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1. (a) There are 10 buses running between two towns X and Y. In how many ways can a man gofrom X to Y and return by a different bus.(b) Each section in first year of plus two course has exactly 30 students. If there are 3 sections, inhow many ways can a set of 3 students representatives be selected from each section.

2. (a) How many numbers are there between 100 and 1000 such that every digit is either 2 to 9?(b) How many odd numbers less than 1000 can be formed using the digits 0, 2, 5, 7 (repetition ofdigits are allowed).

3. (a) A coin is tossed three times and outcomes are recorded. Use the product rule to determinethe number of possible outcomes. Then list all the outcomes.(b) Two persons go in a railway carriage where there are 6 vacent seats. In how many differentways can they seat themselves?

4. (a) In how many ways can 2 prizes be awarded to 9 contestants provided no contestant gets boththe prizes.(b) There are three mathematics teachers in a college in which there are 6 classes, in how manydifferent ways can they choose the classes, provided one teaches one class only.

5. How many numbers are there between 100 and 1000 such that 7 is in the units place.

6. (a) How many automobile license plates can be made if the inscription on each contains twodifferent letters followed by three different digits.(b) How many 2-digit numbers can be formed from the digit 8, 1, 3, 5 and 4 assuming

(i) repetition of digits is allowed?(ii) repetition of digit is not allowed?

7. (a) There are 12 true-false questions in an examination. How many sequences of answers arepossible?(b) To pass an examination a student has to pass in each of the 3 papers. In how many ways canstudent fail in the examination?

8. (a) How many seven-digit phone numbers are possible if 0 and 1 cannot be used as the first digit

and the first three digits cannot be 555, 411, or 936?(b) There are five routes for journey from station A to station B. In how many different ways can aman go from A to B and return, if for returning(i) any of the routes is taken (ii) the same route is taken (iii) the same route is not taken.

9. (a) For a set of five true-or false questions, no student has written the all-correct answers, and notwo students have given the same sequence of answers. What is the maximum number ofstudents in the class, for this to be possible.(b) In how many of the permutation of n things taken r at a time will 5 things (i) always occur, (ii)never occur?

10. (a) Prove that the number of ways in which n books can be placed on a shelf when two particularbooks are never together is (n 2) x (n 1)!.

(b) In how many ways can 6 boys and 4 girls be arranged in a straight line to that no two girls areever together.

11. (a) How many different numbers can be formed by permuting the digits 1, 3, 5, 7, 9, when takenall at a time, and what is their sum?(b) In how many ways can the letters of the word INDIA be arranged?

12. (a) How many signals can be made by hoisting 2 blue, 2 red and 5 yellow flags on a pole at thesame time?(b) A coin is tossed 6 times. In how many different ways can we obtain 4 heads and 2 tails?

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13. (a) How many numbers can be formed with digits 1, 2, 3, 4, 3, 2, 1, so that odd digits always

occupy the odd places?(b) In how many ways can the letters of the word ARRANGE be arranged? If the two Rs do notoccur together, then how many arrangements can be made? If besides the two Rs, the two Asalso do not occur together, how many permutations will be obtained.

14. (a) In how many ways can 5 prizes be given away to 4 boys, when each boy is eligible for all theprizes.(b) How many numbers each containing four digits can be formed?

15. (a) 20 persons were invited for a party. In how many ways can they and the host be seated at acircular table? In how many of these ways will two particular persons be seated on either side ofthe host?(b) In how many ways can a party of 4 boys and 4 girls be seated at a circular table so that no 2boys are adjacent?

16. Find the number of ways in which 10 different flowers can be strung to form a garland so that 4particular flowers are never separated.

17. (a) In how many ways can 7 persons sit around a table so that all shall not have the sameneighbours in any two arrangements.(b) A round table conference is to be held between delegates of 20 countries. In how many wayscan they be seated if two particular delegates are.

(i) always together (ii) never together?

18. (a) If out of 6 flags any number of flags can be shown at a time, find how many different signalscan be made out of them.(b) In how many of the permutations of n things taken r together will three given things

(i) not occur, (ii) always occur?

