Unit 1

NUMBER SYSTEMS

Q. Define rational number. 11301001

Q. Define irrational number. 11301002

Example 2: Prove that is an irrational number. 11301003

Example 3: Prove is an irrational number. 11301004

EXERCISE 1.1

1. Which of the following sets have closure property w.r.t. addition and multiplication?

11301005

(i) {0} 11301006 (ii) {1} 11301007

(iii) {0, -1} 11301008 (iv) {1, -1} 11301009

2. Name the properties used in the following equations. Letters, here used, represent real numbers. 11301010

3. Name the properties used in the following inequalities: 11301011

(i) -3 < -2 Þ 0 < 1 11301012

(ii) -5 < -4 Þ 20 > 16 11301013

(iii) 1 > - 1 Þ - 3 > - 5 11301014

(iv) a < 0 Þ -a > 0 11301015

(v) a > b Þ < 11301016

(vi) a > b Þ -a < -b 11301017

4. Prove the following rules of addition:

11301018

(i) + = (Board 2008) 11301019

(ii) + = 11301020

5. Prove that - - =

11301021

6. Simplify, justify each step. 11301022

(i) 11301023 (ii) 11301024

(iii) 11301025 (iv) 11301026

Q. What is Complex Numbers? 11301027

Article1: Prove that (a, b) = a + ib 11301028

Article 2: Prove that i² = –1 11301029

EXERCISE 1.2

1. Verify the addition properties of Complex numbers. 11301030

2. Verify the multiplicative properties of Complex numbers. 11301031

3. Verify the distributive law of Complex numbers 11301032

(a,b)×[(c,d)+(e, ¦)]=(a, b) (c, d)+ (a, b) (e, ¦)

4. Simplify the following 11301033

(i) i⁹ 11301034 (ii) i¹⁴11301035

(iii) (-i)¹⁹11301036 (iv) (-1) 11301037

5. Write in terms of i 11301038

(i) b 11301039 (ii) 11301040

(iii) 11301041 (iv) 11301042

Simplify the following:

6. Simplify the following (7, 9) + (3, -5)

11301043

7. Simplify (8, -5) - (-7, 4) 11301044

8. Simplify (2, 6) . (3, 7) 11301045

9. Simplify (5, -4) . (-3, -2) 11301046

10. Simplify (0, 3) . (0, 5) 11301047

11. Simplify (2, 6) ¸ (3, 7) 11301048

12. (5, - 4) ¸ ( -3, - 8) 11301049

13. Prove that the sum as well as the product of any complex number and its conjugate is a real number. 11301050

14. Find the multiplicative inverse of each of the following numbers: 11301051

(i) (-4,7) 11301052 (ii) 11301053

(iii) (1,0) 11301054

15. Factorize the following 11301055

(i) a² + 4b² 11301056 (ii) 9a² + 16b²11301057

(iii) 3x² + 3y² 11301058

16. Separate into real and imaginary parts (write as a simple complex number): 11301059

(i) 11301060 (ii) 11301061

(iii) 11301062

Q. What is Complex Plane? 11301063

Q. What is modulus of a complex number?

11301064

Q. What is Polar form of a complex number? 11301065

Example 4: 11301066

Express the complex number in polar form.

Q. State De Moivre’s Theorem? 11301067

Example 5: 11301068

Find out real and imaginary parts of each of the following complex numbers.

(i) (ii)

EXERCISE 1.3

1. Graph the following numbers on the complex plane: 11301069

(i) 2 + 3i 11301070 (ii) 2 - 3i 11301071

(iii) -2 -3i 11301072 (iv)-2 + 3i 11301073

(v) -6 11301074

(vi) i 11301075

(vii) - i 11301077

(viii) -5 - 6i 11301078

2. Find the multiplicative inverse of the following numbers: 11301080

(i) - 3i 11301081 (ii) 1 - 2i 11301082

(iii) -3 -5i 11301083 (iv) (1, 2) 11301084

3. Simplify 11301085

(i) i¹⁰¹ (Board 2008) 11301086

(ii) (-ai)⁴, a Î R 11301087

(iii) i^-3 11301088

(iv) i^-1011301089

4. Prove that = z iff z is real 11301090

5. Simplify by expressing in the form a+bi

11301091

(i) 5 + 2 11301092

(ii) 11301093

(iii) 11301094

(iv) 11301094

6. (i) Show that " Z Î C z² + () is a real number. 11301096

(ii) Show that (z - ) is a real number for all z Î C. 11301097

7. Simplify the following: 11301098

(i) 11301099

(ii) 11301200

(iii) 11301201

(iv) (a + bi)2 11301202

(v) (a + bi) , (a ¹ 0, b ¹ 0) 11301203

(vi) (a + bi)3 11301204

(vii) (a – bi)3 11301205

(viii) 11301206

Unit 2

SETS, FUNCTIONS AND GROUPS

Q. Write ways of describing a set. 11302001

Q. What is power set? 11302002

Q. What is proper subset and improper subset? 11302003

EXERCISE 2.1

1. Write the following sets in set builder notation: 11302004

(i) {1, 2, 3, …., 1000} 11302005

(ii) {0, 1, 2, …., 100} 11302006

(iii) {0, ±1, ±2, ….. , ±1000} 11302007

(iv) {0, -1, -2, … ,- 500} 11302008

(v) {100, 101, 102, …., 400} 11302009

(vi) {-100, -101, -102, … , -500} 11302010

(vii){Peshawar, Lahore, Karachi, Quetta}

11302011

(viii) {January, June, July} 11302012

(ix) The set of all odd natural numbers 11302013

(x) The set of all rational numbers 11302014

(xi) The set of all real numbers between 1 and 2 11302015

(xii) The set of all integers between -100 and 1000. 11302016

2. Write each of the following sets in the descriptive and tabular forms: 11302017

(i) {x | x Î N Ù x £ 10} 11302018

(ii) {x | x Î N Ù 4 < x < 12} 11302019

(iii) {x | x Î Z Ù -5 < x < 5} 11302020

(iv) {x | x Î E Ù 2 < x £ 4} 11302021

(v) {x | x Î P Ù x < 12} 11302022

(vi) {x | x Î O Ù 3 < x < 12} 11302023

(vii) {x | x Î E Ù 4 £ x 10} 11302024

(viii) {x | x Î E Ù 4 < x < 6} 11302025

(ix) {x | x Î O Ù 5 £ x £ 7} 11302026

(x) {x | x Î O Ù 5 £ x < 7} 11302027

(xi) {x | x Î N Ù x + 4 = 0} 11302028

(xii) {x | x Î Q Ù x² = 2} 11302029

(xiii) { x | x Î R Ù x = x} 11302030

(xiv) {x | x Î Q Ù x = - x} 11302031

(xv) {x | x Î R Ù x ¹ 2} 11302032

(xvi) {x | x Î R Ù x Ï Q} 11302033

3. Which of the following sets are finite and which of these are infinite? 11302034

(i) The set of students of your class. 11302035

(ii) The set of all schools in Pakistan. 11302036

(iii) The set of natural numbers between 3 and 10. 11302037

(iv) The set of rational numbers between 3 and 10. 11302038

(v) The set of real numbers between 0 and 1. 11302039

(vi) The set of rationals numbers between 0 and 1. 11302040

(vii) The set of whole numbers between 0 and 1. 11302041

(viii) The set of all leaves of trees in Pakistan. 11302042

(ix) P(N) 11302043

(x) P {a, b, c} 11302044

(xi) {1, 2, 3, 4, …….} 11302045

(xii) {1,2,3, …., 100000000} 11302046

(xiii) {x | x Î R Ù x ¹ x} 11302047

(xiv) {x | x Î R Ù x² = -16} 11302048

(xv) {x | x Î Q Ù x² = 5} 11302049

(xvi) {x | x Î Q Ù 0 £ x £ 1} 11302050

4. Write two proper subset of each of the following sets: 11302051

(i) {a, b, c} 11302052 (ii) {0, 1} 11302053

(iii) N 11302054 (iv) Z 11302055

(v) Q 11302056 (vi) R 11302057

(vii) W 11302058

(viii) {x | x Î Q Ù 0 < x £ 2} 11302059

5. Is there any set which has no proper sub set? If so name that set. 11302060

6. What is the difference between {a, b} and {{a, b}}? 113020561

7. Which of the following sentences are true and which of them are false? 11302062

(i) {1, 2} = {2, 1} 11302063

(ii) Æ Í {{a}} 11302064

(iii) {a} Ê {{a}} 11302065 (iv) {a} Î {{a}} 11302066

(v) a Î {{a}} 11302067 (vi) Æ Î {{a}} 11302068

8. What is the number of elements of the power set of each of the following sets?

11302069

(i) { } 11302070 (ii) {0, 1} 11302071

(iii) {1, 2, 3, 4, 5, 6, 7} 11302072

(iv) {0, 1, 2, 3, 4, 5, 6, 7} 11302073

(v) {a, {b, c}} 11302074

(vi) {{a, b}, {b, c}, {d, e}} 11302075

9. Write down the power set of each of the following sets: 11302076

(i) {9, 11} 11302077 (ii) {+ , - , ´, ¸} 11302078

(iii) { Æ } 11302079 (iv) {a, {b, c}} 11302080

10. Which pairs of sets are equivalent? which of them are also equal? 11302081

(i) {a, b, c}, {1, 2, 3} 11302082

(ii) The set of the first 10 whole members, {0, 1, 2, 3, …..9} 11302083

(iii) Set of angles of a quadrilateral ABCD, set of the sides of the same quadrilateral. 11302084

(iv) Set of the sides of a hexagon ABCDEF, set of the angles of the same hexagon;

11302085

(v) {1, 2, 3, 4, …..}, {2, 4, 6, 8,…..} 11302086

(vi) {1, 2, 3, 4,…..}, 11302087

(vii) {5, 10, 15, 20, ….. 55555},

{5, 10, 15, 20, ….} 11302088

TYPES OF SETS

Q. What are disjoint sets? 11302089

Q. What are overlapping sets? 11302090

EXERCISE 2.2

1. Exhibit A È B and A Ç B by Venn diagram in the following cases: 11302091

(i) A Í B 11302092 (ii) B Í A 11302093

(iii) A È A¢ 11302094

(iv) A and B are disjoint sets 11302095

(v) A and B are overlapping sets 11302096

2. Show A - B and B - A by Venn diagram when: 11302097

(i) A and B are overlapping sets 11302098

(ii) A Í B 11302099 (iii) B Í A 11302100

3. Under what conditions on A and B are the following statements true? 11302101

(i) A È B = A 11302102

(ii) A È B = B 11302103

(iii) A - B = A 11302104

(iv) A Ç B = B 11302105

(v) n (A È B) = n (A) + n(B) 11302106

(vi) n(A Ç B) = n(A) 11302107

(vii) A - B = A 11302108

(viii) n(A Ç B) = 0 11302109

(ix) A È B = U 11302110

(x) A È B = B È A 11302111

(xi) n(A Ç B) = n(B) 11302112

(xii) U - A = Æ 11302113

4. 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: 11302114

(i) A^c11302115 (ii) B^c 11302116

(iii) A È B 11302117 (iv) A - B 11302118

(v) AÇC 11302119 (vi) A^cÈC^c11302120

(vii) A^c È C 11302121 (viii) U^c11302122

5. Using the Venn diagrams, if necessary, find the single sets equal to the following: 11302123

(i) A^c 11302124 (ii) A Ç U 11302125

(iii) A È U 11302126 (iv) A È Æ 11302127

(v) Æ Ç Æ 11302128

6. Use Venn diagrams to verify the following: 11302129

(i) A - B = A Ç B^c 11302130

(ii) (A - B)^c Ç B = B 11302131

Q. State and prove De Morgan’s Laws. 11302132

Ans: Statement: If A & B are two sets, then

(i) = A^c Ç B^c 11302133

(ii) = A^c È B^c11302134

EXERCISE 2.3

1. Verify the commutative properties of union and intersection for the following pairs of sets: 11302135

(i) A = {1, 2, 3, 4, 5}, B = {4, 6, 8, 10} 11302136

(ii) N, Z 11302137

(iii) A = {x | x Î R Ù x ³ 0}, B = R 11302138

2. Verify the properties for the sets A, B and C given below: 11302139

(i) Associativity of union 11302140

(ii) Associativity of intersection. 11302141

(iii) Distributivity of union over intersection. 11302142

(iv) Distributivity of intersection over union. 11302143

(a) A = {1, 2, 3, 4},B = {3, 4, 5, 6, 7, 8}, C ={5, 6, 7, 9, 10} 11302144

(b) A = Æ, B = {0} , C = {0, 1, 2} 11302145

(c ) N, Z, Q 11302146

3. Verify De Morgan’s Laws for the following sets:

U = {1, 2, 3 …., 20}, A = {2, 4, 6, ……. 20} and B = {1, 3, 5, ….., 19} 11302147

4. Let U=The set of the English alphabet A={x|x is a vowel},B={y|y is a consonant} Verify De Morgan’s Laws for these sets.

11302148

5. With the help of Venn diagram, verify the two distributive properties in the following cases w.r.t. union and intersection 11302149

(i) A Í B, A Ç C = Æ and B and C are overlapping. 11302150

(ii) A and B are overlapping, B and C are overlapping but A and C are disjoint. 11302151

6. Taking any set, say A = {1, 2, 3, 4, 5} verify the following: 11302152

(i) A È Æ = A 11302153 (ii) A È A = A 11302154

(iii) A Ç A = A 11302155

7. If U = {1, 2, 3, 4, 5, …..20} and 11302156

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

(i) A È A¢ = U 11302157 (ii) A Ç U = A 11302158

(iii) A Ç A¢ = Æ 11302159

8. From suitable properties of union and intersection deduce the following results.

11302160

(i) A Ç (A È B) = A È (A Ç B) 11302161

(ii) A È (A Ç B) = A Ç (A È B) 11302162

9. Using Venn diagrams, verify the following results. 11302163

(i) A Ç B¢ = A if A Ç B = Æ 11302164

(ii) (A - B) È B = A È B 11302165

(iii) (A - B) Ç B = Æ 11302166

(iv) A È B = A È (A¢ Ç B¢) 11302167

11302167(a)

Q. Prepare a truth table for conjunction of two statements.

11302167(b)

Q. Prepare a truth table for disjunction of two statements.

11302167(c)

Q. Prepare a truth table for conditional of two statements.

Example 2: Prepare a truth table of

11302168

Note:

Let p ® q be a given conditional, then

(i) q ® p is called converse of p ® q;

(ii) ~p ® ~q is called inverse of p ® q;

(iii) ~q ® ~p is called contrapositive of p®q.

