49 sin q (cosec q – sin q) = 11909082

(a) cosq (b) sinq

Q.50 (1 – sinq) (1 + tanq) = 11909083

(a) 0 (b) 1

Q.51 (1 – cosq) ( 1 + cotq) = 11909084

(a) tanq (b) 0

Q.52 is equal to: 11909085

(a) cot A tan B (b) tan A cot B

Q.53 In the triangle with sides as shown in the figure, cos A equals to: 11909086

(a)

(b)

(c)

(d)

Q.54 In DABC with measures as shown, cosA is equal to: 11909087

(a)

(b)

(c)

(d)

Q.55 The area A of the sector AOP of radius r is given by

A = ---------, with q in radians. 11909088

(a) r q

(b) rq

(d) r q

Q.56 From given figure sin q = -------11909089

(a)

(b)

(c)

(d)

Q.57 From given figure cos q = --------11909090

(a)

(b)

(c)

(d)

Q.58 From given figure tan q = --------11909091

(a)

(b)

(c)

(d)

Q.59 From given figure cosec q = -----11909092

(a)

(b)

(c)

(d)

Q.60 From given figure sec q = --------11909093

(a)

(b)

(c)

(d)

Q.61 From given figure cot q = --------11909094

(a)

(b)

(c)

(d)

Q.62 The angle between 0° and 360° and co-terminal with - 620° is: 11909095

(a) 100° (b) 200°

Q.63 If s denotes the length of the arc intercepted on a circle of radius r by a central angle of a radians, then: 11909096

(a) s = r + a (b) s = r a

Q.64 The number of radians in the angle subtended by an arc of a circle at the center = 11909097

(a) (b) radius ´ arc

Q.65 The values of trigonometric-ratios for the angles (n ´ 360° + q) where n Î Z, will be the same as those for: 11909098

(a) n ´ q (b) k ´ q

Q.66 If x = a sec q and y = b tan q, then the value of – is: 11909099

(a) 1 (b) – 1

Q.67 If sin q + cosec q = 2, then sinq + cosecq = 11909100

(a) 0 (b) 2

Q.68 – = ------------- 11909101

(a) of a clockwise revolution.

(b) of a counterclockwise

revolution.

(d) of a counterclockwise revolution.

Q.69 – 72 = ------------- 11909102

(a) of a clockwise revolution

(b) of a counterclockwise revolution

(d) None of these.

Q.70 = ------------- 11909103

(a) of a counterclockwise revolution

(b) of a counterclockwise revolution

(d) of a counterclockwise revolution

Q.71 1440 = ------------- 11909104

(a) 4 counterclockwise revolutions

(b) 5 counterclockwise revolutions

(d) 7 counterclockwise revolutions

Unit	Fundamental of Trigonometry
09

Q. 1 What is sexagesimal system? 11909001

Q. 2 What is circular system? 11909002

Q. 3 What is a degree? 11909003

Q. 4 What is a radian? 11909004

Q. 5 Find a relation between degree and radian. 11909005

Q. 6 Prove that = rq 11909006

Q.7 An arc subtends an angle of 70° at the center of a circle and its length is 132 m.m. Find the radius of the circle. 11909007

Q.8 Find the length of the equatorial arc subtending an angle of 1o at the centre of the earth, taking the radius of the earth as 6400 km. 11909008

Q.9 Convert radians into degrees. 11909009

Q. 10 What is the circular measure of the angle between the hands of a watch at 4 O’clock? 11909010

Q. 11 Find l, when: q = 65°20¢, r=18 mm.

(i) Sol: r = 18 mm 11909011

Q.12 What is the length of the arc intercepted on a circle of radius 14 cm by the arms of a central angle of 45°? 11909012

Q.13 A railway train is running on a circular track of radius 500 meters at the rate of 30 km per hour. Through what angle will it turn in 10 sec. 11909013

Q. 14 Show that the area of a sector of a circular region of radius r is rq, where q is the circular measure of the central angle of the sector. 11909014

Q. 15 Two cities A and B lie on the equator such that their longitudes are 45°E and 25°W respectively. Find the distance between the two cities, taking radius of the earth as 6400 kms. 11909015

Q.16 What is angle in standard position?

11909016

Q.17 If cosec q = and , find the values of the remaining trigonometric ratios. 11909017

Q.18 If tan q = and the terminal arm of the angle is not in the III quadrant, find the value of 11909018

Q. 19 If cot q = and the terminal arm of the angle is in the I quadrant, find the value of (Board 2008) 11909019

Q.20 Prove that: 11909020

sin :sin :sin :sin = 1:2:3:4

Q. 21 Find x, if tan45°–cos60°=x sin 45° cos 45° tan 60° 11909021

Q. 22 Find the values of the trigonometric functions of the following quadrantal angles: 11909022

(i) – 2430° ; 11909023

(ii) π ; 11909024

(iii) π 11909025

Q.23 Prove that cos⁴q - sin⁴q

= cos²q - sin²q for all q Î R 11909026

Q.24 Prove that = sec q - tan q,

where q is not an odd multiple of . 11909027

Q.25 sec - cosec = tan - cot 11909028

Q.26 2cosq - 1 = 1 - 2sinq 11909029

Q.27 + cot q = cosec q 11909030

Q.28 = (Board 2014) 11909031

Q.29 tan q + sec q 11909032

(Board 2008)