19. (a) How many different words beginning and ending with a consonant, can be made out of theletters of the word EQUATION?

(b) (i) How many different numbers of six digits can be formed with the digits 3, 1, 7, 0, 9, 5?(ii) How many of them are divisible by 10?(iii) How many of them will have zero in the tens place?

20. (a) How many 5-digit telephone numbers can be formed with digits 0, 1, 2, , 8, 9 if each numberstarts with 35 and no digit appears more than once?(b) How many different words can be formed of the letters of the word MALENKOV so that

(i) no two vowels are together,(ii) the relative positions of the vowels and consonants remain unaltered,(iii) the vowels may occupy odd places?

21. (a) How many signals can be made by hoisting 5 flags of different colours?(b) Find the number of permutations of the letters of the word

(i) INDIA (ii) ALLAHABAD (iii) CHANDIGARH (iv) COMMISSION

22. (a) There are 5 red, 4 white and 3 blue marbles in a bag. They are drawn one by one andarranged in a row. Assuming that all the 12 marbles are drawn, determine the number of differentarrangements.(b) How many 7-digit numbers can be formed using the digits 1, 2, 0, 2, 4, 2 and 4?

23. (a) (i) Find how many arrangements can be made with the letters of the wordMATHEMATICS?

(ii) In how many of them the vowels occur together?

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(b) In how many different ways can the letters of the word SALOON be arranged?(i) If the two Os must not come together;(ii) If the consonants and vowels must occupy alternate places.

24. In how many ways can 3 letters be posted in four letter boxes in a village? If all the three lettersare not posted in the same letter box, find the corresponding number of ways of posting.

25. (a) A committee of 11 members sits at a round table. In how many ways can they be seated if thePresident and the Secretary choose to sit together?(b) There are 5 gentlemen and 4 ladies to dine at a round table. In how many ways can they seatthemselves so that no two ladies are together?(c) Find the number of ways in which n things of which r are alike, can be arranged in a circularorder.

26. (a) In how many ways 6 gentlemen and 3 ladies can be seated round a table so that everygentleman way have a lady by his side.(b) The letters of the word ZENITH are written in all possible orders. How many words arepossible if all these words are written out as in a dictionary? What is the rank of the wordZENITH?(c) In how many ways can 4 persons be selected from amongst 9 persons? How many times will

a particular person be always selected?

27. (a) Find the number of diagonals that can be drawn by joining the angular points of a heptagon.(b) A committee of 4 is to be selected from amongst 5 boys and 6 girls. In how many ways canthis be done so as to include (i) exactly one girl, (ii) at least one girl?(c) There are 5 questions in question paper. In how many ways can a boy solve one or morequestions?

28. (a) Prove that from the letters of the sentence, Daddy did a deadly deed, one or more letters canbe selected in 1919 ways.(b) In how many ways can 15 things be divided into 3 groups containing 8, 4 and 3 thingsrespectively?(c) In how many ways can 18 different books be divided equally among 3 students?

29. (a) In how many ways can 52 playing cards be placed in 4 heaps of 13 cards each? In how manyways can they be dealt out to four players giving 13 cards each?(b) How many different words, each containing 2 vowels and 3 consonants can be formed with 5vowels and 17 consonants?(c) Find the number of (i) combinations (ii) permutations of four letters taken from the wordEXAMINATION.

30. (a) In how many ways can a committee of 4 be selected out of 12 persons so that a particularperson may (i) always be taken (ii) never be taken?(b) In how many ways can a team of 11 players be selected from 14 players when two of themcan play as goalkeepers only?(c) A person has got 12 friends of whom 8 are relatives. In how many ways can he invite 7 guests

such that 5 of them may be relatives?

31. (a) How many diagonals are there in a polygon of (i) 8 sides, (ii) 10 sides?(b) Sixteen clerkships are vacant in a merchants office. How many different batches of men canbe chosen out of 20 candidates? How often may any particular candidate be selected?(c) In how many ways can a committee, consisting of a chairman, secretary, treasurer and fourordinary members be chosen from eight persons?

32. (a) In how many ways can a student choose 5 courses out of 9 courses if 2 courses arecompulsory for every student?