Q. Define the following: 11302169

(i) Tautology 11302170 (ii) Absurdity 11302171

EXERCISE 2.4

1. Write the converse, inverse and contra positive of the following conditionals: 11302172

(i) ~ p ® q 11302173 (ii) q ® p 11302174 (iii) ~p ® ~q 11302175

(iv) ~ q ® ~ p 11302176

2. Construct truth tables for the following statements: 11302177

(i) (p ® ~p) Ú (p ® q) 11302178 (ii) (p Ù ~ p) ® q 11302179 (iii) ~(p ® q) « (p Ù ~q) 11302180

3. Show that each of the following statements is a tautology: 11302181

(i) (p Ù q) ® p 11302182 (ii) p ® (p Ú q) 11302183

(iii) ~(p ® q) ® p 11302184

(iv) ~qÙ(p®q) ® ~p 11302185

4. Determine whether each of the following is a tautology, a contingency or an absurdity: 11302186

(i) p Ù ~p 11302187 (ii) p ® (q ® p) 11302188

(iii) q Ú (~q Ú p) 11302189

5. Prove that p Ú (~p Ù ~q) Ú (p Ù q) = p Ú (~p Ù ~q) 11302190

Example 1: 11302191

Give logical proofs of the following theorems:

i) =

The last two columns of the equality of the two sides of eq. (1)

EXERCISE 2.5

Convert the following theorems to logical and prove them by constructing truth tables: 11302192

1. (A Ç B)¢ = A¢ È B¢ 11302193

2. (A È B) È C = (A È B) È C 11302194

3. (A Ç B) Ç C = A Ç (B Ç C) 11302195

4. A È (B Ç C) = (A È B) Ç (A È C)

11302196

Q. What are linear and quadratic functions?

11302197

Q. Define inverse of a function. 11302198

EXERCISE 2.6

1. 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. 11302199

(i) {(x, y) | y = x} 11302200

(ii) {(x, y) | y + x = 5} 11302201

(iii) {(x, y) | x + y < 5} 11302202

(iv) {(x, y) | x+y>5} 11302203

2. Repeat Q.1 when A = R, the set of real numbers. Which of the real lines are functions. 11302204

3. Which of the following diagrams represent functions and of which type?11302205

Fig. (1)

Fig. (2)

Fig. (3)

Fig. (4)

4. Find the inverse of each of the following relations. Tell whether each relation and its inverse is a function or not: 11302206

(i) {(2,1), (3,2), (4,3), (5,4), (6,5)} 11302207

(ii) {(1,3), (2,5), (3,7), (4,9), (5,11)} 11302208

(iii) {(x, y) | y = 2x + 3, x Î R} 11302209

(iv) {(x, y) | y² = 4ax, x ³ 0} 11302210

(v) {(x, y) | x² + y² = 9, | x| £ 3, | y | £ 3} 11302211

Q. What are

(i) Unary operation 11302212

(ii) Binary operation ? 11302213

Theorem: 11302214

Example 6: 11302215

Give the table for addition of elements of the set of residue classes module 5.

Example 7: 11302216

Give the table for addition of elements of the set of residue classes modulo 4.

Example 8: 11302217

Give the table for multiplication of elements of the set of residue classes modulo 4.

EXERCISE 2.7

1. Complete the table, indicating by a tick mark those properties which are satisfied by the specified set of numbers. 11302218

2. What are the field axioms? In what respect does the field of real numbers differ from that of complex numbers?

11302219

3. Show that the adjoining table is that of multiplication of the elements of the set of residue classes modulo 5. 11302220

	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

4. Prepare a table of addition of the elements of the set of residue classes module 4. 11302221

5. Which of the following binary operations shown in tables (a) or (b) is commutative?

	a	b	c	d		a	b	c	d
a	a	c	b	d	a	a	c	b	d
b	b	c	b	a	b	b	d	b	a
c	c	d	b	c	c	b	b	a	c
d	a	a	b	b	d	d	a	c	d

(a) (b) 11302222

6. Supply the missing elements of the third row of the given table so that the operation may be associative. 11302223

	a	b	c	d
a	a	c	b	d
b	b	a	c	d
c	-	-	-	-
d	d	c	c	d

7. What operation is represented by the adjoining table? Name the identity element of the relevant set, if it exists. Is the operation associative? Find the inverses of 0, 1, 2, 3, if they exist. 11302224

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

Q. Define Groupoid. 11302225

Q. Define Semi-group. 11302226

Q. Define Monoid. 11302227

Q. Define Group. 11302228

Example: 11302229

Let upon which operation has been performed as shown in the following table show that S is an abelian group under .

	0	1	2
0	0	1	2
1	1	2	0
2	2	0	1

Example: 11302230

Consider the set set up the multiplication table and show that the set S is an abelian group under multiplication.

Q. If a,b being elements of a group G, solve the following equations: 11302231

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

Q. State and prove reversal law of inverses.

11302232

Theorem: 11302233

If (G, ) is a group with e its identity, then e is unique.

Theorem: 11302234

If (G, ) is a group and a Î G then there is a unique inverse of a in G.

EXERCISE 2.8

1. Operation Å performed on the two member set G = {0,1} is shown in the adjoining table. Answer the questions: 11302235

(i) Name the identity element if it exists? 11302236

(ii) What is the inverse of 1? 11302237

(iii) Is the set G, under the given operation a group? Abelian or non-Abelian? 11302238

Å	0	1
0	0	1
1	1	0

2. The operation Å as performed on the set {0, 1, 2, 3} is shown in the adjoining table, show that the set is an Abelian group? 11302239

Å	0	1	2	3
0	0	1	2	3
1	1	2	3	0
2	2	3	0	1
3	3	0	1	2

3. For each of the following sets, determine whether or not the set forms a group with respect to the indicated operation. 11302240

Set Operation

(i) The set of rational numbers ´ 11302241

(ii) The set of rational numbers + 11302242

(iii) The set of positive rational numbers ´ 11302243

(iv) The set of integers + 11302244

(v) The set of integers ´ 11302245

4. Show that the adjoining table represents the sums of the element of the set {E, O}.

What is the identity element of this set? Show that this set is an abelian group. 11302246

Å	E	O
E	E	O
O	O	E

5. Show that the set {1, w, w²}, when w³ = 1 is an abelian group w.r.t. ordinary multiplication. [L. B. 2008 G-II)] 11302247

6. If G is a group under the operation * and a, b Î G, find the solutions of the equations: a x = b, x a = b 11302248

7. Show that the set consisting of elements of the form a + b (a, b being rational), is an abelian group w.r.t. addition. 11302249

8. Determine whether, (P(S), ), where stands for intersection is a semi-group, a monoid or neither. If it is a monoid, specify its identity. 11302250

9. Complete the following table to obtain a semi-group under 11302251

	a	b	c
a	c	a	b
b	a	b	c
c	x	y	a

10. Prove that all 2 ´ 2 non-singular matrices over the real field form a non-abelian group under multiplication. 11302252

Unit 3

MATRICES AND DETERMINANTS

Q. Define a matrix. 11303001

Q. Define a diagonal matrix. 11303002

Q. Define identity matrix 11303003

Q. What are singular and non singular matrices? 11303004

Q. What is Adjoint and inverse of a 2´2 Matrix? 11303004(a)

EXERCISE 3.1

Q.1 If A = and B = ,

then show that 11303005

(i) 4A–3A=A 11303006

(ii) 3B–3A=3(B–A) 11303007

Q. 2 If A= , show that A=I . 11303008

Q. 3 Find x and y if 11303009

(i) = 11303010

(ii) = 11303011

Q.4 If A = and

B= , find the following matrices; 11303012

(i) 4A–3B 11303013 (ii) A+3(B – A) 11303014

Q.5 Find x and y if 11303015

+ 2 =

Q.6 If A = [a_ij]₃_´3 , show that 11303016

(i) l(mA)=(lm)A 11303017

(ii) (l+m)A=lA+mA 11303018

(iii) lA - A = (l - 1)A 11303019

Q.7 If A = [a_ij]₂_´3and B = [b_ij]₂_´3 , show that l(A + B) = lA + lB. 11303020

Q.8 If A = and A = , find the values of a and b. 11303021

Q.9 If A= and A2 = ,
find the values of a and b. 11303022

Q.10 If A = and

B = ,

then show that (A + B) = A + B . 11303023

Q.11 Find A if A = 11303024

Q.12 Find the matrix X if; 11303025

(i) X = 11303026

(ii) X = 11303027

Q.13 Find the matrix A if, 11303028

(i) A = 11303029

(ii) A = 11303030

Q.14 Show that

= rI₃11303031

EXERCISE 3.2

Q.1 If A = [a ] , then show that 11303032

(i) I A = A 11303033 (ii) AI = A 11303034

Q. 2 Find the inverses of the following matrices 11303035

(i) 11303036 (ii) 11303037

(iii) 11303038 (iv) 11303039

Q.3 Solve the following system of linear equations. 11303040

(i) 11303041

(ii) 11303042

(iii) 11303043

Q.4 If A = , B =

and C = , then find 11303044

(i) A – B 11303045 (ii) B – A 11303046

(iii) (A – B) – C 11303047 (iv) A – (B – C) 11303048

Q. 5 If A = , B = and C = , then show that 11303049

(i) (AB)C = A(BC) 11303050

(ii) (A + B) C = AC + BC 11303051

Q.6 If A and B are square matrices of the same order, then explain why in general; 11303052

(i) (A + B)¹ A+ 2AB + B 11303053

(ii) (A – B) ¹ A– 2AB + B 11303054

(iii) (A + B) (A – B) ¹ A– B 11303055

Q.7 If A = ,

then find AA and AA 11303056

Q. 8 Solve the following matrix equations for X: 11303057

(i) 3X – 2A = B if

A = and B = 11303058

(ii) 2X – 3A = B if

A = and B = 11303059

Q.9 Solve the following matrix equations for A: 11303060

(i) A– = 11303061

(ii) A – = 11303062

Q. (Property 2): Prove that the value of a determinant changes sign if any two rows (columns) are interchanged. 11303063

Q. (Property 4): Prove that any two rows (columns) of a determinant are identical the value of determinant is zero. 11303064

Q. (Property 5): Prove that if any row (column) of a determinant is multiplied by a non-zero number k, the value of the new determinant becomes equal to k times the value of original determinant. 11303065

Q. (Property 7): Prove that if any row (columns) of a determinant is multiplied by a non-zero number k and the result is added to the corresponding entries of another row (column), the value of the determinant does not change. 11303066

Example: 11303067

If and then verify that

EXERCISE 3.3

Q.1 Evaluate the following determinants. 11303068

(i) 11303069

(ii) 11303070

(iii) 11303071

(iv) 11303072

(v) 11303073

(vi) 11303074

Q. 2 Without expansion show that 11303075

(i) = 0 11303076

(ii) = 0 11303077

(iii) = 0 11303078

Q. 3 Show that 11303079

(i) 11303080

= +

(ii) = 9 (Board 2008) 11303081

(iii) = l(3a + l) 11303082

(iv) = 11303083

(v) = 4abc 11303084

(vi) = a+ b 11303085

(vii) = r 11303086

(viii) = a³+ b³+ c³–3abc

11303087

(ix)= l(a + b + c + l)

11303088

(x) = (a – b) (b – c) (c – a)

11303089

(xi) (Board 2008) 11303090

= (a + b + c) (a – b) (b – c) (c – a)

Q.4 If A = and

B = , then find; 11303091

(i) A , A , A and 11303092

B , B , B and 11303093

Q.5 Without expansion verify that 11303094

(i) = 0 11303095

(ii) = 0 11303096

(iii) = 0 11303097

(iv) = 0 11303098

(v) = 0 11303099

(vi)= 11303100

(vii) = 0 11303101

(viii)= 11303102

(ix) L.H.S = 11303103

Q. 6 Find values of x if 11303104

(i) = –30 11303105

(ii) = 0 11303106

(iii) = 0 11303107

Q.7 Evaluate the following determinants:

11303108

(i) 11303109

(ii) 11303110

(iii) 11303111

Q.8 Show that

= (x + 3)(x – 1) 11303112

Q. 9 Find and if 11303113

(i) A = 11303114

(ii) A = 11303115

Q.10 If A is a square matrix of order 3,

then show that |kA| = k³ |A|. 11303116

Q.11 Find the values of l if A and B are singular. 11303117

(i) A = 11303118

(ii) B = 11303119

Q.12 Which of the following matrices are singular and which of them are non-singular. 11303120

(i) 11303121

(ii) 11303122

(iii) 11303123

Q.13 Find the inverse of

and show that A^-1 A = I₃. 11303124

Q.14 Verify that (AB)^-1 = B^-1 A^-1 if 11303125

(i) A = , B = 11303126

(ii) A = , B = 11303127

Q.15 Verify that (AB)^t = B^tA^tand if

A = and B = 11303128

Q.16 If A = verify that (A^-¹)^t = (A^t)^-¹11303129

Q.17 If A and B are non singular matrices, then show that 11303130

(i) (AB)^-1 = B^-1A^-1 11303131

(ii) ^-1 = A 11303132

(ii) = A 11303133

Q. What are symmetric and skew symmetric matrices? 11303134

Q. What are hermitian and skew hermitian matrices? 11303135

Q. Define the Rank of a Matrix. 11303135(a)

EXERCISE 3.4

Q.1 If A = and

B = ,

then show that A + B is symmetric 11303136

Q.2 If A = , show that

(i) A + A^t is symmetric 11303137

(ii) A - A^t is skew-symmetric 11303138

Q.3 If A is any square matrix of order 3, show that (Board 2008) 11303139

(i) A + A^t is symmetric 11303140

(ii) A - A^t is skew-symmetric 11303141

Q.4 If the matrices A and B are symmetric and AB = BA, show that AB is symmetric. (Board 2008) 11303142

Q.5 Show that AA^t and A^tA are symmetric for any matrix of order 2 ´ 3. 11303143

Q.6 If A = , show that 11303144

(i) A + () is Hermitian 11303145

(ii) A – () is skew Hermitian. 11303146

Q.7 If A is symmetric or skew symmetric show that A² is symmetric. (Board 2008)

11303147

Q.8 If A = , find A () 11303148

Q.9 Find the inverses of the following matrices. Also find their inverses by using row and column operations. 11303149

(i) 11303150

(ii) 11303151

(iii) 11303152

Q.10 Find the rank of the following matrices 11303153

(i) 11303154

(ii) 11303155

(iii) 11303156

EXERCISE 3.5

Q.1 Solve the following systems of linear equations by Cramer’s rule. 11303157

(i) 2x + 2y + z = 3 11303158

3x – 2y – 2z = 1

5x + y – 3z = 2 (Board 2008)

(ii) 2x – x + x = 5 (Board 2008) 11303159

4x + 2x + 3x = 8

3x – 4x – x = 3

(iii) 2x₁ - x₂ + x₃ = 8 11303160

x₁ + 2x₂ + 2x₃ = 6

x₁ - 2x₂ - x₃ = 1

Q.2 Use matrices to solve the following systems 11303161

(i) x – 2y + z = –1 11303162

3x + y – 2z = 4

y – z = 1

(ii) 2x₁ + x₂ + 3x₃ = 3 11303163

x₁ + x₂ - 2x₃= 0

-3x₁ -x₂ + 2x₃= - 4

(iii) x + y = 2 11303164

2x – z = 1

2y – 3z = – 1

Q.3 Solve the following systems by reducing their augmented metrics to the echelon form and the reduced echelon forms. 11303165

(i) x₁ – 2x₂ - 2x₃= -1 11303166

2x₁ + 3x₂ + x₃ = 1

5x₁ - 4x₂ - 3x₃ = 1

(ii) x + 2y + z = 2 11303167

2x + y + 2z = -1

2x + 3y - z = 9

(iii) x₁ + 4x₂ + 2x₃ = 2 11303168

2x₁ + x₂ - 2x₃ = 9

3x₁ + 2x₂ - 2x₃ = 12

Q.4 Solve the following systems of homogeneous linear equations. 11303169

(i) x + 2y - 2z = 0 11303170

2x + y + 5z = 0

5x + 4y + 8z = 0

(ii) x₁ + 4x₂ + 2x₃ = 0 11303171

2x₁ + x₂ - 3x₂ = 0

3x₁ + 2x₂ - 4x₃ = 0

(iii) x₁ - 2x₂ - x₃ = 0 11303172

x₁ + x₂ + 5x₃= 0

2x₁ - x₂ + 4x₃ = 0

Q.5 Find the value of l for which the following system has a non-trivial solution. Also solve the system for the value of l.11303173

(i) x + y + z = 0 11303174

2x + y - lz = 0

x + 2y - 2z = 0

(ii) x₁ + 4x₂ + lx₃ = 0 11303175

2x₁ + x₂ - 3x₃ = 0

3x₁ + lx₂ - 4x₃ = 0

Q.6 Find the value of l for which the following system does not possess a unique solution. Also solve the system for the value of l. 11303176

x₁ + 4x₂ + lx₃ = 2

2x₁ + x₂ - 2x₃ = 11

3x₁ + 2x₂ - 2x₃ = 16

Unit 4

QUADRATIC EQUATIONS

Q1. Define Quadratic equation. 11304001

Q2. Writ the ways of solving a Quadratic Equations. 11304002

Example: 11304003

Solve the equation by using the quadratic formula.