Q.30 + = 2sec q 11909033

(Board 2014)

Unit 10

MULTIPLE CHOICE QUESTIONS

q Each question has four possible answers. Select the correct answer and encircle it

Q.1 sin ( a + b ) = 11910107

(a) sin a sin b + csc a cos b

(b) sin a sin b + cos a cos b

(d) sin a cos b + cos a sin b

Q.2 cos ( a + b ) = 11910108

(a) cos a cos b – sin a sin b

(b) cos a sin b + sin a cos b

(d) cos a cos b + sin a sin b

Q.3 sin ( a – b ) = 11910109

(a) sin a cos b + cos a sin b

(b) sin a cos b – cos a sin b

(d) cos a cos b + sin a sin b

Q.4 cos ( a – b ) = 11910110

(a) cos a cos b + sin a sin b

(b) cos a cos b – sin a sin b

(d) sin a cos b – cos a sin b

Q.5 2 sin a cos b = (Board 2009) 11910111

(a) sin (a + b ) – sin (a – b)

(b) cos (a + b ) + cos (a – b)

(d) cos (a + b ) – cos (a – b)

Q.6 2 cos a sin b = 11910112

(a) cos (a + b) + cos (a – b)

(b) sin (a + b) + sin (a – b)

(d) cos (a + b) – cos (a – b)

Q.7 – 2 sin a sin b = 11910113

(a) sin (a + b) + sin (a – b)

(b) cos (a – b) – cos (a + b)

(d) sin (a + b) – sin (a – b)

Q.8 2 cos a cos b = 11910114

(a) sin (a + b) – sin (a – b)

(b) cos (a + b) – cos (a – b)

(d) sin (a + b) + sin (a – b )

Q.9 2 sin 12° sin 46° = (Board 2009) 11910115

(a) cos 34° + cos 58°

(b) sin 34° - sin 58°

(d) cos 34° - cos 58°

Q.10 tan ( a – b ) = 11910116

(a) (b)

Q.11 tan ( a + b ) = 11910117

(a) (b)

Q.12 cos q + cos f = 11910118

(a) 2 sin sin

(b) 2 cos cos

(d) 2 cos sin

Q.13 cos q – cos f = 11910119

(a) 2 cos cos

(b) 2 cos sin

(d) 2 sin cos

Q.14 sin q + sin f = 11910120

(a) 2 cos cos

(b) – 2 sin sin

(d) 2 sin cos

Q.15 sin q – sin f = 11910121

(a) – 2 sin sin

(b) 2 cos sin

(d) 2 cos cos

Q.16 sin 2q = 11910122

(a) (b)

Q.17 cos 2q = 11910123

(a) (b)

Q.18 sin 3q = 11910124

(a) 3 sin q – 4 cosq

(b) 4 cosq – 3 sin q

(d) 3 sin q – 4 sinq

Q.19 cos 3q = 11910125

(a) 4 cosq – 3 cos q

(b) 4 sinq – 3 cos q

(d) 3 sin q – 4 sinq

Q.20 tan 3q = 11910126

(a)

(b)

(c)

(d)

Q.21 sin² = 11910126

( a )1 + cos q ( b ) 1 – cos q

( c ) ( d )

Q.22 cos² = 11910127

( a ) 1 + cos q ( b ) 1 – cos q

( c ) ( d )

Q.23 sin = 11910128

( a ) ± ( b ) ±

( c ) ( d )

Q.24 cos = 11910129

( a ) ± ( b ) ±

( c ) ( d )

Q.25 tan = 11910130

( a ) ± ( b )

( c )± ( d )

Q.26 tan = 11910131

( a ) ( b )

( c ) ( d ) None of these.

Q.27 tan = 11910132

( a ) ( b )

( c ) ( d ) None of these.

Q.28 If A + B = 45°, then tan (A+B+C) =

11910133

(a) (b)

Q.29 tan 40° = ………… 11910134

(a)

(b)

(c)

(d)

Q.30 sin40° = ………… 11910135

(a) (b)

Q.31 tan = 11910136

(a) (b)

Q.32 tan = ………… 11910137

(a) (b)

Q.33 The angles 90 ± q, 180 ± q, 270 ± q, 360 ± q are the: 11910138

(a) composite angles

(b) half angles

(d) allied angles

Q.34 A reference angle q is always: 11910139

(a) 0 < q < (b) < q < p

Q.35 sin , where q is a basic angle, will have terminal side in: 11910140

(a) quad. I (b) quad. II

Q.36 tan , where q is a basic angle, will have terminal side in: 11910141

(a) quad. I (b) quad. II

Q.37 cos = ………… 11910142

(a) cos q (b) sin q

Q.38 tan = ………… 11910143

(a) – cot q (b) – tan q

Q.39 sin = ………… 11910144

(a) cos q (b) cos

Q.40 sin equals: (Board 2009) 11910145

(a) cos q (b) sin q

Q.41 csc , where q is a basic angle, will have terminal side in: 11910146

(a) quad. I (b) quad. II

Q.42 sec , where q is a basic angle, will have terminal side in: 11910147

(a) quad. I (b) quad. II

Q.43 tan (–135°) = 11910148

(a) 0 (b) 1

Q.44 = 11910149

(a) cos 0 (b) cos

Q.45 If an angle a is allied to an angle b, then a ± b = ------------- 11910150

(a) 90

(b) multiple of 90

(d) multiple of 180

Q.46 is equal: 11910151

(a) (Board 2014)

