Magnus Academy (Maths Specialist): March 2021

Tuesday, 30 March 2021

PERMUTATION (A- Z)(C)

EXERCISE -A

1) ⁿP₂ = 12 Then n =
a) 2 b) 3 c) 4 d) 5

2) ⁿP₅ = 20. ⁿP₃ then find n
a) 7 b) 6 c) 9 d) 8

3) ⁿP₅ : ⁿP₃= 2 :1 then n=
a) 4 b) 5 c) 6 d) 7

4) ⁿP₃ : ⁿP₂=3:1 then n=
a) 4 b) 5 c) 6 d) 7)

5) ᵐ⁺ⁿP₂=56 and ᵐ⁻ⁿP₂=12 find m,n

a) 6,4 b) 6,6 c) 4,6 d) 6,2

6) ¹⁰Pᵣ₋₁: ¹¹Pᵣ₋₂=30:11 find r
a) 7 b) 8 c) 9 d) 11

7) ⁵⁶Pᵣ₊₆ : ⁵⁴Pᵣ₊₃ = 30800 :1 find r
a) 49 b) 41 c) 51 d) 59

8) ⁿP₁₃ : ⁿ⁺¹P₁₂ = 3:4find n
a) 14 b) 16 c) 15 d) 17

9) ²ⁿ⁺¹Pₙ₋₁ : ²ⁿ⁻¹Pₙ= 3:5
a) 4 b) 6 c) 8 d) 10

10) ⁿ⁺ʳP₂ = 110 and ⁿ⁻ʳP₂=270 find n,r
a) 8,4 b) 8,3 c) 4,8 d) 8,5

11) ⁿ⁺¹⁺ʳP₂ =72, ⁿ⁻ʳP₂=12 find n , r
a) 6,6 b) 6,4 c) 6,2 d) 6,1

12) Find H.C.F of 3!,5!,7!
a) 3 b) 6 c) 2 d) 1

13) Find L. C. M of 3!,5!,7!
a) 6! b) 8! c) 7! d) none

14) Compute 8!/((4!)(3!))
a) 56 b) 6 c) 5 d) none)

15) Convert in to factorial 6, 7, 8, 9.

a) ⁹P₄ b) ⁹P₅ c) 9! d) none

16) (n+1)!=12(n-1)! Find n
a) n-1 b) n+2 c) n d) none

17) 1/9! +1/10! =n/11! Find n
a) 111 b) 121 c) 131 d) none

18) What is the largest integer n such that 33! is Divisible by 2ⁿ
a) 32 b) 64 c) 31 d) 29

19) (2+3)!=2!+3!
A) true B) False

20) (2x3)!=(2!)x(3!)
A) True B) False

21) Evaluate. n!/{(r!).(n-r)! }When n=15 and r=12.
a) 455 b) 545 c) 554 d) none

22) (n+2)!=60(n-1) ! find n
a) 3 b) 4 c) 5 d) 6

EXERCISE - B

1) There are 10 trains moving between Calcutta and Delhi. In how many ways can a man go from Calcutta to Delhi and return by a different one ?
a) 100 b) 90 c) 120 d) none

2) There are 26 stations on a railway line. How many different kinds of tickets of class II must be printed in order that a passenger may go from any one station to another by purchasing a ticket.
a) 65. b) 240. c) 650. d) 1300

3) There are four bus lines between A & B and there are 3 bus lines between B & C. In how many ways can a man take round trips by bus from A to C by way of B, if he does not want to use a bus line more than once ?
A) 36. B) 24. C) 72. D) 120 E) none

4) In s class after every student had sent greeting cards to the other student, it was found that 1640 cards were exchanged. Find the number of students in the class.
A) 40. B) 39. C) 41. D) 42

*** You are given the letters of the word “MONDAY”. Find the number of arrangements in the following cases:

5) Without any restriction.
a) 120 b) 144 c) 720 d) 360

6) Words beginning with M.
a) 120 b) 240 c) 360 d) 720

7) Words beginning with Y.
a) 24 b) 120 c) 96 d) 144

8) Words beginning with M & ending with Y.
a) 24 b) 96 c) 144 d) 240

9) Words beginning with M & not ending with Y.
a) 24 b) 96 c) 360 d)144

10) M & Y are at two extremes.
a) 24 b)48 c)96 d)144

11) Vowels are together.
a) 120 b) 144 c) 249 d) 360

12) Vowels are never together.
a) 144 b) 240 c) 360 d) 480

13) Vowels occupy odd places.
a) 480 b) 240 c) 120 d) 144

14) Vowels occupy even places.
a) 240 b) 156 c) 144 d) 360

15) Relative position of the vowels and consonants are to be kept untouched
a) 96 b) 48 c) 56 d) 144

16) Constants are together.
a) 96 b) 120 c) 144 d) 240

17) How many words can be formed by taking four letters at a time.
a) 240 b) 120 c) 360 d) 700

18) In how many of these (given in previous question) M is always included
a) 240 b) 120 c) 360 d)144

19) In how many of these (given on Q. No. xiii) M is excluded.
a) 240 b) 360 c) 120 d) 144

20) Number of rearrangement of word Monday
a) 720 b) 719 c) 360 d) 717

21) In how many words MO will be together.
a) 240 b) 420 c) 360 d) none

22) Find the number of words may formed by the word FATHER. Also find the number of words may formed when words begin with A and end with R ?
a) 720, 48. b) 720, 24
c) 360, 48. c) 360, 24

*** You are given a word DELHI
23) How many arrangements can be formed with the letter DELHI.
a) 120. b) 24. c) 96. d) 48

24) How many of them will begin with D
a) 96. b) 24. c) 48. d) 100

25) How many do not begin with D
a) 48. b) 24. c) 96. d) 120

26) In how many words LH will be together ?
a) 12. b) 24. c) 48. d) 96

27) How many words can be formed of the letters in the word COSTING, the vowel being not separated ?
a) 144. b) 1440. c) 1280. d) 2880

28) In how many ways can the letters of the word LAUGHTER be arranged so that the vowel may never be separated ?
A) 4320. B) 2480. C) 1440.D) 2880

29) How many words can be formed of the letters in the word ARTICLE so that the vowels may occupy only.

A) the even positions B) the odd positions

a) 144, 576. b) 576, 144 c) 280, 144. d) 288, 144

30) Find how many words can be formed of the letters in the word FAILURE is the four vowels.

A) always coming together B) never coming together

a) 576, 576. b) 576, 4464 c) 5764, 4464 d) none

31) If the letter of the word JUXTAPOSED be arranged in all possible different ways, in how many of these will the vowels occur together ?

A) 60480 B) 30240 C) 120960 D) N

32) In how many ways can the letters of the word MOBILE be arranged so that the consonants always occupy the odd places?

A) 24. B) 72. C) 36. D) 144

33) In how many different ways can the letters of the word VALEDICTORY be arranged so that the vowels are

A) never separated B) not together

a) 997680, 38949120 b) 986578, 34569810 c) 997680, 34597870. D) none

34) In how many different ways can the letters of the word STRANGE be arranged so that the vowels are A) never separated B) not together

a) 3600, 1440. b) 1800, 720 c) 1440, 3600. d) 720, 1800

35) How many arrangements of the letters of the word COMRADE can be made

A) if the vowels are never separated

B) if the vowels are to occupy only odd places.

a) 604800, 7200 b) 604800, 14400 c) 302400, 14400. d) none

EXERCISE - C

*** In how many ways can 8 sweets of different sizes be distributed among 8 boys of different ages, so that

1) Largest sweet goes to the youngest boy ?
a) 2520. b) 1240. c) 5040. d) none

2) Smallest sweet goes to the older boy ?
a) 2520. b) 1240. c) 5040. d) none

3) Largest sweet goes to the youngest and smallest sweet goes to the older .
a) 2520. b) 1240. c) 5040. d) none

4) In how many ways can 8 examination papers be arranged in a row, so that the best and worst papers may never come together ?
A) 15120 B) 30240 C)60480 D) n

5) The Number of ways in which 16 different books can be arranged on a shelf so that two particular books shall not be together.
A) 14.15! B) 15.14!
C) 14.14! D) 15. 15!

6) Six papers are set in an examination, of which two are mathematical. In how many different orders can the papers be arranged so that
A) the two mathematical papers are together.
b) the two mathematical papers are not consecutive ?
a) 480, 240 b) 240, 120 c) 360, 120 d) 120, 76

7) In how many ways can 6 plastics beads of different colours be arranged so that the blue and green beads are never placed together?
A) 240. B) 120. C) 360. D) 480

8) In how many ways can 3 boys and 5 girls be arranged in row so that no 2 boys are together?
A) 14400 B) 604800 C) 2880 D) 28800

9) In how many ways can 5 boys and 4 girls be arranged in a row so that the boys and the girls stand alternatively ?
A) 28800 B) 14400 C) 2880 D) 60480

10) In how many ways can 5 boys and 5 girls be arranged in a row so that they stand alternatively.
A) 14400 B) 2880 C) 604800 D) 28800

11) In how many ways 6 boys and 4 girls be arranged in a row so that no girls are together ?
A) 604800 B) 288000 C) 144000 D) 720000

12) A dinner is arranged for 11 guests in which there are 4 children, 1 old man and 6 adults. The 4 children wish to occupy 4 corner seats and the old man refuses to have a child on his either side. In how many can all guests be arranged ?
A) 28800 B) 43200 C) 86400 D) 14400

EXERCISE - D

*** You are given the letters of the word BALLOON. Find the arrangement.

1) Without any restriction
A) 960. B) 1060. C) 1160. D) 1260

2) Two LL will always come together.
a) 720 b) 360 c) 180 d) 120

3) Two LL and two OO will always come together.
a) 360 b) 480 c) 180 d) 120

4) All the O's & the L's will come together.
a) 120 b) 184 c) 144 d) 168

5) Vowels are together
a) 180 b) 120 c) 360 d) 240

6) B & N are together
a) 180 b) 120 c) 360 d) 240

7) B & N are never together.
a) 900 b) 980 c) 160 d) 720

8) B,N & O's are together.
a) 288 b) 142 c) 144 d) 368

9) Two OO's together
a) 288 b) 142 c) 144 d) none

*** Find the number of arrangements that can be made out of the letters of the following words:

10) CALCUTTA
i) 5040 ii) 4050
iii) 6050 iv) 2530

11) ACCOUNTANT
i) 226980 ii) 228600
iii) 226800 iv) 365980

12) CONTACT
i) 1620 ii) 1560
iii) 1260 iv) 3540

13) ATLANTIC
i) 18000 ii) 10080
iii) 18020 iv) 15950

14) MATHEMATICS
i) 4989600 ii) 4998960
iii) 5987590 iv) 4545450

15) INSTITUTION
i) 554499 ii) 445588
iii) 554400 iv) none

16) STATISTICS
i) 5544000 ii) 4978960
iii) 4589600 iv) 4989600

17) ENGINEERING
a) 272700 b) 277200
c) 288750 e) 288770

18) MISSISSIPPI
i) 34650 ii) 35640
iii) 45630 iv) 56340

*** All different words formed by the letters of the word BHARAT
19) How many different words can be formed with the letters of the word BHARAT?
i) 720. ii) 360. iii) 180. iv) 240

20) In how many of these B and H are never together?
i) 360. ii) 180. iii) 240. iv) 120

21) How many of these begin with B and end with T ?
i) 15. ii) 12. iii) 18. iv) 21

22) How many different words can be formed with the letters of the word CAPTAIN? In how many of these C and T are never together?
i) 2520, 1600. ii) 2520, 1890
iii) 2520, 1800. iv) 3250, 1800

23) In How many ways can the letters of the word ALGEBRA be arranged? In how many of these arrangements will the two A's not come together?
i) 2520, 1890 ii) 2520, 1800
iii) 3520, 1800. iv) none

24) Find how many different words can be formed from the letters of the word PEOPLE in which two P's would not remain side by side.
i) 100. ii) 125. iii) 160. iv) 120

25) In how many ways can the letters of the word CONSTITUTION be arranged? How many of these will have the letter N both at the beginning and at the end?
i) 9979200, 151200
ii) 9989920, 152150
iii) 9979000, 151000. iv) none

26) In how many different ways can the letters of the word VIDYAPITH be arranged? How many arrangements begin with V but do not end with H ?
I) 181000, 176000
ii) 181400, 17640
iii) 181440, 17640
iv) 182000, 18600

27) The number of ways in which the letters of the word ARRANGE can be arranged that the two R's do not come together is:
i) 900 ii) 1800. iii) 450. c) 720

28) In how many ways can be letters of the word EXAMINATION be arranged so that all the A's always come together?
i) 907200. ii) 120960
iii) 962000. iv) 288000

29) In how many ways can the letters of the word AGARTALA be arranged?
i) 1600 ii) 1800 iii) 1980 iv) 1680

30) Taking data from Q. 29, in how many of these will the 4 A's
A) come together B) not together
i) 120, 1560. ii) 360, 1620
iii) 300, 1500. iv) 220,1380

31) In how many ways can 5 dots (.) and 3 crosses (x) be arranged in a row ?
i) 36 ii) 46. iii) 66. iv) 56

32) A library has 5 copies of one book, 4 copies of each of the two books, 6 copies of each of the 3 books and single copies of eight books. In how many ways can all the books be arranged ?
i) 39!/{5! 4! 4! 6!}. ii) 39!/{5! 4! 6!}.
iii) none. iv) both of the above

33) In how many ways the letters of the word MULTIPLE can be rearranged without changing the order of the vowels?
A) 30239. B) 3359
C) 33590. D) 32590