EXERCISE 4.1

Solve the following equations by factorization: 11304004

Q.1 3x + 4x + 1 = 0 11304005

Sol: 3x + 4x + 1 = 0

3x + 3x + x + 1 = 0

3x (x + 1) + 1(x + 1) = 0

(x + 1) (3x + 1) = 0

Either x + 1 = 0

Þ x = –1

or 3x + 1 = 0

3x = –1

Þ x = –

Hence solution set is

Q.2 x² + 7x + 12 = 0 11304006

Q.3 9x – 12x – 5 = 0 11304007

Q.4 x² - x = 2 11304008

Q.5 x(x + 7) = (2x – 1) (x + 4) 11304009

Q.6 + = ; x ¹ –1, 0 11304010

Q.7 + = ; x ¹ – 1,–2,– 5

11304011

Q.8 + = a + b ; x ¹ ,

11304012

Solve the following equations by completing the square: 11304013

Q.9 x – 2x – 899 = 0 11304014

Q.10 x + 4x – 1085 = 0 11304015

Q.11 x + 6x – 567 = 0 11304016

Q.12 x – 3x – 648 = 0 11304017

Q.13 x – x - 1806 = 0 11304018

Q.14 2x² + 12x - 110 = 0 11304019

Find roots of the following equations by using quadratic formula. 11304020

Q.15 5x – 13x + 6 = 0 11304021

Q.16 4x + 7x – 1 = 0 11304022

Q.17 15x + 2ax – a = 0 11304023

Q.18 16x² + 8x + 1 = 0 11304024

Q.19 (x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0 11304025

Q.20 x+x+b + c = 0 11304026

Example 1: 11304027

Solve the equation .

Example 2: 11304028

Solve the equation:

EXERCISE 4.2

Solve the following equations: 11304029

Q.1 x – 6x + 8 = 0 11304030

Q.2 x – 10 = 3x 11304031

Q.3 x – 9x + 8 = 0 11304032

Q.4 8x – 19x – 27 = 0 11304033

Q.5 x + 8 = 6 x 11304034

Q.6 = 24 11304035

Q.7 – 880 = 0 11304036

Q.8 (x – 5)(x – 7) (x + 6) (x + 4) – 504 = 0

11304037

Q.9 (x–1) (x – 2) (x – 8) (x + 5) + 360 = 0 11304038

Q.10 (x + 1)(2x + 3)(2x + 5) (x + 3) = 945 11304039

Q.11 (2x – 7) (x – 9) (2x + 5) – 91 = 0 11304040

Q.12 (x + 6x + 8) (x + 14x + 48) = 105 11304041

Q.13 (x + 6x – 27) (x – 2x – 35) = 385 11304042

Q.14 4 × 2 – 9 × 2 + 1 = 0 11304043

Q.15 2 + 2 – 20 = 0 11304044

Q.16 4 – 3 × 2 + 128 = 0 11304045

Q.17 3 – 12.3 + 81 = 0 11304046

Q.18 – 3 – 4 = 0 11304047

Q.19 x + x – 4 + + = 0 11304048

Q.20 + 3 = 0 11304049

Q.21 2x – 3x – x – 3x + 2 = 0 11304050

Q.22 2x + 3x – 4x – 3x + 2 = 0 11304051

Q.23 6x – 35x + 62x – 35x + 6 = 0 11304052

Q.24 x – 6x + 10 – + = 0 11304053

Type-V:

Q. Define radical equation. 11304054

Q. Define extraneous roots. 11304055

Example 1: Solve the equation: 11304056

Example 2: Solve the equation 11304057

…(1)

EXERCISE 4.3

Solve the following equations:

Q.1 3x + 2x – = 3 …(i)

11304058

Q.2 x – – 7 = x – 3 11304059

Q.3 + = 7 11304060

Q.4 = 2 + 11304061

Q.5 + = 11304062

Q.6 – = 1 11304063

Q.7 +

= 11304064

Q.8 + 3

= 11304068

Q.9 +

= 11304066

Q.10 (x+ 4)(x+1)= + 3x + 31

11304067

Q.11 + = 13 11304068

Q.12 – = x – 4

11304069

THREE CUBE ROOTS OF UNITY

Article: Find the cube roots of unity. 11304070

(2) The Sum of Three Cube Roots of Unity is Zero i.e. 1 + w + w² = 0 (Board 2014) 11304071

Example 2: Prove that: 11304072

Example 3: Prove that 11304073

Article: Find four fourth roots of unity. 11304074

EXERCISE 4.4

Q.1(i) Find the three cube roots of 8 11304075

(ii) Find the three cube roots of -8 11304076

(iii) Find the three cube roots of 27 11304077

(iv) Find the three cube roots of –27 11304078

Find the three cube roots of 64 11304079

Q.2 Evaluate: 11304080

(ii) Evaluate: w²⁸ + w²⁹ + 1 11304081

(iii) Evaluate: 11304082

(iv) Evaluate: 11304083

(v) Evaluate: 11304084

Q.3 Show that: 11304085

x- y=

Show that: x+ y+ z– 3xyz

11304086

Show that:

… 2n factors = 1 11304087

Q.4 If w is a root of x + x + 1 = 0, show that its other root is w and prove that w = 1.

11304088

Q.5 Prove that complex cube roots of –1 are and ; and hence prove that + = – 2

11304089

Q.6 If w is a cube root of unity, form an equation whose roots are 2w and 2w². 11304090

Q.7 Find four fourth roots of 11304091

(i) 16 11304092

81 11304093

625 11304094

Q.8 Solve the following equations: 11304095

(i) 2x – 32 = 0 11304096 3y – 243y = 0 11304097

(iii) x³+x²+x+1=0 11304098 (iv)5x–5x = 0 11304099

Q. Define Polynomial Function. 11304100

Article: State and prove Remainder Theorem. 11304101

Example: 11304102

Find the numerical value of k if the polynominal has a remainder of -4, when divided by

Example: 11304103

Show that is a factor of

Q. What is Synthetic Division? 11304104

Example 6: 11304105

If and are factors of . By use of synthetic division find the values of p and q.

EXERCISE 4.5

Use the remainder theorem to find the remainder when the first polynomial is divided by the second polynomial. 11304106

Q.1 x² + 3x + 7, x + 1 11304107

Q.2 x³ - x² + 5x + 4, x - 2 11304108

Q.3 3x⁴ + 4x³ + x -5, x + 1 11304109

Q.4 x³ - 2x² + 3x + 3, x - 3 11304110

Use the factor theorem to determine if the first polynomial is a factor of the second polynomial. 11304111

Q.5 x - 1, x² + 4x - 5 11304112

Q.6 x - 2 , x³ + x² - 7x + 1 11304113

Q.7 w+ 2 , 2w³ + w² - 4w + 7 11304114

Q.8 x-a, xⁿ-aⁿ where n is a positive integer.

11304115

Q.9 x+a , xⁿ + aⁿ ; where n is an odd integer.

11304116

Q.10 When x⁴ + 2x³ + kx² + 3 is divided by x - 2, the remainder is 1. Find the value of k. 11304117

Q.11 When the polynomial x³+2x²+kx+4 is divided by x - 2, the remainder is 14. Find the value of k. 11304118

Note: Use synthetic division to show that x is the solution of the polynomial and use the result to factorize the polynomial completely. In the following Questions. 11304119

Q.12 x³ - 7x + 6 = 0, x = 2 11304120

Q.13 x³ - 28x - 48 = 0 , x = -4 11304121

Q.14 2x⁴+7x³-4x²- 27x-18 , x=2, x = -3

11304122

Q.15 Use synthetic division to find the values of p and q if x + 1 and x - 2 are the factors of the polynomial x³+px² + qx + 6

(Board 2008) 11304123

Q.16 Find the values of a and b if -2 and 2 are the roots of the Polynomial

x³ - 4x² + ax + b. 11304124

Article: Find the relation between the roots and coefficients of a quadratic equation. 11304125

Article: Find the equation whose roots are given. 11304126

EXERCISE 4.6

Q.1 If a, b are the roots of 3x–2x+4=0, find the values of 11304127

(i) + 11304128 + 11304129

a + b 11304130 a + b 11304131

+ 11304132 a – b 11304133

Q.2 If a, b are the roots of (Board 2014)

x – px – p – c = 0,

Prove that (1 + a)(1 + b) = 1 – c 11304134

Q.3 Find the condition that one root of

x + px + q = 0 is 11304135

(i) double the other 11304136

(ii) square of the other 11304137

(iii) Additive inverse of the other 11304138

(iv) The multiplicative inverse of the other.

11304139

Q.4. If the roots of the equation x– px +q=0 differ by unity, prove that p=4q+ 1. 11304140

Q.5 Find the condition that + = 5 may have roots equal in magnitude but opposite in signs. 11304141

Q.6 If the roots of px+ qx + q = 0 are a and b then prove that

+ + = 0 11304142

Q.7 If a, b are the roots of the equation ax+ bx + c = 0 form the equations whose roots are 11304143

(i) a, b 11304144

(ii) , 11304145

(iii) , 11304146

(iv) a, b 11304147

(v) , 11304148

(vi) a + , b + 11304149

(vii) , 11304150

(viii) – , – 11304151

Q.8 If a , b are the roots of 5x– x–2= 0, form the equation whose roots are and .

11304152

Q.9 If a and b are the roots of x– 3x +5= 0 form the equation whose roots are and . 11304153

Article: Discuss the nature of the roots of 11304154

Example 2: 11304155

For what values of m will the following equation have equal roots.

EXERCISE 4.7

Q.1 Discuss the nature of the roots of the following equations: 11304156

(i) 4x + 6x + 1 = 0 11304157

(ii) x – 5x + 6 = 0 11304158

(iii) 2x – 5x + 1 = 0 11304159

(iv) 25x– 30x + 9 = 0 11304160

Q.2 Show that the roots of the following equation will be real: 11304161

(i) x – 2 x + 3 = 0 ; m ¹ 0 11304162

(ii) (b – c)x+(c – a)x+(a – b)=0 ; a, b, c Î Q

11304163

Q.3 Show that the roots of the following equations will be rational: 11304164

(i) (p + q)x – px – q = 0 (Board 2008) 11304165

(ii) px – (p – q)x – q = 0 11304166

Q.4 For what values of m will the roots of the following equations be equal? 11304167

(i) (m + 1)x + 2(m + 3)x + m + 8 = 0 11304168

(ii) x – 2(1 + 3m)x + 7(3 + 2m) = 0 11304169

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

11304170

Q.5 Show that the roots of x + (mx + c)

= a will be equal, if c = a (1 + m). 11304171

Q.6 Show that the roots of

= 4 a x will be equal, if c = ; m ¹ 0 11304172

Q.7 Show that the roots of the

+ = 1 will be equal if

c= am+ b ; a ¹ 0, b ¹ 0. 11304173

Q.8 Show that the roots of the equation

(a – bc)x + 2(b – ca)x + c – ab = 0 will be equal, if either a + b + c = 3abc or b=0 11304174

Q. What are Simultaneous Equations? 11304175

Example: 11304176

Solve the system of equations and

EXERCISE 4.8

Solve the following systems of equations: 11304177

Q.1 2x – y = 4 ; 2x – 4xy – y = 6 11304178

Q.2 x + y = 5 ; x + 2 y = 17 11304179

Q.3 3x + 2y = 7 ; 3 x = 25 + 2 y 11304180

Q.4 x + y = 5 ; + = 2 ; x ¹ 0, y ¹ 0

11304181

Q.5 x + y = a + b ; + = 2 11304182

Q.6 3x + 4y = 25 ; + = 2 11304183

Q.7 + y = 5 ; 2x = y + 6 11304184

Q.8 + = 5 11304185

x+ y + 2x = 9

Q.9 x + = 18 11304186

+ y= 21

Q.10 x+ y+ 6x = 1 11304187

x+ y+ 2 = 3

Example: 11304188

Solve the equations and

EXERCISE 4.9

Solve the following systems of equations: 11304189

Q.1 2x = 6 + 3y ; 3x – 5y = 7 11304190

Q.2 8x = y ; x + 2y = 19 11304191

Q.3 2x – 8 = 5y ; x – 13 = – 2y

11304192

Q.4 x – 5xy + 6y = 0 ; x + y = 45

11304193

Q.5 12x–25xy+12y=0 ; 4x+7y=148 11304194

Q.6 12x – 11xy + 2y = 0 ; 2x + 7xy = 60

11304195

Q.7 x – y = 16 ; xy = 15 11304196

Q.8 x + xy = 9 ; x – y = 2 11304197

Q.9 y – 7 = 2xy ; 2x + 3 = xy 11304198

Q.10 x + y = 5 ; xy = 2 11304199

Example 1: 11304200

Find two consecutive even positive integers where product is 24.

Example 2: 11304201

The sum of a positive number and its reciprocal is . Find the number.

EXERCISE 4.10

Q.1 The product of one less than a certain positive number and two less than three times the number is 14. Find the number. 11304202

Q.2 The sum of a positive number and its square is 380. Find the number. 11304203

Q.3 Divide 40 into two parts such that the sum of their square is greater than 2 times their product by 100. 11304204

Q.4 The sum of a number and its reciprocal is . Find the number. 11304205

Q.5 A number exceeds its square root by 56. Find the number. 11304206

Q.6 Find two consecutive numbers, whose product is 132. 11304207

Q.7 The difference between the cubes of two consecutive even numbers is 296. Find them. 11304208

Q.8 A farmer bought some sheep for Rs.9000. if he had paid Rs. 100 less for each, he would have got 3 sheep more for the same money. How many sheep did he buy, when the rate in each case is uniform?

11304209

Q.9 A man sold his stock of eggs for Rs. 240 he had 2 dozen more, he would have got the same money by selling the whole for Rs. 0.50 per dozen cheaper. How many dozen eggs did he sell? 11304210

Q.10 A cyclist traveled 48 km at a uniform speed. Had he traveled 2 km/hour slower, he would have taken 2 hours more to perform the journey. How long did he take to cover 48 km? 11304211

Q.11 The area of rectangular field is 297 square meters. Had it been 3 meters longer and one meter shorter, the area would have been 3 square meters more. Find its length and breadth. 11304212

Q.12 The length of a rectangular piece of paper exceeds it breath by 5 cm. If a strip 0.5 cm wide be cut all around the piece of paper, the area of the remaining part would be 500 square cms. Find its original dimensions. 11304213

Q.13 A number consists of two digits whose product is 18. If the digits are interchanged, the new number becomes 27 less than the original number. Find the number. 11304214

Q.14 A number consists of two digits whose product is 14. If the digits are interchanged the resulting number will exceed the original number by 45. Find the number. 11304215

Q.15 The area of a right triangle is 210 square meters. If its hypotenuse is 37 meters long. Find the length of the base and the altitude. 11304216

Q.16 The area of rectangle is 1680 square meters. If its diagonal in 58 meters long find the length and the breadth of the rectangle. 11304217

Q.17 To do a piece of work, A takes 10 days more than B. Together they finish the work in 12 days. How long would B take to finish it alone? 11304218

Q.18 To complete a job, A and B take 4 days working together. A alone takes twice as long as B alone to finish the same job. How long would each one alone take to do the job? 11304219

Q.19 An open box is to be made from a square piece of tin by cutting a piece 2 dm square from each cover and then folding the sides of the remaining piece. If the capacity of the box is to be 128 c. dm, find the length of the side of the piece. 11304219

Q.20 A man invests Rs. 100,000 in two companies. His profit is Rs. 1980 from first company. If he receives Rs. 3080 from second company and at the rate of 1% more from the other, find the amount of each investment. 11304220

Unit 5

PARTIAL FRACTIONS

Example 1: 11305001

Resolve into Partial Fractions.