(b)

(c)

(d)

Q.47 is equal to: (Board 2014)

(a) 11910152

(b)

(c)

(d)

(Board 2008) 11310140

Unit	TRIGONOMETRIC IDENTITIES OF SUM AND DIFFERENCE OF ANGLES
10

Q.1 State fundamental law of trigonometry. 11910001

Q.2 Find the value of cos. 11910002

Q.3 Define Allied Angles 11910002

Q.4 Without using the tables, write down the values of 11910003

(i) cos 315° 11910004 (ii) sin 540° 11910005

(iii) tan(-135°)11910006 (iv) sec(- 1300°) 11910007

Q.5 Simplify 11910008

Q.6 Without using the tables, find the values of: 11910009

(i) sin 11910010

(ii) cot 11910011

(iii) cosec 2040° 11910012

(iv) sec 11910013

(v) tan 1110° 11910014

(vi) sin 11910015

Q.7 Prove sin 780° sin 480°+cos 120° sin30° = 11910016

Q.8 Prove that cos 306° + cos 234° + cos162°+ cos 18° = 0 11910017

Q.9 Prove that cos 330° sin 600°+cos 120° sin150° = –1 11910018

Q.10 Prove that 11910019

Q.11 = – 1

Sol: 11910020

Q.12 If a , b , g are the angles of a triangle ABC, then prove that 11910021

(i) sin(a+b) = sin γ 11910022

(ii) cos= sin 11910023

(iii) cos (a + b) = - cos 11910024

11910025

Q.13 Prove that 11910026

sin(a+b)sin(a-b)= sin²a-sin²b= cos²b-cos²a

From (i) and (ii) hence proved sin(a + b) sin(a-b) = sin²a-sin²b = cos²b-cos²a

Q.14 Without using tables, find the values of all trigonometric functions of 75°. 11910027

Q.15 Prove that:= tan 56°. 11910028

Q.16 If a, b, g are the angles of DABC, prove that 11910029

Q.17 Express: 3 sinq + 4 cosq in the form r sin(q + f), where the terminal side of the angle of measure f is in the I quadrant.

11910030

Q.18 Prove that tan = cotq

11910031

Q.19 Find the value 11910032

(i)sin 15° 11910033 (ii) cos 15° 11910034

(iii) tan 15° 11910035 (iv) sin 105°11910036

(v)cos 105° 11910037 (vi) tan105°11910038

(vii) sec 15° 11910038 (viii) sec 105° 11910039

Q.20 Prove that: 11910040

(i) sin = 11910041

(ii) cos= 11910042

= R.H.S.

Q.21 Prove that

(i)tantan= 1 11910043

(ii) tan+ tan = 0 11910044

(iii) sin+ cos= cosq 11910045

(iv) = tan 11910046

(v) = 11910047

Q.22 Show that: 11910048

coscos= cosa – sinb=cosb – sina

Q.23 Show that: 11910049

= tan a

Q.24 Show that: 11910050

(i) cot= 11910051

(ii) cot= 11910052

(iii) = 11910053

Q.25 Prove that = tan37°

11910054

Q.26 Express the following in the form r sin or r sin,where terminal sides of the angles of measures q and f are in the first quadrant: 11910055

(i) 12 sin q + 5 cosq 11910056

(ii) 3 sin q–4 cosq 11910057

(iii) sinq – cosq 11910058

(iv)5 sin q – 4 cosq 11910059

(v) c sin q + cosq 11910060

(vi)3sin q – 5cosq 11910061

(iv) 5 sin q – 4 cosq

Q.27 Prove that 11910062

(i) 11910063

(ii) 11910064

Q.28 Prove that: 11910065

(i) sin 3a = 3 sin a- 4 sin³a 11910066

(ii) cos 3a = 4 cos³a- 3 cosa 11910067

(iii) tan 3a = 11910068

(2002-G-II)

Q.29 Prove that (Board 2014)

11910069

Q.30 Show that 11910070

(i) sin 2q = 11910071

(ii) cos2q = 11910072

Q.31 Reduce cos⁴q to an expression involving only function of multiples of q, raised to the first power. 11910073

Q.32 Find the values of 11910074

(i) sin2a 11910075 (ii) cos2a 11910076

(iii)tan2a when: 11910077

sina = when 0 < a <

Q.33 Prove the identity cota –tana=2cot 2a.

