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) i14 11301035
(iii) (-i)19 11301036 (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 + 16b2 11301057
(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-10 11301089
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) Ac 11302115 (ii) Bc 11302116
(iii) A È B 11302117 (iv) A - B 11302118
(v) AÇC 11302119 (vi)
AcÈCc 11302120
(vii) Ac È C 11302121 (viii)
Uc
11302122
5. Using the Venn diagrams, if necessary, find the
single sets equal to the following: 11302123
(i) Ac 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
Bc 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) = Ac Ç
Bc 11302133
(ii) = Ac È Bc 11302134
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
|
|
|
||
Q. Prepare a truth
table for conjunction of two statements. |
|
||
Q. Prepare a truth
table for disjunction of two statements. |
|
||
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
|
|
0 |
1 |
2 |
3 |
4 |
|
0 |
0 |
0 |
0 |
0 |
0 |
|
1 |
0 |
1 |
2 |
3 |
4 |
|
2 |
0 |
2 |
4 |
1 |
3 |
|
3 |
0 |
3 |
1 |
4 |
2 |
|
4 |
0 |
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
|
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 = [aij]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 = [aij]2´3 and B = [bij]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
= rI3 11303031
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
= BtAt and if
A = and
B = 11303128
Q.16 If A = verify that (A-1)t
= (At)-1 11303129
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 + At
is symmetric 11303137
(ii) A - At is
skew-symmetric 11303138
Q.3 If A is any square matrix of order 3, show that (Board 2008) 11303139
(i) A + At
is symmetric 11303140
(ii) A - At 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 AAt and
AtA 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
|
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
|
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, 21st
and 26th 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, …., a7 11306015
(ii) 1, 3, 12, 60, ……., a6 11306016
(iii) 1, , , , ……. a7 11306017
(iv) 1,1, -3, 5, -7, ……, a8 11306018
(v) 1, -3, 5, -7, 9, -11, ….. a8 11306019
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 5th term
of an A.P. is 13 and 17th term is 49, find an 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 5th term of an A.P. is –16 and the 20th term is – 46,
what is its 12th term? 11306033
Q.4 Find the 13th term of the sequence x, 1, 2 – x, 3 – 2x,
... 11306034
Q.5 Find the 18th term of the A.P., if its 6th term is 19 and the 9th
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 19th
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 rth
term is 3r + 1. 11306077
Q.6 If Sn = 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 8th
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 S9 if 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 6th and 8th terms of an A.P.
is 40 and the product of 4th and 7th 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 9th 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 nth
term of a G.P. 11306103(a)
Example 1: 11306104
Find the 8th term
of G.P. 1, 2, 4, 8,
EXERCISE
6.6
Q.1 Find the 5th
term of the G.P. 3, 6, 12, …
11306105
Q.2 Find the
11th term of the G.P. 11306106
1 + i , 2 , , …
Q.3 Find the
12th term of 1+i, 2i, –2 + 2i , …
(Board 2008) 11306107
Q.4 Find the
11th 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)2 11306231
(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)
nC4 11307083
Q.2 Find
the value of n, when 11307084
(i) nC5 = nC4 11307085
(ii) nC10 = 11307086
(iii) nC12 = nC6 11307087
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 6th 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)1/2 11308111
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 pxp – qxq
(p – q) xp+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
i) 
Q. Prove that: 11310086
ii) 
Example: Show that: 11310087
i) 
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 42o 38¢ 11312006
(vi) tan 25o 34¢
11312007
(vii) sin 18°31¢ 11312008
(viii) cos 52o 13¢ 11312009
(ix) cot 89o 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 = D2 11312122
(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) sec2 q = 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