EXERCISE 5.1

Resolve the following into Partial Fractions 11305002

Q.1 11305003

Q.2 11305004

Q.3 11305005

Q.4 11305006

Q.5 11305007

Q.6 11305008

Q.7 11305009

Q.8 11305010

Q.9 11305011

Q.10 11305012

Q.11 11305013

Example 1: 11305014

Resolve, into partial fractions.

Example 2: 11305015

Resolve, into Partial Fractions.

EXERCISE 5.2

Resolve the following into Partial Fractions: 11305016

Q.1 11305017

Q.2 11305018

Q.3 11305019

Q.4 11305020

Q.5 11305021

Q.6 11305022

Q.7 11305023

Q.8 11305024

Q.9 11305025

Q.10 11305026

Q.11 11305027

Q.12 11305028

Example 1: 11305029

Resolve into Partial Fractions.

EXERCISE 5.3

Resolve the following into Partial Fractions: 11305030

Q.1 11305031

Q.2 11305032

Q.3 11305033

Q.4 11305034

Q.5 11305035

Q.6 (Board 2014) 11305036

Q.7 11305037

Q.8 11305038

Q.9 (Board 2014) 11305039

Q.10 11305040

EXERCISE 5.4

Resolve into Partial Fractions: 11305041

Q.1 11305042

Q.2 11305043

Q.3 11305044

Q.4 11305045

Q.5 11305046

Q.6 11305047

Unit 6

SEQUENCES AND SERIES

Example 2: 11306001

Write first two, 21^st and 26^th terms of the sequence whose general term is

Example 3: Find the sequence if 11306002 and

EXERCISE 6.1

Q.1(a) Write the first four terms of the following sequences, if 11306003

a = 2n – 3 11306004

a = (–1) n 11306005

a = (–1) 11306006

a = 3n – 5 11306007

a = 11306008

a = 11306009

a – a = n + 2, a = 2 11306010

a = n a , a = 1 11306011

a = a , a = 1 11306012

a = 11306013

Q.2 Find the indicated terms of the following sequences; 11306014

(i) 2, 6, 11, 17, …., a₇11306015

(ii) 1, 3, 12, 60, ……., a₆11306016

(iii) 1, , , , ……. a₇11306017

(iv) 1,1, -3, 5, -7, ……, a₈11306018

(v) 1, -3, 5, -7, 9, -11, ….. a₈11306019

Q.3 Find the next two terms of the following sequence. 11306020

(i) 7, 9, 12, 16, …… 11306021

(ii) 1, 3, 7, 15, 31, …….. 11306022

(iii) -1, 2, 12, 40, ……. 11306023

(iv) 1, - 3, 5, - 7, 9, - 11, …….. 11306024

Q. Define an Arithmetic Progression (A.P) 11306024(a)

Article: Prove that 11306024(b)

Example: 11306025

2, 5, 8, 11, . . . . . . is an A.P. because the rule is of addition (adding 3 to the previous no.) i.e. 2 + 3 = 5, 5 + 3 = 8, 8 + 3 = 11 etc.

Example: 11306026

If the 5^th term of an A.P. is 13 and 17^th term is 49, find a_n and a₁₃.

Example: 11306027

Find the number of terms in the an A.P. if; and

Example: 11306024(c)

If find the nth term of the sequence.

EXERCISE 6.2

Q.1 Write the first four terms of the following arithmetic sequences, if 11306028

a = 5 and other three consecutive terms are 23, 26, 29. 11306029

(ii) a = 17 and a = 37 11306030

(iii) 3a = 7a and a = 33 11306031

Q.2 If a = 2n – 5, find the nth term of the sequence. 11306032

Q.3 If the 5^th term of an A.P. is –16 and the 20^th term is – 46, what is its 12^th term? 11306033

Q.4 Find the 13^th term of the sequence x, 1, 2 – x, 3 – 2x, ... 11306034

Q.5 Find the 18^th term of the A.P., if its 6^th term is 19 and the 9^th term is 31. 11306035

Q.6 Which term of the A.P. 5,2,–1,…is– 85?

(Board 2008) 11306036

Q.7. Which term of the A.P.–2,4,10,…is 148? 11306037

Q.8 How many terms are there in A.P. in which a = 11, a = 68, d = 3? 11306038

Q.9 If the nth term of an A.P is 3n – 1, find the A.P. 11306039

Q.10 Determine whether 11306040

(i) –19 11306041 (ii) 2 11306042

are the terms of A.P. 17, 13, 9, … or not.

Let a = 2 and n = ? 11306043

Q.11 If l, m, n are the pth, qth, rth terms of an A.P., show that 11306044

(i) l(q – r) + m(r – p) + n (p – q) = 0 11306045

p(m – n) + q(n – l) + r(l – m) = 0 11306046

Q.12 Find the nth term of the sequence,

, , , … 11306047

Q.13 If , and are in A.P. , Show that
b = 11306048

Q.14 If , and are in A.P. Show that the common difference is 11306049

Article: Find n, A.Ms between 'a, and 'b, 11306049(a)

EXERCISE 6.3

Q.1 Find A.M between 11306050

3 and 5 11306051

x – 3 and x + 5 11306052

1 – x + xand 1 + x + x 11306053

Q.2 If 5 , 8 are two A.Ms between a and b, find a and b. (Board 2008) 11306054

Q.3 Find 6 A.Ms between 2 and 5. 11306055

Q.4 Find four A.M’s between and

Sol: Let A, A , A , A be the four A.M’s 11306056

Q.5 Insert 7 A.M’s between 4 and 8 11306057

Q.6 Find three A.M’s between 3 and 11.

11306058

Q.7 Find n so that may be the A.M. between a and b. 11306059

Q.8 Show that the sum of n A.Ms between a and b is equal to n times their A.M. 11306060

Article: Find the sum of first n terms of an A.P. 11306060(a)

Example: 11306060(b)

Find the 19^th term and the partial sum of 19 terms of the arithmetic series:

Example: 11306061

How many terms of the series amount to 66?

EXERCISE 6.4

Q.1 Find the sum of all the integral multiples of 3 between 4 and 97. 11306062

Q.2 Sum the series 11306063

(i) –3 + (–1) + 1 + 3 + 5 + ××× + a 11306064

+ 2 + + ××× + a 11306065

1.11 + 1.41 + 1.71 + ××× + a 11306066

–8 – 3 + 1 + … + a 11306067

++ + … to n terms

11306068

(vi) + + + … to n terms

11306069

(vii) ++ + … to n terms

11306070

Q. 3 How many terms of the series 11306071

(i) – 7 + (–5) + (–3) +… amount to 65? 11306072

(ii) –7+(–4)+(–1) + … amount to 114? 11306073

Q.4 Sum the series 11306074

(i) 3 + 5 – 7 + 9 + 11 – 13 + 15 + 17 – 19

+ …. to 3n terms. 11306075

1 + 4 – 7 + 10 + 13 – 16 + 19 + 22 – 25

+ …. to 3n terms. (Board 2008) 11306076

Q.5 Find the sum of 20 terms of the series whose r^th term is 3r + 1. 11306077

Q.6 If S_n = n (2n - 1), then find the series.

11306078

Q.7 The ratio of the sums of n terms of two series in A.P. is 3n + 2 : n + 1. Find the ratio of their 8^th terms. (Board 2014) 11306079

Q.8 If S,S, S are the sums of 2n, 3n, 5n terms of an A.P., show that S=5 (S–S ). 11306080

Q.9 Obtain the sum of all integers in the first 1000 integers which are neither divisible by 5 nor by 2. 11306081

Q.10 S₈ and S₉ are the sums of the first eight and nine terms of an A.P., find S₉if 50 S₉ = 63 S₈ and a₁ = 2. 11306082

Q.11 The sum of 9 terms of an A.P. is 171 and its eighth term is 31. Find the series. 11306083

Q.12 The sum of S and S is 203 and
S – S = 49, S and S being the sums of the first 7 and 9 terms of an A.P. respectively. Determine the series. 11306084

Q.13 S and S are the sums of the first 7 and 9 terms of an A.P. respectively. If
= and a = 20, find the series. 11306085

Q.14 The sum of three numbers in an A.P. is 24 and their product is 440. Find the numbers. 11306086

Q.15 Find four numbers in A.P whose sum is 32 and the sum of whose squares is 276. 11306087

Q.16 Find the five numbers in A.P. whose sum is 25 and the sum of whose squares is 135. 11306088

Q.17 The sum of the 6^th and 8^th terms of an A.P. is 40 and the product of 4^th and 7^th terms is 220. Find the A.P. 11306089

Q.18 If a, band care in A.P., show that , , are in A.P. 11306090

EXERCISE 6.5

Q.1 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. 11306091

Q.2 378 trees are planted in rows in the shape of an isosceles triangle, the numbers in successive rows decreasing by one from the base to the top. How many trees are there in the row which forms the base of the triangle? 11306092

Q.3 A man borrows Rs. 1100 and agrees to repay with a total interest of Rs. 230 in 14 installments, each installment being less than the preceding by Rs. 10. What should be his first installment? 11306093

Q.4 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? 11306094

Q.5 A student saves Rs. 12 at the end of the first week and goes on increasing his saving Rs. 4 weekly. After how many weeks will he be able to save Rs. 2100? 11306095

Q.6 An object falling from rest, falls 9 meters during the first second, 27 meters during the next second, 45 meters during the third second and so on. 11306096

How far will it fall during the fifth second? 11306097

How far will it fall up to the fifth second? 11306098

Q.7 An investor earned Rs. 6000 for year 1980 and Rs. 12000 for year 1990 on the same investment. If his earning have increased by the same amount each year, how much income he has received from the investment over the past eleven years? 11306099

Q.8 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. 11306100

Q.9 The prize money Rs. 60,000 will be distributed among the eight teams according to their positions determined in the match series. The award increases by the same amount for each higher position. If the last place team is given
Rs. 4000, how much will be awarded to the first place team? 11306101

Q.10 An equilateral triangular base is filled by placing eight balls in the first row, 7 balls in the second row and so on with one ball in the last row. After this base layer, second layer is formed by placing 7 balls in its first row, 6 balls in its second row and so on with one ball in its last row, continuing this process, a pyramid of balls is formed with one ball on top. How money ball one there in the pyramid? 11306103

Article: Prove that or find the rule for n^th term of a G.P. 11306103(a)

Example 1: 11306104

Find the 8^th term of G.P. 1, 2, 4, 8,

EXERCISE 6.6

Q.1 Find the 5^th term of the G.P. 3, 6, 12, …

11306105

Q.2 Find the 11^th term of the G.P. 11306106

1 + i , 2 , , …

Q.3 Find the 12^th term of 1+i, 2i, –2 + 2i , …

(Board 2008) 11306107

Q.4 Find the 11^th term of the sequence,

1 + i , 2, 2(1 – i), … 11306108

Q.5 If an automobile depreciates in values 5% every year, at the end of 4 years what is the value of the automobile purchased for Rs. 12,000? 11306109

Q.6 Which term of the sequence (Board 2014)

x– y, x + y, , … is ? 11306110

Q.7 If a, b, c, d are in G.P. prove that 11306111

(i) a – b, b – c, c – d are in G.P. 11306112

(ii) a²– b², b²– c², c²– d² are in G.P. 11306113

(iii) a+ b, b+ c, c+ d are in G.P. 11306114

Q.8 Show that the reciprocals of the terms of the geometric sequence a, ar, ar, … form another geometric sequence. 11306115

Q.9 Find the nth term of geometric sequence, if = and a = 11306116

Q.10 Find three numbers in G.P. Whose sum is 26 and their product is 216. 11306117

Q.11 If the sum of the four consecutive terms of a G.P. is 80 and A.M of the second and the fourth of them is 30. Find the terms. 11306118

Q.12 If , and are in G.P. Show that the common ratio is ± 11306119

Q.13 If the numbers, 1, 4 and 3 are subtracted from three consecutive terms of an A.P., the resulting numbers are in G.P. Find the numbers if their sum is 21. 11306120

Q.14 If three numbers in A.P. are increased by 1, 4, 15 respectively, the resulting numbers are in G.P. Find the original numbers if their sum is 6. 11306121

Example 1: 11306122

Find G.M. between -5i and 5i

Example 2: 11306123

Insert two geometric means between 3 and 192.

EXERCISE 6.7

Q.1 Find G.M. between 11306124

(i) –2 and 8 11306125

(ii) –2i and 8i 11306126

Q.2(i) Insert two G.Ms. between 1 and 8

11306127

Q.2(ii) Insert two G.Ms. between 2 and 16

11306128

Q.3 Insert the three G.M’s between 11306129

(i) 1 and 16 11306130

(ii) 2 and 32 11306131

Q.4 Insert four real geometric means between 3 and 96. 11306132

Q.5 If both x and y are positive distinct real numbers, show that the geometric mean between x and y is less than their arithmetic mean. 11306133

Q.6 For what value of n, is the geometric mean between a and b ? 11306134

Q.7 The A.M. of two positive integral numbers exceeds their (positive) G.M. by 2 and their sum is 20, find the numbers. 11306135

Q.8 The A.M between two numbers is 5 and their (positive) G.M. is 4. Find the numbers. 11306136

Example: Show that the series 11306136(a)

is convergent.

Example: Find the sum of the series 8 + 4 + 2 + 1 …… to 6 terms 11306136(b)

EXERCISE 6.8

Q.1 Find the sum of first 15 terms of the geometric sequence

1, , , … 11306137

Q.2 Sum to n terms, the series 11306138

(i) .2 + .22 + .222 + ××× 11306139

(ii) 3 + 33 + 333 + ××× 11306140

Q.3 Sum to n terms the series 11306141

(i) 1 + (a + b) + (a+ ab + b) 11306142

+ (a+ ab + ab+ b ) + ×××

(Board 2008)

r + (1 + k) r+ (1 + k + k) r+ ××× 11306143

Q.4 Sum the series 2 + (1 – i) + + ××× to 8 terms 11306144

Q.5 Find the sum of the following infinite geometric series: 11306145

(i) + + + ××× 11306146

(ii) + + + …… 11306147

(iii) + + 1 + + …. 11306148

(iv) 2 + 1 + 0.5 + …. 11306149

(v) 4 + 2 + 2 + + 1 + …... 11306150

(vi) 0.1 + 0.05 + 0.005 + …. 11306151

Q.6 Find a vulgar fraction equivalent to the recurring decimals. 11306152

(i) 1. 11306153

0. 11306154

0. 11306155

1. 11306156

(v) 0. 11306157

1.1 11306158

Q.7 Find the sum to infinity of the series:

r + (1 + k) r + (1 + k + k)r + ×××

r and k being proper fractions. 11306159

(Board 2014)

Q.8 If y = + x+ x+ ××× and if 0 < x < 2,

then prove that x = 11306160

Q.9 If y = x + x+ x + ××× and

if 0 < x < , then show that x =

11306161

Q.10 A ball is dropped from a height 27 meters and it rebounds two-third of the distance it falls. If it continues to fall in the same way what distance will it travel before coming to rest? 11306162

Q.11 What distance will a ball travel before coming to rest if it is dropped from a height of 75 meters and after each fall it rebounds of the distance it fell? 11306163

Q.12 If y = 1 + 2x + 4x + 8x + ×××× 11306164

(i) Show that x = 11306165

(ii) Find the interval in which the series is convergent. 11306166

(ii) y = 1 + 2x + 4x+ 8x+ ×××× 11306167

Q.13 If y = 1 + + + ×××× 11306168

(i) Show that x = 2 11306169

(ii) Find the interval in which the series is convergent. 11306170

(ii) y = 1 + + + ×××× 11306171

Q.14 The sum of an infinite geometric series is 9 and the sum of the squares of its terms is . Find the series. 11306172

Example: 11306173

A man deposits in a bank Rs.2000 in the first year, Rs.4000 in second year, Rs.8000 in third year and so on. Find the amount he will have deposited in 8 years.