34) In how many ways can the letters of the word PARNECIOUS be arranged without changing the order of the vowels?
A) 33590. B) 30239
C) 14400. D) 28800

35) Find the number of different arrangements that can be made of the seven prismatic colours (Violet, Indigo, Blue, Green, Yellow, Orange and Red) so that the Violet and Red shall never come together?
A) 5040. B) 1440. C) 3600. D) N

36) There are six students of whom 2 are Indians, 2 Americans and the remaining are Russians. They have to stand in a line so that the two Indians are together, the 2 Americans are together, and also the 2 Russians are together. Find the number of ways in which they can do so ?
A) 48. B) 84. C) 8. D) none

37) How many different arrangements can be made out of the letters in the expression x²y⁴z³ when written at full length ?
A) 7200. B) 3600. C) 1260 D)1800

EXERCISE - E

*** With the digits 1,2,3,4,5,6 Find
1) 6 digits numbers
i) 120 ii) 320 iii) 520 iv) 720 v) N

2) 5-digit numbers
i) i) 120 ii) 320 iii) 520 iv) 720 v) N

3) 4-digits numbers
i) 120 ii) 240 iii) 360 iv) 480 v) N

4) 3-digit numbers
i) 120 ii) 240 iii) 360 iv) 480 v) N

5) 2-digit numbers
i) 30 ii) 60 iii) 120 iv) 240 v) N

6) 1-digit numbers
i) 6 ii) 12. iii) 24 iv) 30 v) N

*** With the digits 1, 2, 3, 4, 5 find the numbers.
7) Greater than 30000
i) 96 ii) 120 iii) 144 iv) 1440 v) N

8) Greater than 2000
i) 216 ii) 219 iii) 312 iv) 240 v) N

9) 4-digit numbers greater than 2000
i) 72 ii) 96 iii) 120 iv) 1440 v) N

10) Greater than than 400
i) 24 ii) 120 iii) 144 iv) 1440 v) N

11) 3-digit greater than 400
i) 24 ii) 36 iii) 48 iv) 60 v) N

**** You have the number 2,3,4,5,6 find the following:

12) 5 digit number
i) 120. B) 210. C) 720. D) none

13) 5 digit number greater than 30000
i) 24. ii) 42. iii) 68 iv) 96. v) N

14) Greater than 3000
i) 96. ii) 216. iii) 2160. iv) n

15) 4 digit number greater than 3000.
i) 96 ii) 216. iii) 144. iv) 210 v) N

16) 5-digit number divisible by 5
i) 24. ii) 42. iii) 120 iv) 210 V) N

17) The numbers not divisible by 5
i) 520. ii) 420. iii) 1200 iv) 96 v) n

18) 5-digit numbers not divisible
by 5
i) 1200 ii) 1440 iii) 72 iv) 720 v) n

19) Between 2000 and 4000
i) 24 ii) 48 iii) 96 iv) 144 v) n

20) Greater than 23000
i) 69 ii) 96 iii) 102 iv) 120 v) N

21) Divisible by 4
i) 48 ii) 72. iii) 96 iv) 102 v) N

22) Not divisible by 4
i) 36 ii) 720 iii) 1440 iv) 720 v) N

23) Greater than 3200
i) 321 ii) 498 iii) 498 iv) 984 v) N

24) Less than 3000
i) 60 ii) 600 iii) 6000 iv) 72 v) N

25) Less than 2400
i) 321 ii) 498 iii) 498 iv) 984 v) N

26) How many 5 digits numbers can be formed with the digits 4,5, 6, 7, 8, and 9
A) 120. B) 720. C) 144. D) 480

27) Find the number of numbers Greater than 3000 that can be formed with the digits 2,3,4,5,6
A) 216. B) 219. C) 312. D) 240

28) How many numbers Greater than 5000 can be formed with the digits 2,3,4,5,6 when no digits is repeated ?
A) 216 b) 212 c) 182 d) 180 e) n

29) find the number of numbers Greater than 3000 that can be formed from the digits 1,2,3,5,7
A) 212 b) 192. C) 216 d) 180 e)N

30) Find the number of numbers Greater than 6000 that can be formed with the digits 1,4,6,8,9.
A) 212 b) 180 c) 216 d) 192 e) N

31) How many 6 digit number can be formed with 3,4,5,6,7,8? How many of them
a) Divisible by 5 b) not Divisible by 5
i) 720,120,600 ii) 720,600,120
iii) 780,600,180 iv) 780,180,600

32) How many numbers between 4000 and 5000 can be formed with the digits 2,3,4,5,6,7
A) 40. B) 50. C) 60. D) 80

33) How many numbers Greater than 7000 can be formed from the digits 1,3,5,7,8,9

A) 129 b) 192 c) 178 d) 287 e) N

34) How many even numbers Greater than 300 can be formed with the digits 1,2,3,4,5

A) 121. B) 111 c) 222 d) 124 e) N

35) How many numbers can be formed with the digits 3,4,5,6,7? How many of them are greater than 5600 ?

A) 180. B) 120 c) 160 d) 360 e) N

EXERCISE - F

*** You have the number 0,1,2,3,4 find the following :
1) 5 digit number
i) 48 ii) 96 iii) 24 iv) 120 v) none

2) 3 digit number.

i) i) 48 ii) 96 iii) 24 iv) 120 v) none

3) Number between 400 to 4000
i) 48 ii) 96 iii) 24 iv) 120 v) none

4) Greater than 3000
i) 48 ii) 96 iii) 24 iv) 120 v) none

5) Less than 1000 Divisible by 5
i) 48 ii) 96 iii) 24 iv) 120 v) none

6) Odd numbers
i) 48 ii) 96 iii) 24 iv) 120 v) none

7) Even numbers
i) 48 ii) 96 iii) 24 iv) 120 v) none

8) Divisible by 4
i) 48 ii) 96 iii) 24 iv) 120 v) none

9) Not divisible by 4
i) 48 ii) 96 iii) 24 iv) 120 v) none

10) Even numbers greater than 3000.
i) 48 ii) 96 iii) 24 iv) 120 v) none

11) Less than 3000.
i) 48 ii) 96 iii) 24 iv) 120 v) none

12) Less than 4000 more than 40.
i) 48 ii) 96 iii) 24 iv) 120 v) none

13) How many numbers of four different digits each Greater than 400 can be formed from the digits 1, 3, 4, 6, 7 & 0 ?
A) 160 b) 180 c) 150 d) 240 e) N

14) How many 4 digits numbers can be formed with the digits 0,1, 2, 3, 4,5 and 6
A) 360 b) 480 c) 600 d) 720 e) N

15) How many odd numbers of 6 digits can be formed with the digits 0, 2,3,4,5,6.
A) 144 b) 212 c) 288 d) 480 e) N

16) How many numbers between 100 and 1000 can be formed with the digits 3,4,5,0,7,9
A) 120 b) 100 c) 150 d)180 e) n

17) How many numbers Greater than a million can be formed with the digits 2,3,0,3,4,2,3
A) 240 b) 180 c) 540 d) 360 e) n

18) How many numbers less than 1000 and divisible by 5 can be formed with the digits 0,1,2,3,4,5.. and 9,
A) 145 b) 165 c) 154 d) 194 e) N

19) How many numbers between 300 and 3000 can be formed with the digits 0,1,2,3,4,5
A) 180 b) 120 c) 160 d) 240 e) N

20) How many numbers of 4 different digits Greater than 5000 can be formed from the digits 1,4,5,7,8 and 0 ?
A) 180 b) 360 c) 240 d) 120 e) N

21) How many numbers lying between 100 and 1000 can be formed with the digits 5,0,6,7,9. How many of these are odd?

A) 48,81 B) 48, 18. C) 38, 18
D) 129, 232. E) none of these

EXERCISE - G

1) In how many ways 8 persons can be seated at a round table?

A) 5040 B)40320 C)2020 D)2520

2) In how many ways can 8 persons can be seated at a round table so that 2 particular persons can be together?

A) 180 B)240 C) 360 D) none

3) Find the no of ways in which 5 beads can be arranged to form a necklace.

A) 12 B) 24 C) 36 D) 48

4) In how many ways can 4 MBA & 4 MCA be seated at the round table so that 2 MBA students are adjacent.

A) 12 B) 24 C) 96 D)144

5) In how many ways can 7 Englishman & 6 Indians be seated at the round table so that 2 indians are together.

A) 2488000 B)3628000 C) 3628800 D) none

6) In how many ways can 5 Gentleman & 4 ladies be seated at the round table so that no 2 ladies are together.

A) 14400 B)28800 C) 28880 D)2880

7) 20 persons are invited to a party. In how many ways can they and host be seated at a circular table? In how many of these 2 particular persons be seated side of the host ?

A) (18!), 2(20!) B)(18!), (20!) C) (20!),(18!), D) (20!) (18!)

8) In how many ways can 8 persons form a ring ?

A) 5040 B) 40320 C)1220 D)2880

9) In how many ways can 8 persons can be seated at a round table, with respect to the table ?

A)40320 B) 5040 C) 720 D) none

10) In how many ways , as given in the previous question, do 3 perticular persons sit side by side?

A) 4320 B) 5040 C)720 D)1400

11) In how many ways can 6 ladies and 6 gents be arranged at a round table, if the two particular ladies Miss X and Miss Y refuses to sit next to Mr. Z, all men being separated?

A) 1278 B)1730 C)1729 D)1687

12) In how many ways can 8 stones of different colours be arranged on a ring? In how many of these arrangements red and yellow beads being separated?

A) 2520,900 B) 2520,1800 C) 1800, 2520 D) 1800, 1260

*** A round table conference is to be held for a committee of 7 persons which includes President and Secretary. Find the no of ways the committee can be seated so that

13) The president and Secretary can sit together.

A) 120 B) 240 C) 360 D) 480

14) The secretary sits on the right side of the President

A) 120 B) 240 C) 360 D) 480

15) The President and the secretary do not sit together.

A) 120 B) 240 C) 360 D) 480

16) In how many ways can 5 Gentleman and 5 ladies be seated at a round table so that no 2 ladies are together? What is the number if there are 4 ladies instead of 5 ?

A) 1440,1440 B) 2880,1440 C)1440,2880 D) 2880,2880

17) Find the no of ways in which 8 different flowers can be strung to form a garland of which 4 particular flowers are never separated.

A) 288 B) 144 C) 122 D) 140

18) In how many ways can 7 persons be seated at around table so that all shall not have the same neighbour in any two arrangements.

A) 720 B) 240 C) 360 D) 120

EXERCISE - H

Continue..........

Mg. A. R-1

1) There are 10 trains playing between Kolkata and Delhi. In how many ways can a man go from Kolkata to Delhi and return by different train?

A) 90. B) 120 C) 720 D) 210

2) There are 26 stations on a certain railway line. How many different kinds of tickets of class II must be printed in order that a passenger may go from one station to any other by other by purchasing a ticket.

A) 600 B) 650. C) 660 D) 720

3) Five letters are written and five envelopes directed; in how many ways can the letters be put in the envelopes?

A) 120. B) 72 C) 420 D) 210

4) How many different numbers of 5 digit can be formed with the digits 1, 2, 3, 4, 5, 6, none of the digits being repeated in any of the numbers so formed?

A) 120 B) 216 C) 600 D) 720.

5) Find the total number of numbers greater than 2000 that can be formed with the digits 1, 2, 3, 4, 5, no digit being repeated in any number.

A) 120 B) 216. C) 600 D) 720

6) How many numbers greater than 4000 can be formed with the digits 2, 3, 4, 5, 6 when no digit repeated.

A) 120 B) 192. C) 210 D) 212

7) Find the total number of numbers greater than 3000 that can be formed with the digits 1, 2, 3, 4, 5, no digit repeated in any number.

A) 120 B) 192. C) 210 D) 212

8) How many words can be formed using all the letters of the word MYSORE?

A) 210 B) 420 C) 640 D) 720.

** In how many different ways can the letters of the word

9) MONDAY be arranged?

A) 120 B) 96 C) 720. D) 90

10) How many of these are arrangements begin with M?

A) 120. B) 96 C) 720 D) 90

11) How many begin with M and do not end with N?

A) 120 B) 96. C) 720 D) 90

** Find the number of words that can be formed by considering all possible permutations of the letters of the word

12) FATHER

A) 120 B) 96 C) 720. D) 90

13) How many of these words begin with A and end with R?

A) 120 B) 96 C) 720 D) 24.

** How many arrangement can be formed

14) with the letters of the word DELHI ?

A) 120. B) 24 C) 96 D) 48

15) How many of them will begin with D?

A) 120 B) 24. C) 96 D) 48

16) How many do not begin with D?

A) 120 B) 24 C) 96. D) 48

17) In how many words LH will be together ?

A) 120 B) 24 C) 96 D) 48.

18) How many words can be formed with the letters in the word COSTING, the vowels being not separated ?

A) 1440. B) 1400 C) 1340 D) 1210

19) In how many ways can the letters of the word LAUGHTER be arranged so that the vowels may never be separated ?

A) 1440 B) 3440 C) 4320. D) none

20) In how many ways can 8 sweets of different sizes be distributed among 8 boys of different ages, so that the largest sweet always goes to the youngest boy? (assume that each boy gets a sweet)

A) 4080 B) 5000 C) 5040. D) n

** How many words can be formed with the letters in the word ARTICLE, so that vowels may occupy only

21) The even positions.

A) 144. B) 576 C) 720 D) 800

22) the odd positions?

A) 144 B) 576. C) 720 D) 800

** Find how many words can be formed with the letters in the word FAILURE, if the four vowels

23) always coming together.