11910078

Q.34 Prove the identity=tan a

11910079

Q.35. Prove the identity =tan

11910080

Q.36 Prove the identity

= sec 2a – tan 2a 11910081

Q.37 Prove the identity

= 11910082

Q.38 Prove the identity 11910083

= cot

Q.39 Prove the identity

1 + tan atan 2a = sec 2a 11910084

Q.40 Prove the identity

= tan 2q tan q 11910085

Q.41Prove the identity– =2

11910086

Q.42 Prove theidentity

+ = 4cos 2q 11910087

Q.43 Prove the identity= sec q

11910088

Q.44 Prove the identity

+ = 2cot 2q 11910089

Q.45 Reduce sin⁴q to an expression involving only function of multiples of q, raised to the first power. 11910090

Q.46 Express 2 sin 7qcos 3q as a sum or difference. 11910091

Q.47 Prove without using tables, that

sin19°cos 11°+ sin 71° sin 11° = 11910092

Q.48 Express sin 5x+sin7x as a product.

(Board 2014) 11910093

Q.49 Express cos A + cos 3A + cos 5A + cos 7A as a product. 11910094

Q.50 Express the following products as sums or differences: 11910095

(i) cossin 11910096

(ii) coscos11910097

(iii) sinsin 11910098

Q.51 Express the following sums or differences as products: 11910099

sin + sin

Q.52 Prove the following identities: 11910100

(i)= cot 2x 11910101

(ii) = tan 5x 11910102

(iii) = 11910103

Q.53 Prove that: 11910104

(i) cos 20° + cos100° + cos 140° = 0 11910105

(Board 2008)

(ii) sinsin= cos 2q 11910106

MULTIPLE CHOICE QUESTIONS

q Each question has four possible answers. Select the correct answer and encircle it.

Q.1 Graphs of trigonometric function within their domains are: 11911020

(a) line segments

(b) sharp corners

(d) smooth curves

Q.2 Period of a trigonometric function is:

(a) any real number 11911021

(b) any negative real number

(d) a least positive number

Q.3 The graph of sin q compared with graph of cos q is: 11911022

(a) same (b) inverted

(d) 90° to the left

Q.4 To solve graphically the equation x cot x = 2, we can draw on the same axes the graphs: 11911023

(a) y = cot x and y = 2x

(b) y = cot x and y =

(d) y = tan x and y =

Q.5 A function of x with amplitude 2 and period p could have as its rule f(x) =11911024

(a) cos (b) 2 cos 2x

Q.6 Amplitude of sin x is: 11911025

(a) R (b) [ – 1, 1]

Q.7 For y = a sin nx: 11911026

(a) Amplitude = a , period =

(b) Amplitude = a , period = 2n p

(d) Amplitude = n , period =

Q.8 A function ¦ (x) is said to be the periodic function if, for all x in the domain of ¦, there exists a smallest positive number p such that

¦ (x + p) = ----------- 11911027

(a) p (b) 2p (c) 3p (d) 4p

Q.34 The period of 7 sec is: 11911053

(a) p (b) 2p

Q.35 The period of y = 5 + sin is:

11911054

(a) p (b) 2p

Q.36 The period of 5 cos is: 11911055

(a) 5p (b) 10p

Q.37 The period of cosec is: 11911056

(a) 2p (b) 4p

Q.38 The period of 5 + sin x is: 11911057

(a) 15p (b) 30p

Q.39 The graph of y = csc x from – 2p to 2p is: 11911058

(a)

(b)

(c)

(d)

Q.40 The following graph represents: 11911059

(a) sin x from – 2p to 2p (b) csc x from – 2p to 2p

Q.41 The following graph represents: 11911060

(a) sin x from – 2p to 2p (b) csc x from – 2p to 2p

Q.42 The following graph represents: 11911061

(a) sin x from – 2p to 2p (b) csc x from – 2p to 2p

Q.43 The following graph represents: 11911062

(a) tan x from – 2p to 2p (b) cos x from – 2p to 2p

Q.44 Period of equals: 11911063

(a) (b) (c) (d)

Unit

TRIGONOMETRIC FUNCTIONS AND

THEIR GRAPHS

Q.1 Write domain and range of 11911001

Ans:

Q.2 What is Periodicity and period?

11911002

Q.3 Prove that sine is periodic and its period is . 11911003

Q.4 Prove that tangent is a periodic function and its period is . 11911003

Q.5 Find the periods of the following functions: 11911004

(i) sin 3x 11911005

(ii) cos 2x 11911006

(iii) tan 4x 11911007

(iv) cot 11911008

(v) sin 11911009

(vi) cosec 11911010

(vii) sin (Board 2014) 11911011

Þ tan = 1 Þ = Þ g =

Q.53 In any triangle ABC, law of sines is:

11912040

(a) cos a =

(b) a c sin b

(d) = =

Q.54 In any triangle ABC, law of cosines is:

(a) cos a = 11912041

(b) a c sin b

(d) = =

Q.55 In any triangle ABC, law of tangents is:

(a) = 11912042

(b) =

(d) All of these.