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(b) In how many ways can we select a cricket eleven from 17 players in which 5 players canbowl? Each cricket team must include 2 bowers.

33. How many committees of 5 members each can be formed with 8 officials and 4 non-officialmembers in the following cases:(a) each consists of 3 officials and 2 non-official members;(b) each contains at least two non-official members;(c) a particular official member is never included;(d) a particular non-official member is always included?

34. How many different groups can be selected for playing tennis out of 4 ladies and 3 gentlemen,there being one lady and one gentleman on each side?

35. (a) Prove that ifn1

C3+n1

C4>4C3then n > 7.

(b) In order to pass an examination, a minimum is to be secured in each of the seven subjects. Inhow many ways can a student fail?(c) In how many ways can 10 marbles be divided between two boys so that one of them may get2 and the other 8?(d) In how many ways can 20 students be divided into four equal groups? In how many ways canthese be sent to four different schools?

(e) To go on a journey 8 persons are to be divided into 2 groups, one group to go by car and theother by train. In how many ways can this be done if there must be at least 3 persons in eachgroup?

36. A table has 7 seats, 4 being on one side facing the window and 3 being on the opposite side. Inhow many ways can 7 people be seated at the table,(a) if 2 people, X and Y, must sit on the same side;(b) X and Y must sit on opposite sides;(c) If 3 people, X, Y and Z must sit on the side facing the window.

37. Seven cards, each bearing a letter, can be arranged to spell the word DOUBLES. How manythree-letter code-words can be formed from these cards? How many of these words(a) contain the letter S; (b) do not contain the letter O;

(c) consist of a vowel between two consonants?

38. How many triangles may be formed by joining any three of the nine points when(i) no three of them are collinear; (ii) five of them are collinear.

39. (a) In how many ways can 3 ladies and 3 gentlemen be seated at a round table, so that any twoand only two of the ladies sit together?(b) Two parallel lines each have a number of distinct points marked on them. On one line thereare 2 points P and Q; on the other there are 8 points.

(i) Calculate the total number of different triangles which could be formed having 3 of the10 points as vertices.

(ii) How many of these triangles have P as a vertex?

40. (a) There are 12 points in a plane, of which 5 are collinear. Find(i) the number of triangles that can be formed with vertices at these points;(ii) the number of straight lines obtained by joining these points in pairs.

(b) 11 persons decide to spend an afternoon in two groups. A group of them decides to go to atheatre and the remaining decide to play tennis. In how many ways can the group for tennis beformed, if there must be at least four persons in each group?

41. A committee of 5 is to be formed from a group of 10 people, consisting of 4 single men, 4 singlewomen and a married couple. The committee is to consist of a chairman, who must be a singleman, 2 other men and 2 women.

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(i) Find the total number of committees possible.(ii) How many of these would include the married couple?

42. (a) A committee of 5 is to be formed from a group of 6 gentlemen and 4 ladies. In how many wayscan this be done if the committee is to include at least one lade?(b) Five balls of different colours are to be placed in three boxes of different sizes. Each box canhold all the five balls. In how many ways can we place the balls so that no box remains empty?

43. Our of 3 books on Economics, 4 books on Political Science and 5 books on Geography, howmany collections can be made, if each collection consists of(i) Exactly one book on each subject, (ii) at least one book on each subject?

44. Find the number of words which can be formed by taking two alike and two different letters fromthe word COMBINATION.

45. Find the number of ways in which the letters of the word, INTERMEDIATE, can be arrangedtaken all at a time so that the vowels are not all together.

46. (a) Prove thatn

n 1 mr r 1 r 1

k m

C C C

(b) How many five-letter words can be formed such that the letters appearing in the odd positionsare taken from the unrepeated letters of the word MATHEMATICS whereas the letters whichoccupy even places are taken from amongst the repeated letters?

47. Find the total number of ways of selecting five letters from the letters of the wordINDEPENDENT.

48. (a) Eighteen guests have to be seated, half on each side of a long table. Four particular guestsdesire to sit on one particular side and three other on other side. Determine the number of waysin which the seating arrangements can be made.(b) Given 5 different green dyes, four different blue dyes and three different red dyes, how manycombinations of dyes can be chosen taking at least one green and one blue dye?