A) 576. B) 4464 C) 10080 D) n

24) never coming together.

A) 576 B) 4464. C) 10080 D) n

25) If the letters of the word JUXTAPOSED be arranged in all possible different ways, in how many of these will be vowels occur together?

A) 12000 B) 120000 C) 120960. D) 130000

26) In how many ways can the letters of the word MOBILE be arranged so that the consonants always occupy the odd places?

A) 36. B) 72 C) 144 D) 288

Mg. A- R .2

1) In how many different orders can 8 examination papers be arranged in a row, so that the best and the worst papers may never come together ?

A) 240 B) 3240 C) 300 D)30240.

2) Find the number of ways in which 16 different books can be arranged on a shelf so that two particular books shall not be together is...

A) 14! B) 15! C) 14. 15! D) none

** Six paper are set in an examination, of which tow are mathematical. In how many different orders can the papers be arranged so that

3) the two mathematical papers are together.

A) 240. B) 480 C) 720 D) 1440

4) the two mathematical papers are not consecutive?

A) 240 B) 480. C) 720 D) 1440

5) In how many ways can 6 plastic beads of different colours be arranged so that the blue and green beads are never placed together?

A) 240 B) 480. C) 720 D) 1440

6) How many four-digits numbers can be formed with 1, 2, 3, 4,5, 6, 0, none of these digits occuring more than once in each number?

A) 60. B) 480 C) 720 D) 1440

7) In how many ways can 6 boys and 4 girls be arranged in a row that so that no two girls are together ?

A) 240. B) 480 C) 720 D)604800

8) In how many ways can 5 girls and 3 boys stand in a row so that there no two boys are together?

A) 240. B) 480 C) 720 D) 14400

** Find the number of arrangements that can be made out of the letters of the following words:

9) CALCUTTA

A) 5040. B) 226800 C) 1260 D) 10080

10) ACCOUNTANT

A) 5040 B) 226800. C)1260 D) 10080

11) CONTACT

A) 5040 B) 226800 C) 1260. D) 10080

12) ATLANTIC

A) 5040 B) 226800 C) 1260 D) 10080.

13) MATHEMATICS

A) 5040. B) 226800 C) 1260 D) 4989600

14) In how many ways can the letters of the word BALLOONS be arranged so that the two O's are always together ?

A) 300. B)360 C)240 D) 12

* BHARAT

15) How many different words can be formed with the letters given above

A) 300 B) 360 C) 240 D) 12

16) In how many of these B and H are never together?

A) 300 B)360 C)240. D) 12

17) how many of these begin with B and end with T ?

A) 300 B)360 C)240 D) 12.

**How many different words can be formed with the letters

18) of the word CAPTAIN ?

A) 2520. B) 1800 C)120 D) none

19) In how many of these C and T are never together?

A) 300 B)360 C)240 D) 1800.

** In how many ways can the letters of the word

20) ALGEBRA be arranged?

A) 2520. B) 1800 C)120 D) none

21) In how many of these arrangements will the two A's not come together?

A) 2520 B) 1800. C)120 D) none

22) Find how many different words can be formed from the letters of the word PEOPLE in which two P's would not remain side by side

A) 100 B) 120. C) 240 D) 300

*** In how many different ways can the letters of the word

23) CONSTITUTION be arranged?

A) 9979200. B) 151200 C) 9989790 D) 151320

24) How many of these will have the latter N both at the beginning and at the end?

A) 9979200 B) 151200. C) 9989790 D)151320

* In how many different ways can the letter of the word

25) VIDYAPITH be arranged?

A) 9979200. B) 151200 C) 9989790 D) 181440.

26) how many arrangements begin with V but do not end with H?

A) 9979200. B) 151200

C) 9989790 D) 17640.

Mg. A- R-3

1) If five coins are tossed, find the number of ways in which atleast one head will turns up.

A) 31. B) 15 C) 144 D) 720

2) In a Senior Secondary examination a candidate is required to pass the four different subject. In how many ways can be fail ?

A) 31 B) 15. C) 144 D) 720

3) In how many different ways can you arrange three girls and 4 boys in a row so that no two boys sit together ?

A) 31 B) 15 C) 144. D) 720

4) In how many ways the letters of the word ECONOMICS be arranged so that no two consonants come together?

A) 31 B) 15 C) 144 D) 720.

5) find the rank of the word MAKE, when its letters are arranged as in a dictionary (with or without meaning)

A) 29. B) 15 C) 144 D) 720

6) How many numbers of four different digits is greater than 4000 can be formed from the digits 2, 3, 4, 6, 7, 0?

A) 31 B) 15 C) 180. D) 720

7) How many numbers greater than 6000 can be formed with the digit 2, 4, 6, 7, 8 no digit being repeated.

A) 31 B) 15 C) 192. D) 720

** How many numbers of 6 digits can be formed from the digits 3, 4, 5, 6, 7, 8 (no digit being repeated). How many of these are

8) divisible by 5.

A) 31 B) 15 C) 120. D) 720

9) not divisible by 5?

A) 31 B) 15 C) 192 D) 600.

10) How many numbers of four digits can be formed from the digits 3, 4, 5, 6 ? find the sum of all such numbers.

A) 23, 119988 B) 24, 119988 C) 25 119988 D) 24, 118899

** In how many different ways can the letters of the VALEDICTORY be arranged so that the vowels

11) never separated

A) 31 B)997680. C) 192 D) 720

12) not together

A)38949120. B) 15 C) 192. D) 720

** In how many ways can the letters of the word STRANGE be arranged so that

13) the vowels and never separated

A) 31 B) 15 C) 192. D) 14403600

14) the vowels never come together

A) 31 B) 15 C) 192. D) 14403600

15) How many odd numbers of 6 digits can be formed with the digits 0, 1, 2, 3, 4, 5 each digits occuring only once.

A) 288. B) 730 C) 440 D) none

16) Suppose a licence plate contains three distinct letters followed by four digit with the first digits not being zero. How many different licence plates can be printed ?

A) 140400000. B) 14040000 C) 1404000 D) 140400

** In how many of the permutation of 8 things taken 4 at a time, willo ne particular thing

17) never occurr.

A) 840. B) 8400 C) 84000 D) none

18) always occur.

A) 840 B) 8400 C) 84000 D) n

19) find the number of arrangement of 10 different things taken all together in which two particular things will never come together.

A) 2903040. B) 340 C) 420 D) n

20) how many numbers lying between 100 and 1000 can be formed with the digits 2, 3, 4, 0, 8, 9, one digit not occuring more than once in any number ?

A) 100. B) 210 C) 320 D) none

** There are four bus line between A and B; and three Bus lines between B and C.

21) in how many ways can a man travel by bus from A to C by way of B ?

A) 12. B) 72 C) 144 D) none

22) in how many ways can a man travel round trip by bus from A to C way of B, if he does not want to use a bus line more than once?

A) 12 B) 72. C) 144 D) none

** Find the number of arrangements that can be made out of the letter of the word following:

23) INSTITUTION.

A) 554400. B) 4989600 C) 277200 D) 34650

24) STATISTICS

A) 554400 B) 4989600 C) 277200 D) 34650

25) ENGINEERING

A) 554400 B) 4989600 C) 277200. D) 34650

26) MISSISSIPPI

A) 554400 B) 4989600 C) 277200 D) 34650.

Mg. A- R-4

1) Find the number of ways in which the letters of the word ARRANGE can be arranged so that the two R's do not come together is

A) 554400 B) 4989600 C) 277200 D) 900.

2) In how many ways can the letters of the word MATHEMATICS be arranged so that the vowels come together?

A) 554400 B) 120960 C) 277200 D) 34650

3) How many ways can the letters of the word EXAMINATION be arranged so that all the A's always together ?

A) 907200. B) 4989600 C) 277200 D) 34650

** In how many ways can the letters of the word

4) AGARTALA be arranged ?

A) 1680. B) 120 C) 1560 D) none

5) In how many ways of these will the 4 A's has come together.

A)1680 B) 120. C) 1560 D) none

6) not come together ?

A) 1680 B) 120 C) 1560. D) none

** How many arrangements of the letter of the word COMRADE can be made

7) If the vowels are never separated.

A) 1680 B) 720 C) 1560 D) none

8) if the vowels are to occupy only odd places.

A) 1680 B) 120 C)576 D) none

9) If the relative positions of vowels and constants are not changed?

A) 1680 B) 144. C) 1560 D) none

** in how many ways can the letters of the word ARRANGE

10) be arranged

A) 1260. B) 360 C) 900 D) 120

11) How many of these arrangements are there in a which the two R's come together.

A) 1260 B) 360. C) 900 D) 120

12) the two R's do not come together.

A) 1260 B) 360 C) 900. D) 120

13) the two R's and the two A's come together ?

A) 1260 B) 360 C) 900 D) 120.

14) A man has to post 5 letters and there are four letter boxes in the locality. in how many ways can he post the letters ?

A) 1024. B) 1240 C) 1440 D) none

15) In how many ways can 8 prizes be given away to three boys, when each boy is eligible for all the prizes

A) 6651B) 6561. C) 6750 D) none

16) In how many ways can 8 boys form a ring ?

A) 5040. B) 6561 C) 5600 D) none

17) In how many ways can 6 boys be arranged at a round table so that two particular boys may be together

A) 504 B) 240. C) 3400 D) none

18) in how many ways can 7 people be arranged at a round table so that two particular persons be together?

A) 240. B) 480 C) 720 D) 1440

19) In how many ways can 8 different beads be string on a necklace ?

A) 2500 B) 2520. C) 2530 D) none

** In how many ways can 5 men and 2 ladies be arranged at a round table if the two ladies

20) sit together.

A) 240. B) 480 C) 720 D) none

21) are separated ?

A) 240 B) 480 C) 720 D) none

22) In all the words formed by the letters of the word LAKE be written out as in dictionary, find the rank of the word.

A) 20. B) 117 C) 120 D) 144

23) if the letter of the word WOMEN be permuted and the words formed arranged as in a dictionary, what will be the rank of the word WOMEN?

A) 20 B) 117. C) 120 D) 124

24) how many numbers less than 1000 divisible by 5 can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, such that each digit does not occur more than once in each number ?

A) 120 B) 144 C)154. D) none

25) in how many ways 4 prizes-- one is one for recitation, one for sports, one for smartness and one for general proficiency by given away 8 boys ?

A) 8⁴ . B) 4⁸ C) 4095 D) 6904

26) How many 3 digit number are there, with distinct digits, with each digit odd ?

A) 30 b) 60 c) 90 d) 120 e) none

Mg. A- R.5

1) Value of 5!

a) 72 b) 110 c) 120 d) 125 e) none

2) 7!/4!.

a) 210 b) 110 c) 120 d) 125 e) none

3) ⁸P₃.

a) 120 b) 210 c) 276 d) 336 e) none

4) If ⁿP₅ = 20. ⁿP₃ then n is

a) 1 b) 4 c) 6 d) 8 e) none

5) If ⁿ⁺ʳP₂ =110 and ⁿ⁻ʳP₂=20, then n & r is

a) 3,8 b) 3,6 c) 8,3 d) 8,6 e) none

6) 1+1. ¹P₁+ 2.²P₂ + 3.³P₃+...(n-1) ⁿ⁻¹Pₙ₋₁ is

a) n b) 0 c) n! d) ⁿPₙ e) none

7) In a compartment of a train 6 seats are vacant. If 3 passengers get on that compartment, In how many different ways can they sit on the empty seats ?

a) 62 b) 120 c) 144 d) 176 e) none

8) In how many ways 5 Students of the first year and 3 students of 2nd year can be seated in a row so that no two students of 2nd year will sit together?

a) 14400 b) 12200 c) 11100 d) 120 e) n

9) How many different words can be formed with the letters of the word PROBLEM taking 4 letters at a time? (the words may not have any meanings)

a) 840 b) 760 c) 720 d) 240 e) none

10) How many permutation can be found by taking together all the letters of the word PROPORTION?

a)14400 b) 12210 c) 13420 d) 151200 e) none

11) There are four letters and 4 envelopes with definite addresses on it. In how many ways the letters can be put in the correct envelope?

a) 120 b) 24 c) 256 d) 1 e) none

12) There are four letters and 4 envelopes without any addresses on it. In how many ways the letters can be put in the envelopes ?

a) 120 b) 24 c) 256 d) 1 e) none

13) There are 4 letters and four letter boxes. In how many ways the letters can be posted ?

a) 120 b) 24 c) 256 d) 1 e) none

14) How many numbers of three digits can be formed with the digit 1, 2, 3, 4, 5, repetition being allowed ?

a) 125 b) 240 c) 120 d) 1 e) none

15) How many even numbers of three digits can be formed with the digit 1, 2, 3, 4, 5, repetition being allowed ?

a) 20 b) 30 c) 40 d) 50 e) none

16) How many numbers of three digits can be formed with the digit 1, 2, 3, 4, 5, repetition is not allowed ?

a) 125 b) 240 c) 120 d) 1 e) none

17) How many even numbers of three digits can be formed with the digit 1, 2, 3, 4, 5, repetition is not allowed ?

a) 20 b) 30 c) 40 d) 50 e) none

18) How many odd numbers of four significant digits can be formed with the digits 0,1,2,3, 4 where digits can be repeated ?

a) 200 b) 300 c) 400 d) 500 e) none

19) How many odd numbers of four significant digits can be formed with the digits 0,1,2,3, 4 where digits can not be repeated ?

a) 200 b) 300 c) 400 d) 500 e) none

20) How many numbers lying between 1000 to 4000 can be formed with the digit 0, 1, 2,3,4 digit can be repeated ?