Q.56 If in a triangle ABC, = = , then the triangle is: 11912043

(a) right angled (b) equilateral

Q.57 When two sides and included angle is given, then area of triangle is given by:

(a) a b sin g (b) a c sin b 11912044

Q.58 If 2s = a + b + c, where a, b, c are the sides of a triangle ABC, then area of triangle ABC is given by: 11912045

(a)

(b)s

(c)

(d)

Q.59 The circum-radius R of a triangle is given by: 11912046

(a) (b)

(c)

(d)

Q.60 The in-radius r of a triangle is given by : 11912047

(a) (b)

Q.61 r rrr= 11912048

(a) D (b) D

Q.62 If 2s = a + b + c, then in any triangle ABC: 11912049

(a) cos =

(b) cos =

(d) All of these

Q.63 If 2s = a + b + c, then in any triangle ABC: 11912050

(a) sin =

(b)sin =

(a) The law of sines

(b)The law of cosines

(d) The fundamental law

Q.72 In any triangle ABC, =

(a) (b) 11912059

Hint: By law of sines: =

Þ = Þ =

Q.73 The circum-radius R of a triangle is given by: 11912060

(a) (b)

Q.74 A circle which touches one side of a triangle externally and the other two produced sides internally is known as:

11912061

(a) in-circle (b) escribed-circle

Q.75 A circle drawn inside a triangle and touching its sides is known as: 11912062

(a) circum-circle (b) in-circle

Q.76 A circle passing through the vertices of a triangle is known as: 11912063

(a) in-circle (b) escribed-circle

Q.77 When two angles and included side is given, then area of a triangle ABC is given by: 11912064

(a) (b)

Q.78 With usual notations for triangle R equals: (Board 2009) 11912065

(a) (b)

Unit	APPLICATION OF TRIGONOMETRY
12

Q. 1 Find the unknown angles and sides of the following triangles. 11912001

(i) b = 90, a = 45, a = 4 11912002

(ii) a = 60° , b = 90° , b = 12 11912003

(iii) b = 90° , b = 5 , c = 10 11912004

(iv) = 90°, a = 40°, a = 8 11912005

(v) a = 56°, g = 90°, c = 15 11912006

(vi) g = 90, a = 8, b = 8 11912007

Q. 2 Solve the right triangle ABC , in which g=90 a = 3720¢, a = 243 11912008

Q. 3 Define angle of elevation and angle of depression 11912009

Q. 4 What is right triangle and sum of angles of a triangle? 11912010

Q. 5 1 A vertical pole is 8 m high and the length of its shadow is 6 m. What is the angle of elevation of the Sun at that moment? 11912011

Q. 6 At the top of a cliff 80 m high, the angle of depression of a boat is 12°. How far is the boat from the cliff? 11912012

Q. 7 A ladder leaning against a vertical wall makes an angle of 24° with the wall. Its foot is 5 m from the wall. Find the length.

Sol: (Board 2014) 11912013

Q. 8 Prove that a² = b² + c²–2bc cos a

Let side of triangle ABC be along the positive direction of the x-axis With vertex A as origin, then ÐBAC will be in the standard position. 11912014

Proof: Q = c and mÐBAC = a

Q. 9 State and prove law of sines

In any triangle ABC, prove that: where a, b, g are the measure of the angles opposite to the sides of lengths a, b, c respectively. 11912015

Q. 10 Write laws of tangents. 11912016

Q. 11 State half angles formulas of sine in terms of sides 11912017

Q. 12 Solve the triangle ABC, if b=60°, g =15° , b = 11912018

Q. 13 Find the smallest angle of the triangle ABC ,when a = 37.34 , b = 3.24, c = 35.06

11912019

Q.14 Find the measure of the greatest angle, if side of the triangle are 16,20,33 11912020

Q. 15 The sides of a triangle are x+ x + 1, 2x + 1 and x– 1. Prove that the greatest angle of triangle is 120 11912021

Q. 16 Prove that Area of triangle ABC = bc sin a = ca sin b = ab sing

11912022

Q. 17 prove that 11912023

Area of triangle =

= =

Q. 18 prove that 11912024

Area of triangle =

Q. 19 State circum-circle and circum-radius. 11912025

Q. 20 Prove that: 11912026

R = = = , with usual notations.

Q. 21 Prove that R = 11912027

Q. 22 Prove that: r = with usual notations 11912028

Q. 23 Prove that: 11912029

r₁ = , with usual notations

r₂ = , with usual notations

and r₃ = , with usual notations

Q. 24 r = s tan 11912030

Q. 25 Prove that r r r r = D²11912031

Q. 26 Prove that r r r = r s 11912032

Q. 27 Show that 11912033

abc (sin a+sin b+sin g) = 4Ds

H.S.

MULTIPLE CHOICE QUESTIONS

q Each question has four possible answers. Select the correct answer and encircle it.