49. (a) There are four balls of different colours and four boxes of colours, same as those of the balls.The number of ways in which the balls, one each in a box, could be placed such that a ball doesnot go to a box of its own colour is .......................(b) A student is to answer 10 out of 13 questions. How many choices has he? How many, if hemust answer the first two questions? How many, if he must answer at least 3 out of the first 5questions?

50. (a) A gentleman invites 13 guests to a dinner and places 8 of them at one table and remaining 5at the other, the tables being round. In how many ways can he arrange the guests?(b) A man has 7 relatives, 4 of them are ladies and 3 gentlemen, his wife has 7 relatives and 3 ofthem are ladies and 4 gentlemen. In how many ways can they invite a differ party of 3 ladies and3 gentlemen so that there are 3 of mans relatives and 3 of wifes relatives.

Answer key1. (a) 90 (b) 27000

2. (a) 8 (b) 323. (a) HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. (b) 304. (a) 72 (b) 120 5. (a) 90 (b) 4,21,2006. (a) 25 (b) 20 7. (a) 4096 (b) 2

3 1 = 7 ways

8. (a) [8 x 10 x 10 3] x 10 x 10 x 10 x 10 = 7970000(b) (i) 5 x 5 = 25 (ii) 5 x 1 = 5 (iii) 5 x 4 = 20

9. (a) 31 (b) (i)rP5x

n5Pr5(ii)

n5Pr 10. (b) 604800

11. (a) 6666600 (b) 60 12. (a) 756 (b) 1513. (a) 18 (b) 900 14. (a) 1024 (b) 9000

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15. (a) 2 x 18! (b) 3! X 4! or 144 16. 864017. (a) 360 (b) (i) 2(18!) (ii) 17(18!) 18. (a) 1956 (b) (i)

n3Pr(ii)

rP3x

n

3Pr3

19. (a) 4320 (b) (i) 600 (ii) 120 (iii) 12020. (a) 336 (b) (i) 14400 (ii) 720 (iii) 288021. (a) 325 (b) (i) 60 (ii) 7560 (iii) 907200 (iv) 226800 22. (a) 27720 (b) 36023. (a) (i) 4989600 (ii) 120960 (b) (i) 240 (ii) 36 24. 64, 60

25. (a) 2(9!) (b) 2880 (c) x = n 1 !

r!

26. (a) 1440 (b) 616 (c) 5627. (a) 14 (b) 325 (c) 31

28. (b) 225225 (c)

3

18 !

6!

29. (a) (i)

4

52!

4! 13 ! (ii)

4

52!

13! (b) 816000 (c) 2454

30. (a) (i) 165 (ii) 330 (b) 132 (c) 33631. (a) (i) 20 (ii) 35 (b) 4845; 3876 (c) 168032. (a) 35 (b) 2200 33. (a) 336 (b) 456 (c) 462 (d) 330

34. 36 35. (b) 127 (c) 90 (d)

420!

4! 5! (e) 182

36. (a) 2160 (b) 2880 (c) 576Solution of triangle

90. In ABC

(i) If a = 3, b = 4, sin A = 3/4, find b.

(ii) If A = 30, C = 90, c = 7 3 , find a, b.

91. In ABC

(i) a = 1, c = 2, B = 60find b.

(ii) If a = 26, b = 30, cos C = 63/65, find c

92. (i) If two sides and included angle of a triangle are 3 + 3 , 3 3 and 60respectively then findits third side.

(ii) IfA 5 B 20

tan ,tan ,2 6 2 37

then showC 2

tan2 5

and a, b, c are in A.P.

93. (i) If a = 5, b = 4, cos(A B) =31

32, then show that c = 6.

(ii) If the sides of a triangle in the ratio 2 : 6: 3 + 1, show that its angles are 45, 60, 75.

94. If in a triangle, the cotangents of half angles are in the ratio 1 : 4 : 15, show that the largest angle

in that triangle is 120.