a) 300 b) 356 c) 374 d) 420 e) none

21) How many numbers lying between 1000 to 4000 can be formed with the digit 0, 1, 2,3,4 digit can not be repeated

a) 300 b) 356 c) 374 d) 420 e) none

22) There are 6 chairs in a row. In how many ways 6 men out of 10 men can be seated in such a way that two particular men ? never sit in that row

a) 20160 b) 20100 c) 22200 d) 23000 e) none

23) There are 6 chairs in a row. In how many ways 6 men out of 10 men can be seated in such a way that two particular men ? always sit in that row ?

a) 20160 b) 20100 c) 22200 d) 23000 e) none

24) How many 6 digit even numbers can be formed by 2, 3, 5, 3, 4, 5 only ?

a) 40 b) 50 c) 60 d) 70 e) none

25) How many odd numbers of 6 significant digits can be formed with the digits 0,1,2,3,4,5. when no digit been repeated in any number ?

a) 120 b) 210 c) 256 d) 288 e) none

26) How many numbers of 4 digits can be formed from the numbers 1,2,3,4?

a) 24 b) 36 c) 42 d) 60 e) none

27) find the sum of the numbers of 4 digits can be formed from the numbers 1,2,3,4?

a) 60000 b) 66000 c) 66600 d) 66660 e) n

Mg. A- R.6

1) In how many ways can the letters of the word MONDAY be arranged ?

a) 120 b) 320 c) 420 d) 720 e) none

2) how many of them will you begin with M but not end with Y from previous question.

a) 60 b) 72 c) 96 d) 100 e) none

3) How many arrangements can be made with all the letters of the word VENUS such that order of the vowels remains unaltered ?

a) 60 b) 72 c) 96 d) 100 e) none

4) Determine the number of ways in which the letters of the word INTERMEDIATE can be arrange so that the vowels do not come together.

a) 19870 b) 19870720 c) 19807200 d) 198762000 e none

5) Find the rank of the word LAND when its letters are arranged as in dictionary.

a) 10th b) 12th c) 14th d) 16th e) none

6) find the values of 6!

a) 420 b) 360 c) 520 d) 720 e) none

7) Value of 9!/6! is

a) 304 b) 404 c) 504 d) 604 e) none

8) Value of ⁷P₄ is

a) 420 b) 840 c) 1680 d) 1 e) none

9) value of n(n-1). ⁿ⁻²Pₓ₋₂ is

a) ⁿPₓ b) n c) n! d) (n-1)! e) none

10) value of ⁹P₃ + 3. ⁹P₂ is

a) ¹⁰P₃ b) ¹¹P₃ c) ¹²P₃ d) ¹³P₃ e) none

11) If ⁿ⁺¹P₃ = 10 . ⁿ⁻¹P₂ then n is

a) 4 b) 5 c) 4 or 5 d) 4 and 5 e) none

12) If ⁿP₄ : ⁿP₆ = 1: 2, then n is

a) 2 b) 3 c) 2 or 3 d) 2 and 3 e) 6

13) If ²ⁿ⁺¹Pₙ₋₁ : ²ⁿ⁻¹Pₙ = 3: 5 then n is

a) 4 b) 5 c) 4 or 5 d) 4 and 5 e) none

14) If 5. ⁴Pₓ = 6. ⁵Pₓ₋₁ then x is

a) 3 b) 4 c) 5 d) 6 e) none

15) If ⁵Pₓ = ⁶Pₓ₋₁ then x is

a) 2 b) 3 c) 4 d) 5 e) none

16) If ⁹P₅ + 5. ⁹P₄ = ¹⁰Pₓ. Then x is

a) 2 b) 3 c) 4 d) 5 e) none

17) If ⁿ⁺ˣ⁺²P₂ = 132 & ⁿ⁻ˣ⁺³P₂ = 20, then Find the value of n and x.

a) 2,4 b) 4, 2 c) 6,4 d) 4, 6 e) none

18) There are 8 different trains run between Kolkata and Shantiniketan. In how many ways can a man travel from Kolkata to Shantiniketan in one train and return in a different train ?

a) 5 b) 6 c) 56 d) 65 e) none

19) There at 12 stations on a railway line. How many different kinds of tickets of 2nd class must be printed in order that a passenger may go from any one station to any other.

a) 1 b) 2 c) 3 d) 12 e) 123

20) There are 6 ways to enter in a hall. In how many ways can three men enter into that hall through different entrances?

a) 12 b) 21 c) 120 d) 210 e) none

21) The number of permutation of 35 different things taken at a time.

a) 35! b) 13! c) 13!/35! d) 35/13! e) none

22) In how many ways can 7 boys in 3 girls be arrange in a rows so that no two girls will come together ?

a) 144 b) 142 c) 1440 d) 1420 e) none

23) In how many ways the letters of the word CHEMISTRY be arranged taken 4 letters at a time ?

a) 302 b) 3024 c) 403 d) 4023 e) none

24) In how many ways that the letter of the word DELHI be arranged taken all together .

a) 120 b) 1200 c) 12 d) 132 e) none

25) how many numbers of four digits can be formed with the digit 3, 5, 7, 9 when each of the number consists of different digits ?

a) 12 b) 24 c) 36 d) 48 e) none

26) How many different permutations can be made by taking all the letters of the word CONTACT

a) 12 b) 60 c) 126 d) 1260 e) none

27) How many different permutations can be made by taking all the letters of the word COLLEGE

a) 12 b) 60 c) 126 d) 1260 e) none

Mg. A -R.7

1) how many numbers of 6 digits can be formed with the digits of the number 234532 ?

a) 18 b) 180 c) 1800 d) 18000 e) none

2) There are 5 copies of one book, four copies of each of two books, 6 copies of each of three books and one copy of each eight books. In how many ways can all the books be arranged ?

a) 39! b) 39!/5! c) 39!/(5! .6!) d) 39!/(5! . (4!)². (6!)³) e) none

3) In the basis of result of the annual examination three prizes, first, second and third, are given in every class of a school. There are 10 students in a class. In how many ways can the prizes be given in that class?

a) 27 b) 72 c) 720 d) 7200 e) none

4) In how many ways can three prizes - one for recitation, one for sports and one for regular attendance be given to 10 students?

a) 10 b) 100 c) 1000 d) 10000 e) none

5) In how many ways can the result (win, defeat, draw) of 5 successive matches be decided ?

a) 2 b) 3 c) 4 d) 243 e) 234

6) A dice is thrown thrice. How many different outcomes are possible ? In how many way the result of 3 throw will be different ?

a) 216,120 b) 120, 216 c) 210, 120 d) 126,216 e) none

7) Find the number of permutations taking 5 things at a time from 10 different things, each of which can be taken up to 5 times repeatedly.

a)10 b) 100 c) 1000 d) 100000 e) 1

8) How many number of 4 digits can be formed with the digit 1, 2, 3, 4, 5, 6 , 7 where digits can be used more than once?

a) 24 b) 21 c) 240 d) 2401 2400

9) How many number of 5 digits can be formed with the digit 0, 2,5,6, 7, no digits being repeated in any number ?

a) 69 b) 96 c) 699 d) 996 e) 969

10) How many numbers lying between 3000 to 4000 can be formed to with the digit 0, 1, 2, 3, 4 where repetition of digit is allowed ?

a) 12 b) 21 c) 124 d) 214 e? 412

11) how many of them are odd(from Q. No. 10)

a) 40 b) 42 c) 50 d) 500 e) 30

12) How many numbers of 5 digits can be made with at least one repeated digits?

a) 62 b) 627 c) 6278 d) 62784 e) none

13) There are 8 questions in a question paper. In how many ways can a student answer 5 questions where the answers of 2 particular questions never occur.

a) 72 b) 720 c) 7200 d) 72000 e) none

14) There are 8 questions in a question paper. In how many ways can a student answer 5 questions where the answers of 2 particular questions always occur.

a) 24000 b) 2400 c) 240 d) 24 e) 42

15) How many number of 6 digits can be formed using the digits of the number 5, 6, 7,7,2,4 ? How many of this number so formed are even ?

a)180,360 b) 360,180 c) 120, 240 d) 240,120 e) none

16) How many numbers greater than 5000 can be found with the four of the digits 3, 4 , 5, 6, 7 when no digit is repeated ?

a) 27 b) 72 c) 36 d) 63 e) 90

17) if none of the figures 2,4,5,7,8,0 be repeated, how many different numbers of 4 digits can be formed with them?

a) 100 b) 200 c) 300 d) 400 e) 500

18) How many even numbers of 5 digits can be formed with the digit 0,1,2,3,4, each digit not occurring more than once in each number? how many of them are divisible by 4 ?

a) 60,30 b) 30,60 c) 27,72 d) 72,27 e) 32, 48

19) How many positive numbers of 3 digits and divisible by 5 can be formed when the digit of each number are different from each other ?

a) 12 b) 23 c) 36 d) 136 e) 144

20) Find the sum of all the numbers which are formed by the digit 2, 3, 4, 5, taking all together. Each digit does not occur more than once in each number.

a) 3324 b) 9334 c) 93324 d) 34342 e) 12000

21) How many arrangements can made by the letters of the word ORANGE ? How many of these will begin with O ? How many of these will not end with E ?

a) 700,120,600 b) 120,720,600 c) 120,600,720 d) 600,120,720 e) 720,600,120

**22) How many arrangements can be made by taking all the letters of the word TRIANGLE without changing the order of vowels ?

a) 720 b) 620 c) 6720 d) 2670 e) 7206

23) In how many ways can the letters of the word SUNDAY be arranged taken all together where the letters S,N and D are in this order ?

a) 120 b) 250 c) 360 d) 480 e) 720

24) Find the number of ways in which the letters of the word DROUGHT can be arranged so that the vowels are always together.

a) 144 b) 1440 c( 120 d) 1200 e) 14400

25) In how many ways can the letters of the word CONSTANT be arranged so that the two vowels always occur together ?

a) 220 b) 252 c) 2520 d) 5220 e) 2340

26) In how many ways can 37 different books be arranged on a shelf so that two particular books are never together ?

a) 35! b) 36! c) 35!.36 d) 35.36! 36!.35!

27) In how many ways can the letters of the word ALGEBRA be arranged in so that the two A are not together ?

a) 18 b) 180 c) 1800 d) 18000 e) 176

Mg. A- R.8

1) In how many ways can the letters of the word ORION be arranged so that the two consonants do not come together ?

a) 24 b) 42 c) 36 d) 63 e) 72

2) How many arrangements can be made out of the letters of the word COMMITTEE taking all at a time, such that the four vowels do not come together ?

a) 43 b) 430 c) 4300 d) 4320 e) 43200

3) How many different words can be formed taking all the letters of the word BALLOON in which two L will not come together ?

a) 90 b) 900 c) 9000 d) 30 e) 300

4) Find the rank of the word NAME when the letters are arranged as in a dictionary.

a) 20 b) 22 c) 24 d) 26 e) 30

5) Everyday there are 6 periods of a class. In how many ways 5 different subjects can be arrange daily ?

a) 5!/6 b) 5x6! c) 5x6!/2 d) 6! e) none

*** How many different arrangements can be made by the letters of the word ARRANGE so that

6) 2 R never come together

a) 900 b) 600 c) 800 d) 700 e) 500

7) 2A's will be together but two R's do not come together

a) 240 b) 420 c) 402 d) 322 e) 180

8) two A's never come together and also 2 R's never come together ?

a) 600 b) 620 c) 640 d) 660 e) 680

9) In how many ways can the letters of the word STRANGE be arranged so that vowels may occupy only the odd places?

a) 142 b) 144 c) 1420 d) 1440 e) 12440

10) In a railway compartment there are two benches on two sides of it. Five can seat on each of the benches. In how many ways can a group of 6 boys and 4 girls can sit if the girls always sit at the end of the bench ?

a) 1220 b) 1720 c) 17280 d) 17720 e) none

11) How many 5 digited telephone numbers with pairwise distinct digits can be formed ?

A) 5⁹ b) 9⁵ c) 12240 d) 1440 e) none

12) In how many ways can the letters of the word BANANA be arranged?

a) 24 b) 32 c) 60 d) 72 e) 84

13) How many different permutations can be made out of the letter the expression x³y²z⁴ when at full length?

a)124 b) 1240 c) 126 d) 1260 e) 1620

14) if ⁴⁻ˣP₂ = 6, find the value of x.

a) 1 b) 2 c) 3 d) 4 e) 24

15) In how many ways can the 10 coins of 10 paise and five coins of 5 paise can be arranged in a line so that two coins of 5 paise do not come together ?

a) 42 b) 46 c) 146 d) 462 e) 236

16) the number of different messages by 5 signals with 3 dots and two dashes is

a) 20 b) 1000 c) 10 d) 7 e) none

17) At the end of the conference each of the representatives exchange their signature with others. If the total number of Signature is 420, how many representatives were there in the conference ?

a) 12 b) 210 c) 120 d) 2104 e) none

Sunday, 28 March 2021

COMPOUND INTEREST (C. A)

1) Find the compound amount and the compound interest of ₹4000 invested for 7 years at 6% compounded annually.
A) 6014.52, 2014.52
B) 5014, 1014
C) 7296, 3296. D) none

2) Find the amount if ₹1800 is invested at 5% compounded semi annually for 8 years. also determine the C. I
A) 2672.10, 800.10
B) 2672.10, 872.10
C) 2600, 800. D) none

3) if ₹1000 is invested at annual rate of interest of 15%, what is the amount after 5 years if the compounding takes place monthly.
A) 2100. B) 2110
C) 2107.18. D) 2701.81

4) To what amount will ₹10,000 accumulate in 6 years if invested at 8% compounded quarterly?
A) 16000. B) 16084.30
C) 15084 D) 16018.03

5) if ₹1000 is invested at an annual rate of 9% compounded continuously. find the amount at the end of 6 years.
A) 1616 B) 1617 C)1715 D) 1716

6) if ₹2000 is deposited in a savings account that earns interest at an annual rate of 6% compounded continuously, what is the value of the account at the end of 3 years?
A) 2390 B) 2390.91
C) 2394.40 D) 2300.40

7) A person deposits ₹4000 in a bank, which pays an interest of 5% per annum compounded continuously. How much amount will be in his account after 4 years ?
A) 4800 B) 4885.60
C) 4785.80. D) 4588.06

8) How long will it take for ₹5000 to amount ₹7,000 if it is invested at 8% compounded quarterly ?
A) 4.5. B) 5. C) 4.25. D 5.5 yrs.