EXERCISE 6.9

Q.1 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. 11306174

Q.2 A man borrows Rs. 32760 without interest and agrees to repay the loan in installments, each installment being twice the preceding one. Find the amount of the last installment, if the amount of the first installment is Rs. 8. 11306175

Q. 3 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? 11306176

Q. 4 The enrollment of a famous school doubled after every eight years from 1970 to 1994. If the enrollment was 6000 in 1994, what was its enrollment in 1970? 11306177

Q.5 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? 11306178

Q.6 Joining the mid points of the sides of an equilateral triangle, an equilateral triangle having half the perimeter of the original triangle is obtained. We form a sequence of nested equilateral triangles in the number described above with the original triangle having perimeter . What will be the total perimeter of all the triangles formed in this way? 11306179

Example 1: 11306180

Prove that A, G, H are in G.P.

Article: Prove that A > G > H for positive real numbers. 11306181

Article: Prove that A < G < H, if a, and b are two distinct negative real number and 11306181(a)

Example 2: 11306182

If the numbers are in H.P. Find the value of K.

EXERCISE 6.10

Q.1 Find the 9th term of the harmonic sequence 11306183

(i) , , , … 11306184

(ii) - , - , -1,…… 11306185

Q.2 Find the 12th term of the following harmonic sequences 11306186

(i) , , , … 11306187

(ii) , , , … 11306188

Q.3 Insert five harmonic means between the following given numbers. 11306189

(i) – and 11306190

(ii) and 11306191

Q.4 Insert four harmonic means between the following given numbers. 11306192

(i) and 11306193

(ii) and 11306194

(iii) 4 and 20 11306195

Q.5 If the 7th and 10th terms of an H.P. are and respectively, find its 14th term. 11306196

Q.6 The first term of an H.P. is – and the fifth term is . Find its 9th term. 11306197

Q.7 If 5 is the harmonic mean between 2 and b, find b. 11306198

Q.8 If the numbers , and are in harmonic sequence, find k. 11306199

Q.9 Find n so that may be H.M. between a and b. 11306200

Q.10 If a, b and c are in A.P., show that a + b , c + a and b + c are in H.P. 11306201

Q.11 The sum of the first and fifth terms of the harmonic sequence is , if the first term is , find the sequence. 11306202

Q.12 If A, G and H are the arithmetic, geometric and harmonic means between a and b respectively, show that G= A H. 11306203

Q.13 Find A, G, H and show that

G = A H, if (Board 2014) 11306204

(i) a = – 2 , b = – 6 11306205

(ii) a = 2i , b = 4i 11306206

(iii) a = 9 , b = 4 11306207

Q.14 Find A, G, H and verify that

A > G > H , if 11306208

(i) a = 2, b = 8 11306209

(ii) a = , b = 11306210

Q.15 Find A, G, H and verify that

A < G < H , if 11306211

(i) a = – 2, b = – 8 11306212

(ii) a = – , b = – 11306213

Q.16 If the H.M. and A.M. between two numbers are 4 and respectively, find the numbers. 11306214

Q.17 If the (positive) G.M. and H.M. between two numbers are 4 and . Find the numbers. 11306215

Q.18 If the numbers , and are subtracted from the three consecutive terms of a G.P, the resulting numbers are in H.P. Find the numbers if their product is .11306216

EXERCISE 6.11

Sum the following series upto n terms.

Q.1 1 ´ 1 + 2 ´ 4 + 3 ´ 7 + …… 11306217

Q.2 1 ´ 3 + 3 ´ 6 + 5 ´ 9 + ……… 11306218

Q.3 1 ´ 4 + 2 ´ 7 + 3 ´ 10 + ……. 11306219

Q.4 3 ´ 5 + 5 ´ 9 + 7 ´ 13 + ……. 11306220

Q.5 1+ 3 + 5 + ….. 11306221

Q.6 2+ 5 + 8 + ……… 11306222

Q.7 2.1 + 4.2 + 6.3 + ……….. 11306223

Q.8 3 ´ 2+ 5 ´ 3+ 7 ´ 4 + ……… 11306224

Q.9 2´4´7 + 3´6´10 + 4´8´13 +… 11306225

Q.10 1´4´6+4´7´10+7´10´14+… 11306226

Q.11 1 + (1 + 2)+(1 + 2 + 3) + ………. 11306227

Q.12 1²+(1²+2²)+(1²+2²+3²)+……. 11306228

Q.13 2+(2+5)+(2+5+8)+……… 11306229

Q.14 Sum the series 11306230

(i) 1²–2²+3²–4²+…+(2n–1)²–(2n)²11306231

(ii) 1– 3 + 5– 7+…+(4n – 3)– (4n – 1)

11306232

(iii) + + + ……. 11306233

Q.15 Find the sum to n terms of the series whose nth terms are given. 11306234

(i) 3n + n + 1 11306235

(ii) n + 4n + 1 11306236

Q.16 Given nth terms of the series, find the sum to 2n terms. 11306237

(i) 3n + 2n + 1 11306238

(ii) n + 2n + 3 11306239

Unit 7

PERMUTATION, COMBINATION

AND PROBABILITY

Example 1: Prove that 0! = 1 11307001

EXERCISE 7.1

Q.1 Evaluate each of the following: 11307002

(i) 4! 11307003

(ii) 6! 11307004

(iii) 11307005

(iv) 11307006

(v) 11307007

(vi) = 11307008

(vii) 11307009

(viii) 11307010

(ix) 11307011

(x) 11307012

(xi) 11307013

(xii) 4! 0! 1! 11307014

Q.2 Write in the factorial form 11307015

Multiply and divided by 3! (i) 11307016

(ii) 12×11×10 11307017

(iii) 20×19×18×17 11307018

(iv) 11307019

(v) 11307020

(vi) 11307021

(vii) n(n – 1)(n – 2) 11307022

(viii) (n + 2)(n + 1)(n) 11307023

(ix) 11307024

(x) n(n – 1)(n – 2)…….(n – r + 1) 11307025

Article: Prove that 11307026

Example 1: 11307027

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

EXERCISE 7.2

Q.1 Evaluate the following: 11307028

(i) P 11307029

(ii) As 11307030

(iii) As 11307031

(iv) As 11307032

(v) As 11307033

Q.2 Find the value of n when: 11307034

(i) nP2 = 30 11307035

(ii) = 11 10 9 11307036

(iii) nP4 : n–1P3 = 9 : 1 11307037

Q.3 Prove that 11307038

(i) nPr = n ´ n–1Pr–1 11307039

(ii) nPr = n–1Pr + r n–1Pr–1 11307040

Q.4 How many signals can be given by 5 flags of different colours, using 3 flags at a time? 11307041

Q.5 How many signals can be given by 6 flags of different colours, when any number of them are used at a time?

(Board 2008) 11307042

Q.6 How many words can be formed from the letters of the following words using all letters when no letter is to be repeated? 11307043

(i) PLANE 11307044

(ii) OBJECT 11307045

(iii) FASTING 11307046

Q.7 How many 3-digit numbers can be formed by using each one of the digits 2, 3, 5, 7, 9 only once? 11307047

Q.8 Find the numbers greater than 23000 that can be formed from the digits 1,2,3,5,6 without repeating the digit. 11307048

Q.9 Find the number of 5-digit numbers

that can be formed from the digits 1,2,4,6,8 (when no digit is repeated), but 11307049

(i) The digits 2 and 8 are next to each others; 11307050

(ii) The digits 2 and 8 are not next to each other. 11307051

Q10. How many 6-digit numbers can be formed, without repeating any digit from the digits 0, 1, 2, 3, 4, 5? In how many of them will 0 be at the tens place? 11307052

Q.11 How many 5 digits multiples of 5 can be formed from the digits 2, 3, 5, 7, 9, when no digit is repeated? 11307053

Q.12 In how many ways can 8 books including 2 on English be arranged on a shelf in such a way that the English books are never together? 11307054

Q13. Find the number of arrangements of 3 books on English and 5 books on Urdu for placing them on a shelf so that the books on the same subject are always together. 11307055

Q.14 In how many ways can 5 boys and 4 girls be seated on a bench so that the girls and the boys occupy alternate seats? 11307056

Example 1: 11307057

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

Example 2: 11307058

In how many ways can 5 persons be seated at a round table.

EXERCISE 7.3

Q.1 How many arrangements of the letters of the following words, taken all together, can be made: 11307059

(i) pakpattan 11307060

(ii) pakistan 11307061

(iii) mathematics 11307062

(iv) assassination 11307063

Q.2 How many permutations of the letters of the word “PANAMA” can be made, if P is to be the first letter in each arrangement? 11307064

Q.3 How many arrangements of the letters of the word “ATTACKED” can be made, if each arrangement begins with C and ends with K? 11307065

Q.4 How many numbers greater than 1000,000 can be formed from the digits 0, 2, 2, 2, 3, 4, 4? 11307066

Q.5 How many 6-digit numbers can be formed from the digits 2,2,3,3,4,4 How many of them will lie between 400,000 and 430,000? 11307067

Q.6 11 members of a club form 4 committees of 3,4,2,2 members so that no member is a member of more than one committee. Find the number of committees. 11307068

Q.7 The D.C.Os of 11 districts meet to discuss the law and order situation in their districts. In how many ways can they be seated at a round table, when two particular D.C.Os insist on sitting together? 11307069

Q.8 The Governor of the Punjab calls a meeting of 12 officers. In how many ways can they be seated at a round table? 11307070

Q.9 Fatima invites 14 people to a dinner. There are 9 males and 5 females who are seated at two different tables so that guests of one sex sit at one round table and the guests of the other sex at the second table. Find the number of ways in which all guests are seated. 11307071

Q.10 Find the number of ways in which 5 men and 5 women can be seated at a round table in such a way that no two persons of the same sex sit together. 11307072

Q.11 In how many ways can 4 keys be arranged on a circular key ring? 11307073

Q.12 How many necklaces can be made from 6 beads of different colours? 11307074

Q. Prove that 11307075

Example 1: If 11307076

Example 2: 11307077

Find the number of the diagonals of a 6-sided figure.

Example 3: 11307078

Prove that

EXERCISE 7.4

Q.1 Evaluate the following: 11307079

(i) C 11307080

(ii) C 11307082

(iii) ⁿC₄11307083

Q.2 Find the value of n, when 11307084

(i) ⁿC₅ = ⁿC₄11307085

(ii) ⁿC₁₀ = 11307086

(iii) ⁿC₁₂ = ⁿC₆11307087

Q.3 Find the values of n and r when 11307088

(i) nCr = 35 , nPr = 210 11307089

(ii) n-1Cr-1 : nCr : n+1Cr+1 = 3:6:11 11307090

Q.4 How many (a) diagonals and (b)_ triangles can be formed by joining the vertices of the polygon having. 11307091

(i) 5 sides 11307092 (ii) 8 sides 11307093

(iii) 12 sides? 11307094

Q.5 The members of a club are 12 boys and 8 girls. In how many ways can a committee of 3 boys and 2 girls be formed? 11307095

Q.6 How many committees of 5 members can be chosen from a group of 8 persons when each committee must include 2 particular persons? 11307096

Q.7 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? 11307097

Q.8 Show that: C+C= C 11307098

Q.9 There are 8m men and 10w women members of a club. How many committees of seven persons can be formed, having; 11307099

(i) Four women 11307100

(ii) At the most four women, 11307101

(iii) At least four women? 11307102

(Board 2008)

Q.10 Prove that: nCr + nCr–1 = n + 1Cr 11307103

Example 1: 11307104

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

Example 2: 11307105

What is the probability that a slip of numbers divisble by 4 are picked from the slips bearing numbers, 1, 2, 3, . . . . . 10?

EXERCISE 7.5

For the following experiments, find the probability in each case: 11307106

Q.1 Experiment: 11307107

From a box containing orange flavoured sweets, Bilal takes out one sweet without looking.

Events Happening:

(i) The sweet is orange flavoured 11307108

(ii) The sweet is lemon flavoured. 11307109

Q.2 Experiment: 11307110

Pakistan and India play a cricket match the result is:

Events Happening:

(i) Pakistan wins 11307111

(ii) India does not lose 11307112

Q.3 Experiment: 11307113

There are 5 green and 3 red balls in a box, one ball is taken out.

Events happening.

(i) The ball is green. 11307114

(ii) The ball is red 11307115

Q.4 Experiment: 11307116

A fair coin is tossed three times.

It shows

Events Happening:

(i) One tail 11307117

(ii) At least one head 11307118

Q.5 Experiment: 11307119

A die is rolled. The top shows

Events Happening:

(i) 3 or 4 dots 11307120

(ii) dots less than 5 11307121

Q.6 Experiment from a box containing slips numbered 1, 2, 3, ×××, 5 one slip is picked up 11307122

Events Happening:

(i) The number on the slip is a prime number. 11307123

(ii)The number on the slip is a multiple of 3. 11307124

Q.7 Experiment.

Two dice, one red and the other blue, are rolled simultaneously. The numbers of dots on the tops are added. The total of the two scores is: 11307125

Events Happening:

(i) 5 11307126

(ii) 7 11307127

(iii) 11 11307128

Q.8 Experiment: 11307129

A bag contains 40 balls out of which 5 are green, 15 are black and the remaining are yellow. A ball is taken out of the bag.

Events Happening

(i) The ball is Black 11307130

(ii) The ball is green 11307131

Favourable out comes {i.e. Green} = 5

Q.9 EXPERIMENT: 11307132

One chit out of 30 containing the names of 30 students of a class of 18 boys and 12 girls is taken out at random, for nomination as the monitor of the class.

Events Happening:

(i) The monitor is a boy 11307133

(ii) The monitor is a girl 11307134

Q.10 EXPERIMENT: 11307135

A coin is tossed four times. The tops show

Events Happening:

(i) All heads 11307136

(ii) 2 heads and 2 tails 11307137

Favourable out comes are

THHT, HHTT, HTHT, HTTH, THTH, TTHH i.e. 6.

EXERCISE 7.6

Q.1 A fair coin is tossed 30 times, the result of which is tabulated below. Study the table and answer the questions given below the table. 11307138

Event	Tally Marks	Frequency
Head	~~\|\|\|\|~~ ~~\|\|\|\|~~ \|\|\|\|	14
Tail	~~\|\|\|\|~~ ~~\|\|\|\|~~ ~~\|\|\|\|~~ \|	16

(i)How many times does “head” appear? 11307139

(ii) How many times does “tail” appear? 11307140

(iii) Estimate the probability of the appearance of head? 11307141

(iv) Estimate the probability of the appearance of tail? 11307142

Q.2 A die is tossed 100 times. The result is tabulated below. Study the table and answer the questions given below the table: 11307143

Event	Tally Marks	Frequency
1	~~\|\|\|\|\|~~ ~~\|\|\|\|\|~~ \|\|\|\|	14
2	~~\|\|\|\|\|~~ ~~\|\|\|\|\|~~ ~~\|\|\|\|\|~~ \|\|	17
3	~~\|\|\|\|\|~~ ~~\|\|\|\|\|~~ ~~\|\|\|\|\|~~ ~~\|\|\|\|\|~~	20
4	~~\|\|\|\|\|~~ ~~\|\|\|\|\|~~ ~~\|\|\|\|\|~~ \|\|\|	18
5	~~\|\|\|\|\|~~ ~~\|\|\|\|\|~~ ~~\|\|\|\|\|~~	15
6	~~\|\|\|\|\|~~ ~~\|\|\|\|\|~~ ~~\|\|\|\|\|~~ \|	16

(i) How many times do 3 dots appears?