Q.1 If x is positive or zero, then the principal value of any inverse function of x, if it exist, lies in the interval:

11913015

(a) [0 , 1] (b)

Q.2 Inverse sine function is written as:

11913016

(a) (sin x) (b) sin x

Q.3 The graph of x = sin y is obtained by reflecting the graph of y = sin x about the line: 11913017

(a) x-axis (b) y-axis

Q.4 y = sinx if and only if x = sin y, where: 11913018

(a) - 1 £ x £ 1 and – p £ y £ p

(b) - 1 £ x £ 1 and - £ y £

(d) - 1 £ y £ 1 and - £ x £

Q.5 The domain of principal sine function is: 11913019

(a) (b)

Q.6 The range of principal sine function is:

(a) (b) 11913020

Q.7 The domain of y = sinx is: (Board 2014)

11913021

(a) [- p , p] (b)

Q.8 The range of y = sinx is: 11913022

(a) (b)

Q.9 The graph of y = cosx is obtained by reflecting the graph of y = cos x about:

(a) x-axis (b) y-axis 11913023

Q.10 y = cosx Û x = cos y, where:

(a) - 1 £ x £ 1 and 0 £ y £ p 11913024

(b) - 1 £ y £ 1 and 0 £ x £ p

(d) - 1 £ y £ 1 and - £ x £

Q.11 The domain of principal cosine function is: 11913025

(a) (b)

Q.12 The range of principal cosine function is: 11913026

(a) (b)

Q.13 The domain of y = cosx function is: 11913027

(a) (b)

Q.14 The range of y = cosx function is:

(a) (b) 11913028

Q.15 The graph of inverse tangent function y = tanx is obtained by reflecting the the graph of y = tan x about: 11913029

(a) x-axis (b) y-axis

Q.16 y = tanx if and only if x = tan y, where: 11913030

(a) - 1 < x < 1 and – p < y < p

(b) - ¥ < y < ¥ and - < x <

(d) - ¥ < x < ¥ and - < y <

Q.17 The domain of principal tangent function is: 11913031

(a) ]0 , p[ (b)

Q.18 The range of principal tangent function is: 11913032

(a) ]0 , p[ (b)

Q.19 Domain of the function y=tanx is:

(a) ]0 , p[ (b) 11913033

Q.20 Range of the function y=tanx is:

(a) ]0 , p[ (b) 11913034

Q.21 The principal value of sin^-1 is:

(a) (b) – 11913035

Q.22 The principal value of sin^-1 is:

(a) (b) 11913036

Q.23 The principal value of sin^-1 is:

(a) (b) 11913037

Q.24 The principal value of sin^-1 is:

(a) (b) 11913038

Q.25 If ¦(x) = arccos x, then: 11913039

(a) - 1 £ ¦(x) £ 1 (b) ¦(x) =

Q.26 sin = 11913040

(a) sin x (b) cosec

Q.27 cos = 11913041

(a) secx (b) cos x

Hint: cos = q Þ cos q =

Þ x = sec q Þ q = secx

Q.28 tan = 11913042

(a) cot (b) cotx

Q.29 sin(– x) = 11913043

(a) – sinx (b) sinx

Q.30 cos(– x) = 11913044

(a) p + cos x (b) p – cosx

Q.31 tan(– x) = 11913045

(a) tanx (b) cotx

Q.32 If sin = – x, then x = 11913046

(a) 0 (b)

Hint: sin = – x

Þ = sin

Þ = cos x Þ x =

Q.33 If cos = – sinx, then x =

(a) (b) 11913047

Q.34 If sin^-1 a = - cos^-1 b, then:

(a) a = + b (b) a = – b 11913048

Hint: sina = – cosb

Þ a = sin

Þ a = cos = b

Q.35 cos ( p – sinx ) = 11913049

(a) (b) –

Q.36 cos ( 2 sinx ) = 11913050

(a) 1 – 2x (b) 1 + 2x

Q.37 tan ( p + tanx ) = 11913051

(a) x (b) p + x

Q.38 tan ( p + cotx ) = 11913052

(a) (b) p –

Q.39 cos (tan¥) = 11913053

(a) 0 (b) ¥

Q.40 tan= 11913054

(a) – (b)

Q.41 tanA – tanB = 11913055

(a) tan (b) tan

Q.42 tanA + tanB = 11913056

(a) tan (b) tan

Q.43 sinA + sinB = 11913057

(a)sin^-1

(b) sin^-1

(d) sin^-1

Q.44 sinA – sinB = 11913058

(a) sin

(b) sin

(d) sin

Q.45 cosA + cosB = 11913059

(a) cos

(b) cos

(d) cos

Q.46 cosA – cosB = 11913060

(a)cos

(b)cos

(c)cos

(d) cos

Q.47 tan(- ) is: 11913061

(a) (b)

Q.48 If x = tan , y = tan, then x + y will be: 11913062

(a) tan^-1 (b) tan^-11

Q.49 tan = 11913063

(a) (b)

Unit	INVERSE TRIGONOMETRIC FUNCTION
13

Q.1 Define Principal sine Function. 11913001

Principal sine Function

Q.2 Define inverse principal sine function 11913002

Q.3 Define inverse cosine function. 11913003

Q.4 Define inverse tangent function. 11913004

Q.5 Show that cos^–1 11913005

Q.6 2 cos = sin 11913006

Q.7 sin 11913007

Q.8 Prove that: 11913008

sin^–1 A + sin^–1 B

= sin^–1

Q.9 tan = 2 cos 11913009

Q.10 sin – sin = cos 11913010

Q.11 tan^-1 + tan^-1 - tan^-1 =

11913011

Q.12 2tan + tan = 11913012

Q.13 Show that 11913013

sin (–x) = – sinx

Q.14 Show that tan (sinx) =

11913014

MULTIPLE CHOICE QUESTIONS

Each question has four possible answers. Select the correct answer and encircle it.