9) How long will it take for a principal to double if money is worth 12% compounded monthly
A) 5.84 B) 5. C) 7.5 D) 5.5 yrs

10) Approximately how long will it take to triple an investment at 10% compounded annually ?
A) 16.5 B)15 C) 11.5 D)15.5 yrs

11) At what annual rate of interest, compounded yearly, will money double in 8 years ?
A) 6.5% B) 7.5% C) 8.5% D) 9%

12) How long will it take for ₹4000 to amount to ₹7000 if it is invested at 7% compounded continuously ?
A) 6 B) 5 C) 7. D) 8 yrs

13) How long will it take for a principled double money if money is worth 6% compounded continuously?
A) 16.5 B)15 C) 11.5 D) 14.5yrs

14) A person deposited ₹1000 in a bank at 5% compounded annually after 5 years, the rate of interest was increased to 6% and after 4 more years, the rate was further increased to 7%. The money was withdrawn at the end of 12 years. find the amount.
A) 1973.86. B) 1937.86
C) 1930.59. D) 1903.95

15) find the amount of ₹2000 after 10 years at 8% converted quarterly for the first 4 years and at 6% converted monthly there after.
A) 3978.31. B) 3931.78
C) 3391.87. D) 3900.78

16) Find the amount of ₹5000 after 8 years if invested at 6% compounded semi annually for the first two years, at 8% compounded quarterly for the next 3 years. and at 7% compounded continuously thereafter.
A) 8808.49. B) 8840.94
C) 8480.79. D) 8804.94

17) A person deposited ₹4000 in a bank at 6% compounded continuously, after 3 years, the rate of interest was increased to 7% and after 5 more years, the rate was further increased to 8%.   the money was withdrawn at the end of 10 years. find the amount.
A) 7947.58.          B) 7794.85
C) 7974.85.          D) 7900.85

18) find the effective rate that is equivalent to a nominal rate of 12% compounded monthly.
A)12 % B)11.68% C)12.68% D)13%

19) A money-lender charges interest at the rate of 10 rupees per 100 rupees per half-year, payable in advance. what effective rate of interest does he charge per annum ?
A) 26.5% B) 25%
C) 23.5% D) 25.5%

20) To what sum will ₹6000 accumulate in 8 years if invested at an effective rate of 8% ?
A) 11005.40. B) 11000
C) 10105.40. D) 11005.40

21) how many years will it take for money to double at the effective rate of 6% ?
A) 12. B) 15. C) 11.9. D) 12.3

22) find the compound interest when principal is ₹3000, rate of interest is 5% per annum in two years.
A) 307 B) 300
C) 307.50 D) 290.85

23) calculate the compound interest that Ramsukhlal will get by investing ₹1000 for 3 years at the rate of 10% per annum.
A) 500 B) 300. C) 231. D) 331

24) Find the compound interest on ₹1000 for 4 years at 5% p.a
A)225. B) 215. C)200. D) 315

25) calculate the amount of principal is ₹8000, compound interest rate is 15% per annum after 3 years.
A) 12500. B) 12100
C)12167. D) 11167

26) If ₹1000 is invested for 10 years at 10% compound interest calculated annually ____ will be the amount that would be received.
A) 2500 B)3594 C)2894 D)2594

27) find the compound interest on ₹1000 for 5 years at 10% per annum. how much is it more than simple interest?
A) 600 , 100. B) 610, 110
C) 610.60, 110.60 D) none

28) Compute the compound interest on ₹8000 for 1 years at rate of 12.5% per annum compounded annually.
A) 500 B) 800 C) 1100 D) 1000

29) Find the amount that Shreya would receive if she invests ₹4096 for 18 months at 12.5% per annum, the interest being compounded half-yearly.
A) 4500 B)5000 C)4813 D)4913

30) Find the amount and the compound interest to when P=₹2000, R=8% CI per annum, n=1 year and interest is payable quarterly.
A) 2164.86, 164.86
B) 2264.86, 264.86
C) 2160.86, 160.86. D) none

31) Find the compound interest on ₹6950 for 3 years if interest is payable half-yearly, the rate of the first two years being 6% per annum.
A)1500 B)1600 C)1580 D)1589

32) The difference between simple interest and the compound interest on ₹60 for 1 year at 10% per annum, reckoned half yearly
A) ₹2 B) ₹1. C) ₹1.50. D) ₹2.50

33) when a girl is born, ₹500 is placed to her credit in an account that pays at the rate of 6% compounded monthly. If the account is not disturbed, what amount will there be to her credit on her twentieth birthday ?
A)1500 B) 1626 C) 1699 D)1659

34) in the previous question, if the interest is compounded quarterly, then the amount on her twentieth birthday will be
A) 1600 B)1626 C)1,699 D)1659

35) find the effective rate equivalent to the nominal rate of 8% converted quarterly.
A) 8%. B)8.14%. C)8.24% D) 8.5%

36) What is effective rate of interest corresponding to the rate of 10 % per annum, if interest is compounded half yearly.
A) 10.25% B) 10.50%
C) 10.00% D) 10.15%

37) Find the compound interest and effective rate of interest if an amount of ₹10000 is deposited in a bank for 1 year at the rate of 8% per annum compounded semi annually.
A) ₹816, 8.16% B) 854, 8.54%
C) ₹816, 8.54% D) ₹895, 8.95%

38) which is better investment?
a. 9% p.a compounded quarterly
b) 9.1 % per year simple interest
A) option b B) option a
C) either A or B. D) none

39) which yield more interest ?
a) 7.8% compounded semiannually
b) 8% compounded quarterly
A) option b B) option a
C) both with the same interest
D) none of the above.

40) The present population of a town is 8000. If it increases at the rate of 5% per annum, what will be its population after 3 years.
A)9000 B)9200 C) 9261 D)9276

41) The population of a town is 20000. If the annual birth rate is 4% and annual death rate is 2%, the population after 2 years is:
A) 20000 B) 21000
C) 20808 D) 21808

42) the population of a city in year 2003 was 424000. What will its population in 2005. if it increases 5% per year ?
A) 4678600. B) 467460
C) 459870. D) 462890

43) The population of a city was 160000 three years ago. if it has increased by 3%, 2.5% and 5%. respectively in the last three years, then the present population is..
A) 177300. B) 177499
C) 177549. D) 177366

44) A rare species of Deer was kept in an experimental Jungle and reared. its population increased by 10% in the first year and 5% in the second year, but decreased by 20% in the third year. if the number of animals in the beginning was 500, find their number at the end of 3rd year.
A) 362 B) 598 C) 462 D) none

45) The value of residential flat constructed at a cost of ₹100000 it depreciating at the rate of 10% per annum. Its value after three years of construction would be.
A)92700. B) 72900
C) 80000. D) 85600

46) A Maruti Swift costs ₹360000. Its price depreciates at the rate of 10% a year during first 2 years and at the rate of 20% in the third year. What will be the price of the car after three years ?
A) 233000. B) 235987
C) 245698. D) 233280

47) taking data from the previous question, what would be the total depreciation in the three years?
A) 126000. B) 125498
C) 145984. D) 126720

48) What amount of money lent out at compound interest will amount to ₹1936 in 2 years at 10% per annum Interest Compounded being charged annually.
A) 1600 B) 1660 C)1500 D) none

49) compound interest on a sum of money were ₹450 and ₹477 in two consecutive years. Find the rate percent if the sum of the money is compounded annually.
A) 4.5% B)5.5% C)6% D)4%

50) A sum of money Doubles itself in 5 years. in how many years will it become four fold (if interest is compounded)
A) 15. B) 10. C) 20. D) 12

51) A certain sum of money Doubles itself at a certain rate of compound interest in 3 years, in how many years will the ratio of their principal to the Compound Interest be 1:3?
A) 5. B) 6. C) 9yrs. D) none

52) solve the above question assuming simple interest.
A) 5yrs B)6yrs C) 9yrs D) none

53) A sum of money placed at compound interest Doubles itself in 3 years. in how many years will it amount to 8 times itself ?
A)9yrs. B)8yrs C)27yrs D) 7yrs

54) An amount of money grows up to ₹8000 in 2 years and up to ₹8500 in 3 yrs. find the compounded rate of interest.
A) 6% B)6.25% C)6.67% D) 5.25%

55) the difference between compound interest and simple interest on a certain sum of money is ₹40 for 1st two years and ₹122 for the first three years. find the sum, if the rate is same in both the cases.
A) 8000. B) 12000
C) 15000. D) 16000

56) the compound interest on a certain sum for 2 years is ₹70.40 and the simple interest is ₹70. what is the rate of interest ?
A) 3% B) 4% C) 5% D) 6%

57) if the difference between compound and the simple interest on a certain sum of money for 3 years at 10% per annum is ₹248, find the sum
A) 6000. B)7000 C)8000. D) 9000

58) the population of a city is 2000000. If the annual birth rate and the annual death rate are 6% and 3% respectively, then calculate the population of the city after 2 years.
A) 212090. B) 206090
C) 212000. D) 212180

59) Find the least whole number of years in which a sum of money invested at 20% compound interest per annum will more than double itself.
A) 4 B) 3. C) 2. D) 5

60) A certain sum invested at compound interest becomes ₹34560 in 8 years and ₹24000 in 6 years. What is the amount after 5 years ?
A) ₹12413 B) ₹22500
C) ₹10000. D) 20000

61) the interest on a certain sum invested at a certain rate of Compound Interest is ₹23205 in 4 years and ₹10500 in 2 years. find the amount at the end of the first year.
A) 55000. B) 12000
C) 40000. D) 15000

62) the compound interest earned on a sum of money at a rate of 10% per annum for the first year and 12% per annum for the second year is what percent of the sum lent ?
A)21% B)22% C)23.2% D) 24%

63) A sum triplets itself in n years at a certain rate of compound interest compounded annually. if it becomes 5 times of itself in (2n-m) years, find the approximate value of m and in terms of n. given log 3: log 5= 0.6825
A) 0.683n. B) 0.535n
C)0.355n. D) 0.699n

64) if the rate of interest is 20% per annum, find the difference in the interests accrued for one year on a sum of ₹100000 in cases when interest is compounded half-yearly and when it is compounded quarterly.
A) 550.60. B) 450.55
C) 828.50. D) 719.15

65) the compound interest on a certain sum for two years is ₹1520 and the corresponding simple interest is ₹1500. find the compound interest on the same sum for 3 years approximately.
A) 800. B)816. C)2569 D) 2326

Saturday, 13 March 2021

TRIGONOMETRICAL RATIOS(J)

1) If sinx= 3/5, find the values of other trigonometric ratios.

2) If cosx= 4/5, find the values of other trigonometric ratios.

3) If sinx= 1/√2, find the values of other trigonometric ratios.