95. (i) If a : b : c = 7 : 8 : 9, then prove that cos A : cos B : cos C = 14 : 11 : 16.(ii) If b + c : c + a : a + b = 11:12:13, then prove that cos A : cos B : cos C = 7:19:25.

96. If the angles of a triangle are x2+ x + 1, 2x + 1 and x

2 1, where x > 1, prove that the greatest

angle is 120.

97. IfA B C

cot : cot : cot2 2 2

= 3 : 5 : 7, then show that a : b : c = 6 : 5 : 4.

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98. Show that:2 2 2 2 2 2

asin(B C) bsin(C A) c sin(A B) 1

2Rb c c a a b

99. In ABC if2 2 2 2 2 2

2 2

2 2 2

b c c a b asin A sin B ,

a b k

find k.

100. Show thata b c A B

tan .tan

a b c 2 2

.

101. If a, b, c are in A.P., show thatA C

3 tan tan 12 2

102. IfA 5 C 2

tan , tan2 6 2 5

, then show that a, b, c are in A.P.

103. If a 2 2C A 3b

cos c cos2 2 2

, prove that a, b, c are in A.P.

104. If the length of the side of an equilateral triangle is 10cm, find its circumradius.

105. If a = 3, b = 4, c = 5, find its circumradius

106. If a = 5, b = 12, c = 13, find its circumradius

107. If the sides of a right angled triangle are in A.P, find the ratio of its sides Find the circum diameter

of the triangle whose sides are 61, 60, 11cms.

108. If2 2

2 2

a b sinC,

sin(A B)a b

prove that ABC is a right angled triangle.

109. In ABC, if a = (b + c)costhen show that sin=2 bc A

cos .b c 2

110. In ABC, D is the midpoint of BC. If AD is perpendicular to AC, show that cos A.cos C =

2 22 c a3ac

111. If 1 1 3a c b c a b c

then prove that C = 60.

112. Show that:2 2

2 2

1 cos(A B)cosC a b

1 cos(A C)cosB a c

113. Prove that

(i) (b + c) sinA B C

a cos2 2

(ii) (b c) cos

A B Casin

2 2

.

114. Show that (b a) cos C + c(cos B cos A) = c sinA B A B

cosec2 2

.

115. Show that a3cos(B C) + b

3cos(C A) + c

3cos(A B) = 3abc.

116. Show that

(i)2a

2(cotB cotC)

(ii)

2 2(a b )sin AsinB

2 sin(A B)

(iii) b2sin 2C + c

2sin 2B = 2bc sin A = 4

(iv) a cos A + b cos B + c cos C = 4R sin A sin B sin C =2

R

117. Show that

(i) (b2+ c

2 a

2)tan A = (c

2+ a

2 b

2) tan B = (a

2+ b

2 c

2)tan C = 4

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(ii) cot A + cot B + cot C =2 2 2a b c

4

118. If tan A, tan B, tan C are in H.P, prove that a2, b

2, c

2are in A.P.

Show that

(a) a2cot A + b

2cot B + c

2cot C =

abc

R (b) a cos

2A

2+ b cos

2 B

2+ c cos

2 Cs

2 R

119. (i) If c = 60, prove thata b

1b c c a

(ii) If c = 60, prove that2 2 2 2

b a0

c a c b

(iii) If b + c = 3a, prove thatB C

cot cot 22 2

120. Ifa

sinb c

, then show that2 bc A

cos cosb c 2

121. If a = (b c) sec , prove that2 bc A

tan sinb c 2

122. Show thatA B C

cot ,cot ,cot2 2 2

are in A.P. iff a, b, c are in A.P..

123. Prove that 2 2 2A B Csin , sin , sin2 2 2

are in H.P iff a, b, c are in H.P.

124. If a cos A = b cos B, show that the triangle is either isosceles or right angled.

125. Ifa b c

cosA cosB cosC , prove that ABC is an equilateral triangle.

126. IfA b c

cot2 a

, then find angle B.

127. If cos2A + cos

2B + cos

2C = 1 show that the triangle is right angled.

128. If 8R2= a

2+ b

2+ c

2prove that the triangle is right angled.