Q.1 Reference angles is always in 11914014

(a) IQ (b) IIQ

Q.2 General angles of inverse trigonometric functions are written by using their

(a) Domain (b) Range 11914015

Q.3 Trigonometric equation has __________ solutions. 11914016

(a) unique (b) finite

Q.4 For solving a trigonometric equation, first find the solution over the interval whose length is equal to its ____.

(a) domain (b) range 11914017

Q.5 The general solution of sin x = cos x is ____________. 11914018

(a) nπ (b) 2nπ

Q.6 If sin x + cos x=0, then x=____ 11914019

(a) (b)

Q.7 The solutions of in the interval [0, π] are ________. 11914020

a) (b)

Q.8 If , then x = ______. 11914021

(a) (b)

Q.9 The general solution of 1 + cos x = 0 is ________. 11914022

(a) (b)

Q.10 The solution set of in the interval is ________. 11914023

(a) (b)

Q.11 The solution set of is _________. 11914024

(a) finite set (b) infinite set

Q.12 The solution set of in is _________. 11914025

(a) 0

(b)

(d) all of these

Q.13 Solutions of in [0, 2π] are 11914026

(a) (b)

Q.14 There is a solution of the equation

2 sin q + 1 = 0 in the quadrants: 11914027

(a) 1 and 2 (b) 1 and 3

Q.15 The number of solutions to the equation: sin q cos q = 1 (0 £ q £ 2p) is

(a) 0 (b) 1 11914028

Hint: sin q cos q = 1 Þ 2sin q cos q = 2 Þ sin 2q = 2 > 1, so there is no solution.

Q.16 The number of solutions to the equation: sin q + cos q = 0 (0 £ q £ 2p) is 11914029

(a) 0 (b) 1

Q.17 The number of solutions to the equation: sin q cos q = 0 (0 £ q £ 2p) is

(a) 0 (b) 1 11914030

Hint: Solutions are 0° , 90°, 180°, 270° , 360°

Q.18 Principal solution of sin q = is

(a) - (b) 11914031

Q.19 Secondary solution of sin q = is

(a) - (b) 11914032

Q.20 Principal solution of sin q = - is

(a) - (b) 11914033

Q.21 Principal solution of cos q = is

(a) (b) 11914034

Q.22 Principal solution of cos q = - is

(a) (b) 11914035

Q.23 Principal solution of 11914036

tan q = is

(a) (b)

Q.24 Principal solution of 11914036

tan q = - is

(a) (b)

Q.25 Which trigonometric equation has secondary solution? 11914037

(a) sin q = 1 (b) cos q = 1

Hint: Principal solution of

tan q = 1 is and

secondary solution is .

Q.26 The general solution of the equation sin x = 0 is 11914038

(Board 2009)

(a) x = , nÎZ (b) x = np, n Î Z

Q.27 The general solution of the equation cos x = 0 is 11914039

(a) n , n Î Z (b) (2n+1) ,nÎZ

Q.28 The general solution of the equation tan q = 0 is 11914040

(a) q = , n Î Z (b) q=2np , nÎZ

Q.29 The general solution of the equation sin x = 1 is 11914041

(a) + n , n Î Z (b) + np , n Î Z

Q.30 The general solution of the equation cos q = 1 is 11914042

(a) q ¹ 2np , n Î Z (b) q=2np , nÎZ

Q.31 The general solution of the equation cos q = - 1 is 11914043

(a) q ¹ (2n + 1)p , n Î Z

(b) q = (2n + 1)p , n Î Z

(d) q = np , n Î Z

Q.32 Given tan q = 1 11914044

(a) q lies in quadrants 1 and 4

(b) cos q =

(d) the first quadrant solution is q =

Q.33 The general solution of the equation tan x = – 1 is 11914045

(a) + np , n Î Z (b) + 2np , n Î Z

Q.34 The general solution of the equation tan x = is 11914046

(a) + np , n Î Z (b) + 2np , n Î Z

Q.35 The general solution of the equation cos 2x = is 11914047

(a) È , n Î Z

(b) È , n Î Z

(d) None of these.

Q.36 Solution set of the equation

cos²x + sin x = 1 is 11914048

(a) È , n Î Z

(b) È , n Î Z

(d) È , n Î Z

Q.37. If then equals: (Board 2014)

(a) 11914049

(b)

(c)

(d)