4) If cot x= 9/40, find the values of cosec x, sec x. 41/40, 41/9

5) If tanx = 8/15 find sinx. 8/17

6) If sinx = 8/17 find tanx. 8/15

7) If cosx =1/2 find the value of 2sec x/(1+tan²x). 1

8) If tanx = 1/√2 find the value of (cosec²x- sec²x)/(cosec²x+ cot²x) 3/10

9) If 5 tanx = 12 find the value of (2cosx+sinx)/(sinx - cosx). 22/7

10) If 3tanx = 4 find the value of (5sinx-3cosx)/(5sinx+2cosx). 11/26

11) If 3 cot x = 4 find the value of (5cosx - 2sinx)/(5cosx+3sinx). 14/29

12) If tanx = 12/5 find the value of (1+ sin x)/(1- sinx). 25

13) If sinx = 3/5 find the value of (tanx + sec x)². 4

14) If tanx = 3/4 find the value of (1- cosx)/(1+cosx). 1/9

15) If cosx = 3/5 find the value of cot x + cosec x. 2

16) If cosec A= √2 find (2sin²A + 3cot²A)/(4tan²A- cos²A). 8/7

17) If tanx = 12/5 find (1+sinx)/(1- sinx). 25

18) If cot B = 1/√3 find the value of (1- cos²B)/(2- sin²B). 3/5

19) If sec x = 13/5 find (2sinx - 3cosx)/(4sinx - 9cosx). 3

20) If 5tanx = 6 find (8sinx+ 3cosx)/(8sinx - 3cosx). 21/11

21) If 4cot x =5 find (5sinx - 3cosx)/(5sinx+ 2cosx). 1/6

22) If tanx = 3/4 find (4sinx - 2cosx)/(4sinx + 3cosx). 1/6

23) If tanA = m/n find m sinA + n cos A)/(m sinA - n cosA). (m²+n²)/(m² - n²)

24) If cot x = √3 find (cosec²x+ cot²x)/(cosec²x - sec²x). 21/8

25) If 3 cosx = 1 find (6sin²x+tan²x)/4 cosx. 10

26) If 4tanx = 3 find (4sinx-2cosx)/(4sinx+3cosx). 1/6

27) If √3tanx = 3 sinx find sin²x - cos²x. 1/3

28) If cosec x = 13/12 find (2sinx - 3cosx)/(4sinx - 9 cosx). 3

29) If tanx = 1/√7 find (cosec²x- sec²x)/(cosec²x+ sec²x). 3/4

30) If sinx = a/√(a²+b²) find cosx and tanx. b/√(a²+b²), a/b

31) If sec x = x + 1/4x show that secx + tanx = 2x or 1/x

Tuesday, 9 March 2021

FACTORIZATION A - Z(1)

EXERCISE -A

1) 5x +20. 5(x+4)

2) 6ab - 9bc. 3b(2a -3c)

3) 12a+15. 3(4a+5)

4) 28k - 42. 14(2k-3)

5) 18m + 16n. 2(9m +8n)

6) xyz - 2xy. xy(z-2)

EXERCISE - B

1) 3a²+ 6ab. 3a(a + 2b)

2) 2x²+3x. x(2x+3)

3) 9a² + 12ab. 3a(3a+4b)

4) 2πr² - 4πr. 2πr(r -2)

5) 4x² -12y. 4(x²-3y)

6) 5x² - 20xy. 5x(x-4y)

7) 18x²y - 24xyz. 6xy(3x - 4z)

8) 5xy - 25x³y². 5xy(1- 5x²y)

9) 35x²y²- 21xy. 7xy(5xy-3)

10) 21py²- 56py. 7py(3y -8)

11) 12b - 9b². 3b(4-3b)

EXERCISE - C

1) x²y - xy². xy(x-y)

2) 12a²b +15ab². 3ab(4a + 5b)

3) 24x³- 32x². 8x²(3x-4)

4) 4x³ - 6x². 2x²(2x - 3)

5) 8xy³ + 12x²y². 4xy²(2y + 3x)

6) 3x²y - 6xy². 3xy(x - y)

7) 15ax³ - 9ax². 3ax²(5x -3)

8) 15ab² - 21a²b. 3ab(5b - 7a)

9) 18x³y - 45x²yz. 2x²y(2x- 5z)

10) 24x³ -32x². 8x²(3x-4)

11) 25abc² - 15a²b²c. 5abc(5c - 3ab)

12) 27a³b³-45a⁴b². 9a³b²(3a-5b)

13) 28p²q²r - 42pq²r². 14pq²r(2p - 3r)

14) 35x³y -49xy³. 7xy(5x²- 7y²)

15) 36a³b - 60a²bc. 12a²b(3a-5c)

16) 18x³- 27x²y. 9x²(2x -3y)

EXERCISE - D

1) 6xy²+ 9x²y - 21xy. 3xy(2y +3x -7)

2) 8x³ - 6x²+10x. 2x(4x²-3x+5)

3) 14mn+ 22m - 62p. 2(7mn+ 11m-31p)

4) 14x²y² - 10x²y + 8xy². 2xy(7xy-5x+4y)

5) 18x²y² - 24xy² + 30x²y. 6xy(3xy - 4y+ 5x)

6) 18x⁵y⁷z⁷ - 81x⁷y⁵z⁷+ 5x⁷y⁷z⁵. x⁵y⁵z⁵ (18y²z² - 81x²z²+ 5x²y²)

7) 27x³y³ - 18x²y³ + 75x³y². 3x²y²(9xy - 6q+25x)

8) x²yz + xy²z+ xyz². xyz(x+y+z)

9) 24x²y - 18xy²+ 12xy. 6xy(4x-4y+2)

10) 46x²+ 2xy + 10y². 2(23x²+ xy+ 5y²)

11) 7x³y - 21x²y² + 35y². 7y(x³- 3x²y +5y)

EXERCISE -E

1) 2x(y+z) - 3(y+z). (y+z)(2x-3)

2) 2a(m+n) - 3b(m+n). (m+n)(2a-3b)

3) 2a(x²+y²)+ 4b(x²+y²). 2(x²+y²)(a+2b)

4) 2x(a+b) -3y(a+b). (a+b)(2x-3y)

5) 2x(a²+b²) +4y(a²+b²). 2(a²+b²)(x+2y)

6) 3x(y+2z) + 5a(y +2z). (y+z)(3x+ 5y)

7) 5x(y+z) -7y(y+z). (y+z)(5x-6y)

8) a(2p -3q) -10b(2p -3q). (2p -3q)(a -10b)

9) 5x(2a - 3b)+ 3y(2x- 3b). (2a-3b)(5x+3y)

EXERCISE - F

1) 15a(2p -3q) -10b(2p -3q). 5(2p -3q)(3a - 2b)

2) 3a(x² +y²) + 6b(x²+ y²). 3(x²+y²)(a+ 2b)

3) 2X(x²+ y²) - 4y(x²+ y²). 2(x²+ y²)(x - 2y)

EXERCISE - G

1) a(b-5)+c(5-b). (b-5)(a-c)

2) a(x-2)+b(2-x). (x-2)(a-b)

3) y(y -z)+ 9(z - y). (y- z)(y -9)

4) p²(q-r) - q(r-q). (q-r)(p²+q)

5) 25a(x-y) + 30b(y-x). 5(x-y)(5a-6b)

6) 30a(b - c) - 25(c - b). 5(b -c)(6a+5)

7) a²(a²+ b²- c²) - b²(c² - a² - b²). (a²+b²-c²)(a²+ b²)

EXERCISE - H

1) a(b-c)² - b(b-c). (b-c)(ab- ac-b)

2) x(y-z)²+ y(y-z). (y-z)(xy - xz + y)

3) (x+y)- (x+y)². (x+y)(1-x-y)

4) 3x(2y -z)² - (2y-z). (2y-z)(6xy -3xz -1)

5) x(2a+b)² - y(2a+b). (2a+b)(2ax+bx-y)

EXERCISE - I

1) 4(a+b)-6(a+b)². 2(a+b)(2-3a-3b)

2) 6(3a+4b) - 8(3a+4b)². 2(3a+4b) (3-12a- 16b)

3) 8(3x-2y)² - 10(3x-2y). 2(3x-2y)(12x- 8y -5)

4) 6(2x+3y)² - 8(2x+3y). 2(2x+3y)(6x+9y-4)

5) 16(4a+5b)²- 24(4a+ 5b). 8(4a+5b) (8a +10b -3)

EXERCISE - J

1) (x-y)³+2(x-y). (x-y)(x²+y²-2xy +2)

2) 6(x+ 2y)³+ 8(x+2y)². 2(x+2y)²(3x+6y +4)

3) 14(a -3b)³-21p(a -3b). 7(a -3b){2(a -3b)²- 3p}

4) a(a+b)³ - 3a²b(a+b). a(a+b)(a²+b²- ab)

5) a(a-b)³+3a²b(a-b). a(a-b)(a²+ b² + ab)

6) 10a(2p+q)³- 15b(2p +q)²+ 35(2p+q). 5(2p+q){2a(2p +q)²- 3b(2p+ q)+7)

7) 10(p -2q)³+6(p -2a)²-20(p -2q). 2(p - 2q){5(p -2a)²+ 3(p - 2q) -10}

8) x(x²+ y²-z²)+ y(x²- y²+ z²) - z(x²+ y² -z²). (x²+ y² - z²)(x - y- z)

9) (a+ b)(x + y) + (2a + 3b)(x + y) + (3a + 4b)(x+ y). 2(x+y)(3a +4b)

10) 2a(x -y)+ 3b(5x - 5y) + 4c(2y - 2x). (x -y)(2a+ 15b - 8c)

EXERCISE - K

1) ab + bc + ax + cx. (a+c)(b+x)

2) ab - a - b +1. (b-1)(a-1)

3) 1- a - b - ab. (1- a)(1- b)

4) ax - ay + bx - by. (x -y)(a + b)

5) xy - pq + qy - px. (y- p)(x+q)

6) 6xy + 6 - 9y - 4x. (3y-2)(2x-3)

7) 2a - 4b - xa + 2bx. (a - 2b)(2- x)

8) 3ax - 6ay - 8by + 4bx. (3a+4b)(x-2y)

9) xy - ab + bx - ay. (y+b)(x-a)

10) 3ax - 6ay - 8by +4bx. (x-2y)(3a - 4b)

11) 5ph - 10qk + 2rph - 4qrk. (5+2r)(ph-2qk)

12) 6xy - 4y + 6 - 9x. (3x-2)(2y-3)

EXERCISE - L

1) 1+ x²y² + x²+ y². (1+ x²)(1+ y²)

2) 1 + p + pq + p²q. (1+p)(1+ pq)

3) x² +4x+ x+4. (x+4)(x+1)

4) a²+ bc+ ab+ ac. (a+ b)(a+ c)

5) ax²+ by²+ bx²+ ay². (x²+y²)(a+b)

6) ap² + bp² + aq² + bq². (a+ b)(p²+ q²)

EXERCISE - M

1) a² + b - ab - a. (a-1)(a-b).

2) a² - b + ab - a. (a-1)(a+b)

3) x² + xy - x - y. (x+ y)(x -1)

4) y² - yz - 5y + 5z. (y -z)(y - 5)

5) x² + y - xy - x. (x-y)(x-1)

6) a²b - ab² + 3a - 3b. (a- b)(ab+3)

7) a²+ ab- 2ac - 2bc. (a+b)(a-2c)

8) 5xy +7y - 5y² - 7x. (x-y)(5y-7)

9) 6xy - y² + 12xz - 2yz. (y+2z)(6x-y)

10) 6ab - b² + 12ac - 2bc. (6a-b)(b-2c)

11) 5p² -8pq -10p +16q. (5p -8q)(p-2)

12) 6xy²- 3xy - 10y +5. (2y-1)(3xy-5)

13) x² - 2ax - 2ab + bx. (x+b)(x-2a)

14) 2x² + yz - 2xy - xz. (2x-z)(x+y)

15) x² - 11xy - x +11y. (x-11y)(x-1)

16) 4x² - 10xy - 6xz +15yz. (2x-5y)(2x -3z)

17) 3x²y - 3xy + 12x - 12. 3(x-1)(xy+4)

18) a²xy + abx²+ b²xy + aby². (ax + by)(bx + ay)

EXERCISE - N

1) x³ + 2x²+5x+10. (x+2)(x²+5)

2) x³ + 2x²+x+2. (x+2)(x²+1)

3) x³+ x - 3x²-3. (x²+1)(x-3)

4) x³-3x +x -3. (x-3)(x²+1)

5) x³y - x²y + 5xy - 5y. y(x-1)(x²+5)

6) 4a³ - 8a² + 3a - 6. (a-2)(4a²+ 3)

7) x³ - x²y - xy + y². (x - y)(x² -y)

8) x³ - x² + x -1. (x -1)(x²+1)

EXERCISE - O

1) a(a - 2b - c) + 2bc. (a - 2b)(a - c)

2) x(x+y- z) - yz. (y-z)(x+y)

3) x(x- 2y- z) + 2yz. (y-z)(x-2y)

4) a(a+b-c) - bc. (a+b)(a-c)

5) a(a - 2b-c) + 2bc. (a-2b)(a-c)

EXERCISE - P

1) (4x -1)²- 8x +2. 2(4x-1)(2x-1)

2) m² -(a+b)m+ ab. (m-a)(m-b)

3) x² - (a+ 2b)x+ 2ab. (x-a)(x- 2b)

4) ab² +(a -1)b - 1. (b+1)(ab -1)

5) abx² + (ay - b)x - y. (bx+y)(ax-1)

6) a²x² +(ax²+1)x +a. (a+x)(ax²+1)

7) x³ + xy(1-2x) - 2y². (x-2y)(x²+y)

8) x² + xy(1+ y)+ y³. . (x+ y)(x + y²)

9) y² - xy(1- x) - x³. (y - x)(x + x²)

10) x³ + xy(1- 3x) - 3y². (x²+ y)(x - 3y)

11) abc² + (ac - b)c - c. c(b +1)(ac -1)

EXERCISE - Q

1) ab(x²+y²) - xy(a²+b²). (bx -ay)(ax- by).

2) ab(x²+1)+ x(a²+b²). (ax+b)(bx+a).