11307144

(ii) How many times do 5 dots appears?

11307145

(iii) How many times do an even number of dots appears? 11307146

(iv) How many times does a prime number of dots appear? 11307147

(v) Find the probability of each one of the above cases. 11307148

Q.3 The eggs supplied by a poultry farm during a week broke during transit as follows: 11307149

1% ,2%, 1 %, %, 1%, 2%, 1%.

Find the probability of the eggs that broke in a day. Calculate the number of eggs that will be broken in transiting the following number of eggs:

(i) 7,000 11307150 (ii) 8,400 11307151

(iii) 10,500 11307152

EXERCISE 7.7

Q.1 If sample space = {1, 2, 3, …, 9}, Event A = {2, 4, 6, 8} and

Event B = {1, 3, 5}, find P(AÈB) 11307153

Q.2 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. 11307154

Q.3 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? 11307155

Q.4 A card is drawn from a deck of 52 playing cards. What is the probability that it is a diamond card or an ace? 11307156

Q.5 A die is thrown twice. What is the probability that the sum of the number of dots shown is 3 or 11? 11307157

Q.6 Two dice are thrown. What is the probability that the sum of the number of dots appearing on them is 4 or 6 11307158

Q.7 Two dice are thrown simultaneously. If the event A is that the sum of the number of dots shown is an odd number and the event B is that the number of dots shown on at least one die is 3. Find P (AÈB). 11307159

Q.8 There are 10 girls and 20 boys in a class. Half of the boys and half of the girls have blue eyes. Find the probability that one student chosen as monitor is either a girl or has blue eyes. 11307160

EXERCISE 7.8

Q.1 The probability that a person A will be alive 15 years hence is and the probability that another person will be alive 15 years hence is . Find the probability that both will be alive 15 years hence. 11307161

Q.2 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: 11307162

P (EÇ E) = P (E) . P (E)

Q.3 Determine the probability of getting 2 heads in two successive tosses of a balanced coin. 11307163

Q.4 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. 11307164

Q.5 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. 11307165

Q.6 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: 11307166

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

(ii) Both the cards are faced cards i.e. king, queen, jack. 11307168

Q.7 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? 11307169

Q.8 Find the probability that the sum of dots appearing in two successive throws of two dice is every time 7. 11307170

Q.9 A fair die is thrown twice. Find the probability that a prime number of dots appear in the first throw and the number of dots in the second throw is less than 5. 11307171

Q.10 A bag contain 8 red, 5 white and 7 black balls 3 balls are drawn from the bag. What is the probability that the first ball is red, the second ball is white and the third ball is black, when every time the ball is replaced.” 11307172

Unit 8

MATHEMATICAL INDUCTION AND BINOMIAL THOREM

Example 1: Prove that 11308001

Example 2: 11308002

Show that represents an integer

EXERCISE 8.1

Use mathematical induction to prove the following formulae for every positive integer n. 11308003

Q.1 1 + 5 + 9 + … + (4n – 3) = n(2n – 1) 11308004

Q.2 1 + 3 + 5 + … + (2n – 1) = n 11308005

Q.3 1+4+7+…+(3n – 2) = 11308006

Q.4 1 + 2 + 4 + … + 2 = 2 – 1 11308007

Q.5 1 + + + … + = 2 11308008

Q.6 2 + 4 + 6 + … + 2n = n(n + 1) 11308009

Q.7 2 + 6 + 18 + … + 2 ´ 3 = 3 – 1 11308010

Q.8 1´3+2´5+3´7+… + n ´ (2n + 1) 11308011

= (Board 2008)

Q.9 1 ´ 2 + 2 ´ 3 + 3 ´ 4 + … + n (n + 1)

= 11308012

Q.10 1 ´ 2 + 3 ´ 4 + 5 ´ 6 + … +(2n – 1)´2n = 11308013

Q.11 + + + … +

= 1 – 11308014

Q.12 + + + …

+ = 11308015

Q.13 + + + …

+ = 11308016

Q.14 r + r+ r+… + r = ; (r ¹ 1)

11308017

Q.15 a + (a + d) + (a + 2d) + …

+ = 11308018

Q.16 1+2+3+…+n= – 1

11308019

Q.17 a = a + d where

a , a+ d , a + 2 d, … form an A.P. 11308020

Q.18 a = ar when a , a r , ar, … form a G.P. 11308021

Q.19 1+3+5+…+(2n – 1) =

11308022

Q.20 + + + … + = 11308023

Q.21 Prove by mathematical Induction that for all positive integral values of n. 11308024

(i) n+ n is divisible by 2. 11308025

(ii) 5 – 2 is divisible by 3. 11308026

(iii) 5 – 1 is divisible by 4. 11308027

(iv) 8 ´ 10 – 2 is divisible by 6. 11308028

(v) n– n is divisible by 6. 11308029

Q.22 + + … + = 11308030

Q.23 1– 2+ 3– 4 + … + n

= 11308031

Q.24 1+3+5+…+(2n–1)= n

11308032

Q.25 x + 1 is a factor of x – 1 ;

11308033

Q.26 x – y is a factor of x– y ;

11308034

Q.27 x + y is a factor of x + y

; 11308035

Q.28 Use mathematical induction to show that 1 + 2 + 2+ 2+ ×××+ 2= 2 – 1 for all non-negative integers n. 11308036

Q.29 If A and B are square matrices and AB = BA, then show by mathematical induction that AB = BA for any positive integer n. 11308037

Q.30 Prove by the principle of mathematical induction that n – 1 is divisible by 8 when n is an odd positive integer. 11308038

Q.31 Use the principle of mathematical induction to prove that ℓn x = n ln x for any integer n ³ 0 if x is a positive number.

11308039

Q.32 n! > 2 –1 for integral values of n ³ 4

11308040

Q.33 n> n + 3 for integral values of n ³ 3

11308041

Q.34 4 > 3 +2 for integral values of n³2

11308042

Q.35 3ⁿ < n! for integral values of n > 6.

11308043

Q.36 n! >n for integral values of n³4.11308044

Q.37 3+5+7+…+(2n+5)=(n+2) (n + 4) for integral values of n ³ –1 11308045

Q.38 For any positive number ‘x’, (1+x)>1+nx for any positive integer n³2 11308046

Example1: Expand 11308047

Example 2: Find T₅ in 11308048

Example 3: 11308049

Find the specified term in the expansion of

(i) independent of x

Example 4: 11308050

Find the middle term in the expansion of

EXERCISE 8.2

Q.1 Using binomial theorem expand the following: 11308051

(i) 11308052

(ii) 11308053

(iii) 11308054

(iv) 11308055

(v) 11308056

(vi) 11308057

Q.2 Calculate by means of binomial theorem. 11308058

(i) (0.97) 11308059

(iii) 11308061

(iv) (21) 11308062

Q.3 Expand and simplify the following: 11308063

(i) + …(i) 11308064

(ii) + 11308065

(iii) – 11308066

(iv) + 11308067

Q.4 Expand in ascending powers of x: 11308068

(i) (2+ x – x²) 11308069

(ii) 11308070

(iii) 11308071

Q.5 Expand in descending powers of x, 11308072

(i) 11308073

(ii) 11308074

Q.6 Find the term involving: 11308075

(i) x in the expansion of (3 – 2x) 11308076

(ii) x in the expansion of 11308077

(iii) a in the expansion of 11308078

(iv) yin the expansion of 11308079

Q.7 Find the coefficient of 11308080

(i) x in the expansion of 11308081

(ii) x in the expansion of 11308082

Q.8 Find 6^th term in the expansion of

11308083

Q. 9 Find the term independent of x in the following expansions. 11308084

(i) (Board 2008) 11308085

(ii) 11308086

(iii) (1 + x) 11308087

Q.10 Determine the middle term in the following expansions: 11308088

(i) 11308089

(ii) 11308090

(iii) 11308091

Q.11 Find (2n + 1)^th term from the end in the expansion of 11308092

Q.12 Show that the middle term of is 2 x 11308093

Q.13 Show that 11308094

+ + + … + = 2

Q.14 Show that

+ + + +….+ = 11308095

Example 1: 11308096

Expand to three terms and apply it to evaluate correct to two places of decimal.

Example 2: 11308097

Evaluate correct to three places of decimal.

Example 3: 11308098

If x is so small that its cube and higher power can be neglected, show that

Example 4: 11308099

For

Show that

EXERCISE 8.3

Q.1 Expand the following up to 4 terms, taking the values of x such that the expansion in each case is valid. 11308100

(i) 11308101

(ii) (1 + 2x) 11308102

(iii) (1 + x) 11308103

(iv) (4 – 3x) 11308104

(v) (8 – 2x) 11308105

(vi) (2 – 3x) 11308106

(vii) 11308107

(viii) 11308108

(ix) 11308109

(x) (1 + x – 2x²) 11308110

(xi) (1 - 2x + 3x²)^1/211308111

Q.2 Using Binomial theorem find the value of the following to three places of decimals. 11308112

(i) 11308113

(ii) 11308114

(iii) 11308115

(iv) 11308116

(v) 11308117

(vi) 11308118

(vii) 11308119

(viii) 11308120

(ix) 11308121

(x) 11308122

(xi) 11308123

(xii) 11308124

Q.3 Find the coefficient of x in the expansion of 11308125

(i) 11308126

(ii) 11308127

(iii) 11308128

(iv) 11308129

(v) 11308130

Q.4 If x is so small that its square and higher powers can be neglected, then show that 11308131

(i) » 1 – x 11308132

(ii) » 1 + x 11308133

(iii) » – x 11308134

(iv) » 2 + x 11308135

(v) » 4 11308136

(vi) » – x 11308137

(vii) » – 11308138

Q.5 If x is so small that its cube and higher powers can be neglected, show that 11308139

(i) » 1 – x – x 11308140

(ii) » 1 + x + x 11308141

Q.6 If x is very small nearly equal 1 then prove that px^p – qx^q(p – q) x^p+q 11308142

Q.7 If p – q is small when compared with p or q, show that 11308143

Q.8 Show that 11308144

» – Where n and N are nearly equal.

Q.9 Identify the following series as binomial expansion and find the sum in each case 11308145

(i) 1– + + + ××× ¥ 11308146

(ii) 1 – + – + ××× ¥ 11308147

(iii) 1 + + + + ××× ¥ 11308148

(iv) 1 – . + – + ××× ¥

11308149

Q.10 Use binomial theorem to show that

1 + + + + …¥ = 11308150

Q.11 If y = + + + ××× ¥,

prove that y+ 2y – 2 = 0 11308151

Q.12 If 2y = + + + ×××,

prove that 4y+ 4y – 1 = 0 11308152

Q.13 If y=+ + +××× ¥ ,

show that y+ 2y – 4 = 0 11308153

Unit 9

FUNDAMENTALS OF TRIGONOMETRY

Article: Prove that l = rq 11309001

Where l is one length, r radius of circle & q is central angle measured in radians.

Article: Prove that p radians = 180° 11309002

Example: Convert 54° 45^/ into radians 11309003

Example: 11309004

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.

EXERCISE 9.1

Q.1 Express the following sexagesimal measures of angles in radians. 11309005

(i) 30° 11309006 (ii) 45° 11309007

(iii) 60° 11309008 (iv) 75° 11309009

(v) 90° 11309010 (vi) 105° 11309011

(vii) 120° 11309012 (viii) 135° 11309013

(ix) 150° 11309014 (x) 10°15¢ 11309015
(xi) 35°20¢ 11309016 (xii) 75°6¢30¢¢ 11309017

(xiii) 120¢40¢¢ 11309018 (xiv) 154°20¢¢ 11309019

(xv) 0° 11309020 (xvi) 3¢¢ 11309021

Q.2 Convert the following radian measures of angles into the sexagesimal measures of system: 11309022

(i) 11309023 (ii) 11309024

(iii) 11309025 (iv) 11309026

(v) 11309027 (vi) 11309028

(vii) 11309029 (viii) 11309030

(ix) 11309031 (x) 11309032

(xi) 11309033 (xii) 11309034

(xiii) 11309035 (xiv) 11309036

(xv) 11309037

Q.3 What is the circular measure of the angle between the hands of a watch at 4 O’clock? 11309038

Q.4 Find q, when: 11309039

(i) l = 1.5 cm, r = 2.5 cm 11309040

(ii) l = 3.2 m, r = 2 m 11309041

Q.5 Find l, when: 11309042

(i) q = p radians, r = 6 cm 11309043

(ii) q = 65°20¢, r = 18 mm 11309044

Q.6 Find r, when: 11309045

(i) l = 5 cm, q = radian 11309046

(ii) l = 56 cm, q = 45° 11309047

Q.7 What is the length of the arc intercepted on a circle of radius 14 cm by the arms of a central angle of 45°? 11309048

Q.8 Find the radius of the circle, in which the arms of a central angle of measure 1 radian cut off an arc of length 35cm. 11309049

Q.9 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. 11309050

Q.10 A horse is tethered to a peg by a rope of 9 meters length and it can move in a circle with the peg as centre. If the horse moves along the circumference of the circle, keeping the rope tight, how far will it have gone when the rope has turned through an angle of 70°? 11309051

Q.11 The pendulum of clock is 20 cm long and it swings through an angle of 20° each second. How far does the tip of the pendulum move in 1 second? 11309052

Q.12 Assuming the average distance of the earth from the sun to be 148 ´ 10⁶ km and the angle subtended by the sun at the eye of a person on the earth of measure 9.3 ´ 10^-3 radian. Find the diameter of the sun. 11309053

Q.13 A circular wire of radius 6 cm is cut straightened and then bent so as to lie along the circumference of a hoop of radius 24 cm. Find the measure of the angle which it subtends at the centre of the hoop. 11309054

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. 11309055

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. 11309056

Q.16 The moon subtends an angle of 0.5° at the eye of an observer on earth. The distance of the moon from the earth is 3.844 ´ 10⁵ km approx. What is the length of the diameter of the moon? 11309057

Q.17 The angle subtended by the earth at the eye of a spaceman, landed on the moon, is 1° 54¢. The radius of the earth is 6400 km. Find the approximate distance between the moon and the earth. 11309058

Q. Write the trigonometric identities. 11309058(a)

Example: 11309059

Find the value of other five trigonometric functions of q, if and the terminal side of the angle is not in the I quadrant.