Unit	SOLUTIONS OF TRIGONOMETRIC EQUATIONS
14

Q.1 Define trigonometric equations 11914001

Q.2 What is the Solution of trigonometric equation? 11914002

For example sin x = 11914003

QQ.3 What is the Solution set of trigonometric equation? 11914004

Q.4 Solve the equation sin x = 11914005

Q.5 Solve the equation: 1+cosx=0

Solution: 1 + cos x = 0 11914006

Q.6. Find the solution set of: 11914007

Q.7 Find the solutions of the equation

sin x = – which lie in [0, 2p] 11914008

Q.8 Find the solutions of the equation

cot q = which lie in [0, 2p] 11914009

Q.9 Find the values of q satisfying the equation 2 sin q + cosq – 1 = 0 11914010

Q.10 Find the values of q satisfying the equation 2 sinq – sin q= 0 11914011

Q.11 What is the reference angle of tan q= -? 11914012

Q.12 The solution of cos x = - lies in which quadrants. 11914013

Causative

11th Math Objective + Short Questions

17. Modulus of complex number z = a+ib is: 11901075

18.Modulus of 15 i + 20 is: 11901076

31. If a, b, c and d Î R. Then a = b,c = d Þ

33. a> b Þ –a < –b Name of the property used in the above inequality is: 11901091

34. a> b Þ<, a ¹ 0 , b ¹ 0Name of the property used in the above in equation is:

35.For all x Î R, x = xWhat is above property called? 11901093

37. The identity element with respect to addition is: 11901095

2. Distinct objects means: 11902113

3. The objects in a set are called: 11902114

4. A set can be described by: 11902115

5. If a set is described in words, the method is called: 11902116

6. If a set is described by listing its elements within brackets is called:

7. If a set is described as

{ x | x N Ù x < 100} is the: 11902118

11. The well defined collection of disjoint object is a: 11902122

17. If A = {1, 2, 3, ...., 99}, B = {x | x Î N,

0 < x < 100} then: 11902128

18. The number of elements of the set

{x : xÎ N, x= 1}, where N is the set of all natural numbers, is: 11902129

20. The proper subset E of a set F is denoted by: 11902131

21. An Improper subset of a set F is: 11902132

22. If A È B = Æ, and A = Æ then: 11902133

25. For any two sets A = B if and only if

A È B = 11902136

42. A Ç (B – A) = 11902153

43. If A ËB , B Ë A and A and B have at least one element common, they are called: 11902154

48. The ordered pairs (4,5) and (5 , 4) are:

53. If A = and B = , then:

59. S = {1, –1, 2, –2} is a group under:

60. S = {1 ,w , w} where w is a cube root of unity form an abelian group with respect to: 11902171

61. S = {1, – 1, i, – i} where i = form an abelian group with respect to: 11902172

84. The proposition (p® q) Ù (q ® p) is shortly written as: 11902194

86. Which of the following sentences is equivalent to ~(p Ú q)? 11902196

87. Additive inverse of is: 11902197

88. If R = A ´ B then R is an onto function if: 11902198

89. R is a relation from A to B if and only if R Í 11902199

90. The phrase, “For all x in S”, is abbreviated as: 11902200

91. The phrase, “There exist an x in S”, is abbreviated as: 11902201

92. If two sets P and Q are equivalent, they are denoted by: 11902202

93.If A Ì B, then A – B = 11902203

94. Set A is proper subset of B is denoted by: 11902204

95. If (x – 2, 2) = (3, 2) , then: 11902205

96. An element b Î S is said to be an inverse of a ÎS w. r. t * if: 11902206 (a) a * b = b * a = e

(b) a * b = b * a = 0

97. In a binary relation, the set consisting of all the first elements of the ordered pairs is called: 11902207

98. In a binary relation, the set consisting of all the second elements of the ordered pairs is called: 11902208

99. In the conditional p ® q, p is called:

100. In the conditional p ® q, q is called:

101. A statement which is true for all possible values of the variables involved in it, is called a: 11902211

102. A compound statement of the form “if p then q” is called an: 11902212

103. A groupoid (S) is called ------------ if it is associative in S. 11902213

104. The inverse of the linear function {(x, y): y = mx + c} is: 11902214

105. The graph of the quadratic function is a: 11902215

106. Q = {x | x = where p, q Z Ù q ¹ 0} is a set of: 11902216

Q. 14. Find x and y if 11903020

+ 2 =

Q. 38. Find values of x if 11903060

Q. 5. Solve x(x + 7) = (2x – 1) (x + 4) 11904007

x ¹ – 1,–2,– 5 11904008

Q. 7. Solve + = a + b ;

x ¹ , 11904009

Q.10. Solve by quadratic formula?

15x + 2ax – a = 0 11904012

11904023

Q.14 If a1 is the first term and d is the common difference of an A.P. then its (2m + 1)th term is: 11906087

Q.34 If a , b , c are in A.P., then , , are in:

Q.73 Find the 9th term of the harmonic sequence , , , … (L.B. 2014) 11906064

Q.21 Use binomial theorem to show that

Q.1 sin ( a + b ) = 11910107

Q.12 cos q + cos f = 11910118

Q.13 cos q – cos f = 11910119

Q.14 sin q + sin f = 11910120

Q.15 sin q – sin f = 11910121

Q.23 sin = 11910128

Q.24 cos = 11910129

Q.43 tan (–135°) = 11910148

Q.44 = 11910149

Q.45 If an angle a is allied to an angle b, then a ± b = ------------- 11910150

96. An element b Î S is said to be an inverse of a ÎS w. r. t * if: 11902206
(a) a * b = b * a = e

Q.14 If a₁ is the first term and d is the common difference of an A.P. then its (2m + 1)^th term is: 11906087