3) (mx+ny)² + (nx -my)². (m²+n²)(x²+ y²)

4) (ax +by)² + (bx - ay)². (a²+b²)(x²+ y²)

5) (a² + 2a)² - 3(a²+ 2a) - b(a²+ 2a) + 3b. (a²+2a-3)(a²+2a-b)

6) x² + 1/x² + 2 - 2x - 2/x. (x+ 1/x)(x+ 1/x -2)

7) a² + 1/a² - 2 - 3a + 3/a. (a- 1/a)(a- 1/a - 3)

8) x(x³ - y³)+ 3xy(x - y). x(x-y)(x²+ xy + y² + 3y)

9) 16(a-b)³ - 24(a-b)². 8(a-b)²(2a-2b-3)

EXERCISE - R

1) ax + by + bx + az + ay + bz. a(x+ y+ z)(a+ b)

2) a²b + ab² - abc - b²c + axy + bxy. (a+ b)(ab - bc + xy)

3) ax² - bx²+ ay² - by² + az²- bz². (a- b)(x² + y²+ z²)

4) x - 1 -(x -1)²+ ax - a. (x -1)(2- x + a)

5) x³ + x² + ax + x - a- 1. (x-1)(x²+ a+1)

6) p(x -y)² - qy + qx + 3x -3y. (x- y){p(x - y) + q +3}

7) ax - (ax + by)² + a²x + aby + by. (ax + by)(1+ a - ax - by)

8) a³x + a²(x - y) - a(y +z) - z. (a+1)(a²x - ay - z)

9) (x² - 2x)²- 5(x²- 2x)- y(x²- 2x)+5y. (x²- 2x -5)(x²-2x - y)

MISCELLANEOUS EXERCISE- 1

1) a²+ b² -2(ab - ac + bc). (a- b)(a - b + 2c)

2) x²+ 1/x² + 2 - 5x - 5/x. (x + 1/x)(x + 1/x -5)

3) ax²/b + (a/b + c/d)x + c/d. (x+1)(ax/b + c/d)

4) 2p(a - b) + 3q(5a - 5b) + 4r(2b - 2a). (a- b)(2p + 15q - 8r)

EXERCISE - S

1) 9x² + 12xy + 4y²

2) 1 - 8x +16x²

3) 25a²b² - 20ab²c + 4b²c²

4) x² - 2√5 +5

5) x²/y² + 2 + y²/x²

6) m²+6m+9. (m+3)(m+3)

7) b²+ 18b+ 81. (b+9)(b+9)

8) 1+ 2x + x². (1+x)(1+x)

EXERCISE - T

1) x² - 2xy + y² - x+y

2) 4a²+ 12ab + 9b²- 8a - 12b

3) a² + b²- 2(ab - ac +bc)

4) a²/4b² - 1/3 + b²/9a²

5) 4a² + 25b² + 20ab - 10a - 25b

6) (x - 1/x)² + 6(x - 1/x) + 9

7) 4x² - 4√3 x + 3

8) x⁶ - 8x³ + 16

9) x(x-2)(x-4)+4x - 8

10) 5x³ - 30x² + 45x

11) (x+2)(x²+25) - 10x² -20x

12) x² + 1/x² - 6

13) 2a² + 2√6 ab + 3b²

14) a²/4b² + 1 + b²/a²

15) (x² + 1/x²) - 4(x - 1/x) + 6

16) (a - b+c)² + (b-c+a)² + 2(a-b+c)(b- c+a)

17) a² + b² + 2(ab + bc +ca)

18) 4(x-y)² - 12(x-y)(x+y) + 9(x+y)²

19) 2p(a -b) + 3q(5a- 5b) + 4r(2b - 2a)

20) ab(a²+ b²- c²) + bc(a² + b²- c²) - ca(a² + b² - c²)

21) x(x²+ y²- z²)+ y(- x² - y²+ z²) - z(x² + y² - z²)

22) a³x + a²(x -y) - a(y+z) - z

23) (x² + 3x)² - 5(x² + 3x) - y(x²+3x) + 5y

24) a²+b² - 2(ab - ac + bc)

25) 4a² + 12ab + 9b² - 8a - 12b

26) (5x - 1/x)² + 4(5x - 1/x) +4

27) 4(x+y)² - 28y(x+y)+ 49y²

28) (2a+3b)² + 2(2a+3b)(2a - 3b) + (2a - 3b)²

EXERCISE - U

1) x²- 4y². (x +2y)(x - 2y)

2) 4x² - 25 y². (2x+13y)(2x -13y)

3) 4x² - 169y². (2x+5y)(2x -5y)

4) 9x² -1. (3x+1)(3x -1)

5) 25x²- 36y². (5x+3y)(5x - 3y)

6) 100 - 9x². (10 +3x)(10 -3x)

7) 3x²- 4. (√3x+2)(√3x -2)

8) x² - 1/25. (x +1/5)(x - 1/5)

EXERCISE - V

1) 150 - 6a². 6(5+a)(5-a)

2) 32x² - 18y². 2(4x+3y)(4x -3y)

3) 20x² - 45y². 5(2x+3y)(2x- 3y)

4) πa⁵ - π³ab². πa(a²+πb)(a² - ab)

5) 16y³ - 4y. 4y(2y+1)(2y -1)

6) 49p³q -36pq. pq(7p+6)(7p-6)

7) 63x² -112y². 7(3x+4y)(3x - 4y)

8) 3x²/25 - 27y²/16. 3(x/5 +3y/4)(x/5 - 3y/4)

9) x²y² - 9x⁴y⁴. x²y²(1+ 3xy)(1- 3xy)

10) x³ - 25x. x(x+5)(x -5)

EXERCISE - W

1) (x - y)² - 9. (x- y+3)(x- y-3)

2) 1- (b - c)². (1+ b - c)(1- b + c)

3) 16 - 9(x + y)². (4+3x+3y)(4- 3x- 3y)

4) 9(x + y)² - x². (4x+3y)(2x+3y)

5) 2(x - 2y)² - 50y². 2(x+3y)(x-7y)

6) 3 - 12(a - b)². 3(1+ 2a+ 2b)(1- 2a +2b)

7) 9x² -4(y+2x)². -7(7x+2y)(x +2y)

8) 108x² -3(y- z)². 3(6x+y-z)(6x - y + z)

9) 50x² -2(x -2)². 8(2x+1)(3x -1)

10) 32 -2(x - 4)². 2x(8 - x)

11) 3 - 12(a - b)². 3(1+2a-2b)(1 - 2a -2b)

12) 4(2x -3)² -9(y+1)². (4x+3y -3)(4x - 3y-9)

13) (x-2)(x+2)-3. (x+1)(x-1)

14) x - 2y - x² + 4y². (x -2y)(1- x -2y)

EXERCISE - X

1) 4a²- b²+ 2a + b. (2a+b)(2a - b+1)

2)4a²- 9b²- 2a -3b. (2a+3b)(2a - 3b-1)

3) a(a -2)- b(b -2). (a- b)(a+ b -2)

4) a(a -3)- b(b -3). (a- b)(a+ b -3)

5) a(a -1)- b(b -1). (a-b)(a+ b -1)

6) x² - 2y + xy -4. (x-2)(x+y +2)

7) 9x²- 4a²+ 4ay - y². (3x-y+2a)(3x+ y-2a)

8) 9 - x² + 2xy - y². (3+x- y)(3- x +y)

9) 9x⁴ -(x²+ 2x +1). (3x²+x+1)(3x²- x -1)

10) 9x⁴ - x² - 12x -36. (3x²+x+6)(3x²- x -6)

11) a⁴ - b⁴ + 2b² -1. (a²+b²- 1)(a²- b² +1)

12) x²- 2xy + y²- a² - 2ab - b². (x- y+a+ b)(x - y - a - b)

EXERCISE - Y

1) x³ - 5x² - x + 5. (x+1)(x-1)(x-5)

2) x³ - 3x² - x + 3. (x+1)(x-3)(x-1)

3) p⁴- 81. (p-3)(p+3)(p²+9)

4) 2x⁴ - 32. 2(x²+4)(x +2)(x -2)

4) a²(b + c)- (b +c)³. (b+ c)(a+b+c)(a- b- c)

5) (a+ b)³ - a - b. (a+ b)(a+ b+1)(a+ b-1)

6) 3x⁵- 48x. 3x(x²+4)(x+2)(x-2)

EXERCISE - Z

1) x² + 1/x² - 11. (x -1/x +3)(x - 1/x -3)

2) x⁴ + 5x²+ 9. (x² +x +3)(x² - x +3)

3) a⁴ - b⁴ + 7a²b². (a² +b² +3ab)(a² + b² -3ab)

4) x⁴ + 14x²+ 1. (x² +4x +1)(x² - 4x +1)

5) x⁴+ y⁴ - 11x²y². (x² -y² +3xy)(x² - y² -3xy)

6) x⁴+x²y²+ y⁴. (x² +y² +xy)(x² +y² -xy)

7) x⁴+ 4. (x² +2x +2)(x² - 2x +2)

8) x⁴+ x²+1. (x² +x +1)(x² - x +1)

9) x⁴+ 7x² +16. (x²+ 4+ x)(x²+ 4 -x)

10) a⁸+ a⁴b⁴ + b⁸. (a²+ b²+ ab)(a²+ b² -ab)(a⁴+ b⁴ - a²b²)

MISCELLANEOUS EXERCISE - 2

1) a) (a²- b²)(c² - d²) - 4abcd. (ac - bd + bc + ad)(ac - bd - bc - ad)

b) (1- 4x²)(1- 4y²)+ 16xy. (1+4xy+ 2x -2y)(1+ 4xy - 2x + 2y)

2) 4x² - y² - 3xy + 2x - 2y. (x-y)(4x + y+2)

3) (a+ b+ c)²-(a - b - c)² + 4b² - 4c². 4(b+ c)(a+ b- c)

4) 2(ab + cd) - a² - b² + c²+ d². (c+ d+ a- c)(c + d - a + b)

5) (1- x²)(1- y²)+ 4xy. (xy + x - y +1)(xy - x + y +1)

6) (x²+ y²- z²)². (x+y+z)(x +y-z)(x - y +z)(x - y - z)

7) Show them as difference of two squares

a) (x²-5x +7)(x²+5x +7). (x² +7)²-(5x²)

b) (x²-5x +7)(x²-5x +7). (x² -5x)²- 7²

c) (x²+5x -7)(x²-5x +7). x² -(5x - 7)²

d) (x²-4x +9)(x²+4x -9). (x²)² -(4x - 9)²

8) Simplify with the help of formula:

a) 979² - 21². 958000

b) 99.9² - 0.1². 9980

MIDDLE TERM FACTORIZATION

EXERCISE- A

A) a) x²+ 9x +18. (x+3)(x+6)

b) x²+ 5x +6. (x+2)(x+3)

c) x²+ 14x +45. (x+5)(x+9)

d) x²+ 19x +88. (x+8)(x+11)

B) a) x² +2x -3. (x+3)(x -1)

b) x²+ 6x -7. (x+7)(x-1)

c) x²+ 9x -36. ((x+12)(x -3)

d) x²+ 7x -18. (x+9)(x-2)

C) a) x² -8x +7. (x-1)(x-7)

b) y² - 11y +18. (y-9)(y -2)

c) x²- 24x +108. (x -6)(x -18)

D) a) x²- 3x -54. (x-9)(x+6)

b) x²-7x -18. (x-9)(x+2)

c) x²- 3x -54. (x+6)(x-9)

d) x²- x -132. ((x -12)(x+11)

e) x²- 11x - 42. (x -14)(x +3)

f) x²- 2x -15. (x -5)(x +3)

g) x²- 6xy - 7y². (x+y)(x- 7y)

EXERCISE - B

a) 2x² +13xy - 24y². (x+8y)(2x-3y)

b) 8) 2x²+ x - 45. (x +5)(2x-9)

c) 2x²- 7x +6. (x-2)(2x-3)

d) 2x²- 7x -15. (x-5)(2x+3)

e) 2x²- x - 6. (x- 2)(2x+3)

f) 3x² -10x +8. (3x -4)(x -2)

g) 4y²- 17y -21. (y+1)(4y -21)

h) 4x² -25x +21. (4x -2)(x -1)

i) 5x² +17xy - 12y². (x+4y)(5x-3y)

EXERCISE - C

a) 6x²+ 17x +5. (3x+1)(2x+5)

b) 6x²+13x -5. (3x-1)(2x+5)

c) 6x²+11x -10. (2x+5)(3x-2)

d) 6x²- 7x - 3. (3x+1)(2x-3)

e) 6x² 19x +10. (3x+2)(2x+5)

e) 10x² + 3x - 4. (5x +4)(2x -1)

f) a)12x²- 25x +12. (3x -4)(4x -3)

g) 12x² -7x +1. (4x-1)(3x-1)

EXERCISE - D

a) x²+ x/4 -1/8. (x+1/2)(x-1/4)

b) x²/2 + 3x +4. (1/2)(x+2)(x+4)

c) x²/5 + 2x -15. (1/5)(x -5)(x+ 15)

d) 9x² - 2x - 1/3. (1/3)(9x +1)(3x -1)

EXERCISE - E

1) x²y² - 8xy - 48. (xy+4)(xy - 12)

2) x²y² - 7xy - 30. (xy -6)(2xy +5)

3) 3x²y² - 5xy - 12y². (x -3y)(3x +4y)

4) 6x²- 5xy - 6y². (2x -3y)(3x +2y)

5) 60x²- 70x - 30. 10(2x-3)(3x+1)

EXERCISE - F

a) 1- 18x - 63x². (1-21x)(1+3x)

b) 5- 4x - 12x². (1-2x)(5+6x)

c) 84- 2x - 2x². 2(6- x)(7+x)

EXERCISE - G

1) 7√2x²- 10x - 4√2. (x-√2)(7√2x+4)

2) √3 x² + 5x + 2√3. (√3 x +2)(x + √3)

3) 4 √3 x² + 10x + 2√3. 2(√3 x +1)(2x + √3)

4) 4 √5 x² + 17x - 3√5. (x + √5)(4 √5 x -3)

5) 5 √3 x² -32 x - 7√3. (5x + √3)(√3x - 7)

EXERCISE - H

1) x(12x +7)- 10. (4x+5)((3x -2)

2) x(2x - y)- y². (2x+y)((x -y)

3) (4 - x)² - 2x. (x-8)(x -2)

4) 9x² - (x²-4)². (x+4)(x -1)(1+x)(4-x)

5) (x - y)² -6(x-y) +5. (x-y -5)(x- y-1)

6) (2x - y)² - 11(2x-y) +28. (2x-y -7)(2x- y-4)

7) 4(x - 1)² -4(x-1) -3. (2x -5)(2x -1)

8) (x +4)² -5xy -20y - 6y². (x +y+4)(x -6y +4)

9) (a²- 2a)² - 23(a² - 2a) + 120. (a-4)(a+2)(a-5)(a+3)

10) 8(x - 2y)² - 2x + 4y -1. {2(x -2y) -1}{4(x -2y)+1}

11) 12(x²+ 7x)² - 8(x²+ 7x)(2x -1) - 15(2x -1)². (2x²+ 8x +3)(6x²+52x -5)

12) 8(x+1)² - 2(x +1)(y +2) - 15(y +2)². (2x - 3y -4)(4x + 5y+14)

13) 4(x - y)² - 12(x - y)(y +x) + (y +x)². (x + 5y)(x + 5y)

EXERCISE - I

1) 1- 2a - 2b - 3(a+ b)². (1-3a-3b)(1+a+b)

2) 3- 5a - 5b - 12(a+ b)². (1-3a-3b)(3+ 4a+ 4b)

3) 2x³ + 5x²y -12xy². x(x+4y)(2x -3y)

4) x⁴ - 11x² +10. (x²-10)(x+1)(x-1)

5) x⁴ - 14x²y² -51y⁴. (x- √17y)(x+ √17y)(x² + 3y²).