EXERCISE 9.2

Q.1 Find the signs of the following:

(i) sin 160° 11309060

(ii) cos 190° 11309061

(iii) tan 115° 11309062

(iv) sec 245° 11309063

(v) cot 80° 11309064

(vi) cosec 297° 11309065

Q.2 Fill in the blanks: 11309066

(i) sin = … sin310° 11309067

(ii) cos = … cos 75° 11309068

(iii) tan = … tan 182° 11309069

(iv) cot = … cot 173° 11309070

(v) sec = … sec 216° 11309071

(vi) cosec= … cosec 15° 11309072

Q.3 In which quadrant are the terminal arms of the angle lie when 11309073

(i) sin q < 0 and cos q > 0 11309074

(ii) cot q > 0 and cosec q > 0 11309075

(iii) tan q < 0 and cos q > 0 11309076

(iv) sec q < 0 and sin q < 0 11309077

(v) cot q > 0 and sin q < 0 11309078

(vi) cos q < 0 and tan q < 0 11309079

Q.4 Find the values of the remaining trigonometric function 11309080

(i) sin q = and the terminal arm of the angle is in the quadrant I 11309081

(ii) cos q = and the terminal arm of the angle is in the quadrant IV. 11309082

(iii) cos q = and the terminal arm of the angle is in quadrant III. 11309083

(iv) tan q = - and the terminal arm of the angle is in the quadrant II. 11309084

(v) sin q = - and the terminal arm of the angle is not in the quadrant III. 11309085

Q.5 If cot q = and the terminal arm of the angle is not in quadrant I, find the values of cos q and cosec q. 11309086

Q.6 If cosec q = and, find the values of the remaining trigonometric ratios. 11309087

Q.7 If tan q = and the terminal arm of the angle is not in the III quadrant, find the value of 11309088

Q.8 If cot q = and the terminal arm of the angle is in the I quadrant, find the value of [L.B 2008(G-II] 11309089

EXERCISE 9.3

Q.1 Prove the following:

(i) sin60° cos30° - cos60° sin30° = sin30°

11309090

(ii) sin + sin + tan = 2 11309091

(iii) 2 sin 45° + cosec 45°= 11309092

(iv) sin :sin :sin :sin = 1:2:3:4 11309093

Q.2 Evaluate the following: 11309094

(i) 11309095

(ii) 11309096

Q.3 Verify the following when q=30°,45°

11309097

(i) sin 2 q = 2 sin q cos q 11309098

(ii) cos 2 q = cos q – sin q 11309099

(iii) cos 2 q = 2cos q – 1 11309100

(iv) cos 2 q = 1 – 2 sin q 11309101

(v) tan 2 q = 11309102

Q.4 Find x, if tan45° – cos60° = x sin 45° cos 45° tan 60° 11309103

Q.5 Find the values of the trigonometric functions of the following quadrantal angles: 11309104

(i) π 11309105

(ii) 11309097

(iii) π 11309106

(iv) – π 11309107

(v) – 15 π 11309108

(vi) 1530° 11309109

(vii) –2430° 11309110

(viii) π ; 11309111

(ix) π 11309112

Q.6 Find the values of the trigonometric functions of the following angles: 11309113

(i) 390° 11309114 (ii) – 330° 11309115

(iii) 765° 11309116 (iv) – 675° 11309117

(v) – p 11309118 (vi) p 11309119

(vii) p 11309120 (viii) – p 11309121

(ix) – 1035° 11309122

Example 1: Prove that 11309123

, for all

Example 2: Prove that: 11309124

where q is not an odd multiple of .

Example 3: 11309125

Show that where q is not an integral multiple of .

EXERCISE 9.4

Q.1 tan q + cot q = cosec q sec q 11309126

Q.2 sec q cosec q sin q cos q = 1 11309127

Q.3 cos q + tan q sin q = sec q 11309128

Q.4 cosecq + tanq secq = cosecq sec 11309129

Q.5 sec - cosec = tan - cot 11309130

Q.6 cotq - cosq = cotq cosq 11309131

Q.7 = 1 11309132

Q.8 2cosq - 1 = 1 - 2sinq 11309133

Q.9 cosq - sinq = 11309134

Q.10 = 11309135

Q.11 + cot q = cosec q 11309136

Q.12 = 2 cos q - 1 11309137

Q.13 = 11309138

Q.14 = 11309139

Q.15 = 2 sin q cos q 11309140

Q.16 = 11309141

Q.17 = secq cosec²q 11309142

Q.18 = tan q + sec q

[L.B 2008 G-II] 11309143

Q.19 -

= - 11309144

Q.20 sin q - cos q 11309145

Q.21 sin q - cos q (Board 2014) 11309146

Q.22 sinq + cosq = 1 - 3sinqcosq 11309147

Q.23 + = 2secq 11309148

Q.24 +

= 11309149

Unit 10

TRIGONOMETRIC IDENTITIES OF SUM AND DIFFERENCE OF ANGLES

Distance Formula:

Let and be two points. If “d” denotes the distance between them,

then

Article: State and prove fundamental law of trigonometry. 11310001

EXERCISE 10.1

Q.1 Without using the tables, find the values of: 11310002

(i) sin 11310003

(ii) cot 11310004

(iii) cosec 2040° 11310005

(iv) sec 11310006

(v) tan 1110° 11310007

(vi) sin 11310008

Q.2 Express each of the following as a trigonometric function of an angle of positive degree measure of less than 45°. 11310009

(i) sin 196° 11310010

(ii) cos 147° 11310011

(iii) sin 319° 11310012

(iv) cos 254° 11310013

(v) tan 294° 11310014

(vi) cos 728° 11310015

(vii) sin 11310016

(viii) cos [(Cos (– q) = cosq]11310017

(ix) sin 150° 11310018

Q.3 Prove the following: 11310019

(i) sinsin= –sina cosa

11310020

(ii) sin 780° sin 480°+cos 120° sin 30° =

11310021

(iii) cos 306°+cos 234°+cos 162°+ cos 18° = 0

11310022

(iv) cos 330° sin 600° + cos 120° sin 150° = –1

11310023

Q.4 Prove that 11310024

11310025

(ii)= – 1

11310026

Q.5 If a, b , g are the angles of a triangle ABC, then prove that 11310027

(i) sin = sin γ 11310028

11310029

11310030

11310031

Example: 11310032

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

Example: Prove that: 11310033

Example: 11310034

Express 3 sin q + 4 cos q in the form
r sin(q + f), where the terminal side of the angle of measuring f is in the I quadrant.

EXERCISE 10.2

Q.1 Prove that:

11310035

11310036

11310037

11310038

(v) cos = sin q 11310039

(vi) sin = – cosq 11310040

(vii) tan = tan q 11310041

(viii) cos = cos q 11310042

Q.2 Find the values of the following: 11310043

(i) sin 15° 11310044

(ii) cos 15° 11310045

(iii) tan 15° 11310046

(iv) sin 105° 11310047

(v) cos 105° 11310048

(vi) tan 105° 11310049

Q.3 Prove that: 11310050

(i) sin = 11310051

(ii) cos =

11310052

Q.4 Prove that

(i) tan tan = 1 11310053

(ii) tan + tan = 0 11310054

(iii) sin + cos = cos q 11310055

(iv) = tan 11310056

(v) = 11310057

Q.5 Show that:

coscos= cosa – sinb

= cosb – sina 11310058

Q.6 Show that: 11310059

= tan a

Q.7 Show that: 11310060

(i) cot = 11310061

(ii) cot = 11310062

(iii) = 11310063

Q.8 If sin a = and cos b = where

0 < a < and 0 < b <

Show that sin = 11310064

Q.9 If sina = and sin b = where < a < π and < b < π. 11310065

Find

(i) sin(a + b) 11310066 (ii) cos(a + b) 11310067

(iii) tan(a + b) 11310068 (iv) sin(a – b) 11310069

(v) cos (a – b) 11310070 (vi) tan (a – b) 11310071

In which quadrants do the terminal sides of the angles of measures (a + b) and (a – b) lie?

Q.10 Find sin and cos , given that 11310072

(i) tan a = , cos b = 11310073

and neither the terminal side of the angle of measure a nor that of b is in the I quadrant

(ii) tan a = – , sin b = – 11310074

and neither the terminal side of the angle of measure a nor that of b is in the IV quadrant

Q.11 Prove that = tan 37°

11310075

Q.12 If a, b, γ are the angles of a triangle ABC, show that 11310076

cot + cot + cot = cot cot cot

Q.13 If a + b + γ = 180° , show that cot a cot b + cot b cot γ + cot γ cot a = 1 11310077

Q.14 Express the following in the form
γ sin (q ± f), where terminal sides of the angles of measures q and f are in the first quadrant: 11310078

(i) 12sin q + 5cos q 11310079

(ii) 3sin q – 4cos q 11310080

(iii) sin q – cos q 11310081

(iv) 5sin q – 4cos q 11310082

(v) sin q + cos q 11310083

(vi) 3sin q – 5cos q 11310084

Q. Prove that: 11310085

Q. Prove that: 11310086

ii)

Example: Show that: 11310087

ii)

Example: 11310088

Reduce to an expression involving only function of multiples of q, raised to the first power.

EXERCISE 10.3

Q.1 Find the values of sin2a, cos2a, tan2a, when: 11310089

(i) sin a = when 0 < a < 11310090

(ii) tan a = 2 when 0 < a < 11310091

(iii) cos a = when 0 < a < 11310092

Q.2 cot a – tan a = 2 cot 2a 11310093

Q.3 = tan a 11310094

Q.4 = tan 11310095

Q.5 = sec 2a – tan 2a 11310096

Q.6 = 11310097

Q.7 = cot 11310098

Q.8 1 + tan a tan 2a = sec 2a 11310099

Q.9 = tan 2q tan q 11310100

Q.10 – = 2 11310101

Q.11 + = 4cos 2q 11310102

Q.12 = sec q 11310103

Q.13 + = 2cot 2q 11310104

Q.14 Reduce sinq to an expression involving only function of multiples of q, raised to the first power. 11310105

Q.15 Find the values of sin q and cos q, when q is 11310106

(i) 18° 11310107 (ii) 36° 11310108

(iii) 54° 11310109 (iv) 72° 11310110

Hence prove that:

cos 36° cos 72° cos 108° cos 144° =

Example: 11310111

Express 2 sin 7q cos 3q as a sum or difference.

Example: Show that 11310112

EXERCISE 10.4

Q.1 Express the following products as sums or differences: 11310113

(i) 2 sin 3q cos q 11310114

(ii) 2 cos 5q sin 3q 11310115

(iii) sin 5q cos 2q 11310116

(iv) 2 sin 7q sin 2q 11310117

(v) cos sin 11310118

(vi) cos cos 11310119

(vii) sin 12° sin 46° 11310120

(viii) sin sin 11310121

Q.2 Express the following sums or differences as products: 11310122

(i) sin 5q + sin 3q 11310123

(ii) sin 8q – sin 4q 11310124

(iii) cos 6q + cos 3q 11310125

(iv) cos7q – cos q 11310126

(v) cos 12° + cos 48° 11310127

(vi) sin + sin 11310128

Q.3 Prove the following identities: 11310129

(i) = cot 2x 11310130

(ii) = tan 5x 11310131

(iii) = tan cot

11310132

Q.4 Prove that: 11310133

(i) cos 20° + cos100° + cos 140° = 0 11310134

(Board 2008)

(ii) sin sin = cos 2q 11310135

(iii) = tan 4q

11310136

Q.5 Prove that: 11310137

(i) cos 20° cos 40° cos 60° cos 80° =

(Board 2008) 11310138

(ii) sin sin sin sin = 11310139

(iii) sin 10° sin 30° sin 50° sin 70° =

(Board 2008) 11310140

Unit 11

TRIGONOMETRIC FUNCTIONS AND

THEIR GRAPHS

Q. Define period of a trigonometric function. 11311001

Theorem 1: 11311002

Sine is a periodic function and its period is 2p.

Theorem 2: 11311003

Tangent is a periodic function and its period is p.

EXERCISE 11.1

Find the periods of the following functions: 11311004

Q.1 sin 3x 11311005

Q.2 cos 2x 11311006

Q.3 tan 4x 11311007

Q.4 cot 11311008

Q.5 sin 11311009

Q.6 cosec 11311010

Q.7 sin 11311011

Q.8 Cos 11311012

Q.9 tan (Board 2008) 11311013

Q.10 cot 8x 11311014

Q.11 sec 9x 11311015

Q.12 cosec 10x 11311016

Q.13 3 Sin x 11311017

Q.14 2 cos x 11311018

Q.15 3 cos [L.B 2008 G-I Short] 11311019

EXERCISE 11.2

Draw the graph of each of the following functions for the intervals mentioned against each:

Q.1 (i) Draw the graph of y = - sin x , x Î [ –2p, 2p ]. 11311020

(ii) Draw the graph of y = 2 cos x , x Î [ 0 , 2p ]. 11311021

(iii) Draw the graph of y = tan 2x , x Î [– p , p]. 11311022

(iv) Draw the graph of y = tan x , x Î [– 2p, 2p ]. 11311023

(v) Draw the graph of y = sin , x Î [ 0 , 2p ]. 11311024

(vi) Draw the graph of y = cos , x Î [–p , p]. 11311025

Q.2 On the same axes and to the same scale, draw the graphs, for their complete periods of the pairs of functions defined by the following equations: 11311026

(i) y = sin x , y = sin 2x , x Î [ 0 , 2p ] 11311027

(ii) y = cos x , y = cos 2x , x Î [ 0 , 2p ] . 11311028

Q.3 Solve graphically : 11311029

(i) sin x = cos x, x Î [ 0, p ]. 11311030

(ii) sin x = x, x Î [ 0, p ]. 11311031

UNIT 12

APPLICATION OF TRIGONOMETRY

EXERCISE 12.1

Q.1 Find the value of 11312001

(i) sin 53°40¢ 11312002

(ii) cos 36°20¢ 11312003

(iii) tan 19°30¢ 11312004

(iv) cot 33°50¢ 11312005

(v) cos 42^o 38¢ 11312006

(vi) tan 25^o 34¢ 11312007

(vii) sin 18°31¢ 11312008

(viii) cos 52^o 13¢ 11312009

(ix) cot 89^o 9¢ 11312010

Q.2 (i) Find q, f: Sinq = 0.5791 q=sin^-1q

11312011

(ii) cos q = 0.9316 11312012

(iii) cos q = 0.5257 11312013

(iv) tan q = 1.705 11312014

(v) tan q =21.943 11312015

(vi) sin q = 0.5186 11312016

Exercise 12.2

Q.1 Find the unknown angles and sides of the following triangles. 11312017

(i) b = 90, a = 45, a = 4 11312018

(ii) a = 60° , b = 90° , b = 12 11312019

(iii) g = 90° , b = 5 , c = 10 11312020

(iv) = 90°, a = 40°, a = 8 11312021

(v) a = 56°, g = 90°, c = 15 11312022

(vi) g = 90, a = 8, c = 8 11312023

Q.2 a = 3720¢, a = 243 11312024

Q.3 a = 6240¢ , b = 796 11312025

Q.4 a = 3.28 , b = 5.74 11312026

Q.5 b = 68.4, c = 96.2 11312027

Q.6 a = 5429, c = 6294 11312028

Q.7 b = 50 10¢ , c = 0.832 11312029

Q. Define angles of elevation. 11312030

Q. Define angles of depression. 11312031

Example: 11312032

A string of a flying kite is 200 meters long, and its angle of elevation is 60°. Find the height of the kite above the ground taking the string to be fully stretched.

EXERCISE 12.3

Q.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? 11312033

Q.2 A man 18 dm tall observes that the angle of elevation of the top of a tree at a distance of 12 m from the man is 32°. What is the height of the tree? 11312034

Q.3 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? 11312035

Q.4 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.

11312036

Q.5 A kite flying at a height of 67.2 m is attached to a fully stretched string inclined at an angle of 55° to the horizontal. Find the length of the string. 11312037

Q.6 When the angle between the ground and the sun is 30°, flag pole casts shadow of 40 m long. Find the height of the top of the flag. 11312038

Q.7 A plane flying directly above a post 6000 m away from an anti-aircraft gun observes the gun at an angle of depression of 27°. Find the height of the plane. 11312039

Q.8 A man on the top of a 100 m high light-house is in line with two ships on the same side of it, whose angles of depression from the man are 17° and 19° respecting. Find the distance between the ships. 11312040

Q.9 P and Q are two points in line with a tree. If the distance between P and Q be 30 m and the angles of elevation of the top of the tree at P and Q be 12° and 15° respecting, find the height of the tree.