129. If the perimeter of a triangle is 12 cm and its inradius is 1 cm, then find its area.

130. In an equilateral triangle, find r/R.

In ABC prove the following:131. (i) rr1 = (s b)(s c)

(ii) r1(s a) = r2(s b) = r3(s c) = (iii) (r2+ r3)(r2 r) = a

2

132. (i) 21

2 3

rr Atan

r r 2 (ii) r r1cot

A

2

133. If r1, r2, r3are in H.P, prove that a, b, c are in A.P.

134. If s = 12, A = 90, find the value of r.

135. The area of a triangle is 90 sq. cm and its perimeter is 36 cm find its inradius.

136. Find the value of1 1 1

bc ca ab interms of r, R.

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137. Show that 31 2r rr r r r

sa b c

= r1+ r2+ r3.

138. Show that r + r1+ r2 r3= 4R cos C.

139. In ABC, prove that 1 21 2

1 2

4R r r r r

r r

.

140. In ABC if 1 1

2 3

r r1 1 2

r r

, prove that the triangle is right angled.

141. In ABC prove that: (r1 r)(r2 r)(r3 r) = 4Rr2.

142. In ABC prove that: (r1+ r2)(r2+ r3)(r3+ r1) = 4Rs2.

143. In ABC prove that:3 2 2

1 2 3

1 1 1 1 1 1 abc 4R

r r r r r r r s

In ABC prove the following:

144. (i)

2 2 2

2 2 2 2 21 2 3

1 1 1 1 a b c

r r r r

(ii)1 2 3 2 3 1 3 1 2r r r r r r r r r r r r

bc ca ab

145. (i) (r1+ r2) tan 3C C

(r r)cot c2 2

(ii)1

B C(r r )tan 0

2

146. (i) (r1 + r2)sec2 C

2= (r2+ r3) sec

2A

2= (r3+ r1) sec

2 B

2 (ii) r1r2r3= r

3cot

2 2 2A B C

.cot .cot2 2 2

(iii)A B C

4Rr cos cos cos2 2 2

(iv) 2A B C

r cot .cot .cot2 2 2

147. (i) cos A + cos B + cos C = 1 +r

R (ii) a cot A + b cot B + c cot C = 2(R + r)

(iii) 2 2 2A B C r

cos cos cos 22 2 2 2R

(iv) 2 2 2A B C r

sin sin sin 12 2 2 2R

(v)acos A bcosB ccosC r

a b c R

148. If (a b)(s c) = (b c)(s a), prove that r1, r2, r3are in A.P.

149. If rr2= r1r3, then show that B = 90.

150. If (r2 r1)(r3 r1) = 2r2 r3, show that A = 90.

151. Prove that: r + r3+ r1 r2= 4R cos B

152. Prove that 31 2rr r 1 1

bc ca ab r 2R

153. If p1, p2, p3are the lengths of the altitudes from the vertices of ABC to the opposite sides then,prove that

(i)1 2 3

1 1 1 1

p p p r (ii)

1 2 3 3

1 1 1 1

p p p r (iii) p1p2p3=

2 3

3

(abc) 8

abc8R

Answer key

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90.(ii) a =7

32

, b = 10.5 91. (i) b = 3, (ii) c = 55.56

92.(i) c = 32 99.k =abc

4 104.

10

3 105.

5

2

106.13

2

107.61

2

129. 6 130.1

2

134.r = 12 135.r = 5 136.abc

4s

Inverse

i) Evaluate tan 11 5

cos2 3

. ii) Solve 4sin-1

x + cos-1

x =

.

iii) Evaluate sin-1

( sin6

7

). iv) Evaluate tan 11

2tan45

.

v) Prove that sec2(tan

-12) + cosec

2( cot

-13) = 15.

vi) Show that tan-1

x + cot-1

(1 +x) = tan-1

(x2+ x+ 1) i s an identi ty for x 0.

vii) If u = cot-1 1cos tan cos , prove that sinu = 2tan

2

.

viii ) Show that tan-1 1

2 2

x xsin

aa x

, a > 0.

Answer key

(i)3 5

2

(ii) x = 1/2

(iii) /7 (iv) 9 - 4 5