6) x⁴ +3x²- 28). (x²+7)(x -2)(x +2)

7) (x²- 4x)(x²- 4x -1) - 20. (x -5)(x +1)(x -2)(x -2)

8) (5x - 1/x)² + 4(5x - 1/x) + 4, x≠ 0. (5x - 1/x +2)(5x - 1/x +2)

9) (x² - 2x)² -23(x² -2x) +120. (x +2)(x -4)(x+3)(x -5)

10) (x² - 3x)² -8(x² -3x) -20. (x -5)(x +2)(x-2)(x -1)

11) (x² +x)² + 4(x² + x) - 12. (x²+ x +6)(x +2)(x -1)

MISCELLANEOUS - A

1) 4(2x - 3)² - 3(2x -3)(x -1) - 7(x -1)².

2) (2x² + 5x)²(2x² +5x -19) +84. (x -1)(2x +7)(x+4)(2x -3)

3) (x²- 4x)(x²- 4x -1) -20. (x-5)(x+1) (x+2)²

4) (x-y)²- 7(x²- y²)+ 12(x +y)². 2(3x+4y)(x +2y)

5) 12(x² +7)² -8(x² +7)(2x -1)- 15(2x -1)². (2x² - 6x+17)(6x²+10x +37)

6) 5 - (3x²- 2x)(6- 3x²+ 2x). (1+ 3x)(1- x)(1+ x)(5- 3x).

7) 125a³ - 27b³ + 75a²b - 45ab². (5a+ 3b)(5a+ 3b)(5a - 3b)

8) x⁴ + 2x³y - 2xy³ - y⁴. (x-y)(x+ y)(x +y)(x +y)

9) The area of a rectangle is given by algebraic expression y²+ 5y - 24. Find the possible expressions for the dimensions of the rectangle.

EXERCISE - A

1) x³ + 125. (x+5)(x²-5x+25)

2) a³+ 8. (a+2)(a²- 2a +4)

3) m³ + 216. (m+6)(m²-6m+36)

4) 8x³ + y³. (2x+y)(4x²- 2xy+y²)

5) 27p³ + z³. (3p+z)(9p²-3pz +z²)

6) 8x³+125y³. (2x+5y)(4x²- 20xy+ 5y²)

7) 27y³ +125z³. (3y + 5z)(9y²- 15yz + 25z²)

8) 125a³ + 1. (5a+1)(25a²-5a+1)

9) 64 + a³. (4+a)(16- 4a - a²)

10) 64x³+1. (4x+1)(16x²- 4x +1)

11) 8x³ + 27y³. (2x+ 3y)(4x²- 12xy + 9y²)

12) 1 + 729k³. (1+ 9k)(1 - 9k + 81k²)

13) a³b³ + 8. (ab +2)(a²b² - 2ab + 4)

14) a³b³ + c³d³. (ab- cd)(a²b²- abcd+ c²d²)

9) 729 + p³q³. (9+ pq)(81 - 9pq + p²q²)

10) a³b³ + c³d³. (ab + cd)(a²b² - abcd+ c²d²)

11) 27x³ + c³d³. (3x + cd)(3x - 3xcd+ c²d²)

12) a³b³ + 343x³y³. (ab+ 7xy)(a²b² - 7abxy + 49xy)

13) a³b³c³+ 8. (abc +2)(a²b²c² - 2abc + 4)

14) 216 + 343a³. (6+ 7a)(36 - 42a + 49a²)

15) 8a³b³c³ + 27. (2abc +3)(4a²b²c² - 6abc+ 9)

16) 512a³+ 1/729b³. (8a+ 9b)(64a²- 8a/9b + 1/81b²)

17) 125x³ + 1/216. (5x + 1/6)(25x²- 5x/6 + 1/36)

18) 0.343+ 8a³. (0.7+ 2a)(0.49 - 1.49a + 4a²)

19) 8x³+ 0.125. (2x + 0.5)(4x²- x + 0.25)

EXERCISE - B

1) a³ - 1. (a-1)(a²+a+1)

2) p³ - 27. (p-3)(p²+3p+9)

3) 8 - k³. (2- k)(4+2k+k)

4) 512 - y³. (8-y)(64+8y+ y²)

5) 64x³ - 1. (4x -1)(16x²+4x+1)

6) 729a³ - 8. (9a+2)(81a²+18a+2)

7) 64x³ - 125y³. (4x -5y)(16x²+20xy+ 25y²)

8) ) 8x³ - 125. (2x-5)(4x²+10x+25)

9) x³y³z³ - 27. (xyz-3)(x²y²z²+ 3xyz +9)

10) 8x³ - 1/27y³. (2x -1)3y)(4x²+ 2x/3y + 1/9y²)

11) 27x³y³ -8. (3xy-2)(9x²y²+ 6xy+4)

12) x³ - 27y³/8. (x - 3y/2)(x² + 3xy/2 + 9y²/4)

13) 5√5 a³ - 2√2 b³. (√5 a - √2 b)(5a²+ √10 ab + 2b²)

EXERCISE - C

1) 7a³ +56b³. 7(a +2b)(a²- 2ab+4b²)

2) x² + x⁵. x²(1+ x)(1- x+ x²)

3) 32x⁴ - 500x. 4x(2x-5)(4x²+10x+25)

4) 32x³+ 108y³. 4(2x + 3y)(4x²- 6xy + 9y²)

5) 54x⁶y + 2x³y⁴. 2x³y(3x + y)(⁹x²- 3xy + y²)

6) 3x⁵y³+ 24x². 3x²(xy+2)(x²y²- 2xy +4)

7) 3a⁷b - 24a⁴b⁴. 3a⁴b(a - 2b)(a²+ 2ab+ 4b²)

8) x⁷y + xy⁷. (xy(x²+ y²)(x⁴- x²y² + y⁴)

9) 54x³y - 128y⁴. 2y(3x - 4y)(9x²+ 12xy + 16y²)

EXERCISE - D

1) 27(x+y)³+ 8(2x - y)³. (7x+y)(13x²- 4xy +19y²)

2) a³ + b³+ a+ b. (a+ b)(a²- ab + b²+1)

3) a³ - b³- a+ b. (a- b)(a²+ ab + b²-1)

4) a³- 8b³ + 2ax - 4bx. (a - 2b)(a²+ 2ab + 4b² + 2x)

5) 5a + 20b + a³+ 64b³. (a+ 4b)(5+ a²- 4ab + 16b²)

6) 8a³- 27b³ - 4ax + 6bx. (2a - 3b)(4a²+ 6ab + 9b²- 2x)

7) 2x - 3y - 8x³ + 27y³. (2x - 3y)(1- 4x²- 6xy - 9y²)

EXERCISE - E

1) (x + 2y)³ + (2x + y)³. 9(x + y)(x²+ xy + y²)

2) (2a + 3b)³ - (2a - 3b)³. (18b(4a²+ 3b²)

3) x³+ x +2. (x+1)(x²- x+2)

4) x³- x - 120. (x-5)(x²+5x+24)

5) x³+6x²+12x +16. (x+4)(x²+2x+4)

6) x³ -3x²y +3xy² -2y³. (x-2y)(x²-xy+y²)

7) 2x³+ 16y³ -5x -10y. (x+2y)(2x²- 4xy +8y² -5)

8) x³ - 1/x³ - 2x + 2/x. (x - 1/x)(x²+ 1/x² -1)

9) x² - 8/x. 1/x (x -2)(x²+ 2x +4)

10) 250(a - b)³+ 2. 2(5a -5b+1)(25a²- 50 ab + 25b² - 5a + 5b +1)

11) 32a²x³ - 8b²x³ - 4a²y³+ b²y³. (2a+ b)(2a - b)(2x - y)(4x²+ 2xy + y²)

12) (3x +4)³+ (7- 3x)³. 11(27x²- 27x +37)

13) (2x +1)³ - (x +1)³. x(7x²+ 9x +3)

14) (a/b + b/5)³ - (a/3 - b/5)³. 2b/5 (a²/3) + b²/25)

15) x³- 3x²+ 3x + 7. (x+1)(x²- 4x +7)

EXERCISE - F

1) x⁶ - y⁶. (x + y)(x - y)(x²+ xy + y²)(x²- xy + y²)

2) x⁶ - 1. (x+ 1)(x -1)(x²+ x+ 1)(x²- x + 1).

3) 64a⁶ - 729b⁶. (2a- 3b)(2a+ 3b)(4a²+ 6ab+ 9b²)(4a²- 6ab + 9b²).

4) x⁶/343 + 343/x⁶. (x²/7 +7/x²)(x⁴/49 -1+ 49/x⁴)

5) ) x⁶- 7x³ - 8. (x-2)(x²+ 2x+4)(x+1)(x²- x +1)

6) x⁶ - 26x³- 27. (x -3)(x +1)(x²+ 3x +9)(x²- x +1)

7) x⁸ - x²y⁶. x²(x + y)(x - y)(x²- xy + y²)(x²+ xy + y²)

8) x⁹ - b⁹. (a - b)(a²+ ab+ b²(a⁶+ a³b³+ b⁶)

9) x⁹+ y⁹. (x+ y)(x²- xy+ y²)(x⁶- x³y³ +y⁶)

10) x¹² - y¹². (x + y)(x - y)(x²+ y²)(x²+ xy+ y²)(x²- xy + y²)(x⁴- x²y² + y⁴)

USING FORMULA TO FACTORIZE

EXERCISE - A

1) x³+ 6x²y + 12xy²+ 8y³. (x + 2y)(x + 2y)(x+ 2y)

2) 27x³+ 64y³+ 108x²y +144xy². (3x+4y)(3x+4y)(3x+4y)

3) x³- 27y³- 9x²y + 27xy². (x+3y)(x+3y)(x+3y)

4) 8x³ - 36x²y + 54xy²- 27y³. (2x - 3y)(2x - 3y)(2x - 3y)

5) x³ + 3x²/2 + 3x/4 + 1/8. (x + 1/2)(x + 1/2) (x + 1/2)

6) x³ - x²y + xy²/3 - b³/27. (x - y/3)(x - y/3)(x - y/3)

7) 64x³/27 + 27y³/64 + 4xy² + 9xy²/4. (4x/3 + 3y/4)(4x/3 + 3y/4)(4x/3 + 3y/4)

8) a³/8 + a²b/4 + ab²/6 + b³/27. (a/2+ b/3)(a/2+ b/3) (a/2+ b/3)

9) 8x³/27 - 28x²/3 + 98x - 343. (2x/3 -7) (2x/3 -7) (2x/3 -7)

10) p⁶ - 27q⁶/8 - 9p⁴q²/2 + 27p²q⁴/4. (p²- 3q²/2)(p²- 3q²/2)(p²- 3q²/2)

EXERCISE - B

1) 8x³+ 27y³ + 64z³ - 72xyz. (2x + 3y+ 4z)(4x²+ 9y²+ 16z²- 6xy -12yz - 8xz)

2) 8x³- 27y³ + z³ +18xyz. (2x - 3y+ z)(4x²+ 9y²+ z²+ 6xy + 3yz - 2xz)

3) 27a³+ 125b³ - c³ + 44abc. (3a + 5b -c)(9a²+ 25b²+ c²- 15ab + 5bc + 3ca)

4) x³- 27y³ - 1 - 9xy. (x - 3y-1)(x²+ 9y²+ 16z²+ 3xy - 3y + x+1)

5) -27x³+ y³ - z³ - 9xyz. (-3x + y- z)(9x²+ y²+ z² + 3xy + yz - 3xz)

6) x³/8 - 64y³ + 27z³ +18xyz. (x/2 - 4y+ 3z)(x²/4 + 16y²+ 9z² + 2xy + 12yz - 3xz/2)

7) 2√2x³+ 3√3y³ + √5(5 - 3√6 xy). (√2x + √3y+ √5)(2x²+ 3y²+ 5 - √6xy - √15y - √10x)

8) 8x³ - y³ - 1 - 6xyz. (2x - y -1)(4x²+ y²+ 1 + 2xy -y + 2x)

9) - a⁶+ 8b⁶ + c⁶+ 6a²b²c². (-a²+ 2b²+ c²)(a⁴+ 4b⁴+ c⁴+ 2a²b² - 2b²c² + c²a²)

10) 27x³ - 8y⁶ + 125z³ + 90xy²z. (3x -2y²+ 5z)(9x²+4y⁴+ 25z²+ 6xy² +10y²z - 15xz)

11)