11312041

Q.10 Two men are on the opposite sides of a 100 m high tower. If the measures of the angles of elevation of the top of the tower are 18° and 22° respecting, find the distance between them. 11312042

Q.11 A man standing 60 m away from a tower notices that the angles of elevation of the top and the bottom of a flag staff on the top of the tower are 64° and 62° respectively. Find the length of the flag staff. 11312043

Q.12 The angle of elevation of the top of a 60 m high tower from a point p, on the same level as the foot of the tower, is 25°. Find the angle of elevation of the top of the tower from a point Q, 20 m nearer to A from the foot of the tower. 11312044

Q.13 Two buildings A and B are 100 m apart. The angle of elevation from the top of the building A to the top of the building B is 20°. The angle of elevation from the base of the building B to the top of the building A is 50°. Find the height of the building B. 11312045

Q.14 A window washer is working in a hotel building. An observer at a distance of 20 m from the building finds the angle of elevation of the worker to be of 30°. The worker climbs up 12 m and the observer moves 4 m farther away from the building. Find the new angle of elevation of the worker. 11312046

Q.15 A man standing on the bank of a canal observes that the measure of the angle of elevation of a tree is 60°. On retreating 40 meters from the bank, he finds the measure of the angle of elevation of the tree as 30°. Find the height of the tree and the width of the canal. 11312047

Article: Prove that .

11312048

Article: State and prove Law of Sines.

In any triangle ABC, with usual notations, prove that: 11312049

Article: Prove that

11312050

Article: Prove that 11312051

Exercise 12.4

Solve the triangle ABC, if 11312052

Q.1 b = 60°, g =15° , b = 11312053

Q.2 b = 52° , g = 89°35¢, a = 89.35 11312054

Q.3 b = 125 , g = 53°, a = 47° 11312055

Q.4 c = 16.1 , a = 42° 45¢ , g = 74° 32¢

11312056

Q.5 a = 53 , b = 88° 36¢ , g = 31° 54¢ 11312057

Example: 11312058

Solve the triangle ABC, by using the cosine and sine laws, given that b = 3, c = 5 and a = 120°.

Exercise 12.5

Q.1 b = 45 , c = 34 , a = 52 11312060

Q.2 b = 12.5 , c = 23 , a = 38° 20 11312061

Q.3 a = – 1 , b = + 1, g = 60°11312062

Q.4 a = 3, c = 6 , b = 36° 20 11312063

Q.5 a = 7 , b = 3 , g = 38 13 11312064

Solve the following triangles, using Law of tangent and then Law of sines:

Q.6 a = 36.21 , b = 42.09 , g = 44° 2911312065

Q.7. a = 93, c = 101 , b = 80 11312066

Q.8 b =14.8 , c = 16.1 , a = 42° 45 11312067

Q. 9 a = 319 , b = 168 , g = 110° 22 11312068

Q.10 b = 61, c = 32 and 11312069

Q.11 Measure of two sides of a triangle are in ratio 3 : 2 and they include an angle of measure 57. Find the remaining two angles. 11312070

Q.12 Two forces of 40N and 30N are represented by and which are inclined at an angle of 147° 25¢. Find , the resultant of and . 11312071

Example: 11312072

Solve the triangle ABC, by using the law of cosine when a = 7, b = 3, c = 5.

Example: 11312073

Solve the triangle ABC, by half angle formula, when a = 283, b = 317,

c = 428

Exercise 12.6

Note: Solution of an oblique when three sides are given:

Solve the following triangles, if 11312074

Q.1 a = 7 , b = 7 , c = 9 11312075

Q.2 a = 32, b = 40 , c = 66 11312076

(Board 2008)

Q.3 a = 28.3 , b = 31.7 , c = 42.8 11312077

Q.4 a = 31.9 , b = 56.31 , c = 40.2711312078

Q.5 a = 4584 , b = 5140 , c = 3624 11312079

Q.6 Find the smallest angle of the triangle ABC, when a = 37.34 , b = 3.24, c = 35.06 11312080

Q.7 Find the measure of the greatest angle, if side of the triangle are 16,20,33

11312081

Q.8 The sides of a triangle are x + x + 1, 2x + 1 and x– 1. Prove that the greatest angle of triangle is 120 11312082

(Board 2014)

Q.9 The measures of sides of a triangular plot are 413, 214 and 375 meters. Find the measure of the corner angles of the plot. 11312083

Q.10 Three villages A, B, C are connected by straight roads

6 Km., 9 Km. and 13 Km. What angles these roads makes with each others? 11312084

Article: Prove that area of triangle
= bc sina 11312085

Article: 11312086

In an triangle ABC, with usual notation, prove that:

Area of triangle =

Example: 11312087

Find the area of the triangle ABC, in which

b = 21.6, c = 30.2 and

Example: 11312088

Find the area of the triangle ABC in which
a = 275.4, b = 303.7, c = 342.5

Exercise 12.7

Q.1 Find the area of the triangle ABC, given two sides and their included angle: 11312089

(i) a = 200 , b = 120 , g = 150° 11312090

(ii) b = 37, c = 45, a = 30° 50 11312091

(iii) a = 4.33 , b = 9.25 , g = 56° 44 11312092

Q.2 Find the area of the triangle ABC, given one side and two angles: 11312093

(i) b = 25.4 , g = 36° 41 , a = 45° 1711312094

(ii) c = 32, a = 47° 24 , g = 70° 16 11312095

(iii) a = 4.8 , g = 37° 12 , a = 83° 42 11312096

Q.3 Find the area of the triangle ABC , given three sides: 11312097

(i) a = 18 , b = 24 , c = 30 11312098

(ii) a = 524 , b = 276 , c = 315 11312099

(iii) a = 32.65 , b = 42.81 , c = 64.92 11312100

Q.4 The area of triangle is 2437. If
a = 79 , and c = 97 , then find angle b.

11312101

Q.5 The area of triangle is 121.34. If a=3215, b =65 37,then find c and angle g.

11312102

Q.6 One side of a triangular garden is 30m. If its two corner angles are 22° 112°, find the cost of planting the grass at the rate of Rs. 5 per square meter. 11312103

Q. Define e-circle (Escribed Circle, ex-Circle). 11312104

Prove that: 11312105

Article: 11312106

Prove that: with usual notations.

Example: Show that: 11312107

Example: 11312108

Prove that

Exercise 12.8

Q.1 Show that 11312109

(i) r = 4 R sin sin sin 11312110

(ii) s = 4 R cos cos cos 11312111

Q.2 Show that r = a sin sin sec

= b sin sin sec 11312112

= c sin sin sec

Q.3 Show that 11312113

(i) r1 = 4 R sin cos cos 11312114

(ii) r = 4R sin cos cos 11312115

(iii) r3 = 4R sin cos cos 11312116

Q.4 Show that 11312117

(i) r = s tan 11312118

(ii) r = s tan 11312119

(iii) r = s tan 11312120

Q.5 Prove that 11312121

(i) r r + r r + r r = s²

(ii) r r r r = D²11312122

(iii) r + r + r – r = 4R 11312123

(iv) r r r = r s 11312124

Q.6 Find R, r, r, r and r, if measures of the sides of triangle ABC are 11312125

(i) a = 13 , b = 14 , c = 15 11312126

(ii) a = 34 , b = 20 , c = 42 11312127

Q.7 Prove that in an equilateral triangle,

(i) r : R : r = 1 : 2 : 3 11312128

(ii) r : R : r : r : r₃ = 1 : 2 : 3 : 3 : 3 11312129

Q.8 Prove that

(i) D = r2 cot cot cot 11312130

(ii) r = s tan tan tan 11312131

(iii) D = 4 R r cos cos cos 11312132

(Board 2014)

Q.9 Show that 11312133

(i) = + + 11312134

(ii) + + = 11312135

Q.10 Prove that r =

= = 11312136

Q.11 abc (sin a + sin b + sin g) = 4Ds 11312137

Q.12 Prove that 11312138

(i) (r + r) tan = c 11312139

(ii) (r – r) cot = c 11312140

UNIT 13

INVERSE TRIGONOMETRIC FUNCTION

Example: 11313001

Example: Find the value of: 11313001(a)

i) ii)

iii)

Example: 11313002

Prove that

EXERCISE 13.1

Q.1 Evaluate without using tables:-

(i) sin(1) 11313003

(ii) sin (–1) 11313004

(iii) cos 11313005

(iv) tan 11313006

(v) cos 11313007

(vi) tan 11313008

(vii) cot (–1) 11313009

(viii) cosec 11313010

(ix) sin 11313011

Q.2 Show that:-

(i) tan = sin 11313012

(ii) 2 cos = sin 11313013

(iii) cos = cot 11313014

Q.3 Find the value of each expression:

(i) cos 11313015

(ii) sec 11313016

(iii) tan 11313017

(iv) cosec 11313018

(v) sec 11313019

(vi) tan 11313020

(vii) sin 11313021

(viii) tan 11313022

(ix) sin 11313023

Prove that:

11313024

EXERCISE 13.2

Prove that following:-

Q. 1 sin + sin = cos ^–1 11313025

Q. 2 tan + tan = tan 11313026

Q.3 2 tan = sin 11313027

Q.4 tan = 2 cos 11313028

Q.5 sin^-1 + cot^-1 3 = 11313029

Q. 6 sin + sin = sin 11313030

Q.7 sin ^–1 – sin ^–1 = cos ^–1 11313031

Q. 8 cos + 2 tan = sin 11313032

Q.9 tan^-1 + tan^-1 - tan^-1 =

(Board 2008) 11313033

Q. 10 sin + sin + sin = 11313034

Q.11 tan + tan = tan +tan

11313035

Q. 12 2tan + tan = 11313036

Q. 13 Show that cos =

11313037

Q. 14 Show that sin = 2x

11313038

Q.15 Show that 11313039

cos (2 sin x) = 1 – 2x

Q.16 Show that 11313040

tan(–x) = – tan x

Q. 17 Show that 11313041

sin (–x) = – sinx

Q. 18 Show that cos(–x) = p – cosx

11313042

Q.19 Show that tan (sin x) =

11313043

Q.20 Given that x = sin , find the values of following trigonometric functions: sin x, cos x, tan x, cot x, sec x and cosec x.

11313044

UNIT 14

SOLUTIONS OF TRIGONOMETRIC EQUATIONS

Example: Solve the equation 11314001

Example: Solve the equation: 1 + cos x = 0

11314002

Example: Solve: sinx + cos x = 0 11314003

Example: Solve the equation: sin 2x = cos x

11314004

Þ 2 sin n cos x = cos x

Þ 2 sin x cos x – cos x = 0

Þ cos x(2 sin x – 1) = 0

\ cos x = 0 or 2 sin x – 1 = 0

i. If cos x = 0

Exercise 14

Q. 1 Find the solutions of the following equations which lie in [0, 2p] 11314005

(i) sin x = – 11314006

(ii) cosec q = 2 11314007

(iii) sec x = –2 11314008

(iv) cot q = 11314009

Q. 2 Solve the following trigonometric equations:- 11314010

(i) tanq = 11314011

(ii) cosec² q = 11314012

(iii) sec²q = 11314013

(iv) cot² q = 11314014

Q. 3 Find the values of q satisfying the equations 11314015

3 tanq + 2 tan q + 1 = 0

Q. 4 tanq – sec q – 1 = 0 11314016

Q.5 2 sin q + cos q – 1 = 0 11314017

Q. 6 2 sin q – sin q = 0 11314018

Q. 7. 3 cos²q – 2 sinq cosq–3 sin²q=0 11314019

Q. 8 4 sinq – 8 cos q + 1 = 0 11314020

Q.9 Find the solution sets of the following questions. tanx–seex–1=0 11314021

Q.10 sin 3x = cos 2x 11314022

Q.11 sec 3q = sec q 11314023

Q. 12 tan 2q + cot q = 0 11314024

Q.13 sin 2x + sin x = 0 11314025

Q. 14 sin 4x – sin 2x = cos 3x 11314026

Q.15 sin x + cos 3x = cos 5x 11314027

Q.16 sin 3x + sin 2x + sin x = 0 11314028

Q. 17 sin 7x – sin x = sin 3x 11314029

Q.18 sin x + sin 3x + sin 5x = 0 11314030

Q. 19 sin q + sin 3q + sin 5q + sin 7q=0 11314031

Q.20 cos q + cos 3q + cos 5q + cos 7q = 0 11314032

Causative

11th Math Subjective

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EXERCISE 2.1

EXERCISE 2.2

EXERCISE 2.3

EXERCISE 2.4

EXERCISE 2.5

EXERCISE 2.6

EXERCISE 2.7

EXERCISE 2.8

EXERCISE 3.1

Q.4 If A = and

B= , find the following matrices; 11303012

Q.5 Find x and y if 11303015

+ 2 =

EXERCISE 3.2

A = and B = 11303059

EXERCISE 3.3

(v) = 4abc 11303084

Q. 6 Find values of x if 11303104

EXERCISE 3.4

EXERCISE 3.5

Q.2 Use matrices to solve the following systems 11303161

Q.3 9x – 12x – 5 = 0 11304007

Q.5 x(x + 7) = (2x – 1) (x + 4) 11304009

Q.6 + = ; x ¹ –1, 0 11304010

Q.7 + = ; x ¹ – 1,–2,– 5

Q.8 + = a + b ; x ¹ ,

Q.9 x – 2x – 899 = 0 11304014

Q.10 x + 4x – 1085 = 0 11304015

Q.11 x + 6x – 567 = 0 11304016

Q.12 x – 3x – 648 = 0 11304017

Q.13 x – x - 1806 = 0 11304018

Find roots of the following equations by using quadratic formula. 11304020

Q.16 4x + 7x – 1 = 0 11304022

Q.17 15x + 2ax – a = 0 11304023

Q.19 (x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0 11304025

Q.20 x+x+b + c = 0 11304026

EXERCISE 4.2

Solve the following equations: 11304029

Q.3 x – 9x + 8 = 0 11304032

Q.4 8x – 19x – 27 = 0 11304033

Q.7 – 880 = 0 11304036

Q.10 (x + 1)(2x + 3)(2x + 5) (x + 3) = 945 11304039

Q.19 x + x – 4 + + = 0 11304048

Q.4 = 2 + 11304061

THREE CUBE ROOTS OF UNITY

EXERCISE 4.4

EXERCISE 4.5

EXERCISE 4.6

Q.5 Find the condition that + = 5 may have roots equal in magnitude but opposite in signs. 11304141

EXERCISE 4.7

EXERCISE 4.8

EXERCISE 4.10

EXERCISE 5.1

Q.5 11305007

Q.6 11305008

Q.9 11305011

Q.10 11305012

EXERCISE 5.2

EXERCISE 5.3

EXERCISE 5.4

Q.5 11305046

Q.5 Find the sum of 20 terms of the series whose rth term is 3r + 1. 11306077

Q.1 Find the 9th term of the harmonic sequence 11306183

(i) , , , … 11306184

Q.6 Experiment from a box containing slips numbered 1, 2, 3, ×××, 5 one slip is picked up 11307122

Tally Marks

Frequency

Tally Marks

Frequency

Q.3 1+4+7+…+(3n – 2) = 11308006

Q.16 1+2+3+…+n= – 1

EXERCISE 8.2

EXERCISE 8.3

Q.10 Use binomial theorem to show that

EXERCISE 9.1

EXERCISE 9.2

Q.4 Find the values of the remaining trigonometric function 11309080

Q.5 Find the sum of 20 terms of the series whose r^th term is 3r + 1. 11306077

Q. 7. 3 cos²q – 2 sinq cosq–3 sin²q=0 